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平成 年度 東京理科大学
22
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平成22年度 研究成果報告
量子論から見る情報と生命の研究
量子論から見る
情報と生命の研究
研究成果報告
平成
年３月
23
108330_QBIC研究報告書.indd 1
平成23年3月
東京理科大学総合研究機構
量子生命情報研究センター
10/12/14 18:35
㊂ሶ↢ᖱႎ⎇ⓥࡦ࠲ 2010 ᐕ
ࡦ࠲㐳 ᄢ⍫㓷ೣ
2006 ᐕ 4 ߦ⊒⿷ߒߡએ᧪㊂ሶ↢ᖱႎ⎇ⓥࡦ࠲㧔Quantum Bio-Informatics
Center, QBIC㧕ߪ࠲ࡦ࠴ࠨࠢ࠹ࠗࡂޔᢛᬺ⎇ⓥࡊࡠࠫࠚࠢ࠻ޟ㊂ሶ⺰߆
ࠄࠆᖱႎߣ↢ߩ⎇ⓥߦޠၮߠ߈࠲ࡦޔ㐳ᄢ⍫㓷ೣߩਅޔਃߟߩࠣ࡞ࡊޟᢙℂࠣ
࡞ࡊ↢ޟޔޠࠣ࡞ࡊޟޔޠ㊂ሶࠣ࡞ࡊߩߣޠㅪ៤⎇ⓥࠍផㅴߒߡࠆޕ
2010 ᐕᐲߪ᧲⸰⺪ޔ੩ℂ⑼ᄢቇߦ߅ߡࠨࡑࠬࠢ࡞ࠍ㐿ߒޔᢙℂ↢ޔޔ㊂ሶߩ
3 ࠣ࡞ࡊߩᨒࠍ߃ߡޔQBIC ߩ⋡ᮡࠍᔨ㗡ߦޔઍ⑼ቇߩၮ␆߆ߟᧄ⾰ߦ㑐ࠊࠆࠕࠗ࠺
ࠕ߿㗴ὐߥߤߩ⼏⺰ߩ႐ࠍᜬߞߚޕ
ᐕߢ 5 ᐕ⋡߇ቢੌߒߩߘޔ㑆ߩ⚿ታߪ World Scientific ␠߆ࠄᧄߣߒߡ ߐࠇߡ
ࠆޔ߅ߥޕ2010 ᐕ 4 ߆ࠄ 2011 ᐕ 3 ߹ߢߩ⎇ⓥᚑᨐߩ߁ߜޔਥⷐߥ߽ߩࠍએਅߦ
ߔࠆޕ
(1)
ⴊᶧਛߩDDS(Drug Delivery System)⮎ߩᝄࠆ⥰ࠍ⸥ㅀߔࠆᢙℂࡕ࠺࡞ࠍ᭴▽
ߒߚᦨߪ࡞࠺ࡕߩߎޕㆡߥ⮎ࠨࠗ࠭ߩⓍ߽ࠅߦ↪ߢࠆߣᦼᓙߐࠇࠆޕ
(2)
ࠕࡒࡁ㉄㈩ߦኻߔࠆᣂߒࠕࠗࡔࡦ࠻ᣇᴺ(MTRAP)ࠍឭ᩺ߒߚߩߎޕᣇᴺ߇ᣢ
ሽߩᣇᴺࠃࠅ߽㜞♖ᐲࠍ߽ߟߎߣࠍ⸽ߒߚޕ
(3)
ᖱႎㆡᔕജቇߩᨒ⚵ߺߦ߅ߡឭ᩺ߐࠇߚࠞࠝࠬዤᐲࠍ↪ߔࠆߎߣߦࠃߞߡࠗޔ
ࡦࡈ࡞ࠛࡦࠩ A ࠙ࠖ࡞ࠬߩಽ㘃ࠍⴕߞߚޕ
(4)
⣖ᯏ⢻ߦ㑐ߔࠆ㊂ሶജቇ⊛ߥᢙℂࡕ࠺࡞ࠍቢᚑߒ⼂ޔㆊ⒟⸽ߦᚑഞߒߚޕ
(5)
৻⥸ൻ㊂ሶ࠴ࡘࡦࠣᯏ᪾ߩቯᑼൻࠍⴕߞߚౕߩ߆ߟߊޕ⊛ߥ㗴ߦߟߡ
㊂ሶࠕ࡞ࠧ࠭ࡓࠍឭ᩺ⶄߩ▚⸘ߩߘޔ㔀ߐߦߟߡ⠨ኤߒߚޕ
(6)
⍮ޔᗧᕁቯߦ㑐ㅪߒߚታ㛎ߩ⛔⸘࠺࠲ߦߺࠄࠇࠆ non-Kolmogorovian ᕈࠍ㊂
ሶࡑ࡞ࠦࡈࡕ࠺࡞ߦ߅ߌࠆ⏕₸ㆊ⒟߆ࠄ⺑ߒߚޕ
(7)
Two-player ࠥࡓߦ߅ߌࠆ㊂ሶ⺰⊛ᗧᕁቯࡕ࠺࡞ߩ৻⥸⊛ቯᑼൻࠍⴕޔ㊂ሶ㐿
♽ࠍ⸥ㅀߔࠆ㧳㧷㧿㧸ࡕ࠺࡞ߣߩ㑐ㅪᕈࠍ⠨ኤߒߚޕ
−1−
(8)
ዪᚲൻߐࠇߚࠛࡦ࠲ࡦࠣ࡞࠼ࡑ࡞ࠦࡈㅪ㎮ߩࠛࡦ࠲ࡦࠣ࡞࠼ߩᐲวߦኻߔࠆᐞߟ
߆ߩᖱႎ㊂ࠍ↪ߚ⹏ଔߦߟߡߩ⠨ኤਗ߮ߦࠛࡦ࠲ࡦࠣ࡞࠼ᓇ↪⚛ߩዉ
ߦߟߡ⸛⼏⠨ኤߒߚߩࠣࡦࠖ࠹ࡈޔߦࠄߐޕᔨ↪ߦߟߡࠨࠠࡘࡦ࠻
⁁ᘒߩᐞߟ߆ߩࠢࠬߩࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ߩᐲวߦߟߡ⠨ኤߒߚޕ
(9)
㒢ᰴరߦ߅ߌࠆࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻౮ߣPPT ⁁ᘒߩ㑐ㅪߦߟߡ⠨ኤߒߚޕ
(10)
ࡉ࠙ࡦേജቇᴺࠍขࠅࠇࠆߎߣߢޔዊࡍࡊ࠴࠼ߩࡑࠗࠢࡠ⑽⒟ᐲߩ᭴ㅧ੍᷹ࠪ
ࡒࡘ࡚ࠪࡦ߇น⢻ߣߥߞߚࠣࡦࡊࡦࠨࡉࡦࠕޔߡߒߘޕᴺࠍ↪ߚ⥄↱
ࠛࡀ࡞ࠡ⸘▚ߦࠃࠆࡍࡊ࠴࠼ߩ߳࠶ࠢࠬᒻᚑᯏ᭴ߦߟߡ⎇ⓥࠍ߅ߎߥߞߚޕ
(11)
㊂ሶ⏕₸⺰ࠍၮ␆ߣߒߚࠢࠬ࠲ࠍ↪ߡℂᗐࡏ࠭᳇ߩ☸ሶ♽ߩࠪࡒࡘ
࡚ࠪࡦࠍⴕߞߚ⚿ߩߎޕᨐߪ┙ߩ⾰ࠢࡄࡦ࠲ޔ᭴ㅧ੍᷹ߢ↪ࠄࠇߡࠆࡉ
࠙ࡦേജቇߩᡷ⦟ࡕ࠺࡞ߩ⎇ⓥ߿⣖ߩᢙℂࡕ࠺࡞ߩ᭴▽ߦᓎ┙ߟߣᕁࠊࠇࠆޕ
(12)
േ‛ߩ⣖⚻♽ߣᲧセߒߥ߇ࠄޔᬀ‛ߩᖱႎવ㆐ࡀ࠶࠻ࡢࠢߩ⎇ⓥࠍⷰߒ․ޔ
ߦᵴᕈ㉄⚛⒳ߣࠞ࡞ࠪ࠙ࡓࠗࠝࡦ߇㑐ਈߔࠆᱜߩࡈࠖ࠼ࡃ࠶ࠢᓮᖱႎࡀ࠶࠻ࡢ
ࠢߦߟߡ⺞ᩏߒߚޕ
(13)
ᐔᚑᐕᣣ߆ࠄᣣߩᦼ㑆ޔቴຬᢎߩ+IQT8QNQXKEJ᳁
ࠬ࠹ࠢࠢࡠ࠙ᢙℂ
⎇ⓥᚲࠍ⡜ߒ⥸৻ޔൻ㊂ሶ࠴ࡘࡦࠣᯏ᪾ߩቯᑼൻߦ㑐ߔࠆห⎇ⓥࠍⴕߞߚޕ
߹ߚޔᐔᚑ22ᐕ101ᣣ߆ࠄ1028ᣣߩᦼ㑆ޔቴຬᢎߩAndrei Khrennikov᳁
(Linnaeusᄢቇ)ࠍ⡜ߒޔᗧᕁቯߦ㑐ㅪߒߚታ㛎ߩ⛔⸘ߣߘߩnon-Kolmogorovian
ᕈߦߟߡ⼏⺰ࠍⴕߞߚޕ
−2−
ᐔᚑ ᐕᐲ ࡊࡠࠫࠚࠢ࠻ࡔࡦࡃ
ᢙℂ⎇ⓥࠣ࡞ࡊ
ᄢ⍫㓷ೣ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ᢎ
ᚭᎹ⟤㇢
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ᢎ
ንỈ⽵↵
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ᢎ
ᷰㆺ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ᢎ
⋓᳗◊㇢
ℂᎿቇㇱ
‛ℂቇ⑼
ᢎ
የ┙㦷
ℂᎿቇㇱ
‛ℂቇ⑼
ᢎ
೨↰⼑ᴦ
ℂᎿቇㇱ
㔚᳇㔚ሶᖱႎᎿቇ⑼
ᢎ
᧻ጟ㓉ᔒ
⺪⸰᧲੩ℂ⑼ᄢቇ
⚻༡ᖱႎቇ⑼
ᢎ
⮮ሶ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
⻠Ꮷ
◉ේᥰሶ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
⻠Ꮷ
ጊญ᧲੩ℂ⑼ᄢቇ
㔚ሶᖱႎᎿቇ⑼
⻠Ꮷ
ጊ⡛ผ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ഥᢎ
↰⇌⠹ᴦ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ഥᢎ
ጊᧄ⚔ม
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ഥᢎ
ේ⧷
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ഥᢎ
⎇ⓥઍ⠪
ࠣ࡞ࡊ࠳
↢⎇ⓥࠣ࡞ࡊ
ጊ⊓৻㇢
ၮ␆Ꮏቇㇱ
↢‛Ꮏቇ⑼
ᢎ
ችፒᥓ
⮎ቇㇱ
↢ഃ⮎⑼ቇ⑼
ᢎ
ᱞ↰ᱜਯ
ℂᎿቇㇱ
ᖱႎ⑼ቇ⑼
ᢎ
ᵤᐘ
ℂᎿቇㇱ
ᔕ↪↢‛⑼ቇ⑼
ᢎ
‛ℂቇ⑼
ᢎ
ࠣ࡞ࡊ࠳
㊂ሶ⎇ⓥࠣ࡞ࡊ
ဈ↰⧷
ℂቇㇱ╙৻ㇱ
㜟ᩉ⧷
✚ว⎇ⓥᯏ᭴
⍹ജ
ℂቇㇱ╙৻ㇱ
ᵲ
✚ว⎇ⓥᯏ᭴
ችᎹት
ℂቇㇱ╙৻ㇱ
ᔕ↪‛ℂቇ⑼
ಎᢎ
੫ᄥ㇢
ℂቇㇱ╙৻ㇱ
ᔕ↪‛ℂቇ⑼
ഥᢎ
ട⮮ᜏ
ℂቇㇱ╙৻ㇱ
‛ℂቇ⑼
ഥᢎ
⍹㤥੫テ
ℂቇㇱ╙৻ㇱ
ᔕ↪‛ℂቇ⑼
ഥᢎ
ᢎ
‛ℂቇ⑼
ᢎ
ᢎ
−3−
ࠣ࡞ࡊ࠳
ᄖㇱห⎇ⓥ⠪
Luigi Accardi
ࡠࡑΤᄢቇ
ᢎ
V. P. Belavkin
ࡁ࠶࠹ࠖࡦࠟࡓᄢቇ
ᢎ
Dariusz Chruscinski
࠾ࠦࠬࠦࡍ࡞࠾ࠢࠬᄢቇ
ᢎ
Artur Ekert
ࠤࡦࡉ࠶ࠫᄢቇ
ᢎ
Karl-Heinz Fichtner
ࡈ࠼࠶ࡅ-ࠪࠗࠛ࠽ᄢቇ
ᢎ
Wolfgang Freudenberg
ࡉࡦ࠺ࡦࡉ࡞ࠣᎿ⑼ࠦ࠻ࡉࠬᄢቇ
ᢎ
㘧↰ᱞᐘ
ฬฎደᄢቇ
ฬᢎ
Andrzej Jamiolkowski
࠾ࠦࠬࠦࡍ࡞࠾ࠢࠬᄢቇ
ᢎ
Andrzej Kossakowski
࠾ࠦࠬࠦࡍ࡞࠾ࠢࠬᄢቇ
ᢎ
Andrei Khrennikov
ࡦࡀᄢቇ
ᢎ
W. A. Majewski
Gdansk ᄢቇ
ᢎ
Ryszard Mrugala
࠾ࠦࠬࠦࡍ࡞࠾ࠢࠬᄢቇ
ᢎ
Denes Petz
ࡉ࠲ࡍࠬ࠻ᛛⴚ⚻ᷣᄢቇ
ᢎ
Igor Volovich
ࠬ࠹ࠢࠢࡠ࠙ᄢቇ
ᢎ
ầ⑲᮸
࿖┙᧲੩Ꮏᬺ㜞╬ኾ㐷ቇᩞ
ᢎ
ዊ᎑ᴰ
੩ㇺᄢቇ
ಎᢎ
Si Si
ᗲ⍮⋵┙ᄢቇ
ಎᢎ
Milosz Michalski
࠾ࠦࠬࠦࡍ࡞࠾ࠢࠬᄢቇ
⻠Ꮷ
㋈ᧁᥓౖ
ጟጊᄢቇᄢቇ㒮 කᱤ⮎ቇ✚ว⎇ⓥ⑼
ഥᢎ
Massimo Regoli
ࡠࡑΤᄢቇ
⎇ⓥຬ
Md. Aminul Hoque
ࡑࡗᄢቇ
PD
ᢙℂ⸃ᨆ⎇ⓥᚲ
−4−
⋡ ᰴ
㧚ᐨᢥ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚ࡊࡠࠫࠚࠢ࠻ࡔࡦࡃ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚ᐔᚑ ᐕᐲ 4%,& ቴຬ⡜ߩ⸥㍳ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚ᢙℂ⎇ⓥࠣ࡞ࡊ ᄢ⍫㓷ೣ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ንỈ⽵↵◉ޔේ㧔ችᧄ㧕ᥰሶ⇌↰ޔ⠹ᴦޔጊᧄ⚔มăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ᷰㆺ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
⋓᳗◊㇢ޔᒄశ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
የ┙㦷 ޔăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
የ┙㦷ޔᐔႦ㕏ਭޔචᧁ⑲৻ޔᷰㆻ㓉 ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
೨↰⼑ᴦޔᶏ⠧Ỉ⾫ผ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
᧻ጟ㓉ᔒ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
⮮ሶ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ጊ⡛ผ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ේ⧷ޔ⮮ሶޔᄢ⍫㓷ೣ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
⎇ⓥᬺ❣ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
㧚↢⎇ⓥࠣ࡞ࡊ
ጊ⊓৻㇢ޔᘕޔ㊄ቢޔ⮮ᩰ჻ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ችፒᥓޔᰨᆻᄢޔ㋈ᧁᥓౖ⿒ޔဈᔒᵤ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ᱞ↰ᱜਯ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă
ᵤᐘ᧪ޔ㗇ቁశޔự↰᥍ᐽ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ⎇ⓥᬺ❣ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚㊂ሶ⎇ⓥࠣ࡞ࡊ
ဈ↰⧷ޔട⮮ᜏ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㜟ᩉ⧷ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ᵲޔầ⑲᮸ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ችᎹት ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ੫ᄥ㇢ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ⍹㤥੫テ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă ⎇ⓥᬺ❣ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚ઃ㍳ 4%,& ࠨࡑࠬࠢ࡞ ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă 㧚ઃ㍳ 4%,& ࡒ࠽ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăă −5−
ᐔᚑ㧞㧞ᐕᐲ 㧽㧮㧵㧯ቴຬ⡜⸥㍳
Igor Volovich (䉴䊁䉪䉪䊨䉡ᢙℂ⎇ⓥᚲ)
㩷 ᐔᚑ 22 ᐕ 4 1 ᣣ䈎䉌 4 10 ᣣ䈱ᦼ㑆䇮ቴຬᢎ䈱 Igor Volovich ᳁䉕⡜䈚䇮ห⎇ⓥ䉕ⴕ䈇䇮৻⥸ൻ
㊂ሶ䉼䊠䊷䊥䊮䉫ᯏ᪾䈱ቯᑼൻ䉕ⴕ䈦䈢䇯䈘䉌䈮䇮ฎౖജቇ䈫㊂ሶജቇ䈱⏕₸⊛ข䉍ᛒ䈇䈮㑐䈜䉎ၮ␆⊛䈭
㗴䈮䈧䈇䈩䉅⸛⼏䈚䈢䇯䈘䉌䈮䇮ᐔᚑ 22 ᐕ 5 10 ᣣ䈎䉌 6 6 ᣣ䈱ᦼ㑆䇮Volovich ᳁䉕⡜䈚䇮ห⎇ⓥ
䈍䉋䈶 Springer 䈱 TMP 䉲䊥䊷䉵䈎䉌 ੍ቯ䈱㊂ሶᖱႎ䈮㑐䈜䉎ᧄ䈱ᦨ⚳ᩞᱜ䉕ⴕ䈉䇯ห⎇ⓥ䈪䈲䇮䈇䈒
䈧䈎䈱ౕ⊛䈭㗴䈮䈧䈇䈩㊂ሶ䉝䊦䉯䊥䉵䊛䉕ឭ᩺䇮䈠䈱⸘▚䈱ⶄ㔀䈘䈮䈧䈇䈩⠨ኤ䈚䈢䇯
Andrei Khrennikov (Linnaeus ᄢቇ)
ᐔᚑ22ᐕ101ᣣ߆ࠄ1028ᣣߩᦼ㑆ޔቴຬᢎߩAndrei Khrennikov᳁ࠍห⎇ⓥߩߚ⡜ߒޔ
ᗧᕁቯߦ㑐ㅪߒߚታ㛎ߩ⛔⸘ߣߘߩnon-Kolmogorovian ᕈߦߟߡ⼏⺰ࠍⴕߞߚޔߦࠄߐޕ
ޟExploring the analogy between classical and quantum signalsߩߩ࡞࠻ࠗ࠲߁ߣޠ
QBICࡒ࠽ࠍⴕߞߚޕห⎇ⓥߢߪޔTwo-player ࠥࡓߦ߅ߌࠆ㊂ሶ⺰⊛ᗧᕁቯࡕ࠺࡞ߩ৻⥸
⊛ቯᑼൻ߿㊂ሶ㐿♽ࠍ⸥ㅀߔࠆ㧳㧷㧿㧸ࡕ࠺࡞ߣߩ㑐ㅪᕈࠍ⠨ኤߒߚޔߚ߹ޕ3-player ࠥࡓ߳ߩ
ᔕ↪ߦߟߡ߽⼏⺰ߒߚޕ
Irina Basieva (ࡠࠪࠕ⑼ቇࠕࠞ࠺ࡒ)
ᐔᚑ22ᐕ101ᣣ߆ࠄ1028ᣣߩᦼ㑆ޔIrina Basievaඳ჻ࠍ⡜ߒޟޔDynamics and
entanglement of exciton states in clusters of resonantly interacting fluorescent
particlesߩߩ࡞࠻ࠗ࠲߁ߣޠQBICࡒ࠽ࠍⴕߞߚޔߦࠄߐޕᏗ㘃㊄ዻߩ‛ℂ⊛ᕈ⾰ߦ㑐ߔࠆ
ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ᕈߩ․⾰ߦߟߡ⼏⺰ࠍⴕޔQBICߩ⺖㗴ߣߩ㑐ㅪᕈߦଥࠊࠆ⺖㗴ߦߟߡห
⎇ⓥߩḰࠍⴕߞߚޕ
Dariusz ChruĞciĔski (N. Copernicus ᄢቇ)
㩷 ᐔᚑ 22 ᐕ 10 26 ᣣ䈎䉌 11 12 ᣣ䈱ᦼ㑆䇮ቴຬᢎ䈱 Dariusz ChruĞciĔski ᳁䉕ห⎇ⓥ䈱䈢䉄⡜䈚䇮
㊂ሶ䉣䊮䉺䊮䉫䊦䊄⁁ᘒ䈱⋧㑐䈱ᒝ䈘䉕⹏ଔ䈜䉎ᣂ䈚䈇ᜰᮡ䈫䈚䈩䇮”symmetric discord”䈫䈇䈉᷹ᐲ䉕ዉ䈚䇮
䈠䈱↪ᕈ䉕ᬌ⸛䇮⠨ኤ䈚䈢䇯䉁䈢䇮ᐔᚑ 22 ᐕ 11 4 ᣣ䈎䉌 11 6 ᣣ䈱ᦼ㑆䈮䇮੩ㇺᄢቇ䈱ၮ␆‛ℂ⎇䈪
㐿䈘䉏䈢⎇ⓥળޟ㊂ሶ⑼ቇߦ߅ߌࠆኻᕈߣࠬࠤ࡞ߦޠෳടߒޟޔNon-Markovian dynamics of
quantum systems 䇹䈫䈇䈉䉺䉟䊃䊦䈱⻠Ṷ䉕ⴕ䈦䈢䇯
Andrzej Jamioákowski (N. Copernicus ᄢቇ)
ᐔᚑ 22 ᐕ 10 26 ᣣ䈎䉌 11 12 ᣣ䈱ᦼ㑆䇮ቴຬᢎ䈱 Andrzej Jamioákowski ᳁䉕ห⎇ⓥ䈱䈢䉄
⡜䈚䇮㊂ሶ⛔⸘䈮䈍䈔䉎චಽᕈ䈱㗴䉕䇮᳁䈏ឭ᩺䈜䉎”Quantum Tomography”䈱䉴䉨䊷䊛䈮䈍䈇䈩ౕ⊛
䈭䊝䊂䊦䉕᭴ᚑ䈚䇮ᬌ⸛䈜䉎䈖䈫䉕ⴕ䈦䈢䇯䉁䈢䇮ᐔᚑ 22 ᐕ 11 4 ᣣ䈎䉌 11 6 ᣣ䈱ᦼ㑆䈮䇮੩ㇺᄢቇ䈱ၮ
␆‛ℂ⎇䈪㐿䈘䉏䈢⎇ⓥળޟ㊂ሶ⑼ቇߦ߅ߌࠆኻᕈߣࠬࠤ࡞ߦޠෳടߒޔ
ޟEffective methods in
investigations of positive maps䇹䈫䈇䈉䉺䉟䊃䊦䈱⻠Ṷ䉕ⴕ䈦䈢䇯
−7−
એਅߦޔᐔᚑ㧞㧟ᐕߩ㧟㧣ᣣ㨪㧝㧠ᣣ߹ߢߩᦼ㑆ߦ㐿੍ቯߩ࿖㓙ળ⼏߳ߩ⡜੍ቯ⠪ߩࠬ࠻ࠍ
ߍࠆޕ
QBICࡔࡦࡃߢߪޔ㘧↰ ᱞᐘ㧔ฬฎደᄢቇฬᢎޔฬၔᄢቇฬᢎ㧕ޔዊ᎑ ᴰ㧔੩ㇺᄢቇᢙ
ℂ⸃ᨆ⎇ⓥᚲ㧕ޔSi Si㧔ᗲ⍮⋵┙ᄢቇ㧕ޔLuigi Accardi (䊨䊷䊙IIᄢቇ)䇮Massimo Regoli(䊨䊷䊙IIᄢቇ)䇮
Karl-Heinz Fichtner (䊐䊥䊷䊄䊥䉾䊍䊶䉲䊤䊷䉟䉣䊅ᄢቇ)䇮Wolfgang Freudenberg (䊑䊤䊮䊂䊮䊑䊦䉫Ꮏ⑼䉮䊃䊑䉴
ᄢቇ)䇮Igor Volovich (䉴䊁䉪䉪䊨䉡ᢙℂ⎇ⓥᚲ)䇮Andrei Khrennikov (Linnaeusᄢቇ) 䇮
Viacheslav P. Belavkin䋨䊉䉾䊁䉞䊮䉧䊛ᄢቇ䋩䇮Andrzej Jamioákowski䋨䊆䉮䊤䉴䉮䊕䊦䊆䉪䉴ᄢቇ䋩䇮
Dariusz ChruĞciĔski䋨䊆䉮䊤䉴䉮䊕䊦䊆䉪䉴ᄢቇ䋩䇮Milosz Michalski䋨䊆䉮䊤䉴䉮䊕䊦䊆䉪䉴ᄢቇ䋩╬䇱
QBIC 䊜䊮䊋䊷એᄖ䈮䉅 D. Wanke(Tuebingen University)䇮Wonpil Im(Kansas University)䇮Jacek Jurkowski䋨N.
Copernicus University䋩䇮Ludwing. Streit (BiBoS, University)䇮F. Mukhamedov䋨International Islamic University
Malaysia䋩䇮O. Smolyanov䋨Moscow State University䋩╬䇱䉕⡜䈜䉎੍ቯ䈪䈅䉎䇯
−8−
㩷
㩷
㩷
ᢙℂ⎇ⓥ䉫䊦䊷䊒㩷
㩷
㩷
QBIC ⎇ⓥᚑᨐႎ๔
᧲੩ℂ⑼ᄢቇℂᎿቇㇱᖱႎ⑼ቇ⑼
ᄢ⍫㓷ೣ
ᧄᐕᐲߪ↢ᖱႎߣ㊂ሶᖱႎߩⲢวࠍᔨ㗡ߦ㧘એਅߩࠃ߁ߥ⎇ⓥࠍⴕߞߚ㧚
1㧚Characterization of partial positive transposition states and measures of entanglement
Abstract:
A detailed characterization of partial positive transposition (PPT) states, both
in the Heisenberg and in the Schrödinger picture, is given. Measures of entanglement
are defined and discussed in details. Illustrative examples are provided.
2㧚Mathematical description of drug movement into tumor with EPR effect and estimation of
its configuration for DDS
Abstract:
It is known that Drug Delivery System (DDS) is useful to remedy against
tumors for the reduction of side effects and the effective dosage. However the shape, in
particular, the size of drug (medicine) is empirically decided in the present stage, which
will be related to a question how much medicine should be dosed. Taking a particular
reaction of tumor tissues called the EPR effect into consideration; we try to
mathematically describe the behavior (dynamics) of drug in blood vessel by applying
several techniques used in mathematics and physics. In this paper, we estimate the
configuration of drug which is most effective to remedy for tumors under various
conditions. As a result, this model and its simulation will be useful to design the drug in
nano-level.
3㧚On a Combined Quantum Baker's Map and Its Characterization by Entropic Chaos
Degree
Abstract:
Quantum Baker's map is a theoretical model that exhibits chaos in a quantum
system. In this paper, we introduce a combined map by combining several quantum
Baker's maps. Chaos of such a combined dynamics is studied by the entropic chaos
degree.
−9−
4㧚On Generalized Quantum Turing Machine and Its Applications
Abstract:
Ohya and Volovich discussed a quantum algorithm for the SAT problem with a
chaos amplification process (OMV SAT algorithm) and showed that the number of steps
it performed was polynomial in input size. In this paper, we define a generalized
quantum Turing machine (GQTM) and related computational complexity. Then we
show that there exists a GQTM which recognizes the SAT problem in polynomial time.
Moreover, we discuss the problem of finding the quantum algorithm for a partial
recursive function.
5 㧚 Quantum Markov Model for Data from Shafir-Tversky Experiments in Cognitive
Psychology
Abstract:
We analyze, from the point of view of quantum probability, statistical data
from two interesting experiments, done by Shafir and Tversky [1, 2] in the domain of
cognitive psychology. These are gambling experiments of Prisoner Dilemma type. They
have important consequences for economics, especially for the justification of the
Savage "Sure Thing Principle" [3] (implying that agents of the market behave
rationally). Data from these experiments were astonishing, both from the viewpoint of
cognitive psychology and economics and probability theory. Players behaved irrationally.
Moreover, all attempts to generate these data by using classical Markov model were
unsuccessful. In this note we show (by inventing a simple statistical test — generalized
detailed balance condition) that these data are non-Kolmogorovian. We also show that it
is neither quantum (i.e., it cannot be described by Dirac-von Neumann model). We
proceed towards a quantum Markov model for these data.
6㧚On the Low-Temperature Behavior of the Infinite-Volume Ideal Bose Gas, Infinite
Dimensional Analysis
Abstract:
In Ref. 11 clustering representations of the position distribution of the ideal
Bose gas were considered. In principle that gives rise to possibilities concerning
simulations of the system of positions of the particles. But one has to take into account
that in case of low temperature the clusters are very large and their origins are far from
a fixed bounded volume. For that reason we will consider some estimations of the
influence of these clusters on the behavior of the subsystem of particles located in a
fixed bounded volume. All points in the fixed bounded volume come from a bigger
− 10 −
volume which the estimation (5.2) in Theorem 5.2 gives on average. Several numerical
simulations in dimension two are shown in Sec. 5.
7㧚Quantum Models of the Recognition Process - On a Convergence Theorem
Abstract:
One of the main activities of the brain is the recognition of signals. As it was
pointed out in [22, 25] the procedure of recognition can be described as follows: There is
a set of complex signals stored in the memory. Choosing one of these signals may be
interpreted as generating a hypothesis concerning an "expected view of the world".
Then the brain compares a signal arising from our senses with the signal chosen from
the memory. That changes the state of both signals in such a manner that after the
procedure the signals coincide in a certain sense. Furthermore, measurements of that
procedure like EEG or MEG are based on the fact that recognition of signals causes a
certain loss of excited neurons, i.e. the neurons change their state from "excited" to
"nonexcited". For that reason a statistical model of the recognition process should reflect
both — the change of the signals and the loss of excited neurons. Now, [5] represents the
first attempt to explain the process of recognition in terms of quantum statistics.
According to the general conception of quantum theory, the procedure of recognition
should be described by an operator on a certain Hilbert space. In [5] we proposed two
candidates for such an operator. One of them reflects in a clear sense the mentioned
change of the signals. The other one reflects the loss of excited neurons. We will prove
(cf. Theorem 4) that for sufficiently high intensities of the signals both operators are
approximately equal.
8㧚MTRAP: Pairwise sequence alignment algorithm by a new measure based on transition
probability between two consecutive pairs of residues
Abstract:
BACKGROUND: Sequence alignment is one of the most important techniques
to analyze biological systems. It is also true that the alignment is not complete and we
have to develop it to look for more accurate method. In particular, an alignment for
homologous sequences with low sequence similarity is not in satisfactory level. Usual
methods for aligning protein sequences in recent years use a measure empirically
determined. As an example, a measure is usually defined by a combination of two
quantities (1) and (2) below: (1) the sum of substitutions between two residue segments,
(2) the sum of gap penalties in insertion/deletion region. Such a measure is determined
on the assumption that there is no an intersite correlation on the sequences. In this
− 11 −
paper, we improve the alignment by taking the correlation of consecutive residues.
RESULTS:
We introduced a new method of alignment, called MTRAP by introducing a
metric defined on compound systems of two sequences. In the benchmark tests by
PREFAB 4.0 and HOMSTRAD, our pairwise alignment method gives higher accuracy
than other methods such as ClustalW2, TCoffee, MAFFT. Especially for the sequences
with sequence identity less than 15%, our method improves the alignment accuracy
significantly. Moreover, we also showed that our algorithm works well together with a
consistency-based progressive multiple alignment by modifying the TCoffee to use our
measure.
CONCLUSIONS:
We indicated that our method leads to a significant increase in alignment
accuracy compared with other methods. Our improvement is especially clear in low
identity range of sequences. The source code is available at our web page, whose address
is found in the section "Availability and requirements".
9㧚How to Classify Influenza A Viruses and Understand Their Severity
Abstract:
As an application of the chaos degree introduced in the framework of
information adaptive dynamics, we study the classification of the Influenza A viruses.
What evolutional processes determine the severity and the ability for transmission
among human of influenza A viruses? We performed phylogenetic classifications of
influenza A viruses that were sampled between 1918 and 2009 by using a measure
called entropic chaos degree, that was developed through the study of chaos in
information dynamics. The phylogenetic analysis of the internal protein (PB2, PB1, PA,
NS, M1, M2, NS1, and NS2) indicated that Influenza A viruses adapting to human and
transmitting among human were clearly distinguished from swine lineage and avian
lineage. Furthermore, the HA, NA, and internal proteins of the influenza strain that
caused a pandemic or a severe epidemic with high mortality were phylogenetically
different from those from previous pandemic and severe epidemic strains. We have
come to the conclusion that the internal protein has a significant impact on the ability
for transmission among human. Based of this study, we are convinced that entropic
chaos degree is very useful as a measure of understanding the classification and
severity of an isolated strain of influenza A virus.
− 12 −
正方分割表における正規分布型の対称モデルと対称性の
距離尺度に関する研究
数理研究グループ (東京理科大学理工学部情報科学科)
富澤貞男（教授）, 宮本暢子（講師）
田畑耕治（助教）, 山本紘司（助教）
Abstract. 本研究の Part 1 は，
「正方分割表における正規分布型対称モデル」の研究で
あり，セル確率が周辺分散の等しい 2 変量正規分布の密度関数と同様な構造をもつモデル
を提案した．さらに，潜在分布として周辺分散が等しい二つの 2 変量正規分布の 1 − ε と
ε の割合での混合である ε-汚染正規分布が想定される場合に適切であると考えられるモデ
「対称性からの隔たりを測る距離尺度」の研究であり，対
ルを提案した．また Part 2 は，
称性からの隔たりを測る真の距離尺度を導入した．この尺度は，対称性が成り立たない複
数の分割表を比較するのに有用である．
Part 1: 正規分布型対称モデル
行と列が同じ分類からなる正方分割表において，Agresti (1983) は線形対角パラメータ対称
（LDPS）モデルを提案した．さらに Agresti (1983) は，分割表の行変数と列変数が連続量で，周
辺分散が等しい潜在的な 2 変量正規分布が想定される場合，LDPS モデルはその分割表データに
よく適合するかもしれないことを指摘した．本研究では，セル確率そのものが周辺分散の等しい
2 変量正規分布の密度関数と同様な構造をもつモデルを提案する．
行と列が順序のある同じ分類からなる正方 R × R 分割表において，(i, j) セル確率を pij と
する（i = 1, . . . , R; j = 1, . . . , R）．線形対角パラメータ対称（LDPS）モデル（Agresti, 1983）
は次のように定義される：
pij
= θj−i (i < j).
pji
確率変数 X1 と X2 が，E(X1 ) = µ1 ，E(X2 ) = µ2 ，V ar(X1 ) = V ar(X2 ) = σ 2 ，Corr(X1 , X2 ) =
ρ の 2 変量正規分布に従うとする．その密度関数を f (x1 , x2 ) とすると，
f (x1 , x2 )
(x2 − x1 )(µ2 − µ1 )
= exp
f (x2 , x1 )
(1 − ρ)σ 2
(x1 < x2 )
となる．従って，本来，行変数と列変数が連続量で，周辺分散が等しい潜在的な 2 変量正規分布
が想定される場合，LDPS モデルはその分割表データによく適合するかもしれないことを Agresti
（1983）は指摘した．
密度関数 f (x1 , x2 ) は次のようにも表せる:
(x1 −x2 )2 x1 −x2 (x1 +x2 )2 x1 +x2
a2
b1
b2
,
f (x1 , x2 ) = ca1
ここに，
(1)
(µ1 + µ2 )2
(µ1 − µ2 )2
1
−
exp
−
,
c=
4σ 2 (1 − ρ) 4σ 2 (1 + ρ)
2πσ 2 1 − ρ2
−1
µ1 − µ2
, a2 = exp
,
a1 = exp
4σ 2 (1 − ρ)
2σ 2 (1 − ρ)
−1
µ1 + µ2
, b2 = exp
,
b1 = exp
4σ 2 (1 + ρ)
2σ 2 (1 + ρ)
− 13 −
である．
本研究ではセル確率 {pij } が，(1) 式で表されるような 2 変量正規分布の密度関数と同様な
構造をもつモデルを提案する．
正規分布型対称（NDS）モデルを次のように提案する:
(i−j)2
pij = ξα1
(i+j)2 i+j
β2
α2i−j β1
NDS モデルの下では，
(i = 1, . . . , R; j = 1, . . . , R).
pij
= (α22 )i−j
pji
(i > j)
と表せることから，NDS モデルは LDPS モデルの特別な場合である．正方分割表において行変数
を X ，列変数を Y とする．NDS モデルの下で，α2 > 1 は P(X ≤ i) < P(Y ≤ i)，i = 1, . . . , R − 1
と同値である．すなわち，NDS モデルのパラメータ α2 は，X が Y より確率的に小さい（ある
いは大きい）という推測をするのに役立つ．
ここで，
p∗uv = p u+v , v−u ((u, v) ∈ S ∗ ),
2
2
S ∗ = {(u, v) | u = i − j, v = i + j
(i = 1, . . . , R; j = 1, . . . , R)},
とする．このとき，NDS モデルは
2
2
p∗uv = ξα1u α2u β1v β2v
((u, v) ∈ S ∗ )
のようにも表せる．これは U = X − Y ，V = X + Y とすると，NDS モデルは U と V の準独立
性の構造を示している．
さらに本研究では，正方分割表解析において，潜在分布として周辺分散が等しい 2 つの 2 変
量正規分布の 1 − ε と ε の割合での混合である ε-汚染正規分布が想定される場合に適切であると
考えられるモデルを提案する．
正方分割表において，0 < ε < 1，γ = 1 に対して次のモデルを提案する：
1
µ
pij = (1 − ε)µαi β j ψij + ε (αi β j ψij ) γ
γ
(i = 1, . . . , R; j = 1, . . . , R),
ただし，ψij = ψji ．このモデルは次のようにも表せる：
pij = (1 − ε)θj φij + εη θj φij
1
γ
(i = 1, . . . , R; j = 1, . . . , R),
ただし，φij = φji ．このモデルを汚染正規型対称（CNS）モデルと呼ぶ．CNS モデルは ε-汚染
正規分布の密度関数
g(u, v) = (1 − ε)f (u, v|µ1 , µ2 , σ 2 , ρ) + εf (u, v|µ1 , µ2 , ξ 2 σ 2 , ρ)
= (1 − ε)Aαu β v ψuv + ε
ただし，ψuv = ψvu ,
A=
2πσ 2
α = exp
β = exp
1
A u v
(α β ψuv ) ξ2 ,
2
ξ
1
,
1 − ρ2
µ1 − ρµ2
,
σ 2 (1 − ρ2 )
µ2 − ρµ1
,
σ 2 (1 − ρ2 )
− 14 −
と同様な構造をしている．CNS モデルは特に α = β のとき S モデルとなる．CNS モデルは，一
1
j
θ φij = 1，
η(θj φij ) γ = 1 という制約をおくこと
般性を失うことなくパラメータ間に
ができる．さらに，
(1)
(2)
qij = θj φij , qij = η θj φij
(1)
1
γ
(i = 1, . . . , R; j = 1, . . . , R)，
(2)
とおくと，{qij }，{qij } はそれぞれ LDPS モデルの構造を持つ．すなわち，CNS モデルは 2 つ
の LDPS モデルの重み付きの和の構造をしている．また，Tahata, Yamamoto and Tomizawa
(2009) と Yamamoto, Kurakami, Iwashita and Tomizawa (2008) を参照．
Part 2: 正方分割表における対称性からの隔たりを測る距離尺度
行と列が同じ分類からなる正方分割表データの解析において，分類間の関連性は極めて強
く，統計的独立性は成り立たない場合が多い．そのようなデータに対しては，分類間の独立性に
代わって対称性に関心がある．対称性に関するモデルとして，Bowker (1948) の導入した対称モ
デルがある．また，対称モデルが成り立たない場合に，対称性からの隔たりを測る尺度がいくつ
か提案されている．しかしながら，それらの尺度は距離の条件を満たしておらず，真の距離を測
る尺度ではない．そこで本研究では，名義カテゴリ分割表，順序カテゴリ分割表のそれぞれの場
合に，対称性からの隔たりを測る真の距離尺度を提案する．提案する尺度は複数の分割表を比較
するのに有用である．
行と列が同じ分類からなる正方 r × r 分割表において，(i, j) セル確率を pij とする（i =
1, . . . , r; j = 1, . . . , r）．対称モデルは次のように定義される (Bowker, 1948; Bishop, Fienberg
and Holland, 1975, p.282):
pij = pji
(i = 1, . . . , r; j = 1, . . . , r; i = j).
名義カテゴリ分割表において，対称性からの隔たりを測る尺度が次のように導入されている
(Tomizawa, Seo and Yamamoto, 1998):
Φ(λ) =
λ(λ + 1) (λ) ∗
I ({pij }; {psij })
2λ − 1
ただし
⎡
p∗ij
1
(λ)
∗ ⎣
I (·; ·) =
p
λ(λ + 1) i=j ij
psij
p∗ij =
pij
(i = j),
δ
δ=
pij ,
i=j
psij =
(λ > −1),
λ
⎤
− 1⎦ ,
(p∗ij + p∗ji )
2
(i = j).
また，順序カテゴリ分割表においても，対称性からの隔たりを測る尺度が Tomizawa, Miyamoto
and Hatanaka (2001) によって提案されている．しかしながら，これらの尺度は距離の条件を満
たしていないことに注意する．
名義カテゴリ分割表において，{pij + pji > 0} (i = j) を仮定して，対称性からの隔たりを
測る尺度を次のように提案する:
√
2
2 + 2 ∗
∗
Φ =
pij − psij .
2
i=j
ここに Φ∗ は距離の条件を満たす．すなわち Φ∗ は対称性からの隔たりを測る真の距離尺度である．
尺度 Φ∗ は 0 以上 1 以下の値をとり，(i) 対称性が成り立つための必要十分条件は Φ∗ = 0 で
あり，(ii) 対称性からの隔たりが最大 [ここに任意の i < j に対して，pij = 0 または pji = 0 のと
きと定義] であるための必要十分条件は Φ∗ = 1 である．
− 15 −
また，順序カテゴリ分割表における，対称性からの隔たりを測る真の距離尺度も考えられる
(Yamamoto, Fukuda and Tomizawa, 2010)．
r × r 分割表において，(i, j) セル観測度数を nij とする（i = 1, . . . , r; j = 1, . . . , r）．また，
{nij } は多項分布に従うとし，n =
nij とする．さらに，p̂ij = nij /n，Φ̂∗ を Φ∗ の {pij } を
√
{p̂ij } に置き換えた推定値とする．このときデルタ法により， n(Φ̂∗ − Φ∗ ) は漸近的に平均 0，分
散 σ 2 [Φ∗ ] の正規分布に従う．ここに，
⎛
σ 2 [Φ∗ ] =
ただし，
∆ij = (2 +
√
⎞
1
⎝
pij (∆ij )2 − δ(Φ∗ )4 ⎠ ,
4(Φ∗ )2 δ 2 i=j
⎛
2) ⎝
√
√
√
√
√
( pij + pji )2 + pij ( pij − pji )
2 2pij (pij + pji )
⎞
− 1⎠ .
実際のデータ解析においては，σ 2 [Φ∗ ] の {pij } を {p̂ij } に置き換えることにより得られる分散の
推定値を用いて Φ∗ の近似の信頼区間を構成する．また，Yamamoto, Miyamoto, Tsuboi and
Tomizawa (2008) を参照．
References
[1] Agresti, A. (1983). A simple diagonals-parameter symmetry and quasi-symmetry model.
Statistics and Probability Letters 1, 313-316.
[2] Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analysis:
Theory and Practice. Cambridge, The MIT Press.
[3] Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American
Statistical Association 43, 572-574.
[4] Tahata, K., Yamamoto, K. and Tomizawa, S. (2009). Normal distribution type symmetry
model for square contingency tables with ordered categories. The Open Statistics and
Probability Journal 1, 32-37.
[5] Tomizawa, S., Miyamoto, N. and Hatanaka, Y. (2001). Measure of asymmetry for square
contingency tables having ordered categories. Australian and New Zealand Journal of
Statistics 43, 335-349.
[6] Tomizawa, S., Seo, T. and Yamamoto, H. (1998). Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories. Journal
of Applied Statistics 25, 387-398.
[7]Yamamoto, K., Fukuda, M. and Tomizawa, S. (2010). Distance measure of asymmetry for
square contingency tables with ordered categories, submitted.
[8] Yamamoto, K., Kurakami, H., Iwashita, T. and Tomizawa, S. (2008). Contaminated normal
type symmetry model and decomposition of symmetry for square contingency tables.
Journal of Statistical Theory and Practice 2, 651-661.
[9]Yamamoto, K., Miyamoto, N., Tsuboi, H. and Tomizawa, S. (2008). Distance measure
of departure from symmetry for square contingency tables with nominal categories.
International Journal of Pure and Applied Mathematics 48, 483-489.
− 16 −
㊂ሶᖱႎㅢାߩᢙℂ⊛ၮ␆ߩቯᑼൻߦ㑐ߔࠆ⎇ⓥ
ᢙℂࠣ࡞ࡊ 㧔ᖱႎ⑼ቇ⑼ ᷰㆺ⎇ⓥቶ㧕
ᷰㆺ 㧔ᢎ㧕
Abstract. ᧄ⎇ⓥߪޔ㊂ሶᖱႎㅢାℂ⺰ߩ㊀ⷐߥ⺖㗴ߩਛ߆ࠄޔᤓᐕᐲߦᒁ߈⛯߈㧘
․ߦ(ޔ1) ㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߩᢙℂ⊛⎇ⓥ(ޔ߮ࠃ߅ޔ2)㊂ሶ⋧ࠛࡦ࠻ࡠࡇ
ဳዤᐲߦ㑐ߔࠆᖱႎવㅍല₸ߩ⎇ⓥߦὶὐࠍ⛉ࠅࠍ࠲ࡘࡇࡦࠦޔ㚟ߒߡߘࠇࠄߩ
㗴ߩ⸃ߦขࠅ⚵ߺޔᢙℂ⊛ߥቯᑼൻࠍⴕ߁ၮ␆ઃߌࠍⴕ߁ߎߣࠍ⋡⊛ߣߔࠆ߆ߒޕ
ߒߥ߇ࠄ⋡ޔᮡࠍߚߛℂ⺰᭴▽ߛߌߦ⚳ᆎߔࠆߩߢߪߥߊࠄ߆ߎߘޔᓧࠄࠇߚ⚿ᨐ߇ޔ
ᣂߚߥ㊂ሶㅢା߿㊂ሶ⺰ℂࠥ࠻ߥߤࠍ᧪ታൻߔࠆ㓙ߦ‛ޔℂᎿቇ⊛ߦ߽චಽᓎ┙
ߟ߽ߩߣߥࠆߎߣࠍ⋡ᜰߔ߽ߩߢࠆޕ
1㧚ᐔᚑ㧞㧞ᐕᐲߩ⎇ⓥౝኈ
㊂ሶ⊛ߥ․⾰ࠍᜬߟశሶࠍାภߦ↪ࠆశㅢାㆊ⒟ߩ⎇ⓥߪ㧘1980 ᐕઍ㗃ߦ⋓ࠎߦⴕࠊࠇߡ
ߚ߇㧘㊂ሶࠦࡦࡇࡘ࠹ࠖࡦࠣߩ⎇ⓥ߇ᆎࠄࠇߚߎߣߦ㧘ߢߪ㧘㊂ሶᖱႎߣ߁㧘
ᢙቇ‛ℂቇ⸘▚⑼ቇᖱႎ⑼ቇᖱႎᎿቇߩⶄว㗔ၞߦ߹ߚ߇ࠆᣂߚߥಽ㊁߇ᒻᚑߐࠇ
⇇⊛ߦ⎇ⓥ߇ⴕࠊࠇߡࠆޕ㊂ሶᖱႎㅢାㆊ⒟ߩᢙℂ⊛⎇ⓥߢߪ㧘㊂ሶ⏕₸⺰ࠍࡌࠬߣߒߡ㧘
(1) ㊂ሶ╓ภൻߩቯℂ㧘(2) ㊂ሶࠛࡦ࠻ࡠࡇℂ⺰㧘(3) ㊂ሶ࠴ࡖࡀ࡞ℂ⺰㧘ߥߤߩ᭽⎇ߥޘⓥ
߇ⴕࠊࠇߡࠆޕ
ᧂ⸃ߩ߹߹ߩ㊂ሶ╓ภൻߩቯℂߦߟߡߪ㧘(i)ᖱႎḮ╓ภൻߩቯℂ߇㧘ࠪࡘࡑ࠶ࡂ
ࡍ࠶࠷╬ߦࠃߞߡㇱಽ⊛ߦ⼏⺰ߐࠇߡࠆߦࠄߐޕ㧘(ii) ㊂ሶᖱႎㅢାℂ⺰ߩᦨ߽ਛᔃ⊛ߥ⺖
㗴ߩ৻ߟߢࠆ㊂ሶ࠴ࡖࡀ࡞╓ภൻߩቯℂߪ㧘ାภࠍ⺋ࠅߥߊㅍାߔࠆߚߩၮḰࠍਈ߃㧘࠴
ࡖࡀ࡞ࠍ⸳⸘ߔࠆߢ㊀ⷐߥᓎഀࠍᨐߚߔ߽ߩߢࠆޕ
㊂ሶ࠴ࡖࡀ࡞╓ภൻߩቯℂߩ⸃ߦߪ㧘ㅢᏱߩ⏕₸⺰ࠍࡌࠬߣߒߡቯᑼൻߐࠇߡࠆജቇ
⊛ࠛࡦ࠻ࡠࡇ߿ᐔဋ⋧ࠛࡦ࠻ࡠࡇߩ㊂ሶ♽߳ߩዷ㐿߇ਇนᰳߢࠅ㧘ㇱಽઍᢙߩ᭽ޘ
ߥജቇ⊛ࠛࡦ࠻ࡠࡇ߿ㇱಽ⁁ᘒⓨ㑆ߩᐔဋࠛࡦ࠻ࡠࡇߣᐔဋ⋧ࠛࡦ࠻ࡠࡇ߇ቯᑼൻ
ߐࠇߡࠆޕ
ㄭᐕ㧘శࠍାภߦ↪ࠆశㅢାㆊ⒟߇㊂ሶᖱႎㅢାℂ⺰ߦࠃࠅ⼏⺰ߐࠇ㧘᭽⚿ߥޘᨐ߇ᓧࠄ
ࠇߡࠆߦ․ޕ㧘㊂ሶኒᐲ╓ภൻ߿㊂ሶ࠹ࡐ࠹࡚ࠪࡦߥߤߩ㊂ሶ♽ߦ․ߩ㊂ሶࠛࡦ࠲
ࡦࠣ࡞ࡔࡦ࠻ߩᕈ⾰ࠍ↪ߒߚ㊂ሶ࠴ࡖࡀ࡞ߦኻߔࠆᖱႎવㅍߩല₸ࠍ⺞ߴࠆ⎇ⓥߢߪ㧘ࠪࡖ
ࡁࡦℂ⺰ߣߩ⋧㆑ὐ߇ᜰ៰ߐࠇ㧘㊂ሶᐓᷤᕈࠍㅢାㆊ⒟ߩቯᑼൻߩ⎇ⓥߩᔅⷐᕈ߇ต߫ࠇ
ߡࠆޕ
߹ߚ㧘㊂ሶᖱႎㅢାߩ⎇ⓥߪޔㅢାല₸ߩ෩ኒߥ⎇ⓥࠍㅢߒߡశㅢାߩၮ␆ࠍਈ߃ࠆ߽ߩߣ
ߥߞߡ߅ࠅޔ㊂ሶ㊀ߨวࠊߖ⁁ᘒߣ㊂ሶᐓᷤᕈࠍ↪ߡ㘧べ⊛ߥㅦᐲߢℂ⺰⊛ߦ⸘▚ࠍታⴕߢ
߈ࠆ㊂ሶࠦࡦࡇࡘ࠲ߩၮ␆ߣኒធߦߟߥ߇ߞߡࠆߩߎޕ㊂ሶࠦࡦࡇࡘ࠲ߩ⸘▚ㆊ⒟ߪޔ
㊂ሶ⁁ᘒࠍṶ▚ⷙೣߦᓥߞߡ↪ᗧߐࠇߚ㊂ሶ⺰ℂࠥ࠻ࠍㅢߔߎߣߦࠃߞߡታⴕߐࠇࠆ߇ߎޔ
ߩ㊂ሶ⺰ℂࠥ࠻߽ᢙቇ⊛ߦߪࠛࡀ࡞ࠡሽೣࠍḩߚߔ࡙࠾࠲ᄌ឵ߦࠃߞߡߐࠇߡ
ࠆߩࠄࠇߎޕ㊂ሶ⺰ℂࠥ࠻ࠍߤߩࠃ߁ߥ‛ℂ♽ߢታߔࠆ߆ߣ߁⎇ⓥߪޔޔ᭽ࠕߥޘ
ࡊࡠ࠴߇ࠆ߇ߩߘޔᄙߊߪ‛ޔℂᎿቇࠨࠗ࠼߆ࠄߛߌߩࠕࡊࡠ࠴ߢࠅޔℂ⺰⊛ߦ⸃߆
ߥߌࠇ߫ߥࠄߥ᭽ߥޘ㊀ⷐߥ⺖㗴߇ᱷߐࠇߡࠆߣߞߚ⁁ᴫߢࠆޕ
ᧄ⎇ⓥߢߪޔ㊂ሶ♽ߩࠛࡦ࠻ࡠࡇℂ⺰ߣ㊂ሶ࠴ࡖࡀ࡞ℂ⺰ߩ⎇ⓥࠍၮߦޔቢోߦ㊂ሶൻߐ
ࠇߚᖱႎㅢା
㊂ሶᖱႎㅢାࠍᢙℂ⊛ߦቯᑼൻߔࠆߎߣ⋡ᜰߒߩߘޔቯᑼൻߦᔅⷐߣߥࠆၮ␆
ℂ⺰ࠍߢߪࠆ߇߭ߣߟ߭ߣߟ᭴▽ߒߡߊߎߣࠍ⋡⊛ߣߔࠆޕ
− 17 −
2㧚ᐔᚑ㧞㧞ᐕᐲߩ⎇ⓥᣇᴺ
ᧄ⎇ⓥߢߪޔᤓᐕᐲߦᒁ߈⛯߈㧘․ߦ
ޔ㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߩᢙℂ⊛⎇ⓥࠃ߅ޔ
߮
ޔ㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳዤᐲߦ㑐ߔࠆᖱႎવㅍല₸ߩ⎇ⓥߦࡈࠜࠞࠬࠍቯߡޔᢙℂ
⊛ߥ⎇ⓥߣ⸘▚ᯏࠍ↪ߚࠪࡒࡘ࡚ࠪࡦߥߤߩ⚿ᨐࠍ〯߹߃ߡℂ⺰ߩቯᑼൻࠍⴕ߁ߎߣࠍ
⹜ߺࠆౕޕ⊛ߦߪ⎇ޔⓥઍ⠪ߩࠣ࡞ࡊ߇ⴕߞߡ߈ߚ㊂ሶᖱႎㅢାℂ⺰ߩ⎇ⓥࠍၮߦߒߡޔ
᭽⸃ߥޘᨆ⊛ᚻᴺ߿ᖱႎℂ⺰߅ࠃ߮‛ℂቇߦ߅ߌࠆ⻉ᔨࠍขࠅࠇߡ⎇ⓥࠍታᣉߔࠆ⎇ᧄޕ
ⓥߢߪ㧘㊂ሶ♽ߩࠛࡦ࠻ࡠࡇℂ⺰ߣ㊂ሶ࠴ࡖࡀ࡞ℂ⺰ߩ⎇ⓥࠍၮߦ㧘㊂ሶ࠴ࡖࡀ࡞╓ภൻߩ
ቯℂߩቢోߥ⸽ࠍਈ߃ࠆߎߣࠍᦨ⚳⋡ᮡߣߒ㧘ߘߩቯᑼൻߦᔅⷐߣߥࠆᢙℂ⊛ၮ␆ࠍ߭ߣߟ
߭ߣߟⓍߺߍߡߊߎߣࠍ⋡⊛ߣߔࠆޕ
3㧚ᐔᚑ㧞㧞ᐕᐲߩ⎇ⓥ⸘↹
㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳዤᐲߦ㑐ߔࠆᖱႎવㅍല₸ߩ⎇ⓥ
㊂ሶࠛࡦ࠻ࡠࡇℂ⺰ߩ⎇ⓥߪ ޔᐕߦXQP0GWOCPPߦࠃߞߡዉߐࠇߚ㊂ሶࠛࡦ࠻ࡠࡇ
ߦ┵ࠍ⊒ߒޕ㊂ሶ⋧ኻࠛࡦ࠻ࡠࡇߪ7OGICMK߿.KPFDNCF&ޔQPCNFߥߤߦࠃߞߡޔฎౖ♽
ߩ5JCPPQPဳߩ⋧ኻࠛࡦ࠻ࡠࡇߩኻᔕߢ⎇ⓥߐࠇߩߘޔᓟޔ#TCMKޔ7JNOCPPߥߤߦࠃߞߡޔ
ࠃࠅ৻⥸ߩ㊂ሶ♽߳ᒛߔࠆ⎇ⓥ߇ߥߐࠇߡ߈ߚޔߢࠈߎߣޕᖱႎㅢାㆊ⒟ࠍࠛࡦ࠻ࡠࡇࠍ
↪ߡ⸃ᨆߔࠆߢᦨ߽㊀ⷐߥዤᐲ߇⋧ޔࠛࡦ࠻ࡠࡇߢࠆޔߪߩ߽ߩߘࡇࡠ࠻ࡦࠛޕ
ജ⁁ᘒߩ߽ߟᖱႎߩ㊂ࠍߒߡࠆ߇⋧ޔࠛࡦ࠻ࡠࡇߪޔജߩᖱႎ㊂ߩ߁ߜࡀࡖ࠴ޔ
࡞ࠍㅢߒߡജ♽ߦᱜߒߊવ㆐ߐࠇߚᖱႎߩ㊂ࠍߒߡ߅ࠅޔᖱႎㅢାㆊ⒟ߦ߅ߌࠆᖱႎવㅍ
ߩല₸ࠍ෩ኒߦ⺞ߴࠆߢᭂߡ㊀ⷐߥዤᐲߢࠆߩߎޕ㊂ሶ ߢࠆ㊂ሶ⋧ࠛࡦ࠻ࡠࡇ
ߪ ᐕޔ1J[Cߦࠃߞߡቯᑼൻߐࠇޔ㊂ሶㅢା〝ኈ㊂ߩዉࠍᆎ᭽ߥޘᚑᨐ߇ᓧࠄࠇߡ
ࠆߚ߹ޕ㧘࠴ࡖࡀ࡞╓ภൻߩቯℂߪ㧘ࠛ࡞ࠧ࠼ᕈࠍᜬߟㅢାㆊ⒟ߦ߅ߌࠆᖱႎḮߩࠛࡦ࠻ࡠ
ࡇ߇ࠠࡖࡄࠪ࠹ࠖ㧔ㅢା〝ኈ㊂㧕ࠃࠅዊߐ႐วߦߪ㧘ฃାࡔ࠶ࠫ߆ࠄㅍାࡔ࠶ࠫ
ࠍߊࠄߢ߽㜞♖ᐲߢផቯߔࠆߎߣ߇ߢ߈ࠆߎߣࠍ␜ߒߡࠆߜࠊߥߔޕ㧘ᐔဋ⋧ࠛࡦ࠻
ࡠࡇ߇ᖱႎḮߩࠛࡦ࠻ࡠࡇߦ㒢ࠅߥߊㄭߊߥࠆࠃ߁ߥ╓ภൻࠍㆬᛯߢ߈ࠆߣ߁․ᓽࠍᜬ
ߜ㧘㕖Ᏹߦା㗬ᕈߩ㜞╓ภൻߩሽࠍℂ⺰⊛ߦ⸽ߒߡࠆߩߎޕቯℂߦࠃࠅ㧘ㅢᏱߩᖱႎ
ㅢାℂ⺰ߩ⎇ⓥߪ㧘ല₸ߩ⦟╓ภࠍ↢ᚑߔࠆߚߩℂ⺰㧔╓ภℂ⺰㧕߳ߣዷ㐿ߒߡߊߎߣ
ߦߥࠆޕ㧘㊂ሶᖱႎㅢାℂ⺰ߦ߅ߡ㧘࡚ࠪࠕ㧘ࡌࡀ࠶࠻㧘࠾࡞ࡦ╬ߦࠃࠅࠦࡅ
ࡦ࠻ࠛࡦ࠻ࡠࡇ߿ࡦ࠼ࡉ࠶࠻-࠾࡞ࡦࠛࡦ࠻ࡠࡇߥߤߩ㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳ
ߩⶄ㔀ߐߩዤᐲࠍ↪ߚ㊂ሶ࠴ࡖࡀ࡞╓ภൻߩቯℂߩ⼏⺰߇ߥߐࠇߡࠆࠄ߇ߥߒ߆ߒޕ㧘ߎ
ࠇࠄߩዤᐲ߇㧘⽶ߩ୯ࠍขࠅ㧘ࠪࡖࡁࡦߩၮᧄਇ╬ᑼࠍḩߚߐߥߣߞߚᖱႎㅢାߩዤᐲߣ
ߒߡㇺวߩ⦟ߊߥᕈ⾰ࠍᜬߟߎߣ߇ᧄ⎇ⓥઍ⠪㆐ߦࠃߞߡᜰ៰ߐࠇߡࠆޕ
ᧄ⎇ⓥߢߪ㧘ᦨ߽ㆡಾߥ㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳߩዤᐲࠍᭂ㧘㊂ሶ♽ߩࠪࡖࡁࡦ-ࡈࠔࠗ
ࡦࠪࡘ࠲ࠗࡦߩቯℂ(࠴ࡖࡀ࡞╓ภൻߩቯℂ)ߩ⸽ࠍⴕ߁ߚߩၮ␆ઃߌࠍⴕ߁ߎߣࠍ⋡⊛ߣ
ߔࠆޕ
㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߩᢙℂ⊛⎇ⓥ
ㅢᏱߩᖱႎㅢାℂ⺰ߢߪ㧘ജቇ⊛ࠛࡦ࠻ࡠࡇ(㧷㧿(ࠦࡠࡕࠧࡠࡈ-ࠪ࠽ࠗ) ࠛࡦ࠻ࡠࡇ)ࠍ↪
ߡᐔဋ⋧ࠛࡦ࠻ࡠࡇ㧔ᖱႎ㊂㧕߇ቯࠄࠇ㧘ߘࠇࠍၮߦߒߡ╓ภൻߩቯℂ߇৻⥸ൻߐࠇ
ߡࠆߩߎޕ㧷㧿ࠛࡦ࠻ࡠࡇࠍ㊂ሶ♽ߦᒛߒࠃ߁ߣߔࠆ⹜ߺߪ㧘ࠦࡦ-ࠬ࠻࡞ࡑ㧘ࠛࡓ
ࠪࡘ, ࠦࡦ-࠽ࡦࡈࠜ࠶ࡈࠔ-࠴ࠖࡦࠣ (㧯㧺㨀), ࡄࠢ, ࠕ࠷ࠠ-ࡈࠔࡀࠬ(㧭㧲), ᄢ
⍫
%QORNGZKV[ࠕࠞ࡞࠺ࠖᄢ⍫ᷰㆺ
㧭㧻㨃㧘ࠦࠨࠦ࠙ࠬࠠᄢ⍫ᷰㆺ
㧷㧻㨃╬ߦࠃ
ߞߡߥߐࠇߡࠆߩߎޕജቇ⊛ࠛࡦ࠻ࡠࡇߪ㧘♽ߩജቇ⊛⊒ዷߦߞߡᄌൻߒߚ⁁ᘒ߇ᜬߟ
− 18 −
ᐔဋ⊛ߥᖱႎ㊂ࠍߒߡࠆ⎇ᧄޕⓥઍ⠪㆐ߪ㧘㧭㧻㨃ࠛࡦ࠻ࡠࡇ㧷㧻㨃ࠛࡦ࠻ࡠࡇ
߮ᖱႎജቇߩⶄ㔀ߐߩᔨࠍ↪ߡ㊂ሶᐔဋࠛࡦ࠻ࡠࡇߣ㊂ሶᐔဋ⋧ࠛࡦ࠻ࡠࡇߩቯᑼ
ൻࠍⴕ㧘ߎࠇࠄߩ⋧㑐ଥߦߟߡ⎇ⓥࠍⴕߞߚޕ
ᧄ⎇ⓥߢߪ㧘㊂ሶ♽ߩ࠴ࡖࡀ࡞╓ภൻߩቯℂߩ⸽ࠍߔࠆߚߩၮ␆ઃߌࠍⴕ߁ߚߦ㧘㧷
㧻㨃ࠛࡦ࠻ࡠࡇࠍࡌࠬߦᣂߚߥ৻⥸ൻ㧭㧻㨃ࠛࡦ࠻ࡠࡇࠍዉߔࠆߦࠄߐޕ㧘㊂ሶࠛࡦ
࠲ࡦࠣ࡞ࡔࡦ࠻ߩ⁁ᘒᄌൻࠍขࠅㄟࠎߛ㊂ሶᐔဋ⋧ࠛࡦ࠻ࡠࡇߩቯᑼൻࠍⴕ㧘ߘߩᕈ⾰
ࠍ⺞ߴࠆߦࠄߐޕ㧘KOW ജቇ⊛ࠛࡦ࠻ࡠࡇߩቯᑼൻࠍၮߦ㧘ജߦࠬࠢࠗ࠭࠼శ⁁ᘒࠍ↪
㧘శ㔀㖸࠴ࡖࡀ࡞ߣ㊂ሶࡑ࡞ࠦࡈㆊ⒟ߦࠃߞߡ᭴ᚑߐࠇߚജቇ♽ߦኻߒߡ㧘৻⥸ൻߐࠇߚ
AOW ࠛࡦ࠻ࡠࡇࠍ⸘▚ߒ㧘ജ⁁ᘒߩᄌൻߦኻߔࠆ㧘㊂ሶജቇ⊛ࠛࡦ࠻ࡠࡇߩᝄࠆ⥰
ߦߟߡ෩ኒߦ⺞ߴࠆ੍ቯߢࠆޕ
4㧚ᐔᚑ㧞㧞ᐕᐲߩ⎇ⓥᚑᨐ
㊂ሶ࠴ࡖࡀ࡞ℂ⺰ߪޔ㊂ሶ♽ߩ⁁ᘒᄌൻࠍᛒ߁ℂ⺰ߢࠅޔ㊂ሶ♽ߩᖱႎㅢାㆊ⒟ࠍ⸥ㅀߔࠆ
ߚߦ↪ࠄࠇ৻ޔᣇ߇ฎౖ♽ߩඨฎౖ⊛࠴ࡖࡀ࡞ߩዉ߿ޔ1J[C ߦࠃࠆޔቢోߥ㊂ሶ♽ߦ߅
ߌࠆ㊂ሶജቇ⊛࠴ࡖࡀ࡞ߩᢙቇ⊛ቯᑼൻߣᷫㆊ⒟߿శ㔀㖸ㆊ⒟ߩ࠴ࡖࡀ࡞ࡕ࠺࡞(ᷫ࠴ࡖ
ࡀ࡞శ㔀㖸࠴ࡖࡀ࡞)ߩዉߥߤ᭽⎇ߥޘⓥ߇ⴕࠊࠇߡ߈ߚޔߢࠈߎߣޕᄖ⇇ߩᓇ㗀ࠍ⠨ᘦߔ
ࠆᖱႎㅢାㆊ⒟ߪޔ㧞ߟએߩ♽ࠍขࠅᛒ߁ߎߣ߇ᔅⷐߢࠅ⁁ߩ♽ోޔᘒߩ㊀ⷐߥᕈ⾰(ᱜ୯
ᕈ)ࠍሽߔࠆߚߦߪ⁁ࠆߌ߅ߦ♽ోޔᘒᄌൻࠍ⸥ㅀߔࠆ㊂ሶ࠴ࡖࡀ࡞ߦቢోᱜ୯ᕈߥߤߩᱜ
୯ᕈࠃࠅዋߒᒝ᧦ઙࠍઃߌട߃ࠆߎߣ߇ᔅⷐߢࠆޕ㊂ሶᖱႎㅢାㆊ⒟ߩ⎇ⓥߪޔㅢାല₸
ߩ෩ኒߥ⎇ⓥࠍㅢߒߡశㅢାㆊ⒟ߩၮ␆ࠍਈ߃ࠆ߽ߩߦߥߞߡ߅ࠅߦࠄߐޔ㊂ሶ㊀ߨวࠊߖ⁁
ᘒߣ㊂ሶᐓᷤᕈࠍ↪ߡ㘧べ⊛ߥࠬࡇ࠼ߢℂ⺰⊛ߦ⸘▚߇น⢻ߥ㊂ሶࠦࡦࡇࡘ࠲ߩၮ␆ߣ
ኒធߦߟߥ߇ߞߡࠆޔߚ߹ޕ㊂ሶࠦࡦࡇࡘ࠲ߩ⸘▚ㆊ⒟ߪޔ㊂ሶ⁁ᘒࠍṶ▚ⷙೣߦᓥߞߡ
↪ᗧߐࠇߚ㊂ሶ⺰ℂࠥ࠻ࠍㅢߔߎߣߦࠃߞߡታⴕߐࠇࠆߩߎޕ㊂ሶ⺰ℂࠥ࠻ߪࠛࡀ࡞ࠡ
ሽೣࠍḩߚߔ࡙࠾࠲↪⚛ߦࠃߞߡ᭴ᚑߐࠇࠆ࡙࠾࠲࠴ࡖࡀ࡞ߦࠃߞߡᢙቇ⊛ߦ
ߐࠇߡࠆߩߎޕ㊂ሶ⺰ℂࠥ࠻ࠍߤߩࠃ߁ߥ‛ℂ♽ࠍ↪ߡቯᑼൻߔࠆ߆ߣ߁⎇ⓥߪޔ
ޔ᭽ߛ߹ޔ߇ࠆߡࠇࠊⴕߢ࠴ࡠࡊࠕߥޘℂ⺰⊛ߦ⸃ߒߥߌࠇ߫ߥࠄߥ㗴߇ጊⓍߺ
ߩ⁁ᴫߢࠆޕ
ᧄᐕᐲߪޔએߩ⺖㗴ߩਛ߆ࠄᤓᐕᐲߦᒁ߈⛯߈㧘․ߦ
ޔ㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߦ
㑐ߔࠆ⎇ⓥ
ޔ߮ࠃ߅ޔ㊂ሶᓮㅢାㆊ⒟ߣ㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳዤᐲߦ㑐ߔࠆ⎇ⓥߦὶὐ
ࠍ⛉ࠅޔ⸥ߩ⎇ⓥ⸘↹ߦᓥߞߡ⎇ⓥࠍታᣉߒޔએਅߩᚑᨐࠍᓧߚޕ
㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߩᢙℂ⊛⎇ⓥ
㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇߪ㧘1975 ᐕߏࠈ ࠛࡓࠪࡘߣࠦࡦ-ࠬ࠻࡞ࡑߦࠃߞߡᦨೋߦዉ
ߐࠇ㧘 ᐕߦߪ㧘ࠦࡦ࠽ࡦࡈࠜ࠶ࡈࠔ࠴ࠖࡦࠣ߇㧯♽ߦ߅ߡ㧯㧺㨀ജቇ⊛ࠛ
ࡦ࠻ࡠࡇ
%06F[PCOKECNGPVTQR[ࠍቯ⟵ߒߚߪࠢࡄޕ㧘ߊߟ߆ߩࡕ࠺࡞ߦߟߡ㧘㧯
㧺㨀ജቇ⊛ࠛࡦ࠻ࡠࡇࠍ⸘▚ߒߚߚ߹ޕ㧘 ᐕߦߪ㧘ࠕ࠷ࠠࡈࠔࡀࠬ߇නߩ㒢
↪⚛ಽഀࠍ↪ߡ㧭㧲ജቇ⊛ࠛࡦ࠻ࡠࡇ
#(F[PCOKECNGPVTQR[ࠍቯ㧘ࡂ࠺࠶࠻ߪ⋧
ࠛࡦ࠻ࡠࡇߦ㑐ㅪߒߡജቇ⊛ࠛࡦ࠻ࡠࡇࠍ⺰ߓߚߦࠄߐޕ㧘ࡏࠗࠠࡘࠬࠢߪ㧘⥄↱⏕₸
ࠍ߽ߣߦ৻⥸ൻߐࠇߚㄭૃߩࠕࡊࡠ࠴ࠍࡌࠬߣߒߡ㧯߮㨃ઍᢙߩ⥄Ꮖหဳ౮ߦኻ
ߔࠆജቇ⊛ࠛࡦ࠻ࡠࡇࠍዉߒߚ ޕᐕߦߪ㧘ࠕࠞ࡞࠺ࠖᄢ⍫ᷰㆺ߇㧘㊂ሶࡑ࡞ࠦࡈ
ㅪ㎮ࠍㅢߒߡ㧘㧭㧻㨃ജቇ⊛ࠛࡦ࠻ࡠࡇ
#19F[PCOKECNGPVTQR[ࠍቯ⟵ߒ㧘㊂ሶᖱႎℂ⺰ߣ
㑐ㅪߔࠆߊߟ߆ߩ㊂ሶ࠴ࡖࡀ࡞ߩࡕ࠺࡞ߦኻߒߡജቇ⊛ࠛࡦ࠻ࡠࡇߩ⸘▚ࠍⴕߞߡࠆޕ
ߎߩ㧭㧻㨃ജቇ⊛ࠛࡦ࠻ࡠࡇߪ㧘ᓥ᧪ߩᚻᴺࠍ↪ߡቯࠄࠇߚ㊂ሶ♽ߩജቇ⊛ࠛࡦ࠻ࡠࡇ
㧔㧯㧺㨀㧭㧲㧕ࠃࠅ߽♽ߩ⁁ᘒࠍߒߊಽ㘃ߔࠆߎߣ߇ߢ߈㧘ߐࠄߦജቇ⊛ࠛࡦ࠻ࡠࡇ
− 19 −
ࠍ᳞ࠆ⸘▚߇ઁߩ߽ߩߦᲧߴߡኈᤃߢࠆߣ߁․ᓽࠍᜬߟࠠࠬ࠙ࠦࠨࠦޕ-ᄢ⍫-ᷰㆺߪ㧘
㧭㧻㨃ߣ㧭㧲ࠍࠃࠅ৻⥸⊛ߥ♽ߦኻߒߡቢోᱜ୯౮ߦ㑐ߔࠆ㧷㧻㨃ജቇ⊛ࠛࡦ࠻ࡠࡇ
-19F[PCOKECNGPVTQR[ࠍቯᑼൻߒߚޕ㧷㧻㨃ജቇ⊛ࠛࡦ࠻ࡠࡇߪ㧘㧭㧻㨃ജቇ⊛ࠛࡦ࠻ࡠ
ࡇߦᲧߴߡࠃࠅᄙߊߩࡕ࠺࡞ߦኻߒߡㆡ↪ߔࠆߎߣ߇ߢ߈ࠆޕ
ᧄ⎇ⓥߢߪ㧘ߎࠇࠄߩജቇ⊛ࠛࡦ࠻ࡠࡇߩ㑐ଥࠍ⺞ߴ㧘ߟߊ߆ߩᢙℂࡕ࠺࡞ߦኻߒߡജ
ቇ⊛ࠛࡦ࠻ࡠࡇߩ⸘▚ࠍⴕߞߚߦࠄߐޕ㧘 ᐕߦߪ㧘ᄢ⍫߇㧘㧯ᷙวࠛࡦ࠻ࡠࡇࠍࡌ
ࠬߣߒߡ㊂ሶ♽ߦജቇ⊛ࠛࡦ࠻ࡠࡇߣജቇ⊛⋧ࠛࡦ࠻ࡠࡇࠍቯᑼൻߒ㧘ᄙᲑᷫ࠴ࡖ
ࡀ࡞ߦኻߒߡశᄌ⺞ߩല₸ࠍ෩ኒߦ⺞ߴࠆ⎇ⓥࠍⴕߞߚޕ
㊂ሶ⋧ࠛࡦ࠻ࡠࡇဳዤᐲߦ㑐ߔࠆᖱႎવㅍല₸ߩ⎇ⓥ
㊂ሶ♽ߦ߅ߌࠆ࠴ࡖࡀ࡞ߩ⎇ⓥߢߪ㧘ࡎࡏߦࠃߞߡ㧘ඨฎౖ⊛ (৻ᣇ߇ฎౖ♽ߩ) ࠴ࡖࡀ
࡞߇ዉߐࠇ㧘ߐࠄߦ㧘ᄢ⍫ߦࠃߞߡ㧘㊂ሶജቇ⊛ (ቢోߥ㊂ሶ♽ߦ߅ߌࠆ) ࠴ࡖࡀ࡞߇ቯᑼ
ൻߐࠇߡࠆߦ․ ޕ㧘శㅢାㆊ⒟ߣߩ㑐ㅪߢߪ㧘ᄢ⍫ߦࠃࠆᷫㆊ⒟ࠍߔ࠴ࡖࡀ࡞ߩᢙℂࡕ
࠺࡞ߩቯᑼൻߩ⎇ⓥࠍߍࠆߎߣ߇ߢ߈ࠆߩߎޕ㊂ሶ࠴ࡖࡀ࡞ߩ⎇ⓥߪ㧘৻⥸⊛ߦߪ㕖น឵ߥ
♽߆ࠄ㕖น឵ߥ♽߳ߩᄌ឵ࠍขࠅᛒ߁߽ߩߢࠆ߇㧘㕖น឵♽߇น឵♽ࠍߣ߁ᢙቇ᭴ㅧ
ࠍ⠨߃ࠆߣ㊂ሶ࠴ࡖࡀ࡞ߪඨฎౖ⊛ߥ࠴ࡖࡀ࡞߿ฎౖ⊛ߥ࠴ࡖࡀ࡞ߩ⼏⺰߹ߢࠍ߽ࠎߛࠃࠅ
৻⥸⊛ߥߣ⸒߁ߎߣ߇ߢ߈ࠆⷰߥ߁ࠃߩߎ ޕὐ߆ࠄ㧘ฎౖ♽߆ࠄ㊂ሶ♽߳ߩฎౖ-㊂ሶ࠴
ࡖࡀ࡞߿㧘㊂ሶ♽߆ࠄฎౖ♽߳ߩ㊂ሶ-ฎౖ࠴ࡖࡀ࡞ߥߤ߽㊂ሶ࠴ࡖࡀ࡞ߩࠆ․ߥ႐วߣߒ
ߡขࠅᛒ߁ߎߣ߇ߢ߈㧘ฎౖ-㊂ሶ࠴ࡖࡀ࡞㧘㊂ሶ࠴ࡖࡀ࡞㧘㊂ሶ-ฎౖ࠴ࡖࡀ࡞ߣ߁৻ㅪߩ
વㅍㆊ⒟߇৻⽾ߒߚᢙℂ᭴ㅧࠍ↪ߡ⛔৻⊛ߦ⼏⺰ߔࠆߎߣ߇ߢ߈ࠆߩߢࠆޕ
ᧄ⎇ⓥߢߪ㧘㊂ሶኒᐲ╓ภൻ߿㊂ሶ࠹ࡐ࠹࡚ࠪࡦߥߤߩ㊂ሶࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ࠍ
㊂ሶ࠴ࡖࡀ࡞ߩ․ᓽઃߌࠍⴕ㧘1J[C ⋧ࠛࡦ࠻ࡠࡇߣ㧘%QJGTGPVKPHQTOCVKQP߿
0KGNUGP ߩࠛࡦ࠻ࡠࡇߣߞߚ⋧ࠛࡦ࠻ࡠࡇဳዤᐲߣߩᲧセࠍశ㔀㖸࠴ࡖࡀ
.KPFDNCF
࡞ߢⴕ㧘%QJGTGPVKPHQTOCVKQP߿.KPFDNCF0KGNUGP ߩࠛࡦ࠻ࡠࡇ߇ᛴ߃ࠆ㗴ὐౕޔ
⊛ߦߪ㧘࡚ࠪࠕ, ࡌࡀ࠶࠻, ࠾࡞ࡦ╬ࡉ࠼ࡦ߿ࡇࡠ࠻ࡦࠛ࠻ࡦࡅࠦࠅࠃߦޘ
࠶࠻-࠾࡞ࡦࠛࡦ࠻ࡠࡇߥߤߩዤᐲߪ㧘(i) ⽶ߩ୯ࠍขࠅ㧘(ii) ജᖱႎ㊂ࠃࠅᄢ߈ߊߥ
ࠆߥߤߩਇㇺวߥὐࠍᜰ៰ߒ㧘ᖱႎㅢାߩዤᐲߣߒߡലߢߥߎߣࠍ␜ߒߚߜࠊߥߔޕ㧘1J[C
⋧ࠛࡦ࠻ࡠࡇ߇శ㔀㖸࠴ࡖࡀ࡞ߦኻߒߡ߽ᦨ߽⋧ᔕߒዤᐲߢࠆߎߣࠍ␜ߒߚޕ
− 20 −
ࡏࠬࠕࠗࡦࠪࡘ࠲ࠗࡦಝ❗ේሶߩ
1 ᰴరశᩰሶᝒߣේሶᐓᷤ
ᢙℂࠣ࡞ࡊ㧔ℂᎿቇㇱ‛ℂቇ⑼⋓᳗⎇ⓥቶ㧕
ᒄశ㧔&㧕㧘⋓᳗◊㇢㧔ᢎ㧕
Abstract. ࡏࠬࠕࠗࡦࠪࡘ࠲ࠗࡦಝ❗ߒߚ࠽࠻࠙ࡓේሶࠍ 1 ᰴరశᩰሶߦᝒߒޔ
2 శሶࡑࠗࠢࡠᵄ̆ࠫࠝᵄㆫ⒖ࠍ↪ߡේሶᐓᷤታ㛎ࠍⴕߞߚޕF=1,mF=-1ψF=2,
mF=1 ㆫ⒖ߢ 2 ms ߩࠦࡅࡦࠬᤨ㑆ߦኻߒߡ㞲ᐲ 67%ߩࡓᐓᷤାภࠍᓧࠆߎ
ߣ߇ߢ߈ߚޕ
1㧚ߪߓߦ
ᤓᐕᐲ߹ߢߦࡏࠬࠕࠗࡦࠪࡘ࠲ࠗࡦಝ❗㧔BEC㧕ߒߚ࠽࠻࠙ࡓේሶࠍ⏛᳇࠻࠶
ࡊ߆ࠄ⸃ߒޔේሶᐓᷤ⸘ࠍ᭴ᚑߒ 10 ms ߩᐓᷤᤨ㑆ߢ㞲ᐲ߶߷ 1 ߩᐓᷤାภ߇ᓧࠄࠇࠆ
ࠍ⸽ߒߚ[ޕ1] ߒ߆ߒߣߞ߽᧪ᧄޔ㐳ᐓᷤᤨ㑆߇น⢻ߢࠆߦ߽߆߆ࠊࠄߕޔ㊀ജߩ⪭ਅߩ
ᓇ㗀ߩߚ⺃ዉࡑࡦࡄ࡞ࠬߩᾖ㗔ၞ߆ࠄᄖࠇߡࠇߘޔએߩࠦࡅࡦࠬᤨ㑆ࠍ᷹ቯߔࠆߎ
ߣ߇ߢ߈ߥ߆ߞߚޔߢߎߘޕᐕᐲߪࠬࡏࡓ࠙࠻࠽ޔಝ❗ේሶࠍ 1 ᰴరశᩰሶߦᝒߒޔ2
శሶࡑࠗࠢࡠᵄ̆ࠫࠝᵄ㧔MW-RF㧕[2] ࠍ↪ߡ 1 ᰴࡑࡦࠪࡈ࠻ߩߥ F=1,mF=-1ψF=2,
mF=1 ㆫ⒖ߩಽశߣᐓᷤታ㛎ࠍⴕߞߚޕ1 ᰴరశᩰሶߦࠃࠆᝒߣ 2 శሶ MW-rf ಽశߣᐓᷤታ㛎
ߩ⚿ᨐࠍㅀߴࠆޕ
2㧚ࡏࠬಝ❗ේሶߩ 1 ᰴరశᩰሶᝒ
2㧚1 ৻ᰴరశᩰሶ
శߩේሶߦ߷ߔജߦߪᢔੂജߣᭂሶജ߇ࠆޕ೨⠪ߪේሶߩᷫㅦ߿⏛᳇శቇ࠻࠶ࡊߥߤ
ߦࠊࠇࠆജߢࠆ߇ޔᓟ⠪ߪశᒝᐲߩ൨㈩ߦࠃߞߡ߈ޔේሶࠍశߩᒝᐲᦨᄢ㧔߹ߚߪ㔖㧕ߩ
ὐߦ㐽ߓㄟࠆജߢࠆޕశߩᒝᐲ߿㔌⺞ᵄᢙߦ߽ࠃࠆ߇ޔᄢᄌᒙജߢޔ100 PK એਅߩ᷷
ᐲߢߊࠬࡏޔߡߞ߇ߚߒޕಝ❗ේሶߩ᭽ߥૐ᷷ේሶࠍ㐽ߓㄟࠆߩߦㆡߒߡࠆߩߎޕ႐
ว⿒ᣇ㔌⺞శߢశᒝᐲߩᦨᄢߩߣߎࠈߦޔ㕍ᣇ㔌⺞శߢߪశߩή႐ᚲߦේሶࠍᝒߢ߈ࠆޕ1
ᣇะߦㅴⴕߔࠆశࠍ㓸᧤ߒߚߣ߈ޔὶὐ⟎ߦේሶࠍ㐽ߓㄟࠆ႐วࠍޔశᭂሶ࠻࠶ࡊߎޔ
ߩశࠍ㏜ߢ᛬ࠅߒޔ᧪ߚቯᵄߩඨᵄ㐳Ფߦ➅ࠅߔ⣻ߩ⟎ߦේሶࠍ㐽ߓㄟࠆᣇᑼࠍޔ
శᩰሶ࠻࠶ࡊߣ⒓ߔࠆޕశᩰሶߩࠪࡘ࠲࡞ࠢࡐ࠹ࡦࠪࡖ࡞ߪᰴߩᑼߢਈ߃ࠄࠇࠆ(࿑ 1)ޕ
U U, z § U 2 · 1 cos 2kz
U 0 exp ¨ 2 2 u
2
© Z ¹̧
(1)
0
ߎߎߢޔU0 ߪಽᭂ₸ࠍ㊂ߢࠆޕw0 ߪ㓸᧤
⟎ߢߩࠩߩࡆࡓᓘߢࠆޕ
ゲᣇะ㨦ޔേᓘᣇะǹߘࠇߙࠇߩࡐ࠹ࡦࠪࡖ࡞ߩ
⸘▚⚿ᨐࠍ࿑ 2 ߦ␜ߔޔߢߎߎޕశߪ᳓ᐔᣇะߦવ
៝ߒޔy ᣇะࠍ㊀ജᣇะߣߒߚޔߡߞ߇ߚߒޕy ᣇ
ะߦߪߪ㊀ജߩᓇ㗀߇ࠆߩߢߘߩಽ߽ ⠨ᘦߒߡࠆޔࠄ߆ࠄࠇߎޕᵄ㐳 1.06 Pmޔశᒝᐲ
350 mW, ࡆࡓᓘ 40 Pm ߩ႐วޔ30 ǍK ߩࡐ
࠹ࡦࠪࡖ࡞߇᧪ߡࠆߣⓍ߽ࠇߚޕBEC ߩ᷷
1
− 21 −
࿑ 1 1 ᰴరశᩰሶࡐ࠹ࡦࠪࡖ࡞
ᐲߪ 1 ǍK એਅߢࠆߩߢޔMT ߆ࠄߩේሶࠍᝒߔࠆߦߪචಽߥࡐ࠹ࡦࠪࡖ࡞߇ߢ߈ߡࠆߎ
ߣ߇ࠊ߆ࠆޔߚ߹ޕゲᣇะߣേᓘᣇะߩߘࠇߙࠇߩ࠻࠶ࡊᵄᢙ ƺz, ƺǒ ߪࠇߙࠇߘޔࠄ߆ޔ
࿑ 2 ᚑߒߚశᩰሶࡐ࠹ࡦࠪࡖ࡞
137 kHz ߣ 580Hz ߣ⸘▚ߔࠆߎߣ߇᧪ߚޕ
:z
:U
k
2 D 8P
m 2H 0 c SZ 02
(2)
1 4 D 8P
w m 2H 0 c S w 2
(3)
శᩰሶߩࠪࡘ࠲࡞ࠢࠪࡈ࠻ߩ⸘▚ߣࠪࡘ࠲࡞ࠢࡐ࠹ࡦࠪࡖ࡞ࠍⓍ߽ߞߚ࠻ࡈࠪࠢ࡞࠲ࡘࠪޕ
ߪޔ
Q ac
D Z , e
IL
2H 0 ch
(4)
ߢߐࠇޔǂ ߪಽᭂ₸ޔIL ߪࠩᒝᐲࠍ␜ߒߡࠆ⎇ᧄޕⓥߢߪޔ350 mW ߩࠩࠍ 40
Ǎm ߹ߢ㓸శߔࠆߩߢߩᤨߩߘޔᒝᐲߪޔIL=0..35/(Ǒ2010-6)2§2.8108 [W/m2](ޔ2.3.7)ᑼࠃ
ࠅޔF=1, 2 ߩಽᭂ₸ ǂ ߇ 3.84×10-39[J/(v/m)]ߣⓍ߽ࠄࠇࠆޕF=1, 2 Ḱߩߘࠇߙࠇߩࠪࡘ࠲࡞
ࠢࠪࡈ࠻ߪ(ޔ6.1.1)ᑼࠃࠅޔ-305.656kHzޔ-305.661kHz ߣⓍ߽ࠄࠇߚޕ
2㧚2 శᩰሶߩታ㛎ⵝ⟎
1 ᰴరశᩰሶࠍ↪ߚశቇ♽ࠍ
࿑ 3 ߦ␜ߔޕ1064 nm ߩ YAG
ࠩ(✢ 1 kHz, ജ 500
mW)ߪޔ࠲࠰ࠗࠕޔnj/2ޔnj/4ޔ
㔚᳇శቇᄌ⺞ེ(EOM)ޔశࡆ
ࡓࠬࡊ࠶࠲(PBS)ࠍㅢߒߚᓟޔ
f=500 mm ߩࡦ࠭ߦࠃߞߡ㓸
శߐࠇࠆߞࠃߦࡒޔᤨߩߎޕ
ߡߐߖߥ႐วޔᭂሶ࠻
࠶ࡊࠍ᭴ᚑߢ߈ࠆޔߚ߹ޕf=500
mm ߩಳ㕙㏜ߢߔࠆߎߣߦ
ࠃߞߡޔz ゲᣇะߦᦼࡐ࠹ࡦࠪ
ࡖ࡞ߩశᩰሶࠍ᭴ᚑߒޔේሶࠍ
ᝒߔࠆߎߣ߇ߢ߈ࠆ࠰ࠗࠕޕ
࠲ߪޔᚯࠅశ߇ࠩߦࠄ
2
− 22 −
࿑ 3 శᩰሶᝒታ㛎ⵝ⟎࿑
ߥࠃ߁ߦ㈩⟎ߒߡ߅ࠅޔnj/4 ᧼ߣ EOM ߪࠬ࡞ࠤ࠶ࡐޔലᨐࠍ↪ߚశߩࠬࠗ࠶࠴ࡦࠣࠍⴕ߁
ߚߦ↪ߔࠆޔߚ߹ޕnj/2 ᧼ߣ PBS ߪޔnj/2 ᧼ࠍ⺞ᢛߔࠆߎߣߢޔశᩰሶߩࡄࡢߩ⺞▵ࠍⴕ
߁ߩߦ↪ߩ࡞ࠬࠟޔਛᔃߢᦨᄢ 350 mW ߹ߢജࠍนᄌߔࠆߎߣ߇ߢ߈ࠆࠟޔߚ߹ޕ
ࠬ࡞ߩਛᔃߢޔశᩰሶశߩࡆࡓ࠙ࠚࠬ࠻ߪ w0=40 Ǎm ߢߩᤨߩߘޔ㐳ߪ 4.7 mm ߢ
ࠆⷰޔߚ߹ޕ᷹ߪޔේሶࠍ࠻࠶ࡊߒߚ߹߹ޔ㊀ജᣇะߦࡊࡠࡉశࠍᾖߔࠆߎߣߦࠃࠅޔ
࠻࠶ࡊߐࠇߚේሶࠍ᷹ⷰߔࠆޔߚ߹ޕMW-RF ㆫ⒖ಽశࠍⴕ߁ߚߩ࡞ࠬࠟޔ࿁ࠅߦࡑࠗ
ࠢࡠᵄ↪ߩ࠳ࠗࡐ࡞ࠕࡦ࠹࠽ࠍ㈩⟎ߒࠝࠫߚ߹ޔᵄ↪ߦࠟࠬ࡞ߦ㌃✢ࠍ 10 Ꮞ߈ߒߡ
ࠆޕ
2㧚3 శᩰሶߩᝒ
ಳ㕙㏜ߢߒߥᤨޔMT ߆ࠄᭂሶ࠻࠶ࡊߒߚᤨߩๆ↹ࠍ࿑ 4 ߦ␜ߔߩࠇߙࠇߘޕ
ๆ↹ߪޔMT ߆ࠄ⒖ⴕᓟ 40 ms ߆ࠄ 1s ߹ߢේሶ߇࠻࠶ࡊߐࠇߡࠆ᭽ሶࠍ␜ߒߡࠆޕ
ߒ߆ߒޔゲᣇะߩ࠻࠶ࡊ߇ᒙߚߦᤨޔ㑆߇⚻ߟߦߟࠇޔᐢ߇ߞߡࠆߎߣ߇ࠊ߆ࠆޕ
࿑ 4 శᭂሶ࠻࠶ࡊ
ᰴߦޔಳ㕙㏜ߢ࠻࠶ࡊశࠍ᛬ࠅߔߎߣߦࠃߞߡశᩰሶࠍ↢ᚑߔࠆࠇߐࡊ࠶࠻ޔᤨߩߘޕ
ߚᤨߩ᭽ሶࠍ࿑ 5 ߦ␜ߔޕᭂሶ࠻࠶ࡊߣ㆑ޔหߓ⟎ߦ⇐߹ߞߡ߅ࠅޔේሶߩゲᣇะߩ
㐽ߓㄟ߇᧪ߡࠆߎߣ߇ࠊ߆ࠆޕBEC ߩ࠻࠶ࡊല₸ߪ 50㧑ߢߞߚޕ
࿑ 5 శᩰሶ࠻࠶ࡊ
ߎߩᤨߩ࠻࠶ࡊᤨ㑆ߣේሶᢙߩ㑐ଥࠍ⺞ߴߚޕ1 ߮ࠃ߅ޔ3 ߩ៊ᄬଥᢙࠍ᳞ߚޕi ៊
ᄬଥᢙࠍ Ki ߣߔࠆߣ
dN
dt
K1 N K 3 N n 2
(5)
ࠍ↪ߡޕ߁ⴕࠍࠣࡦࠖ࠹࠶ࠖࡈޔK3 ߪޔኒᐲ߇ 1014 ࠍ߃ߡࠆߣലߡߊࠆ㗄ߢࠆޕ
࿁ߪࠢࡇޔኒᐲ߇⚂ 31015 ⒟ᐲߞߚߩߢ⠨ᘦߔࠆᔅⷐ߇ࠆߩࠇߙࠇߘޔᤨߩߎޕଥᢙߪޔ
K1=(1.99±0.04) s-1, K3=(1.22±0.12)10-30 cm6 s-1 ߣߥߞߚޕK3 ୯ߪޔK3=2.1.210-30cm6s-1[3]
ߦㄭ୯ߣߥߞߡࠆޔߚ߹ޕMT ኼߪ 40 ⑽ߢޔK1 ߪޔ1/40=0.025 s-1 ߣⓍ߽ࠄࠇࠆ߇ޔ୯
߇⇣ߥߞߡࠆߩߪޔK1 ࠍⓍ߽ࠆ㓙ߦኒᐲߩᤨ㑆⊛ᄌൻࠍ⠨ᘦߒߡߥߚ୯߇ᄢ߈ߊߡ
3
− 23 −
ߒ߹ߞߡࠆޕ
3㧚 2 శሶࡑࠗࠢࡠᵄ̆ࠫࠝᵄಽశ
3㧚1 ಽశ
ᰴߦޔశᩰሶߦᝒߐࠇߚේሶߦኻߒߡ 2 శሶࡑࠗࠢ
ࡠᵄ㧙ࠫࠝᵄ㧔MW-RF㧕ㆫ⒖ಽశࠍߒߚޕF=1, mF=-1
ߩ BEC ࠍశᩰሶߦᝒߔࠆߎߣ߆ࠄޔǻ 㧗 㧙ǻ 㧗శ
MW-RF 2 శሶߪޔ࿑ 6 ߩࠃ߁ߦ ޔF=1, mF=-1ńF=2,
mF=+1 ߩㆫ⒖ࠍߎߔޕ㊂ሶൻ⏛႐ߩᄢ߈ߐ 67.7 ǍT ࠍ
ශടߒߚᤨࡦࡑޔḰ㑆ߩࠪࡈ࠻㊂ߪޔ474 kHz
ߢࠆޕF=1, mF=-1 ߩේሶߦኻߒߡޔF=1, mF=-1ńF=1,
mF=+1 ಽశࠍⴕߞߚޕ᷹ቯߩᤨ㑆♽ࠍ࿑ 7 ߦ␜ߔ߹ޕ
ߕޔMT ਛߢ BEC ࠍ↢ᚑߒߚᓟߦ㊂ሶൻ⏛႐ 67.7ǍT ┙
ߜߍߩߘޔᓟචಽߦ⏛႐߇ቯߒߚ 100 ms ᓟߦ 2 శ
ሶ MW-RF ࿑ 6 2 శሶ MW㧙RF ㆫ⒖ㆫ⒖ࠍⴕ߁ߘޕ
ߒߡޔశᩰሶਛߩ F=2 ⁁ᘒߩේሶߩሽ⏕₸ࠍⓍ߽ߞ
࿑ 6 2 శሶ MW㧙RF ㆫ⒖
ߚࡠࠢࠗࡑޕᵄᵄᢙߪࡑࡦᵄᢙ㑆㓒ࠃࠅ 420
kHz ૐߊ⸳ቯߒߚޕMW-RF
శ ࠍ ᾖ ߒ ߚ ᤨ ߦ ޔF=2,
mF=+1 ߦㆫ⒖ߒߚේሶߣ
ㆫ ⒖ ߒ ߥ ߆ ߞ ߚ F=1,
mF=-1 ߩේሶࠍ᷹ቯߒߚޕ
ߎࠇࠄߩේሶߩᲧࠍขࠆߎ
ߣߢ F=2 ߩㆫ⒖߳ߩሽ⏕
₸߇ᓧࠄࠇࠆޕRF ᵄᢙࠍ
ᒁߒߚߣߎࠈޔ࿑ 8 ߩࠃ
߁ߥࠬࡍࠢ࠻࡞߇ᓧࠄࠇޔ
F=2 Ḱ߳ߩㆫ⒖ߩਛᔃ
ᵄᢙߪ ǎc ߪ(419.32±0.33)
kHz ߢ ߞ ߚ ޕF=1,
mF=-1ńF=2, mF=+1 ߳ߩ 2
శሶ MW-RF ㆫ⒖ߩ⏛႐ࠪ
࿑ 7 ᷹ቯ࠳ࠗࠕࠣࡓ
ࡈ࠻㊂ߪ-742 Hz ߢࠆߎ
ߣ߆ࠄޔਇ⏕߆ߐߩ▸࿐ߢޔㆫ⒖ࠍⴕ߃ߚߎߣ߇
ࠊ߆ࠆޕ
F=2, mF=+1 Ḱ䈮ㆫ⒖䈚䈢ේሶᢙ䈱䊃䊤䉾䊒ᤨ㑆
䈎䉌䇮F=2, mF=+1 䈱ኼ䉕᷹ቯ䈚䈢⚿ᨐ䉕࿑ 9 䈮␜
䈜䇯䈠䈱⚿ᨐ䈲䇮(7.7㫧2.4) ms 䈪䈅䈦䈢䇯䈖䈱䉋䈉䈮ኼ
䈏䇮⍴䈇ℂ↱䈲䇮2 䈱䉴䊏䊮✭䈏ᒝ䈒ଐሽ䈚䈩
䈇䉎䈫⠨䈋䉌䉏䉎䇯[4]
㡆ᵄᢙߩਅߢࡆޔᝄേࠍ᷹ቯߒߚ⚿ᨐࠍ
࿑ 10 ߦ␜ߔᦨޔᤨߩߎޕᄢߩㆫ⒖⏕₸ 0.8 ߇ᓧࠄ
ࠇߚޕ1.0 ߢߥℂ↱ߪޔవ߶ߤߩࠬࡇࡦ✭ߣ
ኼ߆ࠄ F=2, mF=+1 ߦㆫ⒖ߒߚේሶ߇✭ߒߡ
࿑ 8 2 శሶ MW-RF ࠬࡍࠢ࠻࡞
ߒ߹߁ߚߦޔᷫߒߚߚߣ⠨߃ࠄࠇࠆޕ
4
− 24 −
࿑ 9
F=2, mF=+1 Ḱߩኼ ࿑ 10 ࡆᝄേ
3㧚2 ේሶᐓᷤ⸘
ᦨᓟߦࡓᐓᷤࠍ᷹ⷰߒߚޕ᷹ቯ
ߩᤨ㑆♽ߪޔF=2㧘mF=+1 Ḱߩኼ
ࠍ⠨ᘦߒޔǕ-T-Ǖ ߇ 0.5-1.5-0.5ms ߦ
⸳ቯߒߚޕᐓᷤᤨ㑆 2ms ߩࡓᐓ
ᷤࠍ᷹ⷰߒߚ⚿ᨐࠍ࿑ 11 ߦ␜ߔޕ᷹ቯ
ߩ㑆㓒ߪ☻߇ޔ500 Hz ߩᦼ߇᷹ⷰ
ߐࠇߡࠆࡓޕᐓᷤߩ൮⛊✢ߩ
ਛᔃ߆ࠄࡇࠢ߇ߕࠇߡࠆߩߪޔ2
శሶࠍ↢ᚑߒߡࠆ࠳ࠗࡐ࡞ࠕࡦ࠹
࠽ߣ RF ࠦࠗ࡞ߩ⋧ߩߕࠇ߆ࠄ↢ߓ
ࠆ߽ߩߢޔ⋧ࠍߘࠈ߃ࠇ߫⸃ᶖߐࠇ
ࠆߣ⠨߃ࠄࠇࠆ⎇ᧄޕⓥߢߪޔF=2,
࿑ 11 ࡓේሶᐓᷤ
mF=1 Ḱߩኼ߇⍴߆ߞߚߚߦޔ
2ms એߩᐓᷤࠍࠆߎߣ߇᧪ߥ߆ߞߚޕᓟߪޔḰߩኼࠍ㐳ߊߔࠆᣇᴺࠍߟߌᦨޔᄢ
1s ⒟ᐲߩᐓᷤࠍ᷹ⷰߢ߈ࠆࠃ߁ߦ⎇ⓥߔࠆᔅⷐ߇ࠆ
4㧚⚿⺰
࠽࠻࠙ࡓࡏࠬಝ❗ේሶࠍశᩰሶ࠻࠶ࡊߦᝒߒߚޕᝒല₸ߪ⚂ ߢߞߚޕᝒኼ
ߪࠬࡏޔಝ❗ේሶߩኒᐲ߇㜞ߚ ⴣ⓭ലᨐ߇ᄢ߈߇⑽ ޔએߩᝒ߇ߢ߈ߚޕ
ᝒේሶࠍ⏛႐ߦࠃࠅ ᰴࡑࡦࠪࡈ࠻߇߈ߦߊ శሶࡑࠗࠢࡠᵄ́ࠫࠝᵄㆫ⒖ࠍ↪ߡޔ
(O( Ḱߦㆫ⒖ߐߖߚޕએߩේሶࠍㆫ⒖ߢ߈ߚ߇ ޔߩࠬࡇࡦ✭ߩߚኼߪ OU
ߢߞߚߩߎޕㆫ⒖ࠍ↪ߡޔᐓᷤᤨ㑆 OU ߩࡓᐓᷤታ㛎ࠍⴕߞߚߣߎࠈޔ㞲ᐲߩ⦟
ᐓᷤ❋ࠍᓧࠆߎߣ߇ߢ߈ߚࠅࠃࠇߎޔࠄ߇ߥߒ߆ߒޕ㐳ᤨ㑆ߢߩᐓᷤߪᓧࠄࠇߥ߆ߞߚޕ㐳
ᐓᷤᤨ㑆ߢߩᐓᷤ❋ࠍࠆߚߦߪ(ޔO( Ḱߩᝒኼࠍ㐳ߊߔࠆᔅⷐ߇ࠆޕ
References
[1] H. Imai and A. Morinaga, J. Phys. Soc. Japan 79, 094005 (2010).
[2] M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998).
[3] A. Grlitz et al., Phys. Rev. Lett. 90, 090401 (2003).
[4] S. Tojo et al., Phys. Rev. A 80, 042704 (2009).
5
− 25 −
Local-potential expansion of the time-dependent
meson exchange NN interaction
-Possible Long Range Components in NN-interactionDepartment of Physics, Faculty of Science and Technology, Tokyo University of Science
Shinsho Oryu
Abstract. The Fourier transform of the N N potential in the many-channel twoparticle Lippmann-Schwinger equation is deduced from the πN N three-body FaddeevLovelace equation. The result represents the Yukawa-type potential with an energy
dependent range. By a proper statistical average for the range this results in the typical
r-space potential: 1/{r(r + a0 )} which signifies the Yukawa potential e−r/a0 /r for distances r a0 , while 1/r2 for distances r a0 . We find that such a long range potential
could generate a countably infinite number of energy levels below the threshold, which
is similar to the Efimov states. Even if it does not lead to such states, the 1/r2 structure
could interfere with the Coulomb potential and the centrifugal potential.
1.
Introduction
In the early 1960’s, Lovelace used Faddeev’s mathematical framework [1] for nonrelativistic
three-particle systems to obtain a practical theory of three-particle states[2]. He obtained the
so called “Lovelace equation or Faddeev-Lovelace equation” which has exactly the form of
the many-channel, two-particle Lippmann-Schwinger equation, if one identifies the Born term
Zαn,βm with the potential and τγs with the propagator. Furthermore, he applied the theory
to formulate the 3N and N ππ systems. Later on, he generalized the theory to the relativistic
case[3][4].
In 1976, Thomas calculated π-d scattering using the Faddeev-Lovelace equation for the
three-body πN N system, and obtained a good fit with the experimental data[5]. Although the
major objection to this work was that it did not provide a covariant theory, Thomas claimed
the issue should not be very important below 100 MeV for the πN N system.
In 1988, Rupp and Tjon calculated the triton binding energy from the three-body BetheSalpeter (BS) equation for phenomenological separable potentials of rank one. They found
that the BS results were about 0.5MeV deeper than the nonrelativistic results relative to the
experimental value: 8.48 MeV[6]. They argued that the effective energy-dependent potential
has essentially the same structure as the Alt-Grassberger-Sandhas (AGS) equations in the
nonrelativistic situation, except for the presence of the additional relativistic variable (the zero
component of the four-momenta): q 0 . Furthermore, they mentioned [6] that the potential
had been used in the reference of Lovelace et al. as a starting point for discussing a possible
relativistic quasipotential theory for three particles[3][4].
In this paper, we would like to evaluate the binding energy of the πN N three-body system,
especially the shallow binding energy where the root mean square radius is rather large. This
means that the higher momentum components are small, so that relativistic effects could be
smaller than in the triton case.
− 26 −
2.
Fourier Transform of the Energy Dependent Effective Potential
As stated in the above discussion, we would like to concentrate on Thomas’ nonrelativistic
approach for the πN N system. The effective energy-dependent potential, or the Born term
of the Faddeev-Loverace equation which is given by Zαn,βm , denotes the N -N potential below
the π-threshold and the propagator τγs corresponds to the two-body propagator or particlequasiparticle two-body free Green’s function, respectively.
The Born term has the form
Zαn,βm (qα , qβ ; E) = gαn (pα )G0 (qα , qβ : E)gβm (pβ )δ α,β
=
E−
gαn (pα )gβm (pβ )δ α,β
2
qα /2mα − qβ2 /2mβ − (qα +
qβ )2 /2mγ
,
(1)
where α, β, and γ are the-three-body channels – channel-1, channel-2 denote N1 -(N2 π) and
N2 -(πN1 ) and channel-3 is π-(N1 N2 ) with the notation δ αβ ≡ 1 − δαβ . In eq.(1) the separable
potentials for the two-body interactions with form factors g1 (p1 ), g2 (p2 ), g3 (p3 ) were utilized
for the two-body relative momenta: p1 , p2 and p3 . Here the subscripts n, m and s are the
physical states for the corresponding channels. m1 , and m2 are the nucleon masses, and m3 is
the π mass. G0 (qα , qβ ; E) is the three-body Green’s function written in terms of the three-body
channel momenta or the three-body Jacobi coordinate momenta qα , qβ .
Tβ = T
−Tα = T
Tγ
Tα
−Tβ = −T
Figure 1.
The Born term diagram which corresponds to the NN-potentials where the solid
lines are nucleons and the dashed line denotes meson.
We would like to point out that the Born term of eq.(1) corresponds to a meson exchange
potential where the meson transfer can be written in terms of the momentum transfer between
the two nucleons. For this purpose, we replace the three-body momenta
qα = −q,
qβ = q ,
qγ = −qα − qβ = q − q
(2)
where qγ is the momentum transfer with respect to the three-body center-of-mass system, i.e.,
qα + qβ + qγ = 0. Therefore, the two body momenta are given by
pα =
pβ =
mγ qβ − mβ qγ
= qβ + qα /Λα = q − q/Λα ≡ p,
mβ + mγ
mα qγ − mγ qα
= −qα − qβ /Λβ = q − q /Λβ ≡ p .
mγ + m α
− 27 −
(3)
(4)
In this way, the Born term can be rewritten as
−2gαn (p)mγ gβm (p )δ α,β
−2gαn (p)mγ gβm (p )δ α,β
.
=
−2mγ E + Λα q 2 + Λβ q 2 − 2qq
2qq (χ − x)
Zαn,βm (−q, q ; E) =
−2mγ E + Λα q 2 + Λβ q 2
;
2qq χ =
(5)
then the partial wave expansion is
Zαn,βm (−q, q ; E) =
∞
l
(2l + 1)Zαn,βm
(q, q ; E)Pl (cos θ),
(6)
l=0
with
qq = qq x,
Λα = (mγ /mα + 1) = 1 + ∆α ,
Λβ = (mγ /mβ + 1) = 1 + ∆β .
(7)
If we adopt for the average nucleon mass MN = 938.9035 MeV and Mπ = 138.0477 MeV for
the average π-meson mass, then
∆α = ∆β ≡ ∆ =
Mπ
= 0.147030765,
MN
Λα = Λβ ≡ Λ =
Mπ
+ 1 = 1.147030765.
MN
(8)
It should be noted that E ≤ 0 because the real π-meson does not appear in the N N potential, where the potential is defined below the pion-threshold.
Furthermore, our N1 N2 π system leads to Λα = Λβ = Λ ≡ 1+∆. Therefore, eq.(5) becomes
χ =
−2mγ E/Λ + q 2 + q 2
σ 2 + q 2 + q 2
Λ
=
Λ
2qq 2qq (9)
with
σ2 ≡ −
2mγ E
≥ 0.
Λ
(10)
In order to simplify, let us adopt constant form factors for the g’s; then the partial wave
expansion of eq.(5) is given by using eq.(6),
l
Zαn,βm
(q, q ; E) ≈ −
=
2gαn mγ gβm
2qq γ
Cαn,βm
Ql (Λχ)
qq 1
−1
2gαn mγ gβm
Pl (x)
dx = −
Ql (χ ),
χ −x
qq ≡ VlN N (q, q ; σ),
(11)
neglecting the π(N1 N2 )-channel, but retaining the N1 (N2 π)-channel and N2 (πN1 )-channel,
resulting in
γ
Cαn,βm
= −2gαn mγ gβm ≡ 8πV0 < 0,
χ
Λ
=
σ2
q2
+ +
2qq q 2
≡ χ.
(12)
(13)
Let us expand Ql (Λχ) with respect to Λ = 1 + ∆,
Ql (Λχ) = Ql (χ + ∆χ)
= Ql (χ) + Ql (χ)(∆χ) +
1
1
Ql (χ) (∆χ)2 + Ql (χ) (∆χ)3 + . . .
2!
3!
(14)
− 28 −
Therefore, eq.(11) becomes
8πV0
Ql (Λχ)
qq VlN N (q, q ; σ) =
N N (0)
N N (1)
= Vl
(q, q ; σ) + Vl
+ Vl
(q, q ; σ) . . . .
N N (3)
N N (2)
(q, q ; σ) + Vl
(q, q ; σ)
(15)
where each term of eq.(15) is given by,
N N (0)
(q, q ; σ) =
N N (1)
(q, q ; σ) =
N N (2)
(q, q ; σ) =
N N (3)
(q, q ; σ) =
Vl
Vl
Vl
Vl
8πV0
Ql (χ)
qq 8πV0
∆χQl (χ)
qq 8πV0 ∆2 2 χ Ql (χ)
qq 2!
8πV0 ∆3 3 χ Ql (χ).
qq 3!
(16)
This expansion means that the three-body kinematics is represented by the two-body kinematics. Therefore, the first order term corresponds to the Yukawa potential. The vector
representation of the first order potential term is given by eq.(16),
V N N (0) (−q, q ; σ) =
N N (0)
(2l + 1)Vl
(q, q ; σ)Pl (x) =
l=0
=
|q
16πV0
− q|2 + σ 2
16πV0
≡ V N N (0) (qα , qβ ; σ).
|qβ + qα |2 + σ 2
(17)
It is found that the momentum term of eq.(17) is the Yukawa potential.
The Fourier transform of the above result in eq.(17) becomes
F{V N N (0) (−q, q ; σ)} = F{V N N (0) (qα , qβ ; σ)}
= < xα |VNN(0) (qα , qβ ; σ)|xβ >
= 16πV0
dqα dqβ eiqα xα e−iqβ xβ
,
(2π)3 (2π)3 |qβ + qα |2 + σ 2
(18)
where we define new coordinate relations
2K ≡ qα + qβ = q − q,
Q = q + q,
Q
Q
+ K = qβ ,
− K = −qα
q =
q=
2
2
Q
2 Q
2
Q2 −K +
+ K = 2 K2 +
q 2 + q 2 =
2
2
4
Q
Q
− K xα −
+ K xβ
qα xα − qβ xβ = −qxα − q xβ = −
2
2
x + x α
β
= −Q
− K(xβ − xβ ) = −QR − Kr
(19)
2
with the center-of-mass coordinate R, and the relative momentum r,
xβ + xα
,
r ≡ xβ − xα ,
(20)
R≡
2
where one finds that R is the center-of-mass coordinate and the corresponding momentum is
Q, and r is the relative coordinate with the corresponding momentum K. In terms of the new
coordinates, the Jacobian is unity, i.e.,
dqα dqβ =
∂(qα , qβ )
dKdQ = dKdQ.
∂(K, Q)
− 29 −
(21)
Therefore, eq.(18) becomes
F{V
N N (0)
(qα , qβ ; σ)} = 4πV0
e−iQR dQ
(2π)3
e−iKr
dK
K2 + (σ/2)2 (2π)3
e−σr/2
≡ VNN(0) (r; σ)
(22)
r
where R is the three-body center-of-mass coordinate, and r is the relative coordinate.
It is not easy to obtain the energy independent potential from the three-body Born term.
We propose to take the “statistical average” by using the probability density function with
respect to the possible energy range,
= V0 δ(R)
Pσ = σ 2γ+1 e−aσ
with
ρ =
∞
∞
0
σ 2γ+1 e−aσ dσ ≡
Γ(2γ + 2)
,
a2γ+2
σ 2γ+1 e−aσ dσ =
0
σ 2γ+1 e−aσ
,
ρ
(23)
(24)
where the weight function σ 2γ+1 is adopted, and e−aσ is the damping factor when the threebody energy |E| increases. Therefore, by using the probability density function, the expectation
value of the energy independent potential is obtained:
VlN N (q, q )
=
=
∞
0
Pσ VlN N (q, q ; σ)dσ
8πV0
ρqq 0
∞
1
=
ρ
∞
0
σ 2γ+1 e−aσ VlN N (q, q ; σ)dσ
σ 2γ+1 e−aσ Ql (Λχ)dσ.
(25)
This is the Laplace transform or the Euler integral of the second kind of eq.(22). Therefore,
by using eqs.(22),(25),(24) the first order potential is
V
N N (0)
(r) =
V0 δ(R)
ρ
∞
0
σ 2γ+1 e−aσ
e−σr/2
V0 δ(R)a2γ+2
dσ =
,
r
r(r/2 + a)2γ+2
(26)
where γ = 3/2 means the Van der Waals potential for r a. Furthermore, if and only if the
weight function satisfies σ 2γ+1 e−aσ = δ(σ − 2µ0 ) with the meson range 1/µ0 , then the potential
is a Yukawa potential [7][8].
In eq.(11), if the form factors are energy (or σ) independent, the weight function σ 2γ+1
could be a constant or monotonic in eq.(25) which means γ = −1/2. Hence, we obtain the
final potential by using 2a = a0
V0 a0
V0 a
=
.
r(r/2 + a)
r(r + a0 )
V N N (0) (r) =
(27)
Therefore, the behavior of this potential is given for small r, or for r a0 , as
V N N (0) (r) =
e−r/a0
V0 a0
≈ V0
.
ra0 (1 + r/a0 )
r
(28)
This potential suggests an important meson exchange potential in which case the range is a0
or the meson mass m = µ0 ≈ 1/a0 for the Yukawa potential.
For large r, or r a0 , one has
V N N (0) (r) =
V0 a0
V0 a0
≈ 2 .
r2 (1 + a0 /r)
r
(29)
The latter potential with a0 r shows a long range decay with r−2 in this simple approximation. It should be noted that the reliability of the first order approximation is about 85% in
terms of the ∆ value.
− 30 −
3.
Properties of the 1/r2 potential
The Schrödinger equation is given by
−
V a
h̄2 l(l + 1)/2m h̄2 d2
0 0
+
ψ
(r)
+
ψl (r) = Eψl (r),
l
2m dr2
r(r + a0 )
r2
(30)
where m = MN /2 is the reduced mass.
In order to look at the long range behavior, we can reduce the equation to
d2
dr2
− κ2 −
λ
χl (r) = 0
r2
(31)
or the Bessel’s function differential equation which is given in text books [9],
d2
dr2
+ β2 −
ν 2 − 1/4 χl (r) = 0,
r2
(32)
where we have κ2 = −β 2 = −2mE/h̄2 > 0 for negative energies which are verified by the
effective potential factor: λ = ν 2 − 1/4 = l(l + 1) − 2m|V0 |a0 /h̄2 < 0 where V0 < 0 and a0 > 0.
Let us define ν 2 which should be negative to obtain the binding energies for this potential,
ν2 =
1
1
1
+ λ = + l(l + 1) − 2m|V0 |a0 /h̄2 = (l + )2 − 2m|V0 |a0 /h̄2 .
4
4
2
(33)
For the case: λ < −1/4, or (l + 12 )2 < 2m|V0 |a0 /h̄2 ,
1
ν = i |λ + 1/4| = i 2m|V0 |a0 /h̄2 − (l + )2 ≡ iµ,
2
2m
1
µ2 =
|V0 |a0 − (l + )2 .
2
h̄2
(34)
(35)
In this case we have the solution for the modified Bessel function [9],
√
√
χ(r) = κrZν (iκr) ≡ κrKiµ (κr).
(36)
Because of the pure imaginary index ν = iµ, the modified Bessel function is
Kiµ (κr) =
π
{I−iµ (κr) − Iiµ (κr)}.
2 sin iµπ
(37)
And, for small κr, it becomes,
Iiµ (κr) =
=
(κr/2)iµ
(κr/2)iµ
+ ... =
+ ...
Γ(1 + iµ)
|Γ(1 + iµ)|ei arg Γ(1+iµ)
ei{µln(κr/2)−arg Γ(1+iµ)}
{1 + o(κr)}.
|Γ(1 + iµ)|
(38)
Therefore we have
Kiµ (κr) =
×
=
1
π
2 sin iµπ |Γ(1 + iµ)|
e−i{µ log(κr/2)−arg Γ(1+iµ)} − ei{µ log(κr/2)−arg Γ(1+iµ)}
1
π
κr
sin µ log
− arg Γ(1 + iµ) {1 + o(κr)}.
sinh πµ |Γ(1 + iµ)|
2
(39)
− 31 −
In order to smoothly connect Kiµ (κr) with the full wave function ψl (κr) and the potential
V (r) = V0 a0 /[r(r + a0 )], the logarithmic derivative should be employed at sufficiently large
r = a. For the modified Bessel function, it will be reduced to, for small κr,
√
1
κr
d{ κrKiµ (κr)}/d(κr)
√
= + µ cot µ ln
− arg Γ(1 + iµ) .
(40)
κr
κrKiµ (κr)
2
2
It is found that the periodicity of “ cot ” leads to the relationship
µ ln
κa
2
= C − nπ
{n = 1, 2, . . .}
(41)
2 (C−nπ)/µ
e
≡ κn .
a
κ =
(42)
Therefore, the binding energy −En (for n=1, 2,. . . ) is
−En =
κ2n
2m
2
e2C/µ e−2πn/µ .
ma2
=
(43)
This means that one needs n different boundary conditions to fit the full wave function
smoothly, corresponding to the different energies En . From this relationship the energy ratio has a typical structure for En and eq.(32),
En
En+1
= e2π/µ ,
(44)
En+1 = En e−2π/µ ,
(45)
where En is the binding energy with the quantum number n.
In order to fix the parameter µ, we will adopt E0 = −2.2246 MeV for the deuteron binding
energy. For this aim, we know V0 = −50.577 MeV·fm, and a0 = 1.4295 fm, then we obtain
V0 a0 = −72.3 MeV·fm2 = −0.36641 fm, and 2m = MN = 938.90 MeV=4.7583 fm−1 for h̄ =
c = 1. Therefore, eq.(35) becomes
1
1
µ2 = 4.7583 × 0.36641 − (l + )2 = 1.7435 − (l + )2 ≥ 0.
2
2
(46)
It suggests that only l = 0 is available, and we have µ = 1.2221 > 1.0 which means that this
is not the Efimov state: µ = 1.0[10]. From the value µ = 1.2221, factors e2π/µ and eπ/µ are
obtained,
e2π/µ = 170.98
π/µ
e
(47)
= 13.076.
(48)
The root-mean-square (rms) radius is calculated by using the wave function of eq.(36).
Let us define
√
∞
πΓ([1 + c]/2)Γ([1 − 2a + c]/2)Γ([1 + 2a + c]/2)
c
2
.
ξ[c] ≡
dxx Ka (x) =
4Γ(1 + c/2)
0
(49)
In order to obtain the rms radius, we adopt ξ[3], and ξ[1], i.e., Therefore, the rms radius is
given for the modified Bessel wave function,
2
< r >n =
< r >n ≡
!∞
∗
2
ξ[3]
2
(1 + µ2 )
2
0! drχl (r)r χl (r)
=
=
(1
+
µ
)
=
,
∞
∗
2
0
drχl (r)χl (r)
< r2 >n =
"
ξ[1]
(1 + µ2 )
=
3m|En |
− 32 −
"
3κ
−3mEn
2(1 + µ2 )
≡ rn .
3MN |En |
(50)
(51)
However, our original potential eq.(27) doesn’t satisfy eq.(50) for small quantum numbers n,
because the modified Bessel function leads to r0 = 5.56619 fm. This value is much larger than
r0 = 2.5155 fm which is obtained by using our potential with eq.(27).
Here, we obtain the ratio relation between the rms radius-ratio and the energy-ratio by using
(50),
En
En+1
=
r
n+1 2
rn
= e2π/µ
rn+1 = rn eπ/µ .
(52)
(53)
From this formula, we find that the rms radius becomes larger, corresponding to the shallower
binding energies in the case eπ/µ = 13.076 ≥ 1.
These formulas for the binding energy and rms radius were discussed by Sawada for the
“nucleon-monopole system” with the 1/r2 potential in 1993 [11]. If the condition e2π/µ ≥ 1 is
satisfied, then the series of binding energies for the bound states will be smaller and smaller,
while the rms radius becomes larger and larger.
4.
Second order correction
The vector form of the second order correction term in eq.(16), when γ = −1/2, is
V N N (1) (−q, q ; σ) = −2πV0 ∆
[K 2 + Q2 /4 + σ 2 /2]
[K 2 + (σ/2)2 ]2
V0 ∆ rσ e−σr/2
≡ V N N (1) (r; σ)
1+
2
8
r
1
1
V0 a0 ∆ +
,
V N N (1) (r) = −
2
2
r(r + a0 ) 4(r + a0 )
F{V N N (1) (−q, q ; σ)} = −
(54)
(55)
(56)
where eq.(54) could easily be calculated by using the center-of-mass condition Q = 0, although
this is a local potential approximation. Then the Fourier transform is straightforward as in
eq.(55). Therefore, the energy average with respect to σ leads to an additional 1/(r + a0 )2 form
which gives also a 1/r2 type potential for a0 r.
5.
Summary and discussion
In order to extract typical three-body characteristics, the complicated aspects of the AGS
equations such as the details of the form factors and small components of the channels can
be replaced at very low energy, where the deuteron binding energy is obtained, by a shape
independent potential. On the other hand, it is well known that the AGS equations can
be thought of as multi-channel Lippmann-Schwinger equations in which the quasi-potential
is represented by the energy and momenta. Obviously, the Fourier transform of the quasipotential can be given in r-space with a proper local potential approximation. In order to
obtain an energy independent potential from such an energy dependent potential, we must
eliminate the energy terms. One promising method is that in which one adopts the statistical
average for the energy. The result that we obtained is the energy independent potential:
V0 a0 /{r(r + a0 )} for a first approximation; the second order correction includes not only the
same form but also −V0 a0 ∆/{8(r + a0 )2 }, which then becomes a 1/r2 -type potential in the
long range region.
This approach was applied to the πN N -system to investigate the basic low energy πN N
potential, although as we noted the major objection was that it was not a covariant theory.
− 33 −
The issue should not be very important near the threshold region of the πN N system, because
the rms radius is very large and the higher momentum components should be negligible.
In the framework of the above discussion, the πN N three-body system, reveals a characteristic
long range property of the πN N interaction. Furthermore, such a potential could have an
infinite number of energy levels near zero energy, which belong to the three-body bound states
for an S-wave πN N -interaction. These bound states are not equivalent to the Efimov states
but they contain such states when µ = 1. In addition, the 1/r2 potential could affect not
only the final-state interaction in the break-up experiments with respect to nn and np but
also could affect the three-nucleon force which should be investigated by adding such a long
range πN N interaction. If such an infinite set of bound states exist in the πN N system, the
first excited state of the deuteron could be E1 ≈ E0 /e2π = −2.2246/170.98 ≈ −13 keV, with
r1 ≈ r0 × eπ = 1.97 × 13.076 ≈ 26 fm. Although, the lower quantum number states have rather
larger errors, because one needs the full wave function, nevertheless, it points out that such
an energy region (|E| ≤ 50 keV) has not been investigated. Moreover, the phase shift is not
measured below the energy 200 keV[12].
Even if the 1/r2 potential does not support bound states, such a potential could interfere with
the Coulomb potential and the centrifugal potential. Furthermore, the three-body force at very
low energy could originate from this long range potential, because the explanation in terms of
the ∆-isobar-origin of the Fujita-Miyazawa three-body force seems to be hard to justify in the
very low energy region.
Finally, this kind of long range 1/r2 type potential structure could universally occur,
not only in three-body systems but also in four-body systems by means of the FIG. 1 type
potentials.
The author would like to express his thanks to Dr Takashi Watanabe and Dr. Tetsuo
Sawada for their valuable discussions.
References
[1] L. D. Faddeev, Zh. Eksp. i Teor. Fiz. 39 (1960) 1459, English transl.; Soviet Phys.-JETP
12 (1961) 1014.
[2] C. Lovelace, Phys. Rev. 135 (1964) B1225.
[3] Z. Freedman, C. Lovelace, and J. M. Namyslowski, Nuovo Cimento 43, (1966) 258.
[4] N. Mishima, S. Oryu and Y. Takahashi, Prog. Theor. Phys. 39 (1968), 1569.
[5] A. W. Thomas, Nucl. Phys. A258 (1976) 417.
[6] G. Rupp and J. A. Tjon, Phys. Rev. C37 (1988) 1729, ibid, C45 (1992) 2133.
[7] T. Sawada, Modern Phys. A11 (1996) 5365.
[8] S. Oryu, Y. Hiratsuka, T. Watanabe and S. Gojuki, Proc. of 20th International Conference
on Few-Body Problems in Physics (Bonn 2009), to be published.
[9] For instance, page 362 formulae (9.1.49), Handbook of Mathematical Functions with Formulas Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun,
Dover Publications, INC., New York (1972).
[10] V. Efimov, Phys. Lett. B33 (1970) 563.
[11] T. Sawada, Foundations of Physics, 23, (1993), 291.
[12] J. R. Bergervoet, P. C. van Campen, W. A. van der Sanden, and J. J. de Swart, Phys.
Rev. C 38, 15 (1988).
− 34 −
陽子ー重陽子３体散乱における核力＋遮蔽クーロンポテ
ンシャル繰り込み処理の信頼性限界について
Reliability of the Renormalization Method for Screened
Coulomb Potentials on pd Scattering Using a Rigorous
Coulomb Treatment
数理情報グループ (理工学部物理学科、尾立ゼミ)
祥
尾立晋祥（嘱託教授）
, 平塚靖久（元院生）
五十木秀一（元院生）, 渡邊隆（非常勤講師）
Abstract. 遮蔽クーロンポテンシャルを用いた繰り込み法は広く用いられているが低エ
ネルギーに於いて問題がある事を指摘した。この方法の主たる原因の一つが「クーロンポ
テンシャルは half-energy-shell 振幅が零になる」とした Haeringen の予言の限界が 10MeV
以下では誤差が生じるためである。
Reliability of the screened Coulomb renormalization method, which was proposed
in an elegant way by Alt-Sandhas-Zankel-Ziegelmann (ASZZ), is discussed on the basis
of “two-potential theory” for the three-body AGS equations with the Coulomb potential.
In order to obtain ASZZ’s formula, we define the on-shell Møller function, and calculate
it by using the Haeringen criterion, i.e. “the half-shell Coulomb amplitude is zero”. By
these two steps, we can finally obtain the ASZZ formula for a small Coulomb phase
shift. Furthermore, the reliability of the Haeringen criterion is thoroughly checked by
a numerically rigorous calculation for the Coulomb LS-type equation. We find that the
Haeringen criterion can be satisfied only in the higher energy region. We conclude that
the ASZZ method can be verified in the case that the on-shell approximation to the
Møller function is reasonable, and the Haeringen criterion is reliable.
1.
Two-Potential Theory in the Charged Three-Body Problem
Recently, we obtained the three-body scattering amplitude in the presence of the Coulomb
force by using the rigorous two-potential theory in which the amplitude is calculated by the
double integral with respect to the intermediate momenta 1,
(C)
Xpd (q, q ; E)
q 2 dq q 2 dq C
C
=
Ωpd (q, q ; E)χsC
pd (q , q ; E)Ωpd (q , q ; E)
2π 2
2π 2
C
+ Xpd
(q, q ; E),
(1)
where the Møller function for the p-d Coulomb potential is given by,
Ωpd (q, q ; E) =
C
2π 2
C
δ(q − q ) + Tpd
(q, q ; E)Gpd
0 (q ; E).
q2
(2)
On the other hand, the amplitude for the screened Coulomb renormalization method takes
a form similar to (1), but the Møller function is replaced by the auxiliary phase factor eiφpd =
R
R , and
eiσpd e−iδpd , with the Coulomb phase shift σpd and the screened Coulomb phase shift δpd
− 35 −
the three-body amplitude with the screened Coulomb potential,
(C)
(R)
C
Xpd (q, q ; E) = eiφpd (q) Xpd (q, q ; E)eiφpd (q ) + Xpd
(Q, q ; E).
(3)
(R)
Let us call (3) the ASZZ ansatz 2,3. This holds if and only if Xpd and φpd are converged
for infinite R, but it is still a question as to how the formula could be obtained from the
two-potential theory (1). We would like to check the reliability of the ASZZ ansatz.
2.
Definition of the On-Shell Møller Function Approximation
Here, if and only if we could adopt the on-shell approximation with respect to the integral
in eq.(1), would it become
(C)
C
C
Xpd (q, q ; E) = Ωpd (q, q; E)χsC
pd (q, q ; E)Ωpd (q , q ; E) + Xpd (q, q ; E).
C
(4)
Let us define the on-shell Møller function for eq.(2) by
2
dq 2π 2
∞
q 2 dq C
C
=1+
Tpd
(q, q ; E)GP0:pd (q ; E)
+ Tpd
(q, q; E)Gδ0:pd (q; E),
2π 2
0
C
Ωpd (q, q; E)
≡
q
C
Ωpd (q, q ; E)
(5)
where we have separated the free p-d Green’s function into the principal part GP0:pd and the
imaginary part Gδ0:pd of eq.(5), which gives Gδ0:pd (q; E) = −iµq/2π with the p-d reduced mass
µ.
3.
The Haeringen Criterion
For the half-shell Coulomb amplitude H. van Haeringen proposed the half-shell amplitudes,
which are very small, in his book 4 page 307:
< p|T C |p >≈ (πq)−2 (πη/ sinh πη)(−y)iη ,
(6)
where q ≡ |p − p |, y ≡ (x + 1)/(x − 1) ≈ 0, x ≈ −1, x2 ≡ 1 + (p2 − k 2 )(p 2 − k 2 )(kq)−2 and η
is the Sommerfeld’s parameter. Let us call it the Haeringen criterion.
C (q, q ) numerically
In order to confirm his criterion, we calculated the half-shell function Tpd
using the Coul-07 method 5. In the energy region where the Haeringen criterion is satisfied
(presumably more than 10 MeV), eq.(5) becomes for the small Coulomb phase shift, i.e. by
adopting cos σpd ≈ 1,
µq C
C
Ωpd (q, q; E) ≈ 1 + Tpd
(q, q; E)Gδ0:pd (q; E) = 1 − i
= 1+
2π
e2iσpd − 1
= eiσpd cos σpd ≈ eiσpd ,
2
−
2π e2iσpd − 1 ·
µ
2iq
(7)
(8)
The result of eqs.(5), (7) and (8) is shown in FIG.1. The system is elastic proton-deuteron
scattering in the energy range from the keV region to the MeV region, and the two-body
angular momentum is l = 0. We compare the on-shell Møller function including all terms of
eq.(5) with the Coulomb renormalization eiσpd times cos σpd in eq.(7) and compare eq.(7) with
the Coulomb renormalization eiσpd in eq.(8) in FIG. 1. We calculate both the real part and
the imaginary part of the on-shell Møller function and the Coulomb renormalization factor,
because the real part has a different pattern than does the imaginary part. FIG. 1 shows that
C
eiσpd ≈ eiσpd cos σpd ≈ Ωpd (q, q; E).
− 36 −
ϭ
ZĞ;юɏΔͿ
/ŵ;юɏΔͿ
Ϭ͘ϱ
ZĞ;ĞΔŝʍͿпĐŽƐ;ʍͿ
/ŵ;ĞΔŝʍͿпĐŽƐ;ʍͿ
ZĞ;ĞΔŝʍͿ
Ϭ
/ŵ;ĞΔŝʍͿ
ͲϬ͘ϱ
Ϭ͘ϬϬϭ
Ϭ͘Ϭϭ
Ϭ͘ϭ
ϭ
ϭϬ
ϭϬϬ
ϭϬϬϬ
ĐŵDĞs
Figure 1. Energy dependence of the on-shell Møller function (5), and the renormalization
factors (7), (8). The real part of (5) is shown by the black solid curve, the imaginary is the
black dashed curve. For the factor eiσpd cos σpd of (eq.(7): the real part is the red dotted
curve, and the imaginary part is the red dashed-dotted curve. For the factor eiσpd of (eq.(8):
the real part is given by the blue circles and the imaginary part by the blue triangles.
Thus, this fact leads to the same condition as ignoring the second term of eq.(5), and
validates the Coulomb renormalization method for energies larger than 10 MeV. Therefore, we
have
(C)
(R)
C
C
C
iσpd sC iσpd
C
C
χpd e
+ Xpd
≈ eiφpd Xpd eiφpd + Xpd
.
Xpd = Ωpd χsC
pd Ωpd + Xpd ≈ e
R
R
R
(9)
(R)
iδpd
where eiσpd = eiφpd eiδpd , and eiδpd χsC
≈ Xpd are used. This is the ASZZ ansatz which is
pd e
only obtained by the above mentioned double and triple approximations, whether it is a clever
method or not. Even if the numerical results were converged, it might be converged to some
mysterious value at much lower energies.
We would like to thank Dr. J. Haidenbauer for providing the EST separable expanded
realistic nuclear potentials: PEST, BEST etc.
References
S. Oryu, Phys. Rev. C73, 054001 (2006). :(referred as Coul-06).
E. O. Alt, W. Sandhas, H. Zankel, and H. Ziegelmann, Phys. Rev. Lett 37, 1537 (1976)
E. O. Alt, W. Sandhas, and H. Ziegelmann, Phys. Rev. C17, 1981 (1978)
H. van Haeringen, Coulomb Press Leyden, Charged-Particle Interactions Theory And Formulas (1985).
S. Oryu, S. Nishinohara, N. Shiiki, and S. Chiba, Phys. Rev. C75, 021001(R) (2007). :
ferred as Coul-07).
− 37 −
(re-
㜞➅ࠅߒᵄᢙࡄ࡞ࠬశḮߦ㑐ߔࠆ⎇ⓥ
ᢙℂࠣ࡞ࡊ㧔ℂᎿቇㇱ 㔚᳇㔚ሶᖱႎᎿቇ⑼ ೨↰⎇ⓥቶ㧕
೨↰ ⼑ᴦ㧔ᢎ㧕ޔᶏ⠧Ỉ ⾫ผ㧔ഥᢎ㧕
Abstract. ㊂ሶ࠹ࡐ࠹࡚ࠪࡦޔ㊂ሶ㎛㈩Ꮣ╬ߢ↪ߐࠇࠆ EPR శሶኻࠍޔ㜞➅ࠅ
ߒᵄᢙߢ⊒↢ߐߖࠆߚߩࡄ࡞ࠬశḮߣߒߡޔ㜞⺞ᵄࡕ࠼หᦼࡈࠔࠗࡃࠩߩ⎇ⓥ
ࠍⴕߥߞߡࠆޕᤓᐕᐲ߹ߢߦࡓ࠙ࡆ࡞ࠛޔᷝടࡈࠔࠗࡃჇེߣඨዉశჇེࠍ૬↪
ߒߚౣ↢ࡕ࠼หᦼࠩࠍ᭴▽ߒޔᢙᤨ㑆ߦᷰࠆቯߒߚࡄ࡞ࠬ⊒ᝄࠍታߒߡࠆޕ
ᧄᐕᐲߪࡓ࠙ࡆ࡞ࠛޔᷝടࡈࠔࠗࡃჇེߣඨዉశჇེߩബࡌ࡞ߣࡄ࡞ࠬቯᕈ
ߣߩ㑐ଥࠍ⚦ߦᬌ⸛ߒߚޕ
ߪߓߦ
శߩ㊂ሶ߽ߟࠇ⁁ᘒߪޔ㊂ሶ࠹ࡐ࠹࡚ࠪࡦߩታ㛎߿ޔ㊂ሶ㎛㈩Ꮣࠪࠬ࠹ࡓߦ↪ߐࠇߡ߅ࠅޔ
EPR శሶኻ[1]ࠍ↢ᚑߔࠆߎߣߦࠃߞߡᓧࠆߎߣ߇ߢ߈ࠆ ߪࠇߎޕ2 శሶㆊ⒟ߦࠃߞߡ↢ᚑߐࠇࠆ
శሶኻߩ㑆ߦሽߔࠆ⋧㑐ᕈࠍ↪ߒߚ߽ߩߢࠆޕ
㕖✢ᒻశቇㆊ⒟ߢࠆ 2 శሶㆊ⒟ߩ㜞ല₸ߩታߦߪޔ㜞ࡄࡢߩࠦࡅࡦ࠻శ߇ᔅⷐߢࠆޕ
߹ߚޔEPR శሶኻࠍ↪ߒߚ㊂ሶ㎛㈩Ꮣߩታߦߪޔశሶኻߩ⋧㑐ᕈߣߣ߽ߦ᷹ⷰߩหᤨᕈ߇᳞
ࠄࠇࠆ߇ߩߎޔหᤨᕈࠍᜂߔࠆߚߦߪ৻ޔቯߩ➅ࠅߒᵄᢙࠍᜬߟࡄ࡞ࠬశḮ߇ᔅⷐߢ
ࠆޕ
ߎߩࠃ߁ߥⷐ᳞ࠍḩߚߔశḮߣߒߡޔᚒߪޘ⢻േ㜞⺞ᵄࡕ࠼หᦼࡈࠔࠗࡃࠩ [2] ߦ⌕⋡ߒޔ
ߘߩቯൻߩ⎇ⓥࠍⴕߥߞߡ߈ߚޕᤓᐕᐲ߹ߢߩ⎇ⓥߩ⚿ᨐࠬ࡞ࡄޔ㜞ំࠄ߉ߩේ࿃ߢࠆࠬ
ࡄࡕ࠼㔀㖸ࠍޔඨዉశჇེࠍᝄེౝߦ↪ࠆߎߣߦࠃߞߡᛥߒ [3]⊒ࠬ࡞ࡄޔᝄࠍ
ᢙಽߦࠊߚߞߡቯൻߢ߈ࠆߎߣࠍታ㛎ߦࠃࠅ␜ߒߚࠩޔߦࠄߐޕേਛߩᾲߦࠃࠆࡄ࡞ࠬ
⊒ᝄߩਇቯൻߦኻߒߡߪޔౣ↢ࡕ࠼หᦼ [4] ߣ߫ࠇࠆᚻᴺࠍ↪ࠆߎߣߦࠃߞߡᛥߔࠆ
ߎߣ߇ߢ߈ޔᢙᤨ㑆ߦᷰࠆቯേࠍᓧࠆߎߣߦᚑഞߒߡࠆޕ
ᧄ㜞⺞ᵄౣ↢ࡕ࠼หᦼࠩߢߪޔჇᇦߣߒߡࠛ࡞ࡆ࠙ࡓᷝടశჇེ (EDFA) ߣඨዉ
ࠩჇེ (SOA) ࠍ૬↪ߒߡࠆ߇ࠬ࡞ࡄߩࡓ࠹ࠬࠪࠩޔൻ߅ࠃ߮ߘߩቯᕈߪࠇߎޔ
ࠄߩബࡌ࡞ߦᄢ߈ߊଐሽߔࠆᧄߢߎߘޕᐕᐲߪߩࠄࠇߎޔჇེ߳ߩബࡌ࡞ߣࡄ࡞ࠬ
ቯᕈߣߩ㑐ଥࠍ⚦ߦ⺞ߴޔฦ⊒ࠬ࡞ࡄߩޘᝄߦ㑐ߔࠆᓎഀࠍផቯߒߚޔߦࠄߐޕ㔚᳇࿁〝ㇱߦ
⋧หᦼ࡞ࡊ (PLL) ࠍ↪ߚౣ↢ࡕ࠼หᦼࠩࠪࠬ࠹ࡓߦ㑐ߔࠆޔၮ␆⊛ߥᬌ⸛ࠍᆎߚޕ
㜞⺞ᵄౣ↢ࡕ࠼หᦼߩේℂߣታᣇᴺ
㜞⺞ᵄౣ↢ࡕ࠼หᦼ
㜞⺞ᵄౣ↢ࡕ࠼หᦼߪޔ㜞⺞ᵄࡕ࠼หᦼߢ↪ࠆᄌ⺞ାภࠍࠬ࡞ࡄߩࠄ⥄ޔൻߦࠃߞߡ↢ߓ
ߐߖࠆ߽ߩߢࠆޕ
࿑ 1(a), (b) ߪߘࠇߙࠇࠆߌ߅ߦࠩࠣࡦࡃࠗࠔࡈޔ㜞⺞ᵄࡕ࠼หᦼ߅ࠃ߮㜞⺞ᵄౣ↢ࡕ
࠼หᦼߩᔨ࿑ߢࠆޕ
㜞⺞ᵄࡕ࠼หᦼߢߪޔᄌ⺞ᵄᢙࠍ⥄↱ࠬࡍࠢ࠻࡞㑆㓒ߩᢛᢙ n ߦߒߡޔᝄེౝߦ n ߩ
ࡄ࡞ࠬ߇⊒↢ߔࠆࠃ߁ߦߔࠆࠅ➅ߩࠬ࡞ࡄޔ߈ߣߩߎޕߒᵄᢙߪᄌ⺞ᵄᢙ fm ߦ╬ߒߊޔ
fm=nufc
(1)
ߢࠆ ߢߎߎޕfc ߪ⥄↱ࠬࡍࠢ࠻࡞㑆㓒ߢࠅޔᢛᢙ n ߪ㜞⺞ᵄᰴᢙߣ߫ࠇࠆޕ
− 38 −
⥄↱ࠬࡍࠢ࠻࡞㑆㓒 fc ߪޔᝄེౝߩశߩ
࿁ᤨ㑆ߩㅒᢙޔ
ߔߥࠊߜޔታലᝄེ㐳ࠍ
fm = n × fc
(a)
fr =fm
శㅦᐲߢഀߞߚ୯ߢࠆ߇ࠈߎߣޕታലᝄ
ེ㐳ߪޔ࿐᷷ᐲ╬ߩⅣႺߦࠃߞߡᄌൻߔࠆޕ
Modulator
․ߦ EDFA ߢߪޔബ㔚ሶߩ㕖⊒శㆫ⒖߹
ߚߪ㕖❗ㅌḰ㑆ߩ✭ߦ߁ᾲ߇⊒↢ߒ
ߡࡈࠔࠗࡃ᷷ᐲࠍᄌൻߐߖޔታലᝄེ㐳ߩ
Oscillator
ᄢ߈ߥᄌൻࠍ߽ߚࠄߔޕㅢᏱߩ㜞⺞ᵄࡕ࠼
หᦼߢߪޔᄌ⺞ེߩ㚟േାภḮߣߒߡᄖㇱߩ
ᱜᒏᵄ⊒ᝄེࠍ↪ࠆ߇ࡍࠬ↱⥄ߩࠩޔ
fm = n × fc
(b)
ࠢ࠻࡞㑆㓒߇ᤨ㑆⊛ߦᄌൻߔࠆߣޔ㜞⺞ᵄࡕ
fr =fm
࠼หᦼߦᔅⷐߥᄌ⺞ᵄᢙߣ⊒ᝄེߩ
ᵄᢙߣߩ㑆ߦߕࠇ߇↢ߓ࠼ࡕߪߡ߇߿ޔห
ᦼߘߩ߽ߩ߇ᱛߔࠆޕ
Modulator
Detector
ౣ↢ࡕ࠼หᦼߪߩߎޔታലᝄེ㐳ߩᄌൻ
ߦࠃࠆਇቯࠍసߔࠆᚻᴺߢࠆࠩޕ
ߩജࠍశᬌེߢ㔚᳇ାภߦᄌ឵ߒ⁜ޔᏪ
Amplifier Filter
ၞߩ㔚᳇ࡈࠖ࡞࠲ߦࠃߞߡ➅ࠅߒᵄᢙ
࿑ 1: ࡈࠔࠗࡃࡦࠣࠩߦ߅ߌࠆ㜞⺞ᵄࡕ࠼หᦼ
ߦ⋧ᒰߔࠆࠬࡍࠢ࠻࡞ࠍᛮ߈ߔࠍࠇߎޕ㜞
(a) ߣౣ↢㜞⺞ᵄࡕ࠼หᦼ (b) ߩᔨ࿑ޕfc : ၮᧄ
ᓧߩ㔚᳇Ⴧེߦജߒߡᄌ⺞ାภࠍᓧ
ᵄᢙ, T : ࿁ᤨ㑆, fm:ᄌ⺞ᵄᢙ, fr:➅ࠅߒᵄᢙޕ
ࠆޕ
㜞⺞ᵄౣ↢ࡕ࠼หᦼࠩߪޔ㔚᳇Ⴧེ
ψశᄌ⺞ེψశᬌེψ㔚᳇ࡈࠖ࡞࠲ψ㔚᳇Ⴧེߩ࡞ࡊࠍᜬߟᏫㆶࠪࠬ࠹ࡓߣߥߔߎߣ߇
ߢ߈ࠆޔߢߎߎޕᏫㆶߦⷐߔࠆᤨ㑆ࠍ⺞ᢛߒߡᱜᏫㆶࠍ↢ߓߐߖ৻ޔᎼવ㆐㑐ᢙߩᓧࠍᱜߦߔ
ࠆߎߣߦࠃߞߡోࡓ࠹ࠬࠪޔ߇⊒ᝄེߣߒߡേߔࠆߩߎޕ႐วߦࠩᧄߪᤨޔ㑆⊛ߦᄌ
ൻߔࠆᵄᢙㆬᛯࡈࠖ࡞࠲ߩᓎഀࠍᨐߚߔࠬ࡞ࡄ߇ࠩޔߪߡߒߣࡓ࠹ࠬࠪޕൻߒ߿ߔᵄ
ᢙࠅ߹ߟޔ㜞⺞ᵄࡕ࠼หᦼߩᵄᢙߦ߅ߌࠆ⥄ബ⊒ᝄ߇↢ߓࠆߣ⸃㉼ߢ߈ࠆ⊒ߩߎޕᝄߪޔ
ࠩߩ⥄Ꮖࡄ࡞ࠬൻᵄᢙߦ߅ߡ↢ߓࠆߚޔ࿐ⅣႺߦࠃߞߡߎߩᵄᢙ߇ᄌൻߒߡ߽⥄ޔ
േ⊛ߦᄌ⺞ᵄᢙ߇ㅊᓥߒ⊒ࠬ࡞ࡄޔᝄ߇⛮⛯ߔࠆޔߚߩߎޕᭂߡ㜞ቯߥࡄ࡞ࠬേ߇ታ
ߢ߈ࠆޕ
⋧หᦼ࡞ࡊࠍ↪ߚ㜞⺞ᵄౣ↢ࡕ࠼หᦼ
㜞⺞ᵄౣ↢ࡕ࠼หᦼࠩߦ߅ߌ
ࠆᄌ⺞ାภߩḰᣇᴺߦߪ⁜ޔᏪၞࡈ
Loop Filter
ࠖ࡞࠲ߣ㜞ᓧჇེߩߺߦࠃߞߡ
Phase
Reference
Comparator
᭴ᚑߔࠆᣇᴺߩ߶߆ߦޔ㔚᳇⊛ߥ⋧
หᦼ࡞ࡊ (PLL) ࠍ↪ߚᣇᴺ߇
ࠆޕએਅޔPLL ࠍ↪ߚᣇᴺߦߟߡ
ㅀߴࠆޕ
࿑ 2: ⋧หᦼ࡞ࡊߩၮᧄ᭴ᚑޕVCO: 㔚ᓮ⊒ᝄེޕ
PLL ߩၮᧄ᭴ᚑࠍ࿑ 2 ߦ␜ߔޕ㔚
ߦࠃߞߡ⊒ᝄᵄᢙࠍᄌ߃ࠆߎߣߩ
ߢ߈ࠆ⊒ᝄེ㧔㔚ᓮ⊒ᝄེ㧦VCO㧕ജߩ৻ㇱߪ⋧ᲧセེߦജߐࠇޔၮḰାภߣᲧセߐ
ࠇࠆޔߢߎߎޕ⋧Ყセེߪޔ2 ߟߩജߩ⋧Ꮕߦᔕߓߚ㔚୯ࠍജߔࠆ⚛ሶߢࠆޕ⋧
Ყセེߩജߪߣ࠲࡞ࠖࡈࡊ࡞ޔ߫ࠇࠆૐၞㅢㆊࡈࠖ࡞࠲ࠍㅢߒߡ VCO ߩᓮାภߣߒߡ
↪ࠄࠇࠆߩߎޕᏫㆶ࿁〝ࠍㆡಾߥ᧦ઙਅߢേߐߖࠆߣޔVCO ߩ⊒ᝄᵄᢙ߇ၮḰାภߩᵄ
ᢙߦ৻⥌ߒߟ߆ޔਔ⠪ߩ⋧Ꮕ߇৻ቯߩᏅࠍߟࠃ߁ߦߥࠆޕ
㜞⺞ᵄౣ↢ࡕ࠼หᦼࠩߩᄌ⺞ㇱߦ PLL ࠍ↪ࠆ႐วߦߪޔశᬌེߢᓧࠄࠇߚࡄ࡞ࠬାภ
ࠍၮḰାภߣߒ ࠍࠇߎޔVCO ജߦࡠ࠶ࠢߐߖࠆࠃ߁ߦߔࠇ߫ࠃޔ߈ߣߩߎޕPLL ߪ⁜Ꮺၞ
ߩᏪၞㅢㆊࡈࠖ࡞࠲ߣߒߡേߔࠆߎߣߦߥࠆޕߩᣇࠍߔࠇ߫ᧄࠩޔߣ VCO ߣ߁
VCO
− 39 −
Optical
Variable
Delay
ISO
EDFA
OBPF
1
1
9
1
ISO
Sampling
Oscilloscope
RF
Splitter
SOA
LN Modulator
ISO
Electrical
Band Pass
Filter
Driver
Splitter
Coupler
ISO
Optical
Spectrum
Analyzer
Photo Detector
RF Amplifier
Electrical
Variable Delay
࿑ 3: ታ㛎ࠪࠬ࠹ࡓޕISO: శࠕࠗ࠰࠲ޔEDFA: ࠛ࡞ࡆ࠙ࡓᷝടశჇེޔSOA: ඨዉశჇེޔ
OBPF: శࡃࡦ࠼ࡄࠬࡈࠖ࡞࠲ޕ
ੑߟߩ⊒ᝄེ߇ද⺞േࠍߔࠆߣ⸒߁ߎߣ߇ߢ߈ࠆޕ
PLL ࠍ↪ߚ㔚᳇Ꮻㆶ࿁〝ࠍ↪ࠆߣߥ࠻ࡦࡅࠦߦᤨ࠻࠲ࠬޔାภߢᄌ⺞߇ਈ߃ࠄࠇࠆߚ
ޔᲧセ⊛ኈᤃߦࡄ࡞ࠬൻߢ߈ࠆߣ߁ὐ߇ࠆޔߒ߆ߒޕᵄᢙนᄌߥ⊒ᝄེ߇ 2 ߟሽߔ
ࠆߚޔਇቯൻߒ߿ߔߣ߁ᰳὐ߽ࠆޕ
ታ㛎
㜞⺞ᵄౣ↢ࡕ࠼หᦼߩቯേ᧦ઙߦ㑐ߔࠆᬌ⸛
S/N
12
10
8
6
4
2
0
SOA Current [mA]
550
500
450
400
350
300
250
200
SOA Current [mA]
࿑ 3 ߦ᭴▽ߒߚࠩ߅ࠃ᷹߮ቯࠪࠬ࠹ࡓࠍ␜ߔޕ
᭴ᚑߪޔᤓᐕᐲߦ᭴▽ߒߚࠩ߅ࠃ᷹߮ቯࠪࠬ࠹ࡓߣห৻ߢࠆࡓ࠙ࡆ࡞ࠛޕᷝടࡈࠔࠗࡃߦ
ߪ 4m ߩࠛ࡞ࡆ࠙ࡓỚᐲ߇㜞ᵄᜬ࠲ࠗࡊ߽ߩࠍ↪ޔ1480nm ߩ೨ᣇะബߣߒߡࠆޕ
ඨዉࠩჇེ (SOA) ߪޔLN శᒝᐲᄌ⺞ེߩ⋥೨ߦ⸳⟎ߒߡࠆޔߚ߹ޕജࠞࡊߩ⋥
೨ߦᏪၞ 1nm ߩ 2 ࠠࡖࡆ࠹ࠖဳశࡃࡦ࠼ࡄࠬࡈࠖ࡞࠲ࠍ↪⥄ޔὼశߩ㒰ߣࠬࡍࠢ࠻࡞
ߩቯൻࠍⴕߞߡࠆޕ
శᬌེߦߪޔᏪၞ⚂ 60GHz ߩ߽ߩࠍ↪৻ߩߘޔㇱߪࠨࡦࡊࡦࠣࠝࠪࡠࠬࠦࡊ (Ꮺၞ
550
FWHM
45
500
40
450
35
400
30
350
25
300
20
250
200
150
250 300 350 400 450 500
EDF Pump LD Current [mA]
150
250 300 350 400 450 500
EDF Pump LD Current [mA]
(a)
(b)
࿑ 4: ࡄ࡞ࠬߩାภኻ㔀㖸Ყ (a) ߅ࠃ߮ࡄ࡞ࠬ (b) ߩബࡌ࡞ଐሽᕈޕ
− 40 −
50GHz) ߦ ዉ ߆ ࠇ ߡ ᤨ 㑆 ᵄ
ᒻߩ᷹ⷰߦߚ߹ޔߩ৻ㇱߪ
ౣ↢ࡕ࠼หᦼߩᏫㆶାภߣ
ߒߡ↪ࠄࠇࠆޕ㔚᳇Ꮻㆶ࿁
〝ߪޔ㜞ᓧߩ㜞ᵄჇེ
ߣਛᔃᵄᢙ 10.66GHzޔᏪ
ၞ⚂ 12MHz ߩ⁜Ꮺၞࡈࠖ
࡞࠲߮ࠃ߅ޔนᄌㆃᑧེߦࠃ
ߞߡ᭴ᚑߐࠇࠆޕ
(a)
(b)
ᧄታ㛎ߢߪޔੑߟߩჇᇦ
࿑ 5: ඨዉశჇེ߳ߩᵈ㔚ᵹࠍ 330mA ߦ࿕ቯߒߚߣ߈ߩജࡄ࡞
㧦EDFA ߣ SOA ߩബ
ࠬߩᤨ㑆ᵄᒻ(ޕa) ࠛ࡞ࡆ࠙ࡓᷝടࡈࠔࠗࡃჇེߩബඨዉࠩ߳
ࡌ࡞ࠍᄌൻߐߖޔജࡄ࡞ࠬ
ߩᵈ㔚ᵹ (a) 350mA, (b) 500mAޕ᳓ᐔゲ: 20ps/div.ޕ
ᵄᒻߩᄌൻࠍ᷹ⷰߒߚޕ࿑
4(a) ߪޔEDFA ߩബࠩ
߳ߩᵈ㔚ᵹߣ SOA ߳ߩᵈ
㔚ᵹߦኻߔࠆޔജࡄ࡞ࠬ
ߩାภኻ㔀㖸Ყࠍ␜ߒߚ߽ߩ
ߢࠆޕᮮゲߪ EDFA ߩബ
ࠩ߳ߩᵈ㔚ᵹ❑ޔゲߪ
SOA ߳ߩᵈ㔚ᵹߢࠅޔା
ภኻ㔀㖸Ყߪࠣࠬࠤ࡞
ߢ␜ߒߡࠆߢߎߎޕ㔀㖸ߪޔ
ฃశེ߅ࠃ߮⽶⩄ᛶ᛫ߩᾲ㔀
㖸࠻࠶࡚ࠪޔ㔀㖸ޔ࿁〝㔀㖸
(a)
(b)
ࠍޔߚ߹ޕ࿑ 4(b) ߪޔ࿑
4(a) ߣหߓ᧦ઙߩਅߢࠪࠝޔ
࿑ 6: ࠛ࡞ࡆ࠙ࡓᷝടࡈࠔࠗࡃჇེߩബඨዉࠩ߳ߩᵈ㔚ᵹࠍ
330mA ߦ࿕ቯߒߚߣ߈ߩജࡄ࡞ࠬߩᤨ㑆ᵄᒻޕඨዉశჇེ߳ߩᵈ
ࡠࠬࠦࡊߦ߅ߡ᷹ⷰߐࠇ
㔚ᵹ (a) 250mA, (b) 500mAޕ᳓ᐔゲ: 20ps/div.ޕ
ߚࡄ࡞ࠬ㧔ඨ୯ో㧕ࠍࠣ
ࠬࠤ࡞ߢ␜ߒߡࠆޕ
ߎߎߢߪ⦡⊕ޔඨ୯ో߇
50ps એ⊒ࠬ࡞ࡄߪߚ߹ޔᝄߒߥ߆ߞߚ႐วࠍߔߩࡊࠦࠬࡠࠪࠝޔ߅ߥޕ㔚᳇Ꮺၞߩ㒢ߦ
ࠃࠅޔ20ps એਅߩࡄ࡞ࠬߪ᷹ⷰߢ߈ߡߥޕ
࿑ 4(a) ࠃࠅ EDF ߩബࡌ࡞߇ᄢ߈߶ߤାภኻ㔀㖸Ყ߇ᄢ߈ߊߥࠆะ߇ಽ߆ࠆޔߪࠇߎޕ
ࡄ࡞ࠬᝄ߇ᄢ߈ߊߥࠆߎߣߦኻᔕߒߡࠆޕ࿑ 5 ߪ SOA ߳ߩᵈ㔚ᵹࠍ 330mA ߦ࿕ቯߒޔ
EDFA ߩബࠩ߳ߩᵈ㔚ᵹࠍ (a) 350mA, (b) 500mA ߣߒߚ႐วߩࡄ࡞ࠬᵄᒻߢࠆࡄޕ
࡞ࠬߪቯߒߡ⊒↢ߒߡ߅ࠅޔEDFA ߩബࡌ࡞ߩ㜞߶ߤᵄ㜞߇㜞ߊߥࠅޔାภኻ㔀㖸Ყ߇
ᡷༀߐࠇߡࠆ৻ޕᣇޔ࿑ 4(b) ࠃࠅޔEDFA ߩബࡌ࡞߇㜞႐วߦߪࠬ࡞ࡄޔ߽Ⴧᄢߒߡ
ࠆߪࠇߎޕ࿑ 5(b) ߦࠄࠇࠆࠃ߁ߦߩࠬ࡞ࡄޔᓟ✼߇િ߮ߡߒ߹߁ߎߣ߇ᢙ୯ߣߥߞߡࠇߚ
߽ߩߢࠆޕ
৻ᣇޔEDFA ߩബࡌ࡞ࠍ 325mA એߣߒߚ႐วߦߪޔSOA ߳ߩᵈ㔚ᵹࠍ 500mA એߦ
ߔࠆߣାภኻ㔀㖸Ყ߇ഠൻߔࠆޕ࿑ 5 ߪ EDFA ߩബࠩ߳ߩᵈ㔚ᵹࠍ 330mA ߦ࿕ቯߒޔ
EDFA ߩബࠩ߳ߩᵈ㔚ᵹࠍ (a) 250mA, (b) 500mA ߣߒߚ႐วߩࡄ࡞ࠬᵄᒻߢࠆޕ࿑
5(a) ߢߪᏀฝ߇߶߷ኻ⒓ߥᵄᒻ߇ᓧࠄࠇߡࠆߩߦኻߒޔ࿑ 5(b) ߢߪࡄ࡞ࠬ೨✼߇✭߿߆ߦߥ
ࠅߚ߹ޔᓟ✼ߪࡌࠬࠗࡦઃㄭߦ߅ߌࠆᝄേࠍ␜ߒߡࠆߩࠄࠇߎޕᵄᒻഠൻ߇ޔ࿑ 4(b) ߦ߅
ߌࠆ SOA ߳ߩᵈ㔚ᵹ߇ᄢ߈㗔ၞߢߩࡄ࡞ࠬߩჇടߩේ࿃ߣߥߞߡࠆޕ
એߩ⚿ᨐࠃࠅޔኻ⒓ߥࡄ࡞ࠬᵄᒻࠍᓧࠆߚߦߪ EDFAޔSOA ߦബࡌ࡞ࠍᛥ߃ࠆᔅⷐ߇
ࠅߩߘޔਛߢࡄ࡞ࠬ㜞߇ᄢ߈ߊߥࠆേὐࠍߟߌߥߌࠇ߫ߥࠄߥߎߣ߇ࠄ߆ߦߥߞߚޕ
ߥ߅ᧄታ㛎ࠪࠬ࠹ࡓߦ߅ߡޔჇᇦࠍ EDFA නߣߒߚ႐วߦߪࠬ࡞ࡄޔൻߩళߪࠄࠇ
ߚ߽ߩߩޔቯߒߚ⊒ᝄߦߪ⥋ࠄߥ߆ߞߚޔߚ߹ޕSOA නߣߒߚ႐วߦߪࡄ࡞ࠬ⊒ᝄߩళߔ
ࠄࠄࠇߥ߆ߞߚߡ߅ߦࡓ࠹ࠬࠪᧄޔߜࠊߥߔޕቯߒߚࡄ࡞ࠬ⊒ᝄࠍᓧࠆߚߦߪޔEDFA
ߣ SOA ࠍ૬↪ߔࠆᔅⷐ߇ࠆޕ
− 41 −
⋧หᦼ࡞ࡊࠍ↪ߚ㔚᳇Ꮻㆶ࿁〝ߩၮ␆⊛ᬌ⸛
ࠪࠬ࠹ࡓㆇォ㐿ᆎᤨ߆ࠄߩ
ࡄ࡞ࠬ⊒ᝄേࠍታߔࠆ
ߎߣࠍ⋡⊛ߣߒߡޔౣ↢ࡕ
Reference
࠼หᦼߩ㔚᳇Ꮻㆶ࿁〝ߦ
Bias
⋧หᦼ࡞ ࡊ (PLL) ࠍዉ
ߔࠆߎߣࠍᬌ⸛ߒޔ࿑ 6
ߦ␜ߔ࿁〝ࠍ⹜ߒߚޕ
࿑ 6: ⹜ߒߚ⋧หᦼ࡞ࡊߩ᭴ᚑޕDBM: ᵄᢙࡒࠠࠨޔVCO: 㔚
10GHz એߩേᵄᢙࠍ
ᓮ⊒ᝄེޕ
ᜬߟ⋧Ყセེߪᚻ࿎㔍
ߥߚޔᵄᢙࡒࠠࠨߦࠃߞߡઍ↪ߒߚޕࠆߡ↪ࠍဳ࠼ࠣߪߦ࠲࡞ࠖࡈࡊ࡞ޕෳᾖ
ାภḮߣߒߡᵄᢙࠪࡦࠨࠗࠩࠍ↪ߚ੍ታ㛎ߢߪࠢ࠶ࡠޔᵄᢙ▸࿐⚂ 1MHz ߩ⋧หᦼ
߇ታߐࠇߚޕᒁ߈ㄟߺ߅ࠃ߮ࡠ࠶ࠢᵄᢙ▸࿐ߩᄢߣࡊ࡞ޔᓧߩ⺞ᢛ߇ᔅⷐߢࠆޕ
DBM
Loop Filter
VCO
߹ߣ
ᧄ⎇ⓥߢߪޔEPR శሶኻࠍ㜞➅ࠅߒᵄᢙߢቯߦ⊒↢ߐߖࠆߚߩࡄ࡞ࠬశḮߣߒߡޔ⢻േ
㜞⺞ᵄࡕ࠼หᦼࡈࠔࠗࡃࠩߦ⌕⋡ߒࡓ࠙ࡆ࡞ࠛޔᷝടࡈࠔࠗࡃჇེߣඨዉశჇེࠍ
૬↪ߒߚ㜞⺞ᵄౣ↢ࡕ࠼หᦼߦࠃࠅ㐳ᤨ㑆ቯൻࠍታߒߚࡓ࠙ࡆ࡞ࠛޕᷝടࡈࠔࠗࡃჇེ
ߣඨዉశჇེߩബࡌ࡞ߣ⊒ࠬ࡞ࡄޔᝄߩะࠍ⺞ᩏߒߚ⚿ᨐޔᏀฝኻ⒓ߥశࡄ࡞ࠬࠍታ
ߔࠆߦߪਔჇེߩബࡌ࡞ࠍᛥ߃ߥߌࠇ߫ߥࠄߥߎߣޔඨዉశჇེ߳ߩㆊᄢߥᵈ
㔚ᵹߪࡄ࡞ࠬߩ⪺ߒᱡߦߟߥ߇ࠆߎߣ߇ࠄ߆ߦߥߞߚޔߚ߹ޕᐕᐲᣂߚߦޔ㔚᳇Ꮻㆶ࿁〝
ߦ⋧หᦼ࡞ࡊࠍ↪ࠆࠪࠬ࠹ࡓߩᬌ⸛ࠍⴕޔ⋧หᦼ࡞ࡊߩ⹜ࠍⴕߞߚޕ
ᓟߩ⺖㗴ߣߒߡޔ⋧หᦼ࡞ࡊߩ㜞ᕈ⢻ൻߣࠪࠬ࠹ࡓ߳ߩᔕ↪߇ߍࠄࠇࠆࠢࡍࠬޔߚ߹ޕ
࠻࡞ቯൻߩ⚵ߺ߽⛮⛯ߒߡᬌ⸛ߔࠆޕ
ෳ⠨ᢥ₂
[1] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical
reality be considered complete?” Phys. Rev., vol. 47, pp. 777-780, 1935.
[2] E. Yoshida and M. Nakazawa, “80-200 GHz erbium doped fibre laser using a rational harmonic
mode-locking technique,” Electron. Lett., vol. 32, pp. 1370-1372, 1996.
[3] C. Wu and N. K. Dutta, “High-repetition-rate optical pulse generation using a rational
harmonic mode-locked fiber laser,” IEEE J. Quantum. Electron., vol. 36, pp. 145-150, 2000.
[4] M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively
modelocked polarization-maintaining erbium fibre ring laser,” Electon. Lett., vol. 30, no. 19, pp.
1603-1605, 1994.
ᧄ⎇ⓥߦ㑐ㅪߔࠆ⊒
[1] Joji Maeda, Daichi Kurotsu, and Satoshi Ebisawa, “Stable pulse generation from regenerative
mode-locked fiber ring laser employing semiconductor optical amplifier,” Technical Digest of 15th
Optoelectronics and Communications Conference, pp. 348-349, Jul. 2010.
[2] Joji Maeda, Kazutoyo Kusama, and Satoshi Ebisawa, “Effects of Fiber Nonlinearity on
Radio-over-Fiber Transmission of DSB-BPSK Signal,” ibid, pp. 716-717, Jul. 2010.
[3] Joji Maeda and Satoshi Ebisawa, “Effect of random local dispersion in ultra-high speed optical
link employing periodical dispersion-compensation,” Technical Digest of OSA topical meeting on
Nonlinear Photonics 2010 (CDROM) Jun. 2010.
[4] 㤥ᵤ ᄢᶏ⠧Ỉ ⾫ผ೨↰ ⼨ᴦ, “ඨዉశჇེࠍ↪ߚౣ↢㜞⺞ᵄࡕ࠼หᦼࡈࠔࠗࡃ
ࡦࠣࠩߩቯൻ,” 㔚ሶᖱႎㅢାቇળᛛⴚ⎇ⓥႎ๔, vol. 110, no. 257, pp. 17-20, Oct. 2010.
− 42 −
㊂ሶࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ߦ㑐ߔࠆ⎇ⓥ
ᢙℂࠣ࡞ࡊ 㧔⺪⸰᧲੩ℂ⑼ᄢቇ⚻༡ᖱႎቇㇱ ᧻ጟ⎇ⓥቶ㧕
᧻ጟ 㓉ᔒ㧔ᢎ㧕
Abstract㧚ᐔᚑ 22 ᐕᐲߪޔㆊ 4 ᐕ㑆ߩᚑᨐࠍၮߦޔ㊂ሶࠛࡦ࠲ࡦࠣ࡞࠼⁁ᘒ߆ࠄ᭴ᚑ
ߐࠇࠆࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝ߩવㅍኈ㊂ߦ㑐ߔࠆ⎇ⓥࠍⴕߞߚޕ
1㧚ࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝ߣߘߩવㅍኈ㊂
1.1 ▵ߢޔ21 ᐕᐲߩ⎇ⓥᚑᨐߩ৻ߟߢࠆ㊂ሶࡌࠗࠫࠕࡦ㑐ଥᑼࠍ◲නߦᓳ⠌ߒߩߘޔ㑐ଥᑼࠍ
ၮߦߒߚࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝ߩવㅍኈ㊂ߦ㑐ߔࠆ⚿ᨐࠍႎ๔ߔࠆޕ
1㧚1 ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻౮ߣ㊂ሶ᧦ઙઃ߈⏕₸↪⚛
৻⥸ߦޔ㊂ሶ♽ߩ⁁ᘒߩᢙℂ⊛ߪ‛ࠆޔℂ㊂ A ߩᓳ᷹ቯࠍⴕߞߚߣ߈ߦᓧࠄࠇࠆᦼᓙ୯
A ߣߩኻᔕ㑐ଥࠍਈ߃ࠆᦼᓙ୯᳢㑐ᢙ M (i.e., M A A )ߦࠃߞߡ⸥ㅀߐࠇࠆ ޔߦ․ޕh ࠍ
Hilbert ⓨ㑆 ޔb h ࠍ h ߩ⇇✢ᒻ↪⚛ߩోߣߔࠆߣ ޔb h ߩᱜⷙ⁁ᘒ U ߦߪޔ
ࠆኒᐲ↪⚛ U ( U t 0, trh U 1) ߇ሽߒߡޔછᗧߩ A b h ߦኻߒ U A trh U A ߇ᚑ┙ߔࠆ(ޕએ
ਅޔห᭽ߦ⁁ޔᘒߦኻᔕߔࠆኒᐲ↪⚛ߪ⁁ߩߘޔᘒߣหߓ⸥ภࠍ↪ߔࠆ)ޕ
߹ޔੑߟߩ Hilbert ⓨ㑆 h , k ߩ࠹ࡦ࠰࡞Ⓧ h
k ߩኒᐲ↪⚛ T ߇ߩࠇߙࠇߘޔㇱಽ♽ߩ
ኒᐲ↪⚛(ㄝ⁁ᘒ)ߩ㓸ว ^ Ui ` , ^V i ` ࠍ↪ߡޔ
T
¦pU
i
i
Vi
i
( pi t 0, ¦ pi
(1)
1)
i
ߣಽ⸃ߢ߈ࠆߣ߈ ޔT ࠍนಽഀߥ⁁ᘒߣ߱ޕEPR(Einstein-Podolski-Rosen)ࡍࠕߦઍߐࠇࠆ
㊂ሶ♽․ߩ⋧㑐(㊂ሶ⋧㑐)ࠍߔࠆ⁁ᘒߪ(ޔ1) ᑼߩࠃ߁ߦนಽഀߥ⁁ᘒߩಲ⚿วߦಽ⸃ߢ߈ߥ
㕖นಽഀߥ⁁ᘒ(ࠛࡦ࠲ࡦࠣ࡞࠼⁁ᘒ)ߣߒߡߐࠇࠆޕ
৻ᣇޔBelavkin ߣ Ohya ߪޔ↪⚛ઍᢙߩ౮ࠍ↪ߡޔ㊂ሶ⁁ᘒߩಽ㘃ࠍਈ߃ߡࠆ=1,2?ޕ
วᚑ♽ h
k㧔ߘߩᰴరߪ㒢ߣߪ㒢ࠄߥ㧕ߩછᗧߩวᚑ⁁ᘒ T ߪ ޔb(k) ߆ࠄ b(h ) 㧔ߎ
ߎߢ ޔb(h ) ߪ b(h ) ߩ೨ኻⓨ㑆(a pre-dual space)ޔi.e. b(h )
(b(h ) ) 㧕߳ߩ౮ I ޔ
ࠆߪ b(h ) ߆ࠄ b(k) ߳ߩ౮ I ࠍ↪ߡޔ
T A
B =trh
kT A
B trh AI B trkI A B
(2)
ߣߐࠇࠆ ޔߢߎߎޕI ޔI ߪ ޔI h , I k ࠍߘࠇߙࠇ h , k ߩᕡ╬↪⚛ߣߔࠆߣޔ
I B { trkT I h
B , I A { trhT A
I k (3)
ߣਈ߃ࠄࠇ ޔI ޔI ࠍࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻౮ߣ߱ޕ
ߎߩߣ߈ޔวᚑ⁁ᘒ T ߪ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔ౮ I ࠍ↪ࠆߣޔᰴߩࠃ߁ߦ᭴ᚑߢ߈ࠆޕ
T
¦e
m
en
I en em
(4)
ߎߎߢ^ ޔen ` ߪޔㇱಽ♽ h ߩᱜⷙ⋥ၮᐩߢࠆ ޔߚ߹ޕT ߩㄝ⁁ᘒߪࠇߙࠇߘޔ
m,n
U I I k =trkT ޔV
I I h =trhT
ߣਈ߃ࠄࠇࠆޕ
߹ ޔb(h ) ࠍ b h ߩᱜߩⷐ⚛ߩ㓸วߣߔࠆ ޕb h ߆ࠄ b k ߳ߩ౮ / ߇ b(h ) ߩⷐ
⚛ࠍ b(k) ߩⷐ⚛ߦ⒖ߔߣ߈(i.e. /(b(h ) ) b(k) )ޔ/ ࠍᱜ౮ߣ߱ޔߚ߹ޕᱜ౮ / ߇ޔ
1
− 43 −
છᗧߩ⥄ὼᢙ n ߦኻߒߡޔᰴߢਈ߃ࠄࠇࠆ౮ / n
/ n : M n (b(h )) o M n (b(k)); [ Aij ] 6 [/( Aij )]
ࠍᏱߦᱜߦߔࠆߣ߈ޔ/ ࠍቢోᱜ౮ߣ߱ޔߢߎߎޕM n (b(h )) ߪޔb h ߩⷐ⚛߆ࠄߥࠆ n u n
ⴕ㧔↪⚛ⴕ㧕ߩోߢࠆޕ
ߎߩߣ߈࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔ౮ߩቢోᱜᕈ㧔or 㕖ቢోᱜᕈ㧕ߪޔวᚑ⁁ᘒࠍᰴߩࠃ߁ߦಽ
㘃ߔࠆ[1,2,3,15]ޕ
ቯℂ㧝㧔1㧕 T ࠍ⚐☴⁁ᘒߣߔࠆߣޔᰴ߇ᚑ┙ޕ
T ߇นಽഀޕЊ I ޔI ߪቢోᱜޕ㧔 T ߪ㕖นಽഀޕЊ I ޔI ߇㕖ቢోᱜޕ㧕
㧔2㧕 T ࠍᷙว⁁ᘒߣߔࠆߣޔᰴ߇ᚑ┙ޕ
T ߇นಽഀޕЈ I ޔI ߪቢోᱜޕ㧔 I ޔI ߇㕖ቢోᱜޕЈ T ߪ㕖นಽഀޕ㧕
ቯℂ 1 ߆ࠄ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔ౮ I ߩቢోᱜᕈ㧔or 㕖ቢోᱜᕈ㧕ߪ⁁☴⚐ޔᘒߦ߅ߡ
นಽഀ⁁ᘒ㧔or ࠛࡦ࠲ࡦࠣ࡞࠼⁁ᘒ㧕ߩᔅⷐචಽ᧦ઙޔᷙว⁁ᘒߦ߅ߡนಽഀ⁁ᘒ㧔or ࠛࡦ
࠲ࡦࠣ࡞࠼⁁ᘒ㧕ߩᔅⷐ㧔or චಽ㧕᧦ઙࠍਈ߃ࠆߎߣ߇ࠊ߆ࠆ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔߦ⥸৻ޕ౮
ߪቢోᱜߣߪ㒢ࠄߥ߇ޔᏱߦቢోᱜ㧔KG౮ [ Aij ] M n (b(h )) 6 [/ ( Aji )] M n (b(k)) ߇
છᗧߩ⥄ὼᢙ n ߦኻߒߡᏱߦᱜ㧕ߣ߁ᒻߩቢోߐࠍߒߡࠆ=1,2?ޕ
એߩ⼏⺰ࠍ〯߹߃ߡޔBelavkin ߣ Ohya ߪߩ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛߥ⊛⥸৻ޔቯ⟵ࠍਈ߃ޔ
ߘߩቯ⟵ࠍ↪ߡวᚑ⁁ᘒࠍᰴߩࠃ߁ߦಽ㘃ߒߚ=1,2?ޕ
ቯ⟵㧞 㧔㧝㧕 trh I I k 1 ߣⷙᩰൻߐࠇߚ b(k) ߆ࠄ b(h ) ߳ߩቢోᱜ౮ I ߩᓎ౮
I ࠍ⁁ޔᘒ U { I I k ߆ࠄ⁁ᘒ V { I I h ߳ߩ㧔৻⥸ൻߐࠇߚ㧕ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻
ߣ߮ోߩ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔࠍ eߣߔ ޔߚ߹ޕI ߦኻᔕߔࠆวᚑ⁁ᘒ TI ߪޔ
ᰴᑼߢਈ߃ࠄࠇࠆޕ
TI
¦e
m
m,n
en
I en em
(4)̉
㧔㧞㧕ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ I ߇ቢోᱜߢߥߣ߈ ޔI ࠍ Iq ߣᦠߡ㨝-ࠛࡦ࠲ࡦࠣ࡞
ࡔࡦ࠻ߣ߮ ޔIq ߩోࠍ eq ߣߔޔߚ߹ޕ㨝-ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ߢߥࠛࡦ࠲ࡦ
ࠣ࡞ࡔࡦ࠻ࠍ㕖㨝-ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ߣ߮ోߩߘޔ enon q { e/ eq ߣߔޕ
৻ᣇޔKossakowski ㆐ߪޔ㊂ሶ᧦ઙઃ߈⏕₸↪⚛ߥࠆ↪⚛ࠍዉߒ[4]ޔ㊂ሶวᚑ⁁ᘒࠍ
᭴ᚑߔࠆᚻᴺࠍ␜ߒߡࠆ߇ޔ21 ᐕᐲߪߩߎޔੑߟߩᚻᴺޔߜࠊߥߔޔቯ⟵㧞ߩ(৻⥸ൻߐࠇߚ)
ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻౮ࠍ↪ߡ(4)̉ᑼߢ᭴ᚑߐࠇࠆวᚑ⁁ᘒߩߣޔKossakowski ㆐ߩ㊂
ሶ᧦ઙઃ߈⏕₸↪⚛ߦࠃࠆวᚑ⁁ᘒߩߩห୯ᕈࠍ␜ߒߚޕએਅޔ㊂ሶ᧦ઙઃ߈⏕₸↪⚛
ߩቯ⟵ࠍᓳ⠌ߒޔ21 ᐕᐲߩ⚿ᨐࠍ◲නߦႎ๔ߔࠆޕ
߹ޔวᚑ⁁ᘒ T ߦኻߒߡ✢ޔᒻ↪⚛ S T b h
k ࠍᰴߩࠃ߁ߦቯ⟵ߔࠆޕ
§ 12
· § 12
·
(5)
U
I
Ik ¸
¨
k ¸T ¨ U
©
¹ ©
¹
trkT ߪᔘታ(i.e. U ! 0 )ߢࠆߎߣࠍቯߔࠆ ޔ߈ߣߩߎޕS T ߪᰴߩੑߟߩᕈ⾰ࠍḩ
ST
ߎߎߢ ޔU
ߚߔޕ
㧔ᕈ⾰㧝㧕 S T t 0 ( ޔᕈ⾰㧞) trkS T
Ih ޕ
ቯ⟵㧟 ✢ᒻ↪⚛ S b h
k ߇ޔᕈ⾰ 1㧘2 ࠍḩߚߔߣ߈ ޔS ࠍ㊂ሶ᧦ઙઃ߈⏕₸↪⚛
(quantum conditional probability operator : QCPO)ߣ߱ޕ
ቯ⟵㧟ߩ S ߇ਈ߃ࠄࠇࠆߣޔછᗧߩᔘታߥㄝ⁁ᘒ(ኒᐲ↪⚛) U ߦኻߒߡޔᚒߪޘวᚑ⁁ᘒ
2
− 44 −
ࠍᰴߩࠃ߁ߦ᭴ᚑߢ߈ࠆޕ
§ 1
· § 1
·
T ¨ U 2
Ik ¸S ¨ U 2
Ik ¸
©
¹ ©
¹
ߎߩߣ߈ޔh ߩㄝ⁁ᘒ U ߪ U
(6)
trkT ߢࠅޔk ߩㄝ⁁ᘒࠍ V ߣ⸥ߔࠇ߫ ߩߎޔV ߪ U
ߣ S ߦࠃߞߡޔᰴߩࠃ߁ߦਈ߃ࠄࠇࠆޕ
V trhT trh S U
I k { / U (7)
ᑼߢቯ⟵ߒߚ / ߪ ޔh ߩ⁁ᘒࠍ k ߩ⁁ᘒߦ౮ߔ౮㧔㊂ሶ࠴ࡖࡀ࡞㧕ߣߥߞߡࠆޕ
ᕈ⾰㧝ޔ㧞ࠍḩߚߔߣ߁ᗧߢ ޔS ߪ̌᧦ઙઃ߈⏕₸̍ߣ߁ฬࠍߒߡࠆ߇(ޔ7)ᑼࠍ⠨
ᘦߔࠇ߫ ޔS ߪ̌࠴ࡖࡀ࡞ኒᐲ̍ߣ߽߮ᓧࠆ߽ߩߢࠆޕ
Kossakowski ㆐ߪޔᱜⷙൻߐࠇߚ b k ߆ࠄ b h ߳ߩቢోᱜ౮ M 㧔i.e. M I k I h 㧕ࠍ↪
ߡޔᕈ⾰ 1㧘2 ࠍḩߚߔ㊂ሶ᧦ઙઃ߈⏕₸↪⚛ S M ࠍᰴߩࠃ߁ߦዉߒߚ[4]ޕ
SM
¦M fn
m,n
fm
fn
(8)
fm
ߎߩߣ߈ߩ⥸৻ޔวᚑ⁁ᘒࠍ(4)’ᑼߢਈ߃ࠆࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻౮ I ߣ QCPO ࠍ(8)ᑼߢਈ߃ࠆ
ᱜⷙൻߐࠇߚቢోᱜ౮ M ߪޔᰴߩ㗴[5]ߦࠃߞߡ㑐ଥઃߌࠄࠇࠆߎߣ߇ಽ߆ࠆޕ
㗴㧠 ㄝ⁁ᘒ U { I I k ࠍਈ߃ࠆࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ I ߪޔᰴߩࠃ߁ߦಽ⸃ߐࠇࠆޕ
1
1
I x U 2 M D t x U 2 (9)
ߎߎߢ ޔt ߪォ⟎ࠍขࠆᠲߦኻᔕߔࠆ౮ߢࠅ ޔM ߪᱜⷙൻߐࠇߚቢోᱜ౮ߢ(ޔ9)ᑼߩ
ಽ⸃ࠍਈ߃ࠆ M ߪ৻ߩߘޔᗧߥ⸃ߣߒߡᰴߩࠃ߁ߦ᳞߹ࠆޕ
M x U
1
2
I D t x U
1
2
(10)
ߔߥࠊߜޔ㗴㧠ߢ ࠻ࡦࡔ࡞ࠣࡦ࠲ࡦࠛޔI ߩಽ⸃ࠍਈ߃ࠆቢోᱜ౮ M ࠍ↪ߡ QCPO ࠍ᭴ᚑ
ߔࠇ߫ޔᰴߩቯℂ߇ᚑ┙ߔࠆ[9]ޕ
ቯℂ㧡 ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ I ࠍ↪ߡޔᱜⷙൻߐࠇߚቢోᱜ౮ M ࠍ M x ߣਈ߃ࠆߣޔᰴ߇ᚑ┙ߔࠆޕ
§ 1
· § 1
·
TI ¨ U 2
I k ¸ S M ¨ U 2
I k ¸
©
¹ ©
¹
U
1
2
t D I x U
1
2
(11)
1㧚2 Ohya ⋧ࠛࡦ࠻ࡠࡇࠍ↪ߚࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝ߩવㅍኈ㊂ߩ⹏ଔ
°
½°
ฎౖ⺰ߩࠬࠠࡓߦ߅ߡߪ⚿ޔว⏕₸ಽᏓ r rij ߣߘߩㄝಽᏓ p ® pi ¦ rij ¾ ߮᧦ޔઙઃ
j
¯°
¿°
^ `
߈⏕₸ಽᏓ ^ p i o j ` ߣߩ㑆ߦߪޔᰴ߇ᚑ┙ߔࠆޕ
rij
pi p i o j (12)
ቯℂ㧡ߦ߅ߌࠆวᚑ⁁ᘒ TI ߣ QCPO S M ߩ㑐ଥᑼ(11)ߪ╬ޔᑼ(12)ߩ㊂ሶ ߣߺࠆߎߣ߽ߢ߈ޔ
(11)ᑼࠍᚒ̌ߪޘ㊂ሶࡌࠗࠫࠕࡦ㑐ଥᑼ̍ߣ߱ޕ5JCPPQP ߩᖱႎℂ⺰ߩࠛ࠶ࡦࠬߪ᧦ޔઙઃ
߈⏕₸ಽᏓ ^ p i o j ` ࠍޔㄝಽᏓ p ߆ࠄઁߩ߽߁৻ߟߩㄝಽᏓ q ߳ߩ࠴ࡖࡀ࡞ / ߣߥߒߚ
½
®q j ¦ pi p i o j ¾ / p )⋧ޔࠛࡦ࠻ࡠࡇࠍޔജ p ߩᖱႎ߇࠴ࡖࡀ࡞ /
i
¯
¿
ࠍㅢߒߡജ q ߦߤࠇߛߌᱜ⏕ߦવࠊߞߚ߆ࠍ␜ߔᖱႎ㊂ߣᝒ߃ࠆߎߣߦࠆޔ߹ޕ㊂ሶ⋧
ࠛࡦ࠻ࡠࡇࠍฎౖ♽ߩቯ⟵ߦḰߓߡޔᰴߩࠃ߁ߦቯ⟵ߒࠃ߁ޕ
ߣ߈
KG q
3
− 45 −
ቯ⟵㧢 (1) ㊂ሶ⋧ࠛࡦ࠻ࡠࡇ ITI U , V 㧦
ITI U , V { trTI log TI log U
V (13)
ߎߩߣ߈(ޔ13)ᑼߢቯ⟵ߒߚ⋧ࠛࡦ࠻ࡠࡇ߇ޔฎౖ♽ߣห᭽ߦޔജ߆ࠄജ߳࠴ࡖࡀ࡞ࠍ
ㅢߒߡવࠊࠆᖱႎ㊂ߣߺߥߖࠆ߆⠨߃ߡߺࠃ߁ޕ
ޔวᚑ♽ TI 㧔 U I I k , V I I h 㧕߇ਈ߃ࠄࠇࠆߣ(ޔ8)ᑼ(ޔ10)ᑼޔ߮(11)ᑼࠍ⠨ᘦߔ
ࠇ߫(ޔ7)ᑼߦḰߓߡࠛࡦ࠲ࠣ࡞࠼ㅢା〝ࠍᰴߩࠃ߁ߦ᭴ᚑߢ߈ࠆޕ
V trhTI trh S I U
I k { /I U (14)
ߎߎߢޔ
1
2
¦ U I D t SI
fm
U
fn
m,n
1
2
fm
(15)
fn ޕ
ࠃߞߡޔฎౖ♽ߣห᭽ߦޔวᚑ♽ TI ࠍജ⁁ᘒ U ߆ࠄࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝 /I ࠍㅢߒߡㅍࠄࠇ
ࠆജ⁁ᘒ /I U ߩ⋧㑐ࠍߔ⁁ᘒߣߺߥߔߎߣ߇ߢ߈ࠆߣߔࠇ߫ޔITI U , V ߪജ߆ࠄജ
߳࠴ࡖࡀ࡞ࠍㅢߒߡવࠊࠆᖱႎ㊂ߣੌ⸃ߐࠇᓧࠆߴ߈߽ߩߢࠆ ߫߃ޔߒ߆ߒޕTI ߣߒߡ⚐ޔ
☴ࠛࡦ࠲ࡦࠣ࡞࠼⁁ᘒࠍขࠆߣ◲ޔනߥ⸘▚߆ࠄᰴߩ⚿ᨐࠍᓧࠆޕ
ITI U , V 2S U (16)
ߔߥࠊߜޔ㊂ሶ⋧ࠛࡦ࠻ࡠࡇ ITI U , V ߪޔㅢା〝ࠍㅢߒߡવㅍߐࠇߚᖱႎ㊂߇ḩߚߔߴ߈ޔ
ᰴߩၮᧄਇ╬ᑼࠍḩߚߐߥࠤࠬࠍ⸵ߔߎߣߦߥࠆޕ
^
ITI U , V d min S U , S /I U `
(17)
ࠃߞߡ ޔITI U , V ߪ࠼࡞ࠣࡦ࠲ࡦࠛޔㅢା〝 /I ࠍㅢߒߡㅍࠄࠇࠆജ⁁ᘒߩᖱႎ㊂ࠍ⸘㊂ߔࠆ
ᜰᮡߣߒߡߪㆡಾߥ߽ߩߣߪ⸒߃ߥޕ
Ohya ߪޔ㊂ሶ࠴ࡖࡀ࡞ / ࠍㅢߒߡવㅍߐࠇࠆജ⁁ᘒ U 䈱ᖱႎ㊂ࠍᱜߒߊ⸘㊂ߔࠆᜰᮡߣߒ
ߡߩ⋧ࠛࡦ࠻ࡠࡇࠍᰴߩࠃ߁ߦዉߒߚ[6]ޕ
^
I O U : / { sup S T E , U
/ U 䈖䈖䈪䇮 T E
¦pE
i
i
E
^Ei `
`
(18)
/ Ei 䇮 ^ Ei ` 䈲䇮ജ⁁ᘒ U 䈱࿕ⓨ㑆䈻䈱 1 ᰴరᓇ↪⚛䈱⚵䈪䈅
䉎䋺i.e.
¦pE,
U
i
i
pi t 0, ¦ pi
(19)
1
Ohya ⋧ࠛࡦ࠻ࡠࡇߪޔၮᧄਇ╬ᑼ(17)ࠍḩߚߔߎߣ߇⍮ࠄࠇߡ߅ࠅ[6]ߡߞࠃޔㅢା〝ࠍ
ߒߡવㅍߐࠇࠆജ⁁ᘒߩᖱႎ㊂ߣߺߥߔߎߣ߇ߢ߈ࠆ࠼࡞ࠣࡦ࠲ࡦࠛޔߢߎߘޕㅢା〝 /I ߦ
ኻߒߡޔOhya ⋧ࠛࡦ࠻ࡠࡇࠍㆡ↪ߔࠆߣޔᰴߩቯℂ߇ᓧࠄࠇࠆ[11]ޕ
ቯℂ ㄝ⁁ᘒ UI ࠍᜬߟ⚐☴㨝ࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻ I KG UI
ࡦࠣ࡞࠼ㅢା〝 /I ߦኻߒߡޔജ⁁ᘒ U s h ߇ ޔUI U
IO U : /
{ S U
I I k ߆ࠄ᭴ᚑߐࠇࠆࠛࡦ࠲
UUI ࠍḩߚߖ߫ޔᰴ߇ᚑ┙ߔࠆޕ
(20)
ቯℂ㧣߆ࠄ⚐☴ࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝 /I ߩવㅍኈ㊂߇ᰴߩࠃ߁ߦਈ߃ࠄࠇࠆ[11]ޕ
♽ C /I { sup I O U ; /I
U s h sup S U log(dim h1 ) U s h ቯℂ㧣ߣ♽ 8 ߦࠃࠅޔᄢ⍫⋧ࠛࡦ࠻ࡠࡇߩᗧߢ࠼࡞ࠣࡦ࠲ࡦࠛ☴⚐ޔㅢା〝 /I ߪജ♽
ߩᖱႎࠍജ♽ߦోߡવㅍߔࠆㅢା〝ߢࠆߎߣ߇ಽ߆ࠆ⚿ߩߎޕᨐߪ⁁࠼࡞ࠣࡦ࠲ࡦࠛ☴⚐ޔ
ᘒߩ⋧㑐ߩᒝߐߦኻᔕߔࠆ߽ߩߣߒߡ⥄ޔὼߥ⸃㉼ߣੌ⸃ߐࠇᓧࠆ߽ߩߢࠆޕ
4
− 46 −
ޔᷙวࠛࡦ࠲ࡦࠣ࡞࠼⁁ᘒ߆ࠄ᭴ᚑߐࠇࠆࠛࡦ࠲ࡦࠣ࡞࠼ㅢା〝ߩવㅍኈ㊂ߦߟߡޔᐞ
ߟ߆ߩวᚑ⁁ᘒߩࠢࠬߦ Ohya ⋧ࠛࡦ࠻ࡠࡇࠍㆡ↪ߔࠆߎߣߢ⸃ߩߘޔᨆࠍ⹜ߺߡࠆޕ
ෳ⠨⺰ᢥ
[1] V. P. Belavkin, M. Ohya, Infin. Dim. Anal. Quantum Probab. Relat. Top. 4 (2001) 137-160
[2] V. P. Belavkin, M. Ohya, Proc. R. Soc. London A 458 (2002) 209-231
[3] ᧻ጟ㓉ᔒ㊂ሶࠛࡦ࠲ࡦࠣ࡞ࡔࡦ࠻́㊂ሶ⋧㑐ߩ․ᓽઃߌࠍ⋡ᜰߒߡ́ޔ࿖㓙㜞╬⎇ႎ๔ᦠ
0801 (2008) 81-129
[4 ]M. Asorey, A. Kossakowski, G. Marmo, E.C.G. Sudarshan, Open Syst. Info. Dyn. 12
(2006) 319-329
[5] V.P. Belavkin, X. Dai, J. Quant. Info. 6 (2008) 981-996
[6] M. Ohya, IEEE. 29 (1983) 770-774
⎇ⓥౝኈߦ㑐ㅪߔࠆᐔᚑ 21 ᐕᐲએ㒠ߩቇⴚ⺰ᢥ (ᛩⓂਛޔḰਛࠍ)
[7] D. Chruscinski, A. Kossakowski, K. Mlodawski and T. Matsuoka, Open Syst. Info. Dyn.
17 (2010) 1-17 : e-print arXiv:1004.1655v1.[quant-ph]
[8] L. Accardi, D. Chruscinski, A. Kossakowski, T. Matsuoka and M. Ohya, “On classical and
quantum liftings” to be published in Open Syst. Info. Dyn.,
[9] D. Chruscinski, A. Kossakowski, T. Matsuoka and M. Ohya, “Entanglement mapping vs.
quantum conditonal probability operator” to be published in OP-PQ Quantum
Probability and White Noise Analysis: Quantum Bio-Informatics IV.
[10] D. Chruscinski, Y. Hirota, T. Matsuoka and M. Ohya, “Remarks on the degree of
entanglement” to be published in OP-PQ Quantum Probability and White Noise
Analysis: Quantum Bio-Informatics IV.
[11] D. Chruscinski, A. Kossakowski, T. Matsuoka and M. Ohya, “Entanglement channel
and its capacity” in preparation.
5
− 47 −
⎇ⓥᚑᨐႎ๔
ᢙℂ⎇ⓥࠣ࡞ࡊ㧔᧲੩ℂ⑼ᄢቇℂᎿቇㇱᖱႎ⑼ቇ⑼㧕
⮮ሶ㧔⻠Ꮷ㧕
ᧄᐕᐲⴕߞߚਥߥ ߟߩ⎇ⓥߦߟߡႎ๔ߔࠆ㧚
1㧚How to classify Influenza A viruses and Understand Their Severity
What evolutional processes determine the severity and the ability for transmission among
humans of influenza A viruses? We performed phylogenetic classifications of influenza A
viruses that were sampled between 1918 and 2009 by using a measure called “Entropic Chaos
Degree”, that was developed through the study of chaos in Information Dynamics.
The phylogenetic analysis of the internal protein (PB2, PB1, PA, NS, M1, M2, NS1, and
NS2) indicated that Influenza A viruses adapting to humans and transmitting among
humans were clearly distinguished from swine lineage and avian lineage. Furthermore, the
HA, NA, and internal proteins of the influenza strain that caused a pandemic or a severe
epidemic with high mortality were phylogenetically different than those from previous
pandemic and severe epidemic strains.
We have come to the conclusion that the internal protein has a significant impact on the
ability for transmission among humans. Based on this study, we are convinced that Entropic
Chaos Degree is very useful as a measure of understanding the classification and severity of
an isolated strain of influenza A virus.
K. Sato, T. Tanabe and M. Ohya, How to Classify Influenza A viruses and Understand Their
Severity, Open Systems & Information Dynamics, Vol.17, 297-310, 2010
2㧚Pairwise sequence alignment algorithm by a new measure based on transition probability
between two consecutive pairs of residues
Sequence alignment is one of the most important techniques to analyze biological systems. It
is also true that the alignment is not complete and we have to develop it to look for more
accurate method. In particular, an alignment for homologous sequences with low sequence
similarity is not in satisfactory level. Usual methods for aligning protein sequences in recent
years use a measure empirically determined. As an example, a measure is usually defined by
a combination of two quantities (1) and (2) below: (1) the sum of substitutions between two
residue segments, (2) the sum of gap penalties in insertion/deletion region. Such a measure is
determined on the assumption that there is no an intersite correlation on the sequences.
To improve the alignment by taking the correlation of consecutive residues, we introduced
a new method of alignment, called MTRAP by introducing a metric defined on compound
systems of two sequences. In the benchmark tests by PREFAB 4.0 and HOMSTRAD, our
pairwise alignment method gives higher accuracy than other methods such as ClustalW2,
TCoffee, MAFFT. Especially for the sequences with sequence identity less than 15%, our
− 48 −
method improves the alignment accuracy significantly. Moreover, we also showed that our
algorithm works well together with a consistency-based progressive multiple alignment by
modifying the TCoffee to use our measure.
T. Hara, K. Sato and M. Ohya, MTRAP: Pairwise sequence alignment algorithm by a new
measure based on transition probability between two consecutive pairs of residues, BMC
BIOINFORMATICS, Vol.11: 235, 1-11, 2010
3㧚Evolution of HIV-1 from the viewpoint of Information Theory
Information of life is stored as a sequence of nucleotides,and the sequence composed of four
bases seems to be a sort of code. Therefore, we can consider that the DNA or gene in each
organism is a code showing its inherent structure. Thus, we ask what kind of structure each
code has. More precisely, we ask what roles the code structure has for the emergence of life
and how it is concerned with the changes of the living body. On such questions, by using the
sequences encoded by various artificial codes in information transmission, we explored the
code structure of the sequences of HIV-1 envelope V3 region obtained longitudinally for each
patient and then studied the evolutionary changes during the course of HIV-1 infectionfrom
the viewpoint of the code structure.
As HIV-1 disease progressed, the V3 region was completely different from the structure of
codes observed at the early stage of HIV-1 infection. The code structure of V3 region showed a
change previous to the decrease of CD4+ T-lymphocyte counts. We consider that the changes
of the code structure are related with the stage of HIV-1 disease.
− 49 −
３次元理想ボーズ気体のクラスター表現のシミュレー
ション
数理グループ (山口東京理科大学工学部電気工学科 井上研究室)
井上 啓（講師）
Abstract. 理想ボーズ気体の位置分布のクラスター表現 [2] を用いると、原理的には、粒
子系のシミュレーションが可能である。しかし、低温ではクラスターが非常に大きく、そ
の基準粒子は有限領域から遠く離れている。このため、先の研究では、有限領域上に位置
する粒子の部分系における振る舞いにおけるクラスターの影響を見積もった [5]。本年度
の研究では，この結果を基にして，３次元理想ボーズ気体のクラスター表現のシミュレー
ションを試みた。
1.
Introduction
In [3] it was shown that to each locally normal state ω of boson systems on Rd one can
associate a distribution P ω of a random point system in Rd that can be interpreted as the
position distribution of the state. If the reduced density matrices ρωn (x1 , . . . , xn , y 1 , . . . , y n )
(n)
of the state ω (cf. [1]) are continuous then a density λP ω (w.r.t. Lebesgue’s measure) of
the n-th factorial moment measure of P ω is given by (cf. [4])
(n)
λP ω (x1 , . . . , xn ) = ρωn (x1 , . . . , xn , x1 , . . . , xn );
x1 , . . . , xn ∈ Rd .
(1)
Now, let us consider the (infinite-volume) ideal Bose gas (cf. [1]) corresponding to the
inverse temperature β > 0 and chemical potential µ < 0. z = exp[µβ] denotes the
activity. The conventional quantum-mechanical Hamiltonian for free particles of mass
2
2
∆ but we now choose units such that
= 1. Then the reduced density
m is −
2m
2m
matrices ρn , n = 1, 2, . . . of the ideal Bose gas exist and (cf. [1])
ρn (x1 , . . . .xn , y 1 , . . . , y n )
+∞
n #
xk − y ik 2
zm
=
exp −
;
(4πβm)d/2
4mβ
k=1 m=1
(i1 ,...,in )
x1 , . . . , xn , y 1 , . . . , y n ∈ Rd
(2)
where (i1 , . . . , in ) denotes a permutation of 1, . . . , n. Consequently, densities of the factorial moment measures of the position distribution P of this ideal Bose gas are given
by
(n)
λP (x1 , . . . , xn ) = ρn (x1 , . . . , xn , x1 , . . . , xn );
x1 , . . . , xn ∈ Rd .
(3)
It was proved in [2] that P is an infinitely divisible point process characterized by (3).
Furthermore, clustering representation of P was considered in [2]. Hence P may be
− 50 −
interpreted as distribution of a random point field in Rd which forms independent clusters
or random point-families the distribution of which is given by the clustering field in [2].
This corresponds to the description of the ideal Bose gas one can find in the physical
literature.
In principle one can use the clustering representations for simulations of the system
of positions of the quantum particle system. But one has to take into account that in
case of low temperature the clusters are very large and their origins are far from a fixed
bounded volume. For that reason we have estimated the influence of these clusters on
the behaviour of the subsystem of particles located in a fixed bounded volume [5]. More
precisely, we estimated the expectation of the number of particles inside a fixed bounded
volume being part of any cluster with origin outside of a certain bigger volume.
In this paper using the estimation, several numerical simulations of the position
distribution of the three dimensional ideal Bose gas are shown in the section 3.
2.
Clustering representations
A random point field Ψ with the position distribution of ideal Bose gas P can be constructed as follows: Starting point is a random point field Φ with distribution Q, i.e., Φ
is a stationary Poisson random point field with intensity λQ = λQ (β, µ) given by
λQ ≡
+∞
n=1
zn
n(4πβn)d/2
Further let γ = (γx )x∈Rd be a family of independent random point fields where γx represents a random offspring of an individual at site x ∈ Rd called cluster with origin x.
Finally we assume that Φ and γ are independent. Then formally we put
(4)
Ψ := Φ(dx) γx ,
i.e., we identify the random point field Ψ representing the system of the positions of the
particles of the ideal Bose gas with the independent superpositions of the offsprings of
all points of the point field Φ called to be the underlying random point field.
2.1.
Underlying Random Point Field
In order to illustrate the behaviour of the ideal Bose gas in the case of low temperature,
and in the case of high temperature we will consider the special case d = 2. Further we
fix
(1)
λP = λ.
Then λQ will be a function λQ (β, λ) of β, λ.
Now, let α(β, λ) be the expectation of the number of points in a cluster. Then it
holds
λ = α(β, λ) λQ (β, λ).
Firstly let us consider the case of low temperature. We have the following limits [5].
lim λQ (β, λ) = 0,
β→+∞
lim α(β, λ) = +∞.
β→+∞
− 51 −
Summarizing we can state that in case of low temperature the underlying point system
is very “thin” and the clusters are very large. Furthermore, all points in a fixed bounded
set are parts of a cluster with origin being far from this set if the temperature is low.
Now, let us consider the case of high temperature. Then we have the following
limits[5].
lim λQ (β, λ) = λ,
β→0
lim α(β, λ) = 1.
β→0
Summarizing we can state that in case of high temperature the system of positions of
the ideal Bose gas coincide with the underlying point system, being a stationary random
Poisson field. That reflects the well-known fact that in case of high temperature the
ideal Bose gas behaves like the classical ideal gas.
2.2.
Random cluster
For each x ∈ Rd , let γx be again a random cluster (or offspring) of an individual at site
x with probability distribution Kx .
Further let ηx be the number of points in the cluster γx , i.e., we have
ηx := γx (Rd ).
or
Pr (ηx = n) =
1
zn
.
λQ n(4πβn)d/2
(5)
(d)
Now we denote by qt (x, y) the densities of the transition probabilities of the d-dimensional
Wiener process (W (t))t≥0 , i.e., we have
%
$
1
x − y2
(d)
,
qt (x, y) = exp −
2t
(2πt)d
x = [x1 , . . . , xd ], y = [y1 , . . . , yd ] ∈ Rd , t ≥ 0.
Using these transition probabilities we define the following functions.
qn (x1 , . . . , xn−1 , x)
:=
1
(d)
q2βn (x, x)
(d)
(d)
q2β (x, x1 )q2β (xn−1 , x)
n−2
#
(d)
q2β (xk , xk+1 )
k=1
x1 , . . . , xn−1 , x ∈ Rd .
Then for all measurable Y ⊆ M we have the following equations.
1
dx · · · dxn−1 qn (x1 , . . . , xn−1 , x)
Pr(γx ∈ Y |ηx = n ) =
n−1
δxk , n = 2, 3, . . . ,
×χY δx +
k=1
Pr(γx ∈ Y |ηx = 1 ) = δδx (Y ).
− 52 −
Now, putting pn = Pr(ηx = n), then Kx can be rewritten as
Kx (Y ) =
+∞
pn Pr (γx ∈ Y |ηx = n ) ,
Y ⊆ M.
n=1
According to (5), ΛKx denotes the intensity measure of the point process Kx .
The following theorem was proven in [5].
Theorem 2..1 Let Nx,r be a normal distribution with expectation vector x ∈ Rd and
d × d covariance matrix r. For each measurable subset B of Rd it holds
n−1
1 zn
N k(n−k) (B).
ΛKx (B) = χB (x) +
λQ n≥2 n(4πβn)d/2 k=1 x,2β n I
2.3.
System of the Positions of the Particles of the Ideal Bose Gas
Let ΦA be the system of points of Φ being inside of measurable subset A of Rd
ΦA := Φ(· ∩ A)
and by QA we denote the point process corresponding to ΦA , i.e., QA is the distribution
of ΦA . Then for the intensity measure ΛQA we get
ΛQA = λQ l(· ∩ A)
where l is Lebesgue’s measure on Rd .
Using the Campbell theorem (cf. [6]) we obtain immediately the following lemma.
Lemma 2..2 Let f be a measurable nonnegative function on Rd . Then it holds
dx f (x).
ΛQA (dx) f (x) = E ΦA (dx)f (x) = λQ
A
Finally we denote by ΨA the system of all points of clusters γx with origin in A
A
Ψ := ΦA (dx)γx .
having the distribution QA
(K) .
Because of the independence of Φ and γ we obtain using lemma 2..2.
ΛQA(K) = λQ
dx ΛKx .
A
Using theorem 2..1 and (6) the following theorem was proven in [5].
Theorem 2..3 Let A, B be measurable subsets of Rd . Then it holds
ΛQA(K) (B) = λQ l(A ∩ B) +
n≥2
n−1 zn
dx Nx,2β k(n−k) I (B)
n
n(4πβn)d/2 k=1 A
− 53 −
(6)
3.
Results on simulations
Without loss of generality we assume that
A = [−a, a]d ,
B = [−b, b]d ,
a ≥ b ≥ 0,
In the followings, by Ac we denote the compliment of A.
The following theorem was proven in [5].
c
Theorem 3..1 Let ΛQAc be the intensity measure of ΨA =
a, b ∈ R.
(K)
ΦAc (dx)γx . Then it holds
ΛQAc (B) ≤ f (a, b, d, β, µ)
(K)
where the function f (a, b, d, β, µ) is given by
%d/2 $
2
(n − 1)z n
(a
−
b)
f (a, b, d, β, µ) := (2b)d
1 − 1 − exp −
.
n(4πβn)d/2
nβ
n≥2
(7)
From theorem 3..1 we can expect that all points in the smaller volume [−b, b]d come
from the bigger volume [−a, a]d on average. Using the formula (7) we can consider the
following problem concerning simulations of the system of positions of the particles: Fix
b, β, λ and find a such that f (a, b, β, µ) = ε for different ε’s.
(1)
Let us consider the special case d = 3 and we also fix λP (β, µ) = λ. Furthermore
let us fix b = 1.0, β = 0.1, λ = 1.0. Figure 1 shows a versus ε for d = 3.
4
3.8
3.6
3.4
a
3.2
3
2.8
2.6
2.4
2.2
0.001 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
ε
Fig.1. a versus ε for d = 3
This result means that all points in [−1, 1]3 come from [−4, 4]3 at β = 0.1 on average.
Similarly one finds that a = 3.5 is enough size at β = 0.09.
In order to simulate the position distribution of the ideal Bose gas on [−1, 1]3 we
first produce sample pictures of only one cluster with origin at point 0. Then using
a = 3.5, 4.0 we show sample pictures of the position distribution of the ideal Bose gas
on [−1, 1]3 at β = 0.09, 0.10, respectively.
− 54 −
1
1
0.5
z
0.5
z
0
-0.5
-1-1
-0.5
x
0
0.5
1 -1
-0.5
0
0.5
0
-0.5
1
-1-1
y
-0.5
x
0
Random cluster (point)
0.5
1 -1
-0.5
0
0.5
1
y
Random cluster (point)
(a) β = 0.09
(b) β = 0.10
Fig.2 one cluster with one point
1
1
0.5
z
0.5
z
0
-0.5
-1-1
-0.5
x
0
0.5
1 -1
-0.5
0
0.5
0
-0.5
1
-1-1
y
-0.5
x
0
Random cluster (point)
0.5
1 -1
-0.5
0
0.5
1
y
Random cluster (point)
(a) β = 0.09
(b) β = 0.10
Fig.3 one cluster with two points
1
1
0.5
z
0.5
z
0
-0.5
-1-1
-0.5
x
0
0.5
1 -1
-0.5
0
0.5
y
1
0
-0.5
-1-1
-0.5
x
0
Random cluster (point)
0.5
1 -1
-0.5
0
Random cluster (point)
(a) β = 0.09
(b) β = 0.10
Fig.4 one cluster with three points
− 55 −
0.5
y
1
1
0.5
z
0
-0.5
-1-1
-0.5
x
0
0.5
1 -1
-0.5
0.5
0
y
1
random cluster
underlying random point field
Fig.5. a sample picture of the position distribution on [−1, 1]3 at β = 0.09
1
0.5
z
0
-0.5
-1-1
-0.5
x
0
0.5
1 -1
-0.5
0.5
0
y
1
random cluster
underlying random point field
Fig.6. a sample picture of the position distribution on [−1, 1]3 at β = 0.1
− 56 −
4.
Conclusion
We have tried to simulate the position distribution of the three dimensional ideal Bose
gas in a bounded volume. In order to simulate the position distribution we used the
estimation (7) in theorem 3.1. Using this estimation we could obtain the bigger volume
from which all points in the fixed bounded volume come. More simulations will give us
intuitively that the position distribution of the ideal Bose gas has a structure such that
at low temperature the underlying point system is thin and the clusters are big.
References
[1] Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics
II, Springer-Verlag, New York-Heidelberg- Berlin, 1981.
[2] Fichtner K.-H., On the position distribution of the ideal Bose gas, Math. Nachr. 151
(1991) 59-67.
[3] Fichtner K.-H., Freudenberg W., Point processes and the position distribution of
infinite boson systems, J. Stat. Phys. 47 (1987) 959-978.
[4] Fichtner K.-H., Freudenberg W., Point processes and states of infinite boson systems, Preprint NTZ d.Karl-Marx-Universität Leipzig (1986).
[5] Fichtner K.-H., Inoue K. and Ohya M., On the low-temperature behavior of the
infinite-volume ideal Bose gas, to appear in Infin. Dimens. Anal. Quan. Probab.
Relat. Top. 13 (2010).
[6] Matthes K., Kerstan J., Mecke J., Infinitely divisible point processes, Wiley,
Chichester-New York, 1978.
− 57 −
QBIC ⎇ⓥᚑᨐႎ๔
᧲੩ℂ⑼ᄢቇℂᎿቇㇱᖱႎ⑼ቇ⑼
ጊ⡛ผ
ᧄᐕᐲߪᰴߩࠃ߁ߥ⎇ⓥࠍⴕߞߚ㧚
㧝㧚On Generalized Quantum Turing Machine and Its Applications
৻⥸ൻ㊂ሶ࠴ࡘࡦࠣᯏ᪾ߩቯᑼൻ߅ࠃ߮㧘⸒⺆ࠢࠬߩ൮㑐ଥߦߟߡ⎇ⓥࠍⴕߞ
ߚ㧚ߐࠄߦ㧘Ꮻ⚊㑐ᢙߦߟߡߩ㊂ሶࠕ࡞ࠧ࠭ࡓ㧘5VCVKQPCTKV[ ࠍᜬߚߥ႐วߩ㊂ሶ
ࠕ࡞ࠧ࠭ࡓߩ⸘▚㊂ࠍⓍߞߚ㧚
㧞㧚Mathematical description of drug movement into tumor with EPR effect and estimation of
its configuration for DDS
&&5 ߩ⸳⸘ߦߟߡ㧘ౝߩ⮎☸ሶߩㆇേߦ㑐ߔࠆ‛ℂࡕ࠺࡞ࠍ᭴ᚑߒ㧘ᦨㆡߥᒻ⁁
ࠍ᳞ࠆࠕ࡞ࠧ࠭ࡓࠍ᭴ᚑߒߚ㧚⣲≌ߩᜬߟ '24 ലᨐࠍ⠨ᘦߒ㧘ߐࠄߦᐔဋ႐ࠍขࠅ
ࠇߚ‛ℂࡕ࠺࡞ࠍ⠨߃㧘⸘▚ᯏࠪࡒࡘ࡚ࠪࡦߦࠃࠅᦨㆡ⸃ࠍዉߒߚ㧚
㧟㧚Quantum Algorithm for EXPTIME Problem and Computational Complexity
ੑੱή㐽〝⍹⟎߈ࠥࡓߦኻߔࠆ㊂ሶࠕ࡞ࠧ࠭ࡓࠍ᭴ᚑߒ㧘⸘▚ߩⶄ㔀ߐࠍ⠨ኤߒߚ㧚
ࠆࠝࠢ࡞ࠍቯߒ㧘ฎౖࠕ࡞ࠧ࠭ࡓߣ㊂ሶࠕ࡞ࠧ࠭ࡓࠍᲧセߒߚ㧚⚿ᨐߣߒߡ㧘
ࠝࠢ࡞߇ਈ߃ࠄࠇߚ႐วߢ߽ฎౖࠕ࡞ࠧ࠭ࡓߪᜰᢙᤨ㑆߆߆ࠆ߇㧘㊂ሶࠕ࡞ࠧ࠭ࡓ
ߪᄙ㗄ᑼᤨ㑆ߢ⸃ߌࠆߎߣࠍ␜ߒߚ㧚
㧠㧚Quantum Mutual Entropy Defined by Liftings
A lifting is a map from the state of a system to that of a compound system, which was
introduced in[1]. The lifting can be applied to various physical processes.
In this paper, we defined a quantum mutual entropy by the lifting. The usual quantum
mutual entropy satisfies the Shannon inequality[10], but the mutual entropy defined
through the lifting does not satisfy this inequality unless some conditions hold.
Introduction
Let us generalize a quantum mutual entropy by using liftings. Let H 1 and H 2
be two separable Hilbert spaces and BH a set of all bounded linear operators on
H . For a set SH of all density operators on H ;
SH £>; > u 0, tr> 1¤ , we call a map from SH 1 − 58 −
to
SH 2 a
channel. If " ' is affine, we call it a linear channel. We denote
" : BH 2 v BH 1 by the dual map of " ' ; i.e., tr" ' >A tr>"A for all
> SH 1 and A BH 2 . If " is a complete positive map (i.e., for all
n
'
'
n N, A j BH 2 , Bk BH 1 holding ! j,k1 Bj " A j A k Bk u 0 ), " '
is called a complete positive channel. Channel is a mathematical tool to describe various
physical processes[12].
Lifting was introduced by Accardi and Ohya in C ' -dynamical systems[1] to
integrate various channels and open system dynamics. Here let our C ' - algebras be
realized on some separable Hilbert spaces H 1 and H 2 . A continuous map E '
from the state space SH 1 to the compound state space SH 1 § H 2 is
called a lifting:
E ' : SH 1 v SH 1 § H 2 .
If E ' is affine and its dual is a completely positive map, we call it a CP linear lifting.
If it maps pure states into pure states, we call it pure. Remark that a pure lifting sends a
mixed state to either a pure or a mixed state. A lifting from SH 1 to
SH 1 § H 2 is called non-demolition for a state
the following condition
>1 SH 1 if
E'
holds
tr2 E ' > 1 > 1
The concept of lifting can be used to understand noncommutative probability[1]. Given
a state >1 SH 1 and a channel " ' : SH 1 v SH 2 , the following
problem is important, that is, to find the standard lifting
E ' : SH 1 v SH 1 § H 2 such that it describe the correlation between of >1
and " ' >1 tr1 E ' >1 . There are several solutions of this problem in the papers
[1,10,12].
Quantum Mutual Entropy
The classical mutual entropy was introduced by Shannon to discuss the transmission of
information from an input system to an output system[5], then Kolmogorov[6], Gelfand
and Yaglom[2] gave a measure theoretic expression for the mutual entropy by means of
the relative entropy defined by Kullback and Leibler. Shannon's expression for mutual
entropy was generalized for the finite-dimensional quantum (matrix) case by
Holevo[3,4] and Levitin[7]. Ohya took the measure theoretic expression of KGY to
define quantum mutual entropy by means of quantum relative entropy[10,12].
Let S be the set of all states in a certain C ' -algebra (or von Neumann algebra)
describing a quantum system, and 6 a measure decomposing the state I into
S . Ohya's definition of QM entropy is
Definition Quantum mutual entropy w.r.t. I and " ' is defined as[10,12]
extremal orthogonal states in
− 59 −
II; " ' q sup
where
;S S Araki" ' F, "' I d6; I ;exS Fd6
S Araki
is Araki's relative entropy[A].
Definition In the case that the C ' -algebra is
density operators, the above definition goes to
I>; " ' q sup
BH and
S
is the set of all
! 5 n S Umegaki"' E n , "' > ; > ! 5 n E n
n
n
where > is a density operator, S Umegaki is Umegaki's relative entropy[U] and
> ! n 5 n E n is the Schatten decomposition. The Schatten decomposition is no
always unique, so we take the supremum over all possible decompositions.
Both are completely quantum input and quantum output cases. When the input is
classical, i.e., the state is a probability distribution, the von Neumann-Schatten
decomposition is unique
> ! 5n -n
n
' '
'
'
and if the channel is written as " 2 1 where 1 is one for quantum
'
coding, i.e., 1 - n > n , then the above mutual entropy generalizes Holevo's one
I>; " ' S" ' > "
! 5 n S" ' > n
Moreover, let > ! k 5 k E k be a Schatten decomposition of > SH where
E k is one-dimensional projection expressed by E k |x k x k | with the
eigenvector |x k for 5 k and let @ E be a compound state of > and " ' >
@E ! 5kEk § "'Ek
k
I>; " ' Theorem [10]The quantum mutual entropy
is
I>; " ' sup S@ E , @ 0 E
@ 0 > § " ' >.
Theorem [10] I>; " ' satisfies the following property:
x If a channel " ' is an i. d. , I>; " ' is equal to S> x If the system is classical, I>; " ' is equal to classical mutual entropy
x (Shannon's inequality) 0 t I>; " ' t min£S> , S" ' > ¤
These entropies are discussed in [13,14].
where
Quantum Mutual Entropy defined by Lifting
In this section, we generalize the quantum mutual entropy by means of liftings. We
− 60 −
study under which conditions this generalized quantum mutual entropy (GQM entropy
for short) satisfies the Shannon inequality.
Let " ' be a complete positive channel from SH 1 to SH 2 and E '
be a lifting from SH 1 to SH 1 § H 2 . Here, we take the following two
marginal conditions:
(M1) For an input state > SH 1 , it holds tr2 E ' > > (non-demolition
property).
(M2) For a given channel " ' , tr1 E ' > " ' >.
We defined the quantum mutual entropy w.r.t. E ' as
IL >; E ' q SE ' >, > § " ' > Taking a supremum of IL >; E ' over some set of the liftings
entropy for a channel " ' is defined as
E ' , the GQM
IL >; " ' q sup£IL >; E ' ; tr2 E ' > >, tr1 E ' > " ' >¤
E'
Let us check whether
IL >; " ' satisfies the Shannon inequality
0 t IL >; " ' t min£S> , S" ' > ¤.
'
For a channel " ' we consider the following three liftings E i ( i 1, 2, 3 )
with M1 and M2:
'
'
1) E 1 > ! k 5 k E k § " E k , where > ! k 5 k E k is a Schatten
decomposition.
'
'
2) E 2 > ! k p k > k § " >k for > ! k p k >k . ! k p k 1, p k u 0
'
3) E 3 is a pure lifting.
Concerning the Shannon inequality, we obtain the results below.
'
Theorem E 1 satisfies the marginal condition M1 and M2 and the Shannon
inequality.
Proof Since
tr2 E '1 > tr1 E '1 > ! 5kEk
>
k
! 5 k "' E k
k
it holds M1 and M2. We compute
− 61 −
" ' >,
SE '1 > "trE '1 > logE '1 >
"tr ! 5 k E k § " ' E k log
k
S> ! 5 k E k § "' E k
k
! 5 k S"' E k k
Then we have
S" ' > " IL >; E ' S> " SE '1 > t 0
and
S" ' > " SE '1 > " ! 5 k S" ' E k S" ' > " S> k
! 5 k tr"' E k log" ' E k " tr ! 5 k " ' E k log" ' > " S> k
k
! 5 k tr"' E k log" ' E k " log" ' > " S> k
! 5 k S" ' E k , "' > " S> k
t
! 5 k SE k , > " S> k
! 5 k trE k logE k " E k log> " S> k
0
where our inequality above comes from the monotonicity of the relative entropy.
Therefore we obtain
0 t IL >; E '1 t min£S> , S" ' > ¤
Theorem
E '2
Proof Since
Let
satisfies M1, M2, and the Shannon inequality.
is a compound state of > and " ' > , it holds M1 and M2.
E '2 >
>k ! 5 k,i E 1k,i
i
and
" ' >k ! 5 k,i "' E 1k,i
i
! 5 k,i 6i,j E 2i,j
i,j
2
and E i,j are
'
orthogonal one-dimensional projections(pure states). We compute SE 2 > as
be Schatten decompositions of
>k
and
" ' >k
− 62 −
where
E 1k,i
SE '2 > "trE '2 > logE '2 >
"tr ! p k 5 k,i 5 k,i 6 i,j logp k 5 k,i 5 k,i 6 i,j E 1k,i § E 2i,j
i,j,k
! p k 5 k,i 5 k,i 6i,j logp k 5 k,i E 1k,i § I
"tr
i,j,k
"
! p k 5 k,i 5 k,i 6i,j log5 k,i 6i,j I § E 2i,j
i,j,k
S> ! p k S"' >k k
Then we have
S> " SE '2 > S> " S> "
! p k S" ' >k k
" ! p k S" ' > k t 0
k
and
S" ' > " SE '2 > S" ' > " S> "
! pk S"' >k k
"tr ! p k " ' >k log" ' > " S> tr ! p k " ' > k log" ' > k
k
k
tr ! p k " ' > k log" ' > k " log" ' > " S> k
! pk S" ' >k , " ' > " S> k
t
! pk S>k , > " S> k
! pk tr>k log>k " >k log> " S> k
" ! p k S> k k
t 0
'
Therefore, E 2 satisfies the Shannon inequality.
From the above two theorems, we may conclude that if the lifting is a separable type,
that is, E ' > is a separable state, then the Shannon inequality is satisfied. On the
contrary, there exists several entangled type pure liftings, that is, E ' > is a pure
entangled state that does not satisfy the Shannon inequality. In the rest of our paper, we
'
give three examples of pure lifting E 3 ; one is of satisfying the Shannon inequality
and two others are of not.
− 63 −
"'
Example In the case that a channel
is written as
" ' > V>V '
where V is a linear operator from H 1 to H 2 , the lifting
E ' > > § V>V ' is pure. Let > ! k 5 k E k be a Schatten decomposition of
> , the lifting
E '3 > E '1
is also pure. This is same as
1
Example Let £e k ¤ and
k
so that it holds the Shannon' inequality.
£e 2k ¤
£e 1k ¤
respectively, such that
! 5 k E k § VE k V '
be two CONSs in
H1
and
gives the Schatten decomposition of
> H2
> :
! 5kEk
k
E k |e 1k e 1k |
We can give a pure lifting
E '3 > This pure lifting
E '3
Proof In this case,
E '3
as
!
5 k |e 1k § e 2k
!
k
5 l e 1l § e 2l |
l
does not satisfy the Shannon inequality.
E '3 >
can be written as
E '3 > |88|
|8 !
5 k |e 1k § e 2k .
k
Since
state
E '3 >
is a pure state, and
SE '3 > 0 . For a general (i.e., pure or mixed)
> , one has
S> S" ' > where
> tr2 E '3 >
Then,
" ' > tr1 E '3 >
− 64 −
IL >; E '3 "SE '3 > S> S" ' > 2S> which does not satisfy the Shannon inequality.
Example Let a linear map V : H 1 v H 2 which defines a channel
"> ' V>V ' ,
E '3
We can define a pure lifting
E '3 > !
as
5 k 5 l |e 1k e 1l | § V|e 1k e 1l |V '
k,l
E '3
does not satisfy the Shannon inequality
'
Proof As the same discussion of example(2), IL >; E 3 Shannon inequality.
Then
does not satisfy the
Conclusion
We generalized a quantum mutual entropy by using liftings, so that we can represent the
relation between input and output precisely. In some cases, there exists pure liftings
which do not satisfy the Shannon inequality make an entangled state.
[1] L.Accardi and M.Ohya (1999) Compound channels, transition expectations, and
liftings, Appl. Math. Optim., 39, 33-59.
[2] I.M.Gelfand and A.M.Yaglom (1959) Calculation of the amount of information
about
a
random
function
contained
in
another
such
function,
Amer.Math.Soc.Transl., 12, pp.199-246
[3] A.S.Holevo (1973) Some estimates for the amount of information transmittable by
a quantum communication channel (in Russian), Problemy Peredachi Informacii, 9,
pp.3-11
[4] R.S.Ingarden (1976) Quantum information theory, Rep.Math.Phys., 43-73
[5] R.S.Ingarden, A.Kossakowski and M.Ohya (1997) Information Dynamics and Open
Systems, Kluwer
[6] A.N.Kolmogorov
(1963)
Theory
of
transmission
of
information,
Amer.Math.Soc.Translation, Ser.2, 33, pp.291--321
[7] L.B.Levitin (1969) On the quantum measure of information, in Proceedings of the
Fourth Conference on Information Theory, Tashkent, 111--116. English translation
in Annales de la Foundation Louis de Broglie 21:3, (1996).
[8] H.Araki (1976) Relative entropy for states of von Neumann algebras, Publ. RIMS
Kyoto Univ., 11, 809-833
[9] H.Umegaki (1962) Conditional expectations in an operator algebra IV(entropy and
information), Kodai Math.Sem.Rep., Vol.14, 59-85
− 65 −
[10] M.Ohya (1983) On compound state and mutual information in quantum
information theory, IEEE Trans. Information Theory, 29, No.5, 770-774.
[11] M.Ohya (1984) Entropy Transmission in C*-dynamical systems, J. Math.
Anal.Appl., 100, No.1, 222-235.
[12] M.Ohya (1989) Some aspects of quantum information theory and their applications
to irreversible processes, Rep. Math. Phys., 27, 19-47.
[13] M.Ohya and D.Petz (1993) Quantum Entropy and its Use, Springer-Verlag,
TMP-series.
[14] M.Ohya (2003) New quantum algorithm for studying NP-complete problems,
Rep.Math.Phys.,52, No.1,25-33
[15] L.Accardi, T.Matsuoka and M.Ohya (2006) Entangled Markov Chains are Indeed
Entangled, Infinite Dimensional Analysis Quantum Probability and Related Topics,
vol 9, no 3, 379-390.
− 66 −
高精度配列アライメント法
数理研究グループ（理工学部情報科学科）
原利英（助教），佐藤圭子（講師），大矢雅則（教授）
Abstract. 本年度は，アミノ酸配列に対するアライメントにおいて，アミノ酸ペア間推
移量を用いるようアルゴリズムを改良することで，生成されるアライメントの精度を向上
させる手法を開発した．この新たに開発したアライメント手法である MTRAP 法と，既
存の手法とのアライメント精度面での比較および検証を行った．その結果，特に配列一致
率の低い相同配列に対するアライメントで新手法は効果を発揮することを確認した．
研究目的・内容
現在，ヒトゲノム計画の進展に代表されるように多種多様な生物のゲノム配列が決定され，扱わ
れるデータ量は指数的に増え続けている．特に，2007 年ごろに次世代ゲノムシーケンサとよば
れる装置が登場してからは，ゲノム配列の読み取りが革新的に高速化し，その後に続く配列解析
の精度および速度の重要性が今までになく高まっている時代であるといえる．
現在までに，様々な配列アライメント法が開発されてきた．FASTA [18] や BLAST [2] に
代表されるデータベース検索を目的としたアライメント法では，その速度に重点が置かれる一
方，マルチプルアライメントを目的としたアライメント法ではその正確さに重点が置かれ開発
されている．現在，こうしたマルチプルアライメントを行うための手法として，ClustalW [21],
DIALIGN [14], T-Coffee [16], MAFFT [11], MUSCLE [7], Probcons [6] などが開発されている．
これらの手法は一般的に各サイトの残基を独立したものとして扱っており，たとえばアミノ酸配
列に対するアライメントであればアミノ酸置換行列をもとにした配列間尺度を利用する．これら
の手法は，配列一致率が 40%以上である相同配列に対しては大変よい結果が得られるが，配列一
致率がこの値以下である相同配列に対しては満足な結果が得られていない [4]．
Anfinsen のドグマ [3] として知られているように，少なくとも球形タンパクにおいてはその
高次構造はそのタンパクを構築するアミノ酸配列により決定づけられていることが知られている．
タンパク質は構造的にはアミノ酸のポリマーであるが，一部のタンパク質は自己組織化やシャペ
ロンの影響により α へリックスや β シートといった特定の立体構造をとるように自動的に折りた
たまれ，全体としては決まった構造をとる．この現象のことをフォールディングといい，タンパ
ク質はフォールディングされることで，酵素などとしての特有の機能を発揮するとされる．つま
り，配列を構成するアミノ酸の種類および前後のアミノ酸とのつながりに高次構造を決定する要
因があると思われる．そしてこのことは，前後のアミノ酸の情報，つまり配列から得られる情報
を含めアライメントを行うことで，タンパク質の立体構造的な対応をより正確に反映したアライ
メントを得られることを示唆する．こうした考えの下，本年度，我々は Transition quantity と呼
ぶ量を定め，これを用いたアライメント法として MTRAP 法を提案した [8,9]．この手法による
アライメント精度について，HOMSTRAD (version November 1, 2008) [13,20]，PREFAB4 [7]
といったデータベースを用いた場合の結果を示し，既存の手法と精度面で比較，検証する．
1.
Transition Quantity を用いた配列間尺度
最初にいくつか記号を定義する．Ω をすべてのアミノ酸の集合，Ω∗ を Ω とギャップ”∗ ” による
集合：Ω∗ ≡ Ω ∪ {∗} とする．Ω の要素を残基と呼び Ω∗ の要素をシンボルと呼ぶ．Ω の直積を
Γ ≡ Ω × Ω とし，同様に Γ∗ ≡ Ω∗ × Ω∗ とする．
− 67 −
ここで，配列長 n の２つの配列，A = a1 a2 · · · an と B = b1 b2 · · · bn ，ai , bj ∈ Ω∗ について考
える．この配列を u1 u2 · · · un , ui = (ai , bi ) ∈ Γ∗ とも表記することにする．以下，ui をサイトと
呼ぶ．
配列間に何らかの関連性がある場合と，配列間に何の関連性もない場合との尤度比はオッズ
比と呼ばれる．
R (A, B) =
p (a1 , a2 , · · · , an ; b1 , b2 , · · · , bn )
p (A; B)
=
.
p (A) p (B)
p (a1 , a2 , · · · , an ) p (b1 , b2 , · · · , bn )
(1)
ここで，p (a) は a の生起する確率をあらわし，p (a; b) は同時確率を表す．式 1 を簡単なも
のにするにあたり，置換は位置独立に起き，サイト間での相関もないものと仮定する．つまり，
&
&
&
p (A) = p (a), p (B) = p (b) and p (A, B) = p (a, b)．このとき，式 1 の対数をとったもの
は独立した項を加算したものとして表せ, これは対数オッズ比と呼ばれる．
log
p (A; B)
=
s (ai , bi ) ,
p (A) p (B)
i
(2)
ここで，
s (a, b) = log
p (a; b)
p (a) p (b)
(3)
はシンボル a, b が何の関連性もなく生起する場合と何かしら関連性を持って生起する場合との対
数尤度比である．この s (a, b) はスコアと呼ばれ，S = (s (a, b)) は置換行列と呼ばれる．ペアワ
イズアライメントで用いられる配列間尺度は一般的にこれらの量（式 2 および式 3）を用いて定
義される [1]．
ここで，置換行列の各要素を正規化したもの（0 から 1 の間の値を取るようにしたもの）を
正規化置換行列として新たに定義し，これを用いて対数オッズ比の別表現として差異 dsub (A, B)
を新たに定める．まず，正規化のための関数 fs : [smin , smax ] → R を次のように定める．
fs (x) ≡
smax − x
, 0 ≤ fs (x) ≤ 1,
smax − smin
$
(4)
%
smax ≡ max max {S (u)} , gap cost ,
$
u∈Γ
%
smin ≡ min min {S (u)} , gap cost .
u∈Γ
この関数を用い，スコア s (a, b) を正規化したものを s̃ (a, b) ≡ fs (s (a, b)) , a, b ∈ Ω とする．ま
た，正規化置換行列を M = (s̃ (a, b)) と定める．このとき，配列 A, B 間の差異は
dsub (A, B) =
s̃ (ai , bi ) .
(5)
i
と表される．差異 dsub (A, B) は配列 A, B が同じときに 0 となる．
この加算的で扱いやすい配列間差異は，上述したように置換は位置独立に起き，サイト間で
の相関もないものとの仮定のもとで導かれたものである．これに対し我々は，サイト間の相関を
考慮する配列間差異を新たに提案した [9]．
我々の提案手法では，既存の配列間尺度 R (A, B) にサイト間推移の効果を加えた，次の尺
度を新たに考える．
Rour (A, B) = R (A, B)1−ε Rt (A, B)ε ,
(6)
ここで，
Rt (A, B) ≡
n−1
#
p (ui+1 \ ui )
i=1
− 68 −
(7)
はサイト間推移の効果を表し，ε はその混合比率を表す．
加算的な差異を導出するにあたり，Transition quantity と呼ぶ正規化した推移量 t̃ (ui , ui+1 )
を次のように定義する．
t̃ (ui , ui+1 ) ≡ ft (t (ui , ui+1 ) ; ui ) ,
t (ui , ui+1 ) ≡ log p (ui+1 \ ui ) ,
(8)
−x
maxv∈Γ∗ {−t(u,v)} ,
(9)
'
ft (x; u) =
1,
if x > 0
otherwise
サイト間推移を下にした配列間差異は Transition quantity の和として次のように与えられる．
dtrans (A, B) =
n−1
t̃ (ui , ui+1 ) .
(10)
i=1
これら２つの差異 dsub と dtrans を合わせた差異を次のように定義する．
dMTRAP (A, B) = (1 − ε) dsub (A, B) + εdtrans (A, B) .
(11)
これは既存のスコアリングシステムと同様に加算的で扱いやすい配列間尺度である．我々の開発
した MTRAP 法 [8] は，この尺度を用いた動的計画法 [17] によるアライメント法である（図 1）．
Figure 1.
MTRAP 法
Transition quantity は PAM [5] や BLOSUM [10] といったアミノ酸置換行列と同様に既知ア
ライメントデータセット上での統計を下にその値を求める．本研究では，SABmark データベー
ス（version 1.63）[22] 上の superfamilies サブセットの配列すべてを用いて値を求めた [9]．
1.1.
アライメント精度の検証
この節では，我々の手法を含む各アライメント法により生成されたアライメントの精度を検証す
る方法について述べる．
タンパクの立体構造上の対応がそろうように整列化されたアミノ酸配列群を構造アライメント
（Structural Alignment）と呼び，構造アライメントが登録されているデータベースを構造アライ
メントデータベースと呼ぶ．構造アライメントデータベースには現在，HOMSTRAD，PREFAB
4.0 といったものが存在する．ここで，これらのデータベースを用いアライメント精度の検証を
行うことを考える．
構造アライメントデータベース上のアライメントを正しく対応がそろえられたアライメント
である仮定し，これを便宜的にリファレンスアライメントと呼ぶことにする．また，リファレン
スアライメントの元となるアライメント前の配列群に対し，各アライメント法を適用し構築した
アライメントをテストアライメントと呼ぶことにする．この２つのアライメントを比較すること
で各アライメント構築法の評価を行う．具体的には次の手順となる．
1. 構造アライメントデータベースからアライメントを取得し，これをリファレンスアライメン
トとする
2. リファレンスアライメントからギャップを取り除いた配列群を作成する
− 69 −
3. 2 で作成した配列群に対し，評価したいアライメント構築法でアライメントを作成する．こ
れがテストアライメントとなる．
4. リファレンスアライメントとテストアライメントを下で述べる指標 Q Score を用いて比較する
5. 以上の 1 から 4 の作業を構造アライメントデータベースに登録されているデータすべてに対
し行う．
6. 以上の 1 から 5 の作業を比較したい手法それぞれにおいて行う．
本論文において利用した構造アライメントデータベースの詳細は以下の通り．
HOMSTRAD HOMSTRAD(HOMologous STRucture Alignment Database) は立体構造既
知な相同タンパクを用いた構造アライメントデータベースである [13]．随時データベースの内容
が更新される．そのため，本論文では 2008 年７月１日時点でのデータベースを利用した．利用
したデータはペアワイズアライメント 630 個であり，これら 630 個の配列一致率の分布は表 1 に
示した通りとなる．
Table 1.
HOMSTRAD
0 ≤%ID< 20
87
20 ≤%ID< 40
273
40 ≤%ID< 60
160
60 ≤%ID< 80
83
80 ≤%ID≤ 100
27
ALL
630
表中の値は，2008 年 7 月 1 日時点での HOMSTRAD におけるペアワイズアライメント全 630 個のそれぞれの%ID
（配列一致率）範囲における個数を表す．
PREFAB4 PREFAB4 はアライメント構築法のひとつである MUSCLE の作者らがアライメ
ント法の評価のために作成したアライメントデータベースである [7]．本論文ではバージョン 4 に
あたる PREFAB4 を用いた．利用したデータはペアワイズアライメント 1682 個であり，これら
リファレンスアライメントの配列一致率の分布は表 2 の通りである．
Table 2.
PREFAB4
0 ≤%ID< 15
423
15 ≤%ID< 30
917
30 ≤%ID< 45
148
ALL
1682
表中の値は，PREFAB4 におけるペアワイズアライメント全 1682 個のそれぞれの%ID（配列一致率）範
囲における個数を表す．
リファレンスアライメントとテストアライメントを比較するにあたり，指標 Q Score [7] を
用いた．Q Score の定義を以下に示す．
Q Score とは， テストアライメントにおける残基ペアがリファレンスアライメント上におい
て同じ列に存在しペアをつくる割合を表す．数式による定義は次の通り．
長さが L である N 本の配列から構成されるテストアライメント {s1 , · · · , sN } が与えられ，
aik ∈ Ω∗ を配列 si における k 番目のシンボルとする．配列 si 上のシンボル aik と対応するリ
ファレンスアライメント上のシンボルの列番号を Iik とする．ただし，aik = ∗ のときは Iik = 0
．このとき，Q Score は以下のように与えられる．
L N
−1 N
Q Score
=
'
∆x,y =
∆aik ,ajk δIik ,Ijk
k=1 i=1 j=i+1
L N −1 N
k=1 i=1 j=i+1
1,
0,
,
x = ∗ and y =
∗
x = ∗ or y = ∗ .
− 70 −
(12)
∆aik ,ajk
(13)
1.2.
各種アライメント構築法との精度の比較
上述の構造アライメントデータベース HOMSTRAD，PREFAB4 を用い，MTRAP のアライメン
ト精度に関して次の一般的に用いられる７つの手法：Needle，ClustalW2，MAFFT，T-Coffee，
DIALIGN，MUSCLE，Probcons と比較を行った．各手法の詳細は以下の通り．
1. Needle: Needle は Needleman-Wunsch アルゴリズム [15] によりグローバルペアワイズアラ
イメントを行う EMBOSS パッケージ [19] のプログラムである．BLOSUM62 アミノ酸置換
行列をデフォルトのアミノ酸置換行列として用いる．EMBOSS ver. 5.0.0 を用いた．
2. ClustalW2: ClustalW2 [12,21] は累進法を実装した代表的なプログラムである．彼らの論文
には明記されていないが，ClustalW2 は入力配列の情報を下に指定したシリーズの中からア
ミノ酸置換行列を選択し用いるアルゴリズムを実装している．GONNET アミノ酸置換行列
群をデフォルトのアミノ酸置換行列として用いる．ClustalW2 ver. 2.0.9 を用いた．
3. MAFFT: MAFFT [11] はフーリエ変換を用いた高速なアルゴリズムを実装するプログラム
であり，ver. 6.240 を用いた．
4. T-Coffee: T-Coffee [16] はマルチプルアライメント構築時の目的関数として配列一致率を下
にしたものを利用する累進法によるマルチプルアライメント構築のための手法及びその手法
を実装したプログラムの名称である．現在，累進法に分類されるアルゴリズムの中では最高
水準の精度を有するとされる．Ver. 5.30 を用いた．
5. DIALIGN: DIALIGN [14] は segment-to-segment アプローチによる手法を用いたプログラ
ムであり，ver. 2.2.1 を用いた．
6. MUSCLE: MUSCLE [7] は Log-Expectation を用いた手法を用いたプログラムであり，ver.
3.7 を用いた．
7. Probcons: Probcons [6] は Probabilistic Consistency を用いたプログラムであり，ver. 1.12
を用いた．
これらのプログラムは基本的にそれぞれのデフォルトパラメータを用いた．
提案手法の評価は次の２つを比較することで可能となる．
1. 提案手法によるアライメントとリファレンスアライメントとの間の Q Score
2. 上にあげた各手法によるアライメントとリファレンスアライメントとの間の Q Score
各構造アライメントデータベース上の複数のデータに対しそれぞれの手法における Q Score の計
算を行うことで，これら２手法の比較を行った．
1.3.
結果と考察
表 3 は MTRAP と代表的なグローバルアライメントプログラムである Needle，ClustalW2 と
の HOMSTRAD を用いた比較結果である．各手法による配列アライメントと HOMSTRAD 上
の全 630 個のアライメントとの類似性は指標 Q score により測った．HOMSTRAD 上の構造ア
ライメントを正しいアライメントだとすると，MTRAP は他の２つの手法にくらべ全範囲にわ
たって良い傾向を示すことがわかる．たとえば，MTRAP は 80%以上の精度（e.g., PAM250 や
BLOSUM622 で 0.817）を有するのに対し，Needle や ClustalW2 は 80%未満の精度（e.g.，Needle
は PAM250 で 0.768，BLOSUM62 で 0.768）となる（表 3）．それ以上に重要な点として，配列
一致率が 30%未満のデータに対し，MTRAP はアライメント精度を大変よく改善している点が
あげられる．例えば，PAM250 行列を用いた MTRAP では配列一致率が 0-15%のデータに対し
0.421，15-30%のデータに対し 0.655 といった精度が得られるのに対し，同様に PAM250 行列を
用いた ClustalW2 では配列一致率が 0-15%のデータに対し 0.234，15-30%のデータに対し 0.528
といった精度にとどまる．
− 71 −
正解とする構造アライメントデータベースとして HOMSTRAD のほかに，PREFAB4 を用
いた検証も行った．ここでは，PREFAB4 上の全 1682 個のデータを用い，HOMSTRAD と同様
指標 Q Score による評価を行った．図 2 は他の各プログラム（Needle および ClustalW2）におけ
る平均 Q Score 値に対する MTRAP の平均 Q Score 値の比を，用いたアミノ酸置換行列ごとに
プロットしたものである．配列一致率が 60%以上のデータではこれら３つの手法はどれも，どの
アミノ酸置換行列を用いた場合においてもほぼ等しいアライメント精度を示す．しかし配列一致
率が 0-60%のデータでは，その値がひくいほど MTRAP が他に比べ高いアライメント精度を有
することがわかる．特に配列一致率が 0-20%のデータに対して，MTRAP は Needle の 1.5∼2.3
倍の平均 Q Score 値をとり，ClustalW2 に対しても PAM 行列の利用時に 1.4 倍，BLOSUM 行
列で 1.3 倍，GONNET 行列で 1.1∼1.2 倍の値をとっている．
以上の HOMSTRAD，PREFAB4 を用いた検証の結果，MTRAP 法は配列類似性の低い相
同配列に対するアライメントで効果を発揮することが分かった．また，どのアミノ酸置換行列を
用いた場合も明らかな改善がみられることから，配列間差異（式 11）は既存のアミノ酸置換行列
のみを用いた配列間尺度（式 2; Sum of pairs）に対し， より良くタンパクを構成するアミノ酸
配列の生物学的特徴をとらえるといえる．
Table 3.
HOMSTRAD を用いた場合における MTRAP 法とその他の手法との比較
Matrix
Method
PAM250
MTRAP
Needle
ClustalW2
BLOSUM62
MTRAP
Needle
ClustalW2
GONNET250∗
MTRAP
ClustalW2
0-15% (25)
Sequence identity （%）
15-30% (207) 30-45% (173)
ALL (630)
0.421
0.226
0.234
0.655
0.548
0.528
0.874
0.837
0.817
0.817
0.763
0.747
0.410
0.223
0.276
0.653
0.556
0.585
0.878
0.843
0.861
0.817
0.768
0.784
0.412
0.313
0.659
0.619
0.879
0.867
0.819
0.800
表中の値は HOMSTRAD における各配列一致率範囲での，平均 Q Score 値を表す．括弧内の数
字は各配列一致率におけるアライメント数を表す．太字は各配列一致率および各アミノ酸置換行
列における一番良い値を表す．
∗ Needle は GONNET アミノ酸置換行列をサポートしない．
次に，一般的に用いられているアライメントプログラムである，T-Coffee，MAFFT，DIALIGN，MUSCLE，ClustalW2，Probcons との精度の比較を上記２つのデータベースを用いて
行った結果を表 4 および 5 に示す．各プログラムはその作者の推奨するデフォルトパラメータで
実行した．どちらのデータベースを用いた場合においても，MTRAP 法は一般的に精度を改善す
ることが見て取れる．特に，配列一致率が 30%以下の配列に対し明らかな精度の改善を示し，配
列一致率が 30%以下の配列に対してはほかの手法にくらべ 4∼10%の改善が見られた．
2.
まとめ
Transition quantity を用いたアライメント法として MTRAP 法を開発し，様々なアライメント
プログラムと精度の比較を行った．その結果，特に配列一致率の低い相同配列に対するアライメ
ントで効果を発揮することを確認した．
マルチプルアライメントを構築するプログラムはペアワイズアライメントを下にしたものが
一般的である．よって，本提案手法をマルチプルアライメント構築時にも用いることで，これら
のマルチプルアライメント法はより高精度なものとなることが期待される．
− 72 −
Figure 2.
平均 Q Score の比率：左の２つの図は MTRAP の Needle に対する平均 Q Score の比
を表し，右の３つの図は MTRAP の ClustalW2 に対する平均 Q Score の比を表す．それぞれの折れ線は
図中に示されたアミノ酸置換行列を利用した場合の結果を表す．
− 73 −
Table 4.
PREFAB4 を用いた場合における MTRAP 法とその他の手法との精度の比較
Method
PREFAB 4.0
0-15%(423) 15-30%(917) 30-45%(148) All(1682)
MTRAPa
0.248
0.674
0.877
0.615
MAFFT
0.170
0.671
0.860
0.568
DIALIGNb
0.133
0.556
0.814
0.518
MUSCLE
0.205
0.632
0.867
0.581
ClustalW2
0.199
0.644
0.859
0.586
Probcons
0.204
0.647
0.875
0.590
T-Coffee
0.198
0.642
0.872
0.585
表中の値は PREFAB4 における各配列一致率範囲での，平均 Q Score 値を表す．括
弧内の数字は各配列一致率におけるアライメント数を表す．太字は各配列一致率にお
ける一番良い値を表す．
a
MTRAP は GONNET250 アミノ酸置換行列を用いた．
b
DIALIGN はいくつかのデータでエラーを起こしたため，正常に計算できたものだ
けでの平均 Q Score を求めた．
Table 5.
の比較
HOMSTRAD を用いた場合における MTRAP 法とその他の手法との精度
Method
MTRAPa
MAFFT
DIALIGNb
MUSCLE
ClustalW2
Probcons
T-Coffee
0-15%(25)
0.412
0.309
0.216
0.337
0.313
0.344
0.341
HOMSTRAD
15-30%(207) 30-45%(173)
0.659
0.879
0.610
0.863
0.546
0.825
0.625
0.868
0.619
0.867
0.650
0.884
0.634
0.872
All(630)
0.819
0.796
0.760
0.802
0.800
0.816
0.809
表中の値は PREFAB4 における各配列一致率範囲での，平均 Q Score 値を表す．各
種表記は表 4 と同様である．
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⎇ⓥᬺ❣㧔⎇ⓥ⠪㧕 ᢙℂࠣ࡞ࡊ
ᄢ⍫㓷ೣ
ቇⴚ⺰ᢥ
1.
Improvement in Accuracy of Sequence Alignment by the MTRAP algorithm, Toshihide
Hara, Keiko Sato, Masanori Ohya, IEICE Technical Report, Vol. 109, No. 357, pp.99-104,
2010 䋨ᩏ⺒ή䋩
2.
A Physical Realization of Quantum Teleportation for Non-Maximal Entangled State,
Masanari Asano, Masanori Ohya, Yoshiharu Tanaka, IEICE Technical Report, Vol.109
No.357, 111-116, 2010 䋨ᩏ⺒ή䋩
3.
Mathematical description of drug movement into tumor with EPR effect and estimation
of its configuration for DDS, Toshihide Hara, Satoshi Iriyama, Kimiko Makino, Hiroshi
Terada, Masanori Ohya, Colloids and Surfaces B: Biointerfaces, Vol. 75, pp.42-46, 2010,
䋨ᩏ⺒䋩
4.
K.-H. Fichtner, K.Inoue, M.Ohya (2010), On the Low-Temperature Behavior of the
Infinite-Volume Ideal Bose Gas, Infinite Dimensional Analysis, Quantum Probability
and Related Topics, Vol.13.1, pp.39-63䋨ᩏ⺒䋩
5.
K.-H.Fichtner, L.Fichtner, W.Freudenberg, M.Ohya(2010), Quantum Models of the
Recognition Process - On a Convergence Theorem, Open Systems and Information
Dynamics, Vol 17.2, 161-187䋨ᩏ⺒䋩
6.
T.Hara, K.Sato, M.Ohya(2010), MTRAP: Pairwise sequence alignment algorithm by a
new measure based on transition probability between two consecutive pairs of residues,
BMC Bioinformatics 2010, 11:235䋨ᩏ⺒䋩
7.
K.Sato, T.Tanabe, M.Ohya, How to Classify Influenza A Viruses and Understand Their
Severity, Open Systems & Information Dynamics, 17.3, pp. 297-310, 2010䋨ᩏ⺒䋩
8.
T.Hara, K.Sato, M.Ohya (2010) New sequence alignment by a measure due to entangled
correlation in two consecutive residues, The 3rd International Symposium on Applied
Sciences in Biomedical and Communication Technologies Conference Proceedings䋨ᩏ⺒
ή䋩
9.
S.Iriyama, M.Ohya (2010) Generalized Quantum Turing Machine and Its Use to Find an
Algorithm Solving NP-Complete, The 3rd International Symposium on Applied Sciences
in Biomedical and Communication Technologies Conference Proceedings䋨ᩏ⺒ή䋩
10. K.Sato, M.Ohya (2010) Evolution of HIV-1 from the viewpoint of Information Theory, ,
The
3rd
International
Symposium
on
Applied
Sciences
Communication Technologies Conference Proceedings䋨ᩏ⺒ή䋩
− 76 −
in
Biomedical
and
11. ේ⧷, ⮮ሶ, ᄢ⍫㓷ೣ (2010) 䊕䉝㑆ផ⒖㊂䉕↪䈇䈢㈩䉝䊤䉟䊜䊮䊃ᴺ, IPSJ SIG
Technical Report,Vol.2010-BIO-22, No.3䋨ᩏ⺒ή䋩
12. ጊ⡛ผ䋬ᄢ⍫㓷ೣ (2010), Quantum Mutual Entropy Defined by Liftings and Violation
of the Shannon Inequality, ᢙℂ⸃ᨆ⎇ⓥᚲ⻠ⓥ㍳, 1705, 88-94䋨ᩏ⺒ή䋩
13. ᵻ㊁⌀⺈䋬ᄢ⍫㓷ೣ (2010), Lifting and Its Application to a Non-Kolmogorovian Model,
ᢙℂ⸃ᨆ⎇ⓥᚲ⻠ⓥ㍳, 1705, 95-102䋨ᩏ⺒ή䋩
14. M.Asano, M.Ohya, A.Khrennikov, Quantum-Like Model for Decision Making Process in
Two Players Game A Non-Kolmogorovian Model, DOI: 10.1007/s10701-010-9454-y, 2010
䋨ᩏ⺒䋩
15. S.Iriyama, M.Ohya, Quantum Mutual Entropy Defined by Liftings, Foundations of
Physics, DOI: 10.1007/s10701-010-9432-4, 2010䋨ᩏ⺒䋩
ᓙ⻠Ṷ
1.
S.Iriyama, M.Ohya , Mathematical Characterization of Quantum Algorithm I(Plenary
Lecture), 14th WSEAS International Conference on Applied Mathematics, Puerto De La
Cruz, Tenerife, Spain, 12/14-16, 2009
2.
S.Iriyama,
M.Ohya,
Computational
Complexity
of
Quantum
Algorithm
and
Applications(plenary), Mathematical Physics & Applications - MPA'10, 8/29-9/4, Samara,
Russia, 2010
3.
S.Iriyama, M.Ohya, Generalized Quantum Turing Machine and Its Use to Find an
Algorithm Solving NP-Complete, ISABEL 2010, Center for Ecclesiastical Services, Rome,
Italy, November 7-10, 2010
4.
K.Sato, M.Ohya, Evolution of HIV-1 from the viewpoint of Information Theory, ISABEL
2010, Center for Ecclesiastical Services, Rome, Italy, November 7-10, 2010
ንỈ⽵↵
ቇⴚ⺰ᢥ
1.
A measure of departure from diagonals-parameter symmetry based on association
measure for square contingency tables. Miyamoto, N., Kato, T., and Tomizawa, S.,
Journal of Statistics: Advances in Theory and Applications, 㩷 Vol. 3, pp. 1-16,
2010, 㩷
䋨ᩏ⺒䋩
2.
Measure of departure from collapsed symmetry for multi-way contingency tables with
ordered categories. Yamamoto, K., Tahata, K., Hatori, A., and Tomizawa, S., Journal of
the Japan Statistical Society,
3.
Vol.40, pp.97-109,
2010, 㩷 䋨ᩏ⺒䋩
Measure for no three-factor interaction model in three-way contingency tables.
Yamamoto, K., Hori, K. and Tomizawa, S.
pp.17-22, 2010, 䋨ᩏ⺒䋩䋮
− 77 −
American Journal of Biostatistics, Vol.1,
4.
On test of marginal point-symmetry in multi-way tables. Tahata, K., Tokuno, H. and
Tomizawa, S., Advances and Applications in Statistical Sciences, Vol. 3,
pp.1-12, 2010 ,
䋨ᩏ⺒䋩䋮.
5.
Extension of the marginal cumulative logistic model and decompositions of marginal
homogeneity for multi-way tables. Kurakami, H., Tahata, K. and Tomizawa, S.
Journal
of Statistics: Advances in Theory and Applications, Vol.3, 135-152, 2010, 䋨ᩏ⺒䋩.
6.
Harmonic, geometric and arithmetic means type uncertainty measures for two-way
contingency tables with nominal categories. Yamamoto, K., Miyamoto, N. and Tomizawa,
S.
7.
Advances and Applications in Statistics,
2010,
to appear.㩷 䋨ᩏ⺒䋩䋮
Measures of proportional reduction in error for two-way contingency tables with nominal
Biostatistics, Bioinformatics and
categories. Yamamoto, K. and Tomizawa, S.
Biomathematics, 2010, to appear.䋨ᩏ⺒䋩
ᷰㆺ
ቇⴚ⺰ᢥ
1.
Masanori Ohya and Noboru Watanabe 䋬 Quantum Entropy and Its Applications to
Quantum Communication and Statistical Physics䋬Entropy, 2010, 12(5), 1194-1245䋬䋨ᩏ
⺒䋩
2.
Noboru Watanabe, Note on Entropies of Quantum Dynamical Systems, to appear in
Foundation of Physics, 2010, DOI10䇯1007/s10701-010-9455-x, 䋨ᩏ⺒䋩
3.
Andrei㩷 Khrennikov,㩷 Masanori㩷 Ohya㩷 and㩷 Noboru㩷 Watanabe CLASSICAL SIGNAL
MODEL FOR QUANTUM CHANNELS, to appear in Journal of Russian Laser Research䇯
䋨ᩏ⺒䋩
4.
Noboru Watanabe, On quantum entropies of quantum dynamical systems, Americal
Institute of Physics, Vol.1232, 162-174, 2010䋨ᩏ⺒䋩
5.
Noboru Watanabe, On entropies of quantum dynamical systems, QP- PQ Quantum
Probability and White Noise Analysis, Quantum Bio-Informatics㩷 III; From Quantum
Information to Bio-Informatics, Vol.26, pp 423-441, 2010䋨ᩏ⺒䋩
6.
Noboru Watanabe, Entropy type complexities in quantum systems, to appear in Americal
Institute of Physics䋨ᩏ⺒䋩
7.
On construction of QFT by using FTM gates with Kerr device including third terms of
interaction Hamiltonian, T. Sato and N. Watanabe, to appear in The Proceedings of the
33nd Symposium on Information Theory and its Applications SITA2010 䋨ᩏ⺒ή䋩
8.
On
Efficiency
of
Information
Transmission
for
composite
Quantum
Control
Communication processes by using squeezed states, T. Yoneda and N. Watanabe, to
appear in The Proceedings of the 33nd Symposium on Information Theory and its
Applications SITA2010 䋨ᩏ⺒ή䋩
− 78 −
9.
ᷫ䉼䊞䊈䊦䈫శ䉼䊞䊈䊦䉕䉃㊂ሶᓮㅢାㆊ⒟䈮䈍䈔䉎ᖱႎવㅍ䈱ല₸䈮䈧䈇䈩䋬⣁ᧄᑧ䋬
ᷰㆺ㩷 䋬ାቇᛛႎ Vol.110, No.137, pp 107-112䋬2010 䋨ᩏ⺒ή䋩
10. 䋲䈧䈱 EPR ⁁ᘒ䈎䉌䈧䈒䉌䉏䉎ജ⁁ᘒ䈮ኻ䈜䉎 Superdense coding 䉕䉃㊂ሶㅢାㆊ⒟䈱ᖱ
ႎવㅍ䈱ല₸䈮䈧䈇䈩䋬ฎ⼱㊁ஜᴦ䋬ᷰㆺ㩷 䋬ାቇᛛႎ Vol.110, No.137, pp 113-118䋬2010
䋨ᩏ⺒ή䋩
11. 䋴䊎䊷䊛䉴䊒䊥䉾䉺䊷䉕↪䈇䈢㊂ሶ⺰ℂ䉭䊷䊃䈱⺋䉍⏕₸䈮䈧䈇䈩䋬㜞የ੫䋬⮮ᒄၮ䋬ᷰㆺ㩷 䋬
ାቇᛛႎ Vol.110, No.137, pp 119-124䋬2010 䋨ᩏ⺒ή䋩
12. 䋳ᰴ䈱㗄䉕⠨ᘦ䈚䈢 FTM 䉭䊷䊃䈱⺋䉍⏕₸䈮䈧䈇䈩䋬ਁਛ⾆ᄦ䋬ᷰㆺ㩷 䋬ାቇᛛႎ Vol.110,
No.137, pp 125-130䋬2010 䋨ᩏ⺒ή䋩
࿖㓙ળ⼏
1.
Entropy Type Complexities in Quantum Dynamical Processes, N. Watanabe,㩷 The 4th
International Conference on Quantum Bio-Informatics Center䋬 Tokyo University of
Science䋬 Noda䋬 Japan䋬 2010-03-10䌾03-13 (2010 ᐕ 3 10 ᣣ䌾3 13 ᣣ)䋮
2.
Entropy Type Complexity in Quantum Systems, N. Watanabe, The International
Conference on AQT – Advances in Quantum Theory, Linnaeus University, Vaxjo,
Sweden, 2010-06-14䌾06-17 (2010 ᐕ 6 14 ᣣ䌾6 17 ᣣ)䋮
3.
A Fair Sampling Test for Ekert Protocol, G. Adenier,
N. Watanabe,㩷 The International
Conference on AQT – Advances in Quantum Theory, Linnaeus University, Vaxjo,
Sweden, 2010-06-14䌾06-17 (2010 ᐕ 6 14 ᣣ䌾6 17 ᣣ)䋮
4.
Some Aspects of Complexities for Quantum Processes, N. Watanabe,㩷 42 Symposium
on Mathematical Physics 䋬
Nicolaus Copernicus University 䋬
Torun 䋬
Poland 䋬
2010-06-19䌾06-22 (2010 ᐕ 6 19 ᣣ䌾6 22 ᣣ)䋮
5.
Some Aspects of Complexities for Quantum Systems, N. Watanabe,㩷 3rd International
Symposium on Applied Sciences in Biomedical and Communication Technologies
(ISABEL 2010)䋬 Center for Ecclesiastical Services䋬 Rome䋬 Italy䋬 2010-11-07䌾11-10
(2010 ᐕ 11 7 ᣣ䌾11 10 ᣣ)䋮
6.
Efficiency of Information Transmission of Quantum Control Communication Process by
using Squeezed States (Poster),
T. Jitsukawa,
N. Watanabe,㩷 The 4th International
Conference on Quantum Bio-Informatics Center䋬 Tokyo University of Science䋬 Noda䋬
Japan䋬 2010-03-10䌾03-13 (2010 ᐕ 3 10 ᣣ䌾3 13 ᣣ)䋮
7.
On Efficiency of Information Transmission of Quantum Communication Process
including Superdense Coding for Input State Generated by Binary EPR States (Poster),
K. Koyano, N. Watanabe,㩷 The 4th International Conference on Quantum Bio-Informatics
Center䋬 Tokyo University of Science䋬 Noda䋬 Japan䋬 2010-03-10䌾03-13 (2010 ᐕ 3
10 ᣣ䌾3 13 ᣣ)䋮
8.
On Quantum Mutual Entropy Type Measures for Two Dimensional Quantum Channels
(Poster), R. Fushimi, N. Watanabe,㩷 The 4th International Conference on Quantum
− 79 −
Bio-Informatics Center䋬 Tokyo University of Science䋬 Noda䋬 Japan䋬 2010-03-10䌾
03-13 (2010 ᐕ 3 10 ᣣ䌾3 13 ᣣ)䋮
9.
On Error Probability of FTM Gate Considering Third Term of Kerr Effect (Poster), T.
Manchu, H. Sato, N. Watanabe, 㩷 The 4th International Conference on Quantum
Bio-Informatics Center䋬 Tokyo University of Science䋬 Noda䋬 Japan䋬 2010-03-10䌾
03-13 (2010 ᐕ 3 10 ᣣ䌾3 13 ᣣ)䋮
10. On Efficiency of Information Transmission for Quantum Control Communication Process
including Attenuation Channel and Noisy Optical Channel (Poster), N. Wakimoto, K.
Yoneda, T. Jitsukawa N. Watanabe,㩷 The 4th International Conference on Quantum
Bio-Informatics Center䋬 Tokyo University of Science䋬 Noda䋬 Japan䋬 2010-03-10䌾
03-13 (2010 ᐕ 3 10 ᣣ䌾3 13 ᣣ)䋮
࿖ౝࠪࡦࡐࠫ࠙ࡓ
1.
ᷫ䉼䊞䊈䊦䈫శ䉼䊞䊈䊦䉕䉃㊂ሶᓮㅢାㆊ⒟䈮䈍䈔䉎ᖱႎવㅍ䈱ല₸䈮䈧䈇䈩䋬⣁ᧄᑧ䋬
ᷰㆺ㩷 䋬㔚ሶᖱႎㅢାቇળIT⎇ⓥળ䋬2010ᐕ7 (Ꮏቇ㒮ᄢ)䋮
2.
䋲䈧䈱EPR⁁ᘒ䈎䉌䈧䈒䉌䉏䉎ജ⁁ᘒ䈮ኻ䈜䉎Superdense coding䉕䉃㊂ሶㅢାㆊ⒟䈱ᖱႎ
વㅍ䈱ല₸䈮䈧䈇䈩䋬ฎ⼱㊁ஜᴦ䋬ᷰㆺ㩷 䋬㔚ሶᖱႎㅢାቇળIT⎇ⓥળ䋬2010ᐕ7 (Ꮏቇ㒮
ᄢ)䋮
3.
䋴䊎䊷䊛䉴䊒䊥䉾䉺䊷䉕↪䈇䈢㊂ሶ⺰ℂ䉭䊷䊃䈱⺋䉍⏕₸䈮䈧䈇䈩䋬㜞የ੫䋬⮮ᒄၮ䋬ᷰㆺ㩷 䋬
㔚ሶᖱႎㅢାቇળIT⎇ⓥળ䋬2010ᐕ7 (Ꮏቇ㒮ᄢ)䋮
4.
䋳ᰴ䈱㗄䉕⠨ᘦ䈚䈢FTM䉭䊷䊃䈱⺋䉍⏕₸䈮䈧䈇䈩䋬ਁਛ⾆ᄦ䋬ᷰㆺ㩷 䋬㔚ሶᖱႎㅢାቇળIT⎇
ⓥળ䋬2010ᐕ7 (Ꮏቇ㒮ᄢ)䋮
5.
䌆䌔䌍䉭䊷䊃䉕↪䈇䈢㊂ሶ䉝䊦䉯䊥䉵䊛䈱ቯᑼൻ䈮䈧䈇䈩䋬⮮ᒄၮ䋬 ᷰㆺ㩷 䋬╙33࿁ᖱႎℂ⺰
䈫䈠䈱ᔕ↪䉲䊮䊘䉳䉡䊛 SITA2010䋬2010ᐕ 1130ᣣ䌾123ᣣ(ାᎺ᧻ઍ䊨䉟䊟䊦䊖䊁䊦(㐳㊁
⋵))䋮
⋓᳗◊㇢
ቇⴚ⺰ᢥ
1.
Measurement of the second-order Zeeman effect on the clock transition in the
weak-magnetic-field region using the scalar Aharonov-Bhom phase, K. Numazaki, H.
Imai, and A. Morinaga, Phys. Rev. A 81, 032124 (2010). (ᩏ⺒)
2.
Interference fringes of m=0 spin states under the Majorana transition caused by rapid
half-rotation of a magnetic field, A. Takahashi㩷 and A. Morinaga,
Phys. Rev. A 81,
042111 (2010).(ᩏ⺒)
3.
Ramsey Spectroscopy and Geometric Operations on Sodium Bose-Einstein Condensates
using Two-Photon Stimulated Raman Transitions H. Imai and A. Morinaga, J. Phys. Soc.
Japan㩷 79䇮094005 (2010). (ᩏ⺒)
4.
Atomic Berry㵭s Phase free from dynamical phase shift in a conical rotating magnetic field
− 80 −
A. Morinaga, K. Toriyama, H. Narui, H. Imai, and T. Aoki㩷 䋨ᛩⓂਛ䋩
5.
Atom interferometry of sodium Bose-Einstein condensates in 1D optical lattice H. Imai, T.
Akatsuka, T. Ode, and A. Morinaga (ᛩⓂḰਛ)
የ┙㦷
ቇⴚ⺰ᢥ
1.
Three-Body pd Scattering with a Possible Long-Range Force S. Oryu, Hiratsuka, T.
Watanabe, T. Sawada, and S. Gojuki Proceedings of the Few-Body 19 Conference, edited
by Ulf-G. Meissner and Evgeny Epelbaum, (Bonn 2010), ශਛ.䋨ᩏ⺒䈅䉍䋩
2.
Three-Body pd Scattering Using a Rigorous Coulomb Treatment ~Reliability of the
renormalization of screened-Coulomb potentials~Y. Hiratsuka, S. Oryu and S. Gojuki,
Few-Body Systems (2010), ශਛ㩷 䋨ᩏ⺒䉍䋩
೨↰⼑ᴦ
ቇⴚ⺰ᢥ
1.
Joji Maeda, Daichi Kurotsu, and Satoshi Ebisawa, “Stable pulse generation from
regenerative mode-locked fiber ring laser employing semiconductor optical amplifier,”
Technical Digest of 15th Optoelectronics and Communications Conference, pp. 348-349,
Jul. 2010 䋨ᩏ⺒䈅䉍䋩.
2.
Joji Maeda, Kazutoyo Kusama, and Satoshi Ebisawa, 㵰Effects of Fiber Nonlinearity on
Radio-over-Fiber Transmission of DSB-BPSK Signal, 㵱
Technical Digest of 15th
Optoelectronics and Communications Conference, pp. 716-717, Jul. 2010㩷 䋨ᩏ⺒䈅䉍䋩.
3.
Joji Maeda and Satoshi Ebisawa, 㵰Effect of random local dispersion in ultra-high speed
optical link employing periodical dispersion-compensation,㵱 Technical Digest of OSA
topical meeting on Nonlinear Photonics 2010 (CDROM) Jun. 2010䋨ᩏ⺒䈅䉍䋩.
4.
㤥ᵤ ᄢ䊶ᶏ⠧Ỉ ⾫ผ䊶೨↰ ⼨ᴦ, 㵰ඨዉశჇེ䉕↪䈇䈢ౣ↢㜞⺞ᵄ䊝䊷䊄หᦼ䊐䉜䉟䊋䊥
䊮䉫䊧䊷䉱䈱ቯൻ,㵱 㔚ሶᖱႎㅢାቇળᛛⴚ⎇ⓥႎ๔, vol. 110, no. 257, pp. 17-20, Oct.
2010㩷 䋨ᩏ⺒䈭䈚䋩.
᧻ጟ㓉ᔒ
ቇⴚ⺰ᢥ
1.
D. Chruscinski, A. Kossakowski, K. Mlodawski and T. Matsuoka, 㵰A class of Bell
diagonal states and entanglement witness㵱Open Syst. Info. Dyn., 17, pp. 1-19, 2010䋨ᩏ⺒
䋩
2.
L. Accardi, D. Chruscinski, A. Kossakowski, T. Matsuoka and M. Ohya, 㵰On classical
and quantum liftings㵱
to be published in Open Syst. Info. Dyn.,䋨ᩏ⺒䋩
− 81 −
3.
D. Chruscinski, A. Kossakowski, T. Matsuoka and M. Ohya, 㵰Entanglement mapping vs.
quantum conditonal probability operator 㵱
to be published in OP-PQ Quantum
Probability and White Noise Analysis: Quantum Bio-Informatics IV. 䋨ᩏ⺒䋩
4.
D. Chruscinski, Y. Hirota, T. Matsuoka and M. Ohya, 㵰Remarks on the degree of
entanglement 㵱 to be published in OP-PQ Quantum Probability and White Noise
Analysis: Quantum Bio-Informatics IV. 䋨ᩏ⺒䋩
࿖㓙ળ⼏
1.
Quantum Correlation & Mutual Information via Entanglement, T. Matsuoka, 42th
Symposium on Mathematical Physics, Torun, Poland, 2010
2.
Entanglement Channel & Its Capacity, T. Matsuoka, The Second International
Conference Mathematical Physics and Its Applications, Samara, Russia, 2010 (ᓙ⻠Ṷ)
3.
Quantum Correlation and Mutual Information via Entanglement, T. Matsuoka, Duality
and Scale in Quantum Physical Science, Kyoto, Japan, 2010
⮮ሶ
ቇⴚ⺰ᢥ
1.
T. Hara, K. Sato and M. Ohya, Significant Improvement of Sequence Alignment can be
Done by Considering Transition Probability between Two Consecutive Pairs of Residues
QP-PQ: Quantum Bio-Informatics III, Vol.26, 443-452, 2010 (ᩏ⺒)
2.
T. Hara, K. Sato and M. Ohya, MTRAP: Pairwise sequence alignment algorithm by a new
measure based on transition probability between two consecutive pairs of residues, BMC
BIOINFORMATICS, Vol.11: 235, 1-11, 2010 (ᩏ⺒)
3.
K. Sato, T. Tanabe and M. Ohya, How to Classify Influenza A viruses and Understand
Their Severity, Open Systems & Information Dynamics, Vol.17, 297-310, 2010 (ᩏ⺒)
ᓙ⻠Ṷ
1.
K. Sato, T. Tanabe and M. Ohya, Evolution of HIV-1 from the viewpoint of Information
Theory, ISABEL 2010, Roma, November 2010
◉ේ㧔ችᧄ㧕ᥰሶ
⎇ⓥ⊒ 㧔࿖ౝ㧕
1.
ጊᧄ㩷 ⚔ม䋬ችᧄ㩷 ᥰሶ䋬ንỈ㩷 ⽵↵ (2010 ᐕ 5 ), ᱜᣇಽഀ䈮䈍䈔䉎ኻ⒓ᕈ䈱㓒䈢䉍䉕᷹䉎
〒㔌ዤᐲ䋬ᣣᧄ⸘▚ᯏ⛔⸘ቇળ㩷 ⛔⸘ᢙℂ⎇ⓥᚲ
2.
ችᧄ㩷 ᥰሶ䋬◉ේ㩷 ⡡ (2010 ᐕ 11 ), Mutually M-intersecting K-arcs 䈫శ⋥╓ภ䈻䈱ᔕ↪,
⎇ⓥ㓸ળ䇸ታ㛎⸘↹ᴺ䈍䉋䈶䈠䈱ㄝ㗔ၞ䈮䈍䈔䉎⚵ว䈞᭴ㅧ䈱⸃䈫䈠䈱ᔕ↪䇹ၔፒᄢળ⼏㙚
3.
Tahata, K., Yamamoto, K., Miyamoto, N. and Tomizawa, S., (2010):,
Analysis of
contingency tables based on measure of departure of proportion of variation,
− 82 −
International Conference in QBIC, Poster session.
ቇⴚ⺰ᢥ
1.
Comparison of square contingency tables using measure of departure from marginal
homogeneity. Tahata, K., Yamamoto, K., Miyamoto, N., and Tomizawa, S., Quantum
Probability and White Noise Analysis: Quantum Bio-Informatics III, Vol. 26, pp. 369-379 ,
2010㩷 䋨ᩏ⺒䋩
2.
A measure of departure from diagonals-parameter symmetry based on association
measure for square contingency tables. Miyamoto, N., Kato, T. and Tomizawa, S., Journal
of Statistics: Advances in Theory and Applications,
Vol.3, pp. 1-16, 2010 (ᩏ⺒)
ቇⴚ⺰ᢥ
1.
On estimation of the position distribution of the ideal Bose gas, K.-H. Fichtner, K. Inoue
and M.Ohya, QP-PQ: Quantum Probability and White Noise Analysis (Quantum
Bio-Informatics 㸉), 111-126, 2010䋨ᩏ⺒䋩
2.
On the low-temperature behavior of the infinite-volume ideal Bose gas, K.-H. Fichtner, K.
Inoue and M. Ohya, Infinite Dimensional Analysis, Quantum Probability and Related
Topics, 2010䋨ᩏ⺒䋩
3.
A completely discrete particle model derived from a stochastic partial differential
equation by point systems, K.-H. Fichtner, K. Inoue and M. Ohya, to appear in QP-PQ:
Quantum Probability and White Noise Analysis (Quantum Bio-Informatics 㸊),㩷 2011䋨ᩏ
⺒䋩
ጊ⡛ผ
ቇⴚ⺰ᢥ
1.
S.Iriyama, M.Ohya (2010) Generalized Quantum Turing Machine and Its Use to Find an
Algorithm Solving NP-Complete, The 3rd International Symposium on Applied Sciences
in Biomedical and Communication Technologies Conference Proceedings䋨ᩏ⺒ή䋩
2.
ጊ⡛ผ䋬ᄢ⍫㓷ೣ (2010), Quantum Mutual Entropy Defined by Liftings and Violation
of the Shannon Inequality, ᢙℂ⸃ᨆ⎇ⓥᚲ⻠ⓥ㍳, 1705, 88-94䋨ᩏ⺒ή䋩
3.
S.Iriyama, M.Ohya, Quantum Mutual Entropy Defined by Liftings, Foundations of
Physics, DOI: 10.1007/s10701-010-9432-4, 2010䋨ᩏ⺒䋩
ᓙ⻠Ṷ
1.
S.Iriyama,
M.Ohya,
Computational
Complexity
of
Quantum
Algorithm
and
Applications(plenary), Mathematical Physics & Applications - MPA'10, 8/29-9/4, Samara,
Russia, 2010
2.
S.Iriyama, M.Ohya, Generalized Quantum Turing Machine and Its Use to Find an
− 83 −
Algorithm Solving NP-Complete, ISABEL 2010, Center for Ecclesiastical Services, Rome,
Italy, November 7-10, 2010
↰⇌ ⠹ᴦ
ቇⴚ⺰ᢥ
1.
On test of marginal point-symmetry in multi-way tables. Tahata, K., Tokuno, H. and
Tomizawa, S., Advances and Applications in Statistical Sciences, Vol. 3, pp.1-12, 2010,䋨ᩏ
⺒䋩.
2.
Extension of the marginal cumulative logistic model and decompositions of marginal
homogeneity for multi-way tables. Kurakami, H., Tahata, K. and Tomizawa, S.
Journal
of Statistics: Advances in Theory and Applications, Vol.3, 135-152, 2010,䋨ᩏ⺒䋩.
3.
Measure of departure from collapsed symmetry for multi-way contingency tables with
ordered categories. Yamamoto, K., Tahata, K., Hatori, A., and Tomizawa, S., Journal of
the Japan Statistical Society, Vol.40, pp.97-109, 2010,䋨ᩏ⺒䋩.
ጊᧄ ⚔ม
ቇⴚ⺰ᢥ
1.
1䋮Measure of departure from collapsed symmetry for multi-way contingency tables with
ordered categories. Yamamoto, K., Tahata, K., Hatori, A., and Tomizawa, S., Journal of
the Japan Statistical Society,
2.
Vol.40, pp.97-109,
2010, 㩷 䋨ᩏ⺒䋩䋮
Measure for no three-factor interaction model in three-way contingency tables.
Yamamoto, K., Hori, K. and Tomizawa, S.
American Journal of Biostatistics, Vol.1,
pp.17-22, 2010, 䋨ᩏ⺒䋩䋮
3.
Harmonic, geometric and arithmetic means type uncertainty measures for two-way
contingency tables with nominal categories. Yamamoto, K., Miyamoto, N. and Tomizawa,
S.
4.
Advances and Applications in Statistics,
2010,
to appear.㩷 䋨ᩏ⺒䋩䋮
Measures of proportional reduction in error for two-way contingency tables with nominal
categories. Yamamoto, K. and Tomizawa, S.
Biostatistics, Bioinformatics and
Biomathematics, 2010, to appear.䋨ᩏ⺒䋩䋮
ේ ⧷
ቇⴚ⺰ᢥ
1.
Hara, T., Iriyama, S., Makino, K., Terada, H. and Ohya, M. (2010): Mathematical
description of drug movement into tumor with EPR effect and estimation of its
configuration for DDS., Colloids Surf B Biointerfaces 75, 42-46.䋨ᩏ⺒䋩
2.
Hara, T., Sato, K. and Ohya, M. (2010): MTRAP: pairwise sequence alignment algorithm
by a new measure based on transition probability between two consecutive pairs of
− 84 −
residues., BMC Bioinformatics 11, 235.䋨ᩏ⺒䋩
3.
Hara, T., Sato, K. and Ohya, M. (2010): Significant Improvement of Sequence Alignment
can be Done by Considering Transition Probability between Two Consecutive Pairs of
Residues,
QP-PQ:Quantum
Probability
and
White
Noise
Analysis
(Quantum
Bio-Informatics III) 26, 443-452.䋨ᩏ⺒䋩
4.
Hara, T., Sato, K. and Ohya, M. (2010): Improvement in Accuracy of Sequence Alignment
by the MTRAP algorithm, IEICE technical report, 109, 99-104.䋨ᩏ⺒ή䋩
5.
ේ⧷, ⮮ሶ, ᄢ⍫㓷ೣ(2010): 䊕䉝㑆ផ⒖㊂䉕↪䈇䈢㈩䉝䊤䉟䊜䊮䊃ᴺ, IPSJ SIG
Technical Reports, 2010-BIO-22 䋨ᩏ⺒ή䋩
6.
Hara, T., Sato, K. and Ohya, M. :㩷 SIGNIFICANT IMPROVEMENT OF SEQUENCE
ALIGNMENT CAN BE DONE BY CONSIDERING TRANSITION PROBABILITY
BETWEEN TWO CONSECUTIVE PAIRS OF RESIDUES, QP-PQ: Quantum Probability
and White Noise Analysis, to appear䋨ᩏ⺒䋩
ᓙ⻠Ṷ
1.
Hara, T., Sato, K. and Ohya, M. : New sequence alignment by a measure due to entangled
correlation in two consecutive residues, Center for Ecclesiastical Services, Rome, Italy,
November 7-10, 2010
− 85 −
㩷
㩷
㩷
↢⎇ⓥ䉫䊦䊷䊒㩷
㩷
ࠗࡦࠪࠦ↢‛ቇࠍ⋡ᜰߒߡ
↢ࠣ࡞ࡊ 㧔ၮ␆Ꮏቇㇱ↢‛Ꮏቇ⑼ ጊ⊓⎇ⓥቶ㧕
ጊ⊓৻㇢㧔ᢎ㧕ޔᘕ㧔ഥᢎ㧕
㊄ቢ㧔ࡐࠬ࠼ࠢ㧕ޔ⮮ᩰ჻
Abstract.ࡉ࠙ࡦേജቇᴺࠍ㐿⊒ޔዊࡍࡊ࠴࠼ߩࡑࠗࠢࡠ⑽⒟ᐲߩ᭴ㅧ੍᷹ࠪࡒࡘ
࡚ࠪࡦ߇น⢻ߣߥߞߚޔߢߎߘޕ࿁ߪࠕࡦࡉࠨࡦࡊࡦࠣᴺࠍ↪ߚ⥄↱ࠛࡀ࡞ࠡ
⸘▚ߦࠃࠆࠬࠢ࠶߳ߩ࠼࠴ࡊࡍޔᒻᚑᯏ᭴ߦߟߡႎ๔ߔࠆߚ߹ޕᯏ⢻੍᷹࠰ࡈ࠻
FCANAL ߩలታࠍᤓᐕห᭽⛮⛯ߒߚ‛↢ࠦࠪࡦࠗޔߦࠄߐޕቇߩᦨ⚳⋡ᮡߣߒߡߩࠪ
ࠬ࠹ࡓࠪࡒࡘ࡚ࠪࡦߩ⏕┙ࠍ⋡ᜰߒߡ৻ޔਣߏߣࡉࡈࡀ⩶ࠍࠪࡒࡘ࡚ࠪࡦߔ
ࠆߴߊ↢ޔൻቇ࠺࠲ߩ㓸ߣࠪࡒࡘ࠲ߩ㜞ᕈ⢻ൻࠍⴕߞߡࠆޕ
1㧚ߪߓߦ
ᧄࡊࡠࠫࠚࠢ࠻ߦ߅ߡ⎇ߩߜߚ⑳ޔⓥቶߢߪ⑼ࡓࡁࠥޔቇㅴዷߩ߅㒶ߢࠄ߆ߦߥߞߚฦ⒳↢
‛ߩᜬߟㆮવᖱႎో߆ࠄߩㆮવሶ↥‛ߢࠆ࠲ࡦࡄࠢ⾰᭴ㅧ੍᷹ߩߘޔ᭴ㅧߦၮߠߊᯏ⢻੍᷹ޔ
ߐࠄߦߪߘࠇࠄᯏ⢻ߩ㓸วߣߒߡߩ↢ࠪࠬ࠹ࡓߩࠪࡒࡘ࡚ࠪࡦࠍ⋡ᜰߒߡࠆޕ
࠲ࡦࡄࠢ⾰ߩ᭴ㅧࠪࡒࡘ࡚ࠪࡦߩߚߦࡦ࠙ࡉޔേജቇࠪࡒࡘ࡚ࠪࡦᴺࠍ㐿⊒ߒޔ
㐳ᤨ㑆ߩࠪࡒࡘ࡚ࠪࡦࠍน⢻ߣߒߚߦ▚⸘ࠡ࡞ࡀࠛ↱⥄ߦࠄߐޔߢߎߘޕㆡ↪ߔࠆߚޔ
ࠕࡦࡉࠨࡦࡊࡦࠣᴺࠍዉߒߚޕ࿁ߪࠡ࡞ࡀࠛ↱⥄ࠆࠃߦࠣࡦࡊࡦࠨࡉࡦࠕޔ
⸘▚ᚻᴺߩലᕈࠍᬌ⸽ߔࠆߚࠬࠢ࠶߳ߩ࠼࠴ࡊࡍޔᒻᚑ⢻ߩቯ㊂⊛Ⓧ߽ࠅࠍታ㛎ߣᲧセ
ߒߡⴕߞߚޕએਅߦߪ⊒ߩߘޔේⓂࠍⷐ⚂ߒߡឝタߔࠆޕ
2㧚ࡉ࠙ࡦേജቇᴺ㧛ࠕࡦࡉࠨࡦࡊࡦࠣߦࠃࠆࡍࡊ࠴࠼ߩ߳࠶ࠢࠬᒻᚑ⢻ߩⓍࠅ
Abstract
To evaluate the enthalpic and entropic contribution of polar residues in alanine-based peptides to Į-helical
propensities of peptides, we applied a Brownian Dynamics (BD) simulation, together with the umbrella
sampling, to two alanine-based 21-residue peptides: one composed of alanine only (AAA), the other
composed of 18 alanines and 3 arginines (ARA). Higher Į-helical propensity of ARA than AAA was
obtained. However, they showed similar conformational stability in enthalpy, considering the contribution
of the solvation energy and the potential energy. As an evaluation of entropic effects the fluctuation of a
dihedral angle, ȥ, was investigated. Arginine residue showed smaller fluctuation than alanine in elongated
states. Higher Į-helical propensity might originate from entropic effects. Further arginine seems to affect
the Į-helical propensity of alanines interacting with arginine.
Introduction
As model substances for the research on the mechanism of ǂ-helical propensity
shown by biomolecules in their structures, alanine-based peptides have been investigated in
detail. A peptide with 10 alanine residues did not seem to form ǂ-helical structure, which
suggests that alanine residue does not intrinsically have so high ǂ-helical propensity. If
several alanine residues in such peptides were replaced with polar residues containing large
side chains, the helical propensity of the peptides increased, and, further, if the polar residues
were uncharged, the ǂ-helical propensity of the peptides increased a little more, of course,
affected by the density of the polar residues and their distributions in the peptides. And
recently by NMR spectroscopy similar result was reported. A research group suggested that
the desolvation of the backbone CO and NH groups induced by polar residues should shift the
− 87 −
conformational preference toward the ǂ-helical structure. A similar result using replica
exchange molecular dynamics (REMD) simulation was reported by another group on the basis
of analysis of the sampled ǂ-helical structures, in which the guanidinium group in arginine
interacted with the backbone CO of four amino acid upstream by a water-bridged manner,
thus the backbone desolvating. Such a series of reports emphasized enthalpic contribution of
polar residues to ǂ-helical propensity. By earlier studies, entropic contributions of amino acid
residues to ǂ-helical propensity have been pointed out where the side chain of a residue was
restricted in conformational space as a result of interaction with other side chains or
backbone chains. Such an entropic contribution has not been analyzed in detail for the
explanation of ǂ-helical propensity shown by polar residues, while entropic as well as
enthalpic contributions have been discussed for that of nonpolar residues. In addition, several
reports suggested that such enthalpic and entropic contributions appeared to be
sequence-dependent. For more comprehensive explanation of the ǂ-helical propensity of
alanine-based peptides with polar residues, not only enthalpic but also entropic contributions
should be evaluated.
In this study we investigated enthalpic and entropic contributions of arginine to
ǂ-helical propensity, using computer simulation. As a tool of simulation the Brownian
Dynamics (BD) simulation was used, in which the solvent molecules were treated as a
continuum medium with a viscosity and a dielectric constant to reduce the computational
time. This made it possible to pursue longer time scale of molecular dynamics. We applied
this BD simulation to two alanine-based 21-residue peptides. One of the two peptides was
composed of alanine only (AAA), Ace-A5-(AAAAA)3-A-Nme, the other, containing three
arginine residues substituting three alanines (ARA), Ace-A5-(AAARA)3-A-Nme. For sampling
over broad conformational space the umbrella sampling was applied. The peptides were
elongated from fully ǂ-helical to elongated structures, by applying an umbrella potential
between both termini of the peptides. The Weighted Histogram Analysis Method (WHAM)
was used for generating the potential of mean force (PMF). ARA showed higher ǂ-helical
propensity than AAA, which is consistent with experimental results. To explain the ǂ-helical
stability difference observed between the two peptides, entropic as well as enthalpic
contributions of arginine were explored and discussed.
Methods
Brownian dynamics algorithm and Force field. The BD algorithm was the same described in
the previous report. The AMBER91 united-atom force field, except for the electrostatic
charges of atoms, was used to describe the intramolecular potential energy, Vintra. Interaction
of biomolecules with the surrounding solvent, the solvation free energy, Gsolv = ƴiǔiSAi, was
introduced, where SAi was the solvent-accessible surface area (SA) of atom i, and ǔi was the
atomic solvation parameter of atom i. In this study, the following atom-type dependent
solvation values were adopted, ǔ(C) = 12 cal/mol/Å2, ǔ(O,N) = ï116 cal/mol/Å2, ǔ(S) = ï18
cal/mol/Å2, ǔ(Oï) = ï175 cal/mol/Å2, ǔ(N+) = ï186 cal/mol/Å2.
Furthermore, the effect of the solvent on the conformation of the molecules was
introduced into the dielectric constant (dž) as distant-dependent dielectric constant (DD) model,
dž = 2rij, where rij was the distance between atoms i and j, and the SA-dependent atomic charge
(qi') as the effective charge (EC) model, qi' = qi0(1 – SAi/Si + DŽ–1SAi/Si), where Si was the SA of
atom i when the atom was alone in solvent, and qi0 was the atomic charge assigned in AMBER
force field. DŽ as a shielding parameter was selected as following, DŽ = 5.
The summation of Vintra and Gsolv was the effective energy, Weff, that is, Weff = Vintra +
Gsolv. Therefore, this effective energy contains the chain enthalpy and solvation free energy
terms. Chain entropic term is not included in Weff.
Simulation condition. Two kinds of alanine-based 21 residue peptides were used and their two
termini were modified with acetyl (Ace) and N-methyl (Nme) groups, respectively, that is,
Ace-A5-(AAAAA)3-A-Nme and Ace-A5-(AAARA)3-A-Nme, where A, alanine residue and R,
arginine. Fully ǂ-helical structure was prepared for each peptide as an initial structure, in
which the distance (Ǐ) between two carbon atoms in both termini of those peptides were about
− 88 −
34 Å. Between these two carbon atoms the umbrella potential U(Ǐ) = kumb(Ǐ – Ǐ0)2 was applied,
where kumb = 8 kcal/mol/Å2, and Ǐ0 was increased by 0.5 Å from 34 Å to 70 Å, totally 73 kinds of
Ǐ0. The sampling of Ǐ at each Ǐ0 was completed by 6,400,000 steps of calculation, which
corresponded to 32 ns of simulation because each step of calculation proceeded by 5 fs, thus
totally about 2.3 Ǎs. And after completion of each simulation at each Ǐ0, Ǐ0 was increased by
0.5 Å to start next simulation at the next Ǐ0. For multiple time step algorithm, short time step
ƦǕ of 5 fs and long time step Ʀt of 40 fs were used. The ǂ-helix content in peptides was
estimated using DSSP software. All calculations were performed using two Intel Dual-Core
Xeon (Woodcrest) 3.0 GHz CPUs in IBM BladeCenter H21.
Potential of mean force calculation. The free energy curves along Ǐ, called potential of mean
force (PMF), were generated using WHAM on the basis of the sampled data of Ǐ. From the
data of Ǐ obtained at each Ǐ0, the first half was discarded by taking them as equilibration at
the Ǐ0, and only the second half was used for the generation of the PMF curve.
Results and Discussion
Potential of mean force for the transformation from ǂ-helix to elongated state of peptides---
First, the PMF curve was obtained using WHAM as described above (Fig. 1). The curve can be
considered to be composed of three regions, 1) the region in which the energy increases with
increase of Ǐ, due to the decrease of the ǂ-helix content in the peptide by elongation, and 2) the
second region in which the energy does not increase so much, 3) the last region in which the
energy increases largely, due to the overstretching. In the second region the peptide is
comparatively flexible with less intramolecular interaction. The values of the free energy at
the second region can be considered as the free energy relating to the stability of the ǂ-helical
structure, which were about 6 kcal/mol of peptide for AAA and 14 kcal/mol of peptide for ARA.
This suggests that ARA has higher ǂ-helical propensity than AAA. The representative
sampled structures in Fig. 1 also supported higher ǂ-helical propensity of ARA, whereas AAA
lost its ǂ-helix content at less elongated state than ARA (also see Fig. 2).
Figure 1. The PMF curves as a function of the
end-to-end distance (Ǐ) of the peptides. Each
PMF curve is composed of three regions, the
first corresponding to the region around lower
end-to-end distance, where the ǂ-helix
contents decrease by elongation, the second
corresponding to the region where the free
energy is almost constant, the ǂ-helical
structure completely disappears in the
peptide and the peptide is mostly free from
intra-molecular interactions, and the last
region around higher end-to-end distance,
fully stretched, where the free energy
increases largely. The value of energy around
the second region was estimated as the free
energy for ǂ-helix stability.
Figure 2. Average values of the ǂ-helix
content and the effective energy of AAA
(upper left) and ARA (upper right), the
potential energy (lower left) and the
solvation energy (lower right) of the two
peptides. With elongation, the ǂ-helix
content decreased, the effective energy
increased, the potential energy was
destabilized while the solvation energy was
stabilized. The ǂ-helix content disappeared
at longer elongation in ARA than AAA.
Error bar represents standard deviation.
Contributions of various energies to the structural stabilities--- The effective energy, Weff,
− 89 −
increased with elongation and reached the maximal value (Fig. 2). The amount of the change
in the energy between the fully helical and the fully
elongated structure seemed similar between both
Table 1. The stability of some
peptides (see also Table 1).
representative
energy
components.
These
were
obtained
by
summing
up the
Weff was composed of the potential energy,
stability
of
each
residue
which
was
Vintra, representing the intramolecular interaction
obtained by subtracting the average of
energy in the peptide, and the solvation energy, Gsolv,
the energies corresponding to ǂ-helix
representing the interaction energy of the peptide
structures from that of elongated
with the surrounding solvent. Vintra increased with
structures. All values are in kcal/mol.
increase of time, that is, with decrease in the ǂ-helix
content, while Gsolv decreased, although Vintra than
Gsolv largely changed. Both energies seemed to reach
maximal or minimal values when the ǂ-helix
disappeared completely.
Vintra was considered as the potential of the
peptide in vacuum. The differences of Vintra between
the two states, fully helical and fully elongated, were
about 68.4 and 64.2 kcal/mol, respectively (Table 1). Such large change of Vintra with
elongation suggests that ǂ-helical structure was mostly stabilized by this potential energy.
Vintra was composed of several parts of energies including the van der Waals energy
and the electrostatic energy, which were the energies changing largely with elongation (Fig. 3
and see also Table 1). (Torsion energy is an important factor for stability of ǂ-helical structure.
However, in AMBER91 united-atom force field, force constants for Ǘ and Ǚ angles are 0
kcal/mol.) Both energies became unstable, with elongation. These van der Waals and
electrostatic interactions would be derived from the interactions among the atoms in the main
chains of the peptides, considering that the change of both energies seemed to be mostly
independent of the peptides.
Figure 3. Average values of the van der Waals
energy (upper left), the electrostatic energy
(upper right), the solvation energies of the two
backbone groups, CO (lower left) and NH
(lower right). They were main contributing
elements to the change in the potential energy
or the solvation energy. Error bar represents
standard deviation.
Gsolv was the summation of the solvation
free energy from all atoms in the peptide, which
were resolved into atom type energies according
to the types of atoms composing the peptide, in
which the solvation energies of CO and NH
residues in the main chain of the peptide
changed largely with the elongation of the
peptide (Fig. 3). Both energies became stabilized
with decrease of the ǂ-helix content, although the
changes in these energies were smaller than
those of the van der Waals and electrostatic
energies.
Enthalpic and entropic contributions to the
higher
helix
stability
of
ARA
than
AAA---Contributions of each energy component
to stability of helical states for both peptides are
listed in Table 1. Significant differences in the
electrostatic (Welec) and solvation (Gsolv) terms
were observed between two peptides. However, the two energy components were almost
counterbalanced by each other in each peptide. Additionally, the van der Waals energy term
(WvdW) was similar between the two peptides. Therefore, the sum of solvation and peptide
potential energy, that is, effective energy (Weff), had similar values (36.4 and 35.6 kcal/mol for
AAA and ARA, respectively). This result indicates that net contribution of chain enthalpy and
solvation free energy to ǂ-helix stability is similar for both peptides.
Figure 1 shows that the ARA peptide shows ~7 kcal/mol more stable in ǂ-helix than
− 90 −
the AAA peptide. However, as shown in Table 1 (Weff), the enthalpic contribution and
solvation energy to such free energy changes is similar in each peptide, clearly demonstrating
that the greater stability of the ARA peptide arises from the chain entropy contribution. Next,
to elucidate a mechanism for entropic contribution of arginine to ǂ-helix stability, the
rotational freedom of a dihedral angle, Ǚ, was investigated. Ǚ of arginine showed smaller
fluctuation compared with that of alanine at elongated state (Fig. 5). Higher ǂ-helical
propensity of ARA might originate from the entropic effect. Presumably, due to the steric
hindrance of the large side chain of arginine, main chain of arginine had limited rotational
freedom at elongated state and this shifted the conformational equilibrium of ARA to ǂ-helical
structure.
Relatively
high
ǂ-helical
Figure 4. Fractional
propensity of alanine, compared with those
potential or solvation
of arginine and lysine, has been suggested.
energy
of
several
residues in both peptides
Such suggestion was based on the
whose differences were
host-guest technique assuming that each
comparatively
large.
residue
in
a
peptide
had
Curves were obtained by
sequence-independent
specific
ǂ-helix
averaging the fractional
propensity. Considering several reports
potential or solvation
suggesting that ǂ-helical propensity of a
energies sampled at each
Ǐ0 under the umbrella
residue was sequence-dependent, and
potential. Panels show
considering our result that alanine
representative examples
positioned near arginine in ARA showed
for larger potential or
different enthalpic behaviors from other
smaller solvation energy
alanines (Fig. 4) and that arginine
change of ARA than AAA
positioned in place of alanine showed
accompanying
with
elongation. The black
different entropic behaviors (Fig. 5),
represents for AAA, the
ǂ-helical propensity of a residue should be
red for ARA. Error bar
sequence-dependent and higher ǂ-helical
represents
standard
propensity
of arginine than alanine was
deviation.
reasonable.
Enthalpic changes accompanied with helix-coil transition for various alanine-based
peptides have been investigated. The values were distributed in the range, ï1.3 ~ ï0.2
kcal/mol/residue. They were derived from main chain interactions, and weakly dependent on
the kind of residue. Roughly the values obtained from calorimetry were distributed in lower
range than those obtained from circular dichroism spectroscopy. Calorimetry directly
measuring the enthalpy
Figure 5. Average and root mean
changes measured about
square fluctuation (RMSF) of a
ï1.0 kcal/mol/residue. In
dihedral angle, Ǚ, of arginine in
our study, Weff changes of
ARA and corresponding alanine
AAA and ARA were ï36.4
in AAA. Ǚ had blank region
around
-120°,
indicating
and ï35.6 kcal/mol, which
forbidden region for Ǚ. The
were obtained by summing
values of ï180° < Ǚ < ï120° were
up the effective energy
translated to 180° < Ǚ < 240° by
changes of the individual
adding up 360 to them because
residues as shown in Fig. 4.
-180° and 180° in Ǚ are the same.
These values were used to
Arginine showed smaller Ǚ angle
fluctuations at elongated state
estimate the fractional
than alanine at the same position
contribution per residue as
in the sequence of AAA,
ï1.95
and
ï1.87
presumably,
because
the
kcal/mol/residue.
These
backbone chain’s rotation around
enthalpic changes were
Cǂ – C bond of the residue was
larger
than
the
hindered by large side chain of
arginine.
experimental results. It
should be noted, however,
− 91 −
that another generalized Born/surface area (GB/SA) implicit REMD simulation estimated the
enthalpic change of similar alanine-based peptide to be about ï2.08 kcal/mol/residue. And the
all-atom explicit REMD simulation estimated those of AAA and ARA as ï0.23, ï0.72
kcal/mol/residue, respectively. Explicit solvent molecules may have important contribution for
the estimation of enthalpic effect for stabilization.
The reports of the free energy change of alanine-based peptides accompanied with
helix-coil transition are few. As ǂ-helical propensity of alanine residue the values were
estimated as ï0.04, or ï0.05 kcal/mol/residue, and as that of uncharged lysine, ï0.14
kcal/mol/residue. In our simulation, alanine’s free energy change was estimated to be ï0.28
kcal/mol/residue, which was calculated from the free energy change (ï6 kcal/mol) divided by
the number of residues (21 residues) in AAA. Another implicit solvation model using
CHARMM force field estimated the ǂ-helical propensity of alanine to be ï0.214
kcal/mol/residue. And all-atom explicit REMD simulation estimated the ǂ-helical propensity
of AAA and ARA to be ï6 and ï9.45 kcal/mol, respectively. The GB/SA implicit REMD
simulation estimated those values to be ï14.28, ï9.74 kcal/mol, which were opposite to the
result obtained by all-atom explicit REMD simulation. Our results, ï6 and ï14 kcal/mol of
AAA and ARA, are consistent with those obtained by all-atom explicit REMD simulation and
consistent with the experimental results that arginine contributes to higher ǂ-helical
propensity of alanine based peptides.
In this study several solvation energy models, DD, EC and atom-type-dependent
solvation energy parameter models, were used together with the modified AMBER
united-atom force field. The optimizations of the parameters introduced for such models
would not be sufficient to achieve adequate balance with the AMBER force field. However, our
result and its comparison with other experimental and computational results suggested that
optimization of our parameters was quite reasonable.
Most of the studies using MD simulation concerning alanine-based peptides having
polar residues were confined to the discussion of enthalpic contributions of the polar residues.
They paid little attention to entropic contribution of the polar residues. Our study, for the
first time, estimated both enthalpic and entropic contribution of the polar residues to ǂ-helical
propensity.
Our atom-type-dependent solvation energy parameter model seemed to express the
solvent effect better than the GB/SA model, considering the MD simulation using GB/SA
model gave opposite result to that of all-atom explicit REMD simulation with respect to
ǂ-helical propensity of AAA and ARA. According to the report, the GB/SA model generated
highly rigid unfolded states with nearly all their residues in the ǂ-helix region of the
Ramachandran plot. This indicates under- or over-estimation of the solvation energy or the
GB energy. The computational speeds were similar with each other.
Conclusion
To evaluate enthalpic and entropic contributions of polar residues introduced in
alanine-based peptide to ǂ-helical propensity of the peptide, we performed our BD simulation
of the two 21-residue peptides, one, named AAA, peptide containing only alanine, the other,
named ARA, peptide containing 18 alanines and 3 arginines. Higher ǂ-helical propensity of
ARA than AAA was observed. It was considered that the two peptides showed similar
conformational stability in enthalpy. Entropically, arginine showed lower dihedral angle
fluctuations than alanine, probably, due to the steric hindrance of larger side chain of
arginine. In addition, arginine contributed indirectly to ǂ-helical propensity of ARA by
affecting the structural stability of alanine positioned near arginine.
− 92 −
ಽሶᖱႎࡀ࠶࠻ࡢࠢߩᢙℂࡕ࠺࡞
ࡃࠗࠝࠣ࡞ࡊ 㧔⮎ቇㇱ↢ഃ⮎⑼ቇ⑼ ↢ᖱႎ⑼ቇ⎇ⓥቶ㧕
ችፒ ᥓ㧔ᢎ㧕ޔᰨ ᆻᄢ㧔ഥᢎ㧕
㋈ᧁᥓౖ㧔ഥᢎ㧕⿒ޔဈᔒᵤ㧔M2㧕
Abstract. ᭽ࡓࡁࠥోߢ‛↢࡞࠺ࡕߥޘ㈩ᖱႎ߇ቯߐࠇߡ߅ࠅޔㆮવሶ߿࠲ࡦࡄࠢ⾰
ߩ৻ᰴ㈩ᖱႎߩ⚦߇ᛠីߢ߈ࠆࠃ߁ߦߥߞߡ߈ߡࠆޕᒰ࠴ࡓߢߪޔㆮવሶ㈩
߆ࠄᆎߡࡓࡁࠥޔ㈩ోߦ㓝ߐࠇߡࠆᖱႎࠍᢥሼᖱႎ߆ࠄߩᢙℂߦࠃߞߡ⺒ߺ⸃
ߊߎߣࠍ⹜ⴕߒߡࠆޕᚒ⎇ߩޘⓥߪ ߩߢ߹ࠇߎޔin silico ⸃ᨆࠍ߃ߡࠥࡁࡓ㈩ᖱႎ
ߛߌ߆ࠄޔㆮવሶ㑆ߩᓮ㑐ଥ߿࠲ࡦࡄࠢ⾰⋧↪ߥߤಽሶࡌ࡞ߩࡀ࠶࠻ࡢࠢࠍ
⸃ᨆߔࠆߚߩၮ⋚⎇ⓥߣ⟎ߠߌߡࠆޔߡߞ߇ߚߒޕ㈩ᖱႎߛߌߦࠃࠆࡄࡦ࠲ޔ
ࠢ⾰ߩ┙᭴ㅧ੍᷹╬߽ᚒ⎇ߩޘⓥኻ⽎ߣߥߞߡߊࠆޕ೨ᐕᐲ߹ߢߩ⎇ⓥߦ߅ߡ⊒ޔ
ᓮ㗔ၞߩ㧝ᰴ㈩᭴ㅧߦߟߡޔcis-element ߩ⒳㘃ߣ⟎ᖱႎ߿㈩ࡄ࠲ࡦߩᄙ᭽
ᕈ⸃ᨆࠍⴕߞߡ߈ߚ⾰ࠢࡄࡦ࠲ޔߚ߹ޕᯏ⢻⸃ᨆߩ৻ㇱߣߒߡޔ㧠ᰴర᭴ㅧ੍᷹ߦ⌕ᚻ
ߒߚߣߎࠈߢߞߚᧄޕᐕᐲߪ⸃ߩߢ߹ࠇߎޔᨆၮ⋚ࠍ߽ߣߦౕࠅࠃޔ⊛ߥߩ⸃
ᨆࠍⴕߞߡࠆࡠࠞޔߦߎߎޕᓮ᧦ઙਅߢ⊒ᄌേߒߡࠆߣ್ᢿߐࠇߚ ㆮવ
ሶࠍኻ⽎ߣߒߚㆮવሶォ౮ᓮߩⷰὐ߆ࠄㆮવሶࡀ࠶࠻ࡢࠢߩ੍᷹ߣࡏ࠷ࠬᲥ⚛
ⶄว᭴ㅧߩ D ဳ HA-70 ߣ㕖Ქ㕖ⴊಝ㓸⚛ NTNHA ߩ┙᭴ㅧߩ੍᷹ߦߟߡႎ๔ߔ
ࠆޕ
1㧚ࠪࠬࡕࠫࡘ࡞࠺࠲ࡌࠬࠍᔕ↪ߒߚㆮવሶォ౮ᓮࡀ࠶࠻ࡢࠢߦ㑐ߔࠆ⎇ⓥ
2003 ᐕߦࡅ࠻ࠥࡁࡓࡊࡠࠫࠚࠢ࠻߇⚳ੌߒࡓࡁࠥ࠻ࡅోޔ㈩߇⸃⺒ߐࠇ 1), 2)ޔᄙߊߩ⎇ⓥ⠪ߦ
ࠃࠅߩޘㆮવሶᯏ⢻ߩ⸃߳ะߌߚ᭽⎇ߥޘⓥ߇ⴕࠊࠇߚ․ޕቯߩ᧦ઙਅߢᗧߦ⊒ᄌേߒ
ߡࠆㆮવሶ⟲ࠍߔߚߦ⤘ޔᄢߥᢙߩㆮવሶߦኻߒߡߘࠇߙࠇߩㆮવሶ⊒ߩᄌേ㊂ࠍ⺞
ᩏߔࠆߩߪല₸⊛ߢߪߥ৻ޔߢߎߘޕᐲߦᢙਁߩㆮવሶ⊒ᖱႎࠍᓧࠄࠇࠆᚻᴺߣߒߡࡑޔ
ࠗࠢࡠࠕࠗ࠺࠲⸃ᨆ߇⌕⋡ߐࠇߡ߈ߚ․ޕቯߩ᧦ઙਅߢᓧࠄࠇߚࡑࠗࠢࡠࠕࠗ࠺࠲ߦኻ
ߒߡㆡಾߥ⛔⸘⸃ᨆᚻᴺࠍㆡ↪ߔࠆߎߣߦࠃߞߡ⊒ᄌേߩࠄࠇࠆㆮવሶࠍߢ߈ࠇ߫ޔ
ࠄ߆ߩㅢᕈࠍᜬߟㆮવሶ⟲ߩᯏ⢻⸃ߦะߌߚᄢ߈ߥᚻ߇߆ࠅߦߥࠆޕ
․ቯߩ᧦ઙਅߢ⊒ߔࠆㆮવሶ⟲߇ሽߔࠆ႐วߩࠄࠇߘޔㆮવሶ⟲ߪㅢߩォ౮࿃ሶߦࠃ
ߞߡᓮߐࠇࠆߎߣ߇੍ᗐߐࠇࠆߚ↪ࠍࠗࠕࡠࠢࠗࡑޔߒ߆ߒޕㆮવሶ⊒⸃ᨆߦ߅ߡߪ
ㆮવሶߩ⊒ࡄ࠲ࡦ߇ᄢ㊂ߦߐࠇࠆߚ↢ޔൻቇ⊛ታ㛎ߩߺߦࠃߞߡ⊒ߦ㑐ਈߔࠆ
ォ౮࿃ሶ⟲ࠍ․ቯߔࠆߎߣߪ࿎㔍ߢࠆ⎇ᧄޔߢߎߘޕⓥߢߪޔォ౮࿃ሶ߇⼂ߔࠆࠪࠬࠛࡔ
ࡦ࠻ߣ߫ࠇࠆ․ቯߩႮၮ㈩ࡄ࠲ࡦߦ⌕⋡ߒߩ࠻ࡦࡔࠛࠬࠪޔㅢᕈࠍ↪ߒߡ 3)⊒ޔ
ߒߡࠆㆮવሶߩࠣ࡞ࡊൻࠍⴕ߁ߎߣߦࠃࠅޔㆮવሶォ౮ᓮࡀ࠶࠻ࡢࠢߩ⸃ᨆࠍ⹜ߺߚޕ
ᧄ⎇ⓥߢߪޔㆮવሶࠣ࡞ࡊൻߩ࠺࠲࠶࠻ߣߒߡࡠࠞޔᓮ᧦ઙਅߢ 31099 ㆮવሶࠍ
ኻ⽎ߣߒߚ࠶࠻ߩ GeneChip ߩ⸃ᨆ⚿ᨐ߆ࠄᓧࠄࠇߚ⊒ᄌേߒߡࠆߣ್ᢿߐࠇߚ 42 ㆮવ
ሶࠍㆬቯߒߚޔߚ߹ޕ2 ߟߩㆮવሶ㑆ߩ⊒ߦ㑐ߔࠆ㑐ㅪᕈߜࠊߥߔޔᓮࡔࠞ࠾࠭ࡓߩㅢᕈ
ߩᐲวࠍ⹏ଔߔࠆߚߦ⋧ޔᖱႎ㊂㧔Mutual Information㧕ࠍណ↪ߒߚ 4)⋧ޕᖱႎ㊂ߣߪޔ
2 ߟߩ⏕₸ᄌᢙ㑆ߩଐሽߩ⒟ᐲࠍ␜ߔዤᐲߢࠅ⎇ᧄޔⓥߦ߅ߡߪߩߎޔ୯߇ᄢ߈߶ߤޔਔ
ㆮવሶߩ⊒ᓮᯏ᭴ߦᒝ㑐ㅪᕈ߇ࠆߎߣࠍ␜ߔ⋧ޕᖱႎ㊂ߪᑼ(1)ߢቯ⟵ߐࠇࠆޕ
MI ( X , Y )
¦P
i 0㧘
1
j 0 ,1
XY ( i , j )
§ P XY (i , j ) ·
¸ " (1)
log 2 ¨
¨P P ¸
X
(
i
)
Y
(
j
)
©
¹
1
− 93 −
ߎߎߢ ޔXޔY ߪޔᲧセኻ⽎ߣߔࠆ 2 ߟߩㆮવሶࠍ␜ߒޔPX(i)ޔPY(j)ߪޔㆮવሶ X ߣ Y ߘࠇߙ
ࠇ߇ォ౮࿃ሶߦࠃߞߡ⁛┙ߦᓮߐࠇࠆ⏕₸ޔPxy(i,j)ߪޔㆮવሶ X ߣ Y ߇หᤨߦᓮߐࠇࠆ⏕₸
ࠍ␜ߔ⎇ᧄޕⓥߢߪޔᑼ(1)ࠍ߽ߣߦޔ42 ㆮવሶߦ㑐ߒߡ⠨߃ࠄࠇࠆߔߴߡߩㆮવሶ㑆㧔42C2㧩861
ㅢࠅ㧕ߦߟߡ⋧ᖱႎ㊂ࠍ⸘▚ߒߚޕ
߹ߚޔฦㆮવሶࠍォ౮ᓮߩⷰὐ߆ࠄࠣ࡞ࡊൻߔࠆߚߦޔฦㆮવሶ㑆ߩ㘃ૃᐲߩᜰᮡߣߒ
ߡ⋧ᖱႎ㊂ࠍ↪ࠣࡦ࠲ࠬࠢޔᚻᴺߣߒߡ 2 ㆮવሶ㑆ߩ࡙ࠢ࠶࠼〒㔌Ward ᴺࠍㆬ
ቯߒߚޕ㓏ጀ⊛ࠢࠬ࠲ࡦࠣߣߪޔᣇߩ㘃ૃᐲ㧔ࠆߪ㕖㘃ૃᐲߢࠆ〒㔌㧕ߦၮߠߡޔ
ᦨ߽ૃߡࠆ߆ࠄ㗅ᰴ㓸ߡࠢࠬ࠲ࡦࠣࠍⴕ߁ᚻᴺߩߎߣߢࠆ⎇ᧄޕⓥߦ߅ߡߪᑼ
(1)ߢ⸘▚ߒߚ 861 ㅢࠅߩ⋧ᖱႎ㊂ߩߘࠇߙࠇࠍ৻ߟߩࠢࠬ࠲ߣ⠨߃ࠆޔೋᦼ⁁ᘒ߆ࠄᰴޘ
ߣࠢࠬ࠲ࠍ⚿วߒߢ߹ࠆߥߦ࠲ࠬࠢߩߟ৻ߦ⊛⚳ᦨޔ㓏ጀ᭴ㅧࠍᚑߒߡߊ⋧ߢߎߎޕ
ᖱႎ㊂ߪฦㆮવሶᵹߦሽߔࠆฦォ౮࿃ሶ߇⼂ߔࠆࠪࠬࠛࡔࡦ࠻ࡄ࠲ࡦߩሽߩή
ࠍὐߣߒߡ⸘▚ߐࠇߚ߽ߩߢࠆߚޔㅢߩᓮ࿃ሶ⟲ߦࠃࠆᓮߦ㑐ਈߔࠆน⢻ᕈߩ㜞
ㆮવሶห჻ࠍࠣ࡞ࡊൻߔࠆߎߣ߇น⢻ߣߥࠆޔߚ߹ޕᖱႎ㊂ࠍ↪ߡࠢࠬ࠲ࡦࠣࠍⴕߞߡ
ࠆߚޔ2 ߟߩㆮવሶ㑆ߩᓮᯏ᭴ߩㅢᕈߩᐲวࠍ᷹ࠆߎߣ߇ߢ߈ࠆޕ
࿑ 1 ߪ⊒߇⏕ߐࠇߚ 42 ઙߩㆮવሶߦᯏ⢻ᧂ⍮ㆮવሶ㧔એਅ unknown ߣߔࠆ㧕ࠍട߃ߚ
43 ઙߩㆮવሶߦኻߒߡࠢࠬ࠲ࡦࠣࠍⴕߞߚ⚿ᨐߢࠆޕฦࠣ࡞ࡊߦዻߔࠆㆮવሶߪޔォ౮
ᓮߩⷰὐ߆ࠄ᧦ࠆޔઙਅߢࠞࡠᓮߦ㑐ࠊࠆㆮવሶߩࡀ࠶࠻ࡢࠢߩਛߢࠇߙࠇߘޔዪ
ᚲ⊛ߥࡀ࠶࠻ࡢࠢࠍᒻᚑߒߡࠆߣ⠨߃ࠄࠇࠆޕunknown ߇ዻߒߡࠆࠣ࡞ࡊԝߩㆮવሶ
ߦߪ”ޔEster hydrolase C11orf54 homolog”“ޔa neurotrophic factor that is involved in neuronal
cell protection and cell survival””ޔF-box and leucine-rich repeat protein 2”ߥߤߩᵈ㉼߇ઃ
ߡ߅ࠅࡊ࡞ࠣޔౝߩฦߩޘㆮવሶߩᯏ⢻߇⇣ߥߞߡࠆ⚿ߩߎޕᨐ߆ࠄޔㆮવሶߩᯏ⢻߇⇣ߥ
ࠆ႐วߢߞߡ߽ޔォ౮ᓮᯏ᭴ߦኻߔࠆㅢᕈࠍ↪ߡޔㆮવሶォ౮ᓮߦ߅ߡ㑐ㅪᕈߩᒝ
ㆮવሶ⟲ࠍࠣ࡞ࡊൻߔࠆߎߣ߇น⢻ߢࠆߎߣ߇␜ໂߐࠇߚޕunknown ߩㆮવሶᯏ⢻ߪਇ
ߢࠆ߇ࡊ࡞ࠣޔԝߦዻߔࠆㆮવሶߣォ౮ᓮᯏ᭴ߩⷰὐ߆ࠄޔㅢᕈࠍᜬߟߎߣ߇੍ᗐߐ
ࠇࠆޕ
ᧄ⎇ⓥߩ⚿ᨐޔォ౮ᓮᯏ᭴ߩⷰὐ߆ࠄㅢᕈࠍᜬߟㆮવሶห჻ߪࡊ࡞ࠣߩߘޔౝߢォ౮
ᓮߦᓇ㗀ࠍ߷ߒߞߡࠆน⢻ᕈ߇ࠆߣ⠨߃ࠄࠇࠆޔߡߞ߇ߚߒޕᯏ⢻ᧂ⍮ߩㆮવሶ߇ሽ
ߔࠆ႐วߢߞߡ߽⋧ޔᖱႎ㊂ࠍ↪ߚㆮવሶߩࠢࠬ࠲ࡦࠣࠍㅢߓߡㅢߩォ౮ᓮᯏ᭴
ࠍᜬߟㆮવሶߩࠣ࡞ࡊࠍߔߎߣ߇น⢻ߢࠆߡ↪ࠍߤߥࠗࠕࡠࠢࠗࡑޔࠅ߹ߟޕㆮવ
ሶ⊒⸃ᨆࠍⴕ߁ߎߣߥߊޔㆮવሶߩᓮᯏ᭴ߩㅢᕈ߆ࠄㆮવሶ㑆ߩォ౮ᓮࡀ࠶࠻ࡢࠢࠍ
੍᷹ߔࠆߎߣ߇น⢻ߢࠆߎߣ߇␜ໂߐࠇߚޕ
ᧄ⎇ⓥߢߪ Cis-DB ߩᖱႎ㊂߇ዋߥ߆ߞߚߚޔᥳቯ⊛ߦ TRANSFAC ߩᓮ࿃ሶᖱႎࠍ↪
ߒߡޔฦㆮવሶߩォ౮ᓮߦ㑐ࠊࠅᓧࠆ߆ߣ߁ߎߣࠍᯏ᪾⊛ߦត⚝ߒߚ߇ߩߎޔᣇᴺߢߪᵹ
㈩ਛߦࡅ࠶࠻ߒߚࠪࠬࠛࡔࡦ࠻ࡄ࠲ࡦ߇ࠪࠬࠛࡔࡦ࠻᭽㈩ߢࠆน⢻ᕈ߇ࠅߚ߹ޔ
ታ㓙ߪォ౮ᓮߦ㑐ࠊߞߡߥᡆ㓁ᕈߩࡅ࠶࠻߽ᄙߊ߹ࠇࠆน⢻ᕈ߇ࠆޔߡߞ߇ߚߒޕ
ᓟޔCis-DB ߩలታࠍ࿑ࠆߎߣ߇ᔅⷐਇนᰳߢࠆޕ
unknown
Ԙ ԙ Ԛ ԛ Ԝ ࿑ 1 43 ㆮવሶߩࠢࠬ࠲ࡦࠣ⚿ᨐ
2
− 94 −
ԝ
ᦝߦޔᯏ⢻ᧂ⍮ㆮવሶ߽ߚㆮવሶߩࠢࠬ࠲ࡦࠣࠍⴕߞߚ߇ޔᯏ⢻ᧂ⍮ㆮવሶߪ✚ߓߡ
ᯏ⢻ᣢ⍮ㆮવሶߣߩ㈩⋧หᕈ߇ߥ߽ߩ߇ᄙޔߡߞ߇ߚߒޕᓟޔ࿁ߩᚻᴺߦࠃߞߡหߓ
ࠣ࡞ࡊߦዻߒߡࠆߣ߁ߎߣ߇੍ᗐߐࠇߚㆮવሶߦߟߡޔห᭽ߩᓮᯏ᭴ࠍᜬߟㆮવሶห
჻ߩ㈩⋧หᕈࠍ⺞ᩏߔࠆ੍ቯߢࠆޕ㈩⋧หᕈߩត⚝ߦࠃߞߡޔ㈩ߦ⋧หᕈ߇ߥߎߣ߇
⏕ߐࠇߚ႐วޔㆮવሶߩ㈩⋧หᕈ߿ᯏ⢻ߦ㘃ૃᕈ߇ߥ႐วߢߞߡ߽ޔォ౮ᓮߩⷰὐ߆
ࠄㆮવሶࠍࠣ࡞ࡊൻߔࠆᣂߚߥᚻᴺࠍឭ᩺ߔࠆߎߣߦߟߥ߇ࠆޕ
߹ߚޔ࿁ኻ⽎ߣߒߚ 42 ઙߩㆮવሶߩࠝ࠰ࡠࠣㆮવሶࠍ⇣ߥࠆ↢‛⒳࠙ࡑ߿࠻ࡅ߫߃ޔ
ࠬ߆ࠄߒޔห᭽ߩࠢࠬ࠲ࡦࠣࠍⴕ߁੍ቯߢࠆ‛↢ޕ⒳㑆ߢࠢࠬ࠲ࠍᲧセߔࠆߎߣߦ
ࠃߞߡޔォ౮ᓮࡀ࠶࠻ࡢࠢߦ⋧㆑߇ࠆ߆ุ߆ࠍᲧセᬌ⸛ߒߚޕ
2. ࡏ࠷ࠬᲥ⚛ⶄว᭴ㅧߩ in silico ⸃ᨆ
ᄙߊߩ↢ౝᔕߦߪ࠲ࡦࡄࠢ⾰ߣ࠲ࡦࡄࠢ⾰ߩ⋧↪߇ᷓߊ㑐ਈߒߡ߅ࠅ↢ޔ⑼ቇߩಽ㊁
ߢߪߩ⾰ࠢࡄࡦ࠲ޔ᭴ㅧߣᯏ⢻ߦ㑐ߔࠆ⎇ⓥ߇⋓ࠎߦⴕࠊࠇߡࠆ♖ߩ⾰ࠢࡄࡦ࠲ޕᛛⴚ߿ X
✢⚿᥏᭴ㅧ⸃ᨆޔNMRޔಽሶ㑆⋧↪⸃ᨆߥߤߩᚻᴺ߇⊒ዷߒߩ⾰ࠢࡄࡦ࠲ޔ᭴ㅧߣᯏ⢻ߦ
㑐ߔࠆ⍮߇ႎ๔ߐࠇߡࠆ߇↢ߩࠄࠇߎޔൻቇ⊛ታ㛎ᚻᴺߩᰳὐߣߒߡᤨޔ㑆߿⾌↪߇߆߆ࠆ
ὐ߇ߍࠄࠇࠆ৻ޕᣇ▚⸘ޔᯏౝߢ࠲ࡦࡄࠢ⾰ߩ᭴ㅧ߿ᯏ⢻ࠍ੍᷹ߒࠃ߁ߣߔࠆ᭴ㅧ੍᷹ࠪࡒࡘ
࡚ࠪࡦᴺ߇㐿⊒ߐࠇ੍ޔ᷹♖ᐲ߇ะߒޔᓟߩ࠲ࡦࡄࠢ⾰᭴ㅧ⎇ⓥߦ߅ߌࠆᒝജߥᚻᴺߦ
ߥࠆߣᕁࠊࠇࠆߢߎߘޕᚒⶄޔߪޘ㔀ߥⶄว᭴ㅧࠍߔࠆߚߦ⸃ᨆ߇㔍ߒࡏ࠷ࠬᲥ⚛
࠲ࡦࡄࠢ⾰ߣ㑐ㅪ࠲ࡦࡄࠢ⾰ࠍ࠲ࠥ࠶࠻ߦߒߡޔᣢሽߩ᭴ㅧ੍᷹ࡊࡠࠣࡓࠍ↪ߚ࠲ࡦࡄࠢ
⾰ⶄว᭴ㅧߩ੍᷹ߦขࠅ⚵ࠎߢࠆޕ
ࡏ࠷ࠬᲥ⚛ߪࡏ࠷ࠬ⩶㧔Clostridium botulinum㧕߇↥↢ߔࠆᒝജߥ⚻Ქ⚛࠲ࡦࡄ
ࠢ⾰㧔Botulinum neurotoxin; BoNT㧕ߢࠆޕBoNT ߩ᛫ේᕈߩᏅ⇣߆ࠄ AޯG ဳߦಽ㘃ߐࠇߡ
ࠆ 5)ޕBoNT ߪઁߩήᲥߥ࠲ࡦࡄࠢ⾰⟲ߣᏂᄢߥⶄว᭴ㅧࠍᒻᚑߒߡࠆޕήᲥ࠲ࡦࡄࠢ⾰
⟲ߪޔ㕖Ქ㕖ⴊಝ㓸⚛㧔Non-toxic non-hemaglutinin; NTNHA; 130kDa㧕߅ࠃ߮ಽሶ㊂ߩ⇣
ߥࠆ 3 ⒳ߩⴊಝ㓸⚛㧔Hemagglutinin; HA㧕HA-70 ޔHA-33 ޔHA-17㧔ߘࠇߙࠇ 70ޔ30ޔ
17kDa㧕ߢ᭴ᚑߐࠇߡ߅ࠅߪࠄࠇߎޔᲥ⚛߇േ‛ߦ⚻ญ៨ขߐࠇߚ㓙ߦ BoNT ࠍᶖൻ㉂⚛߿㉄ᕈ
᧦ઙਅ߆ࠄ⼔ߒޔዊ⣺߆ࠄߩല₸⊛ߥๆߦ㑐ਈߔࠆߣߐࠇߡࠆ 6), 7)ⶄޔߚ߹ޕวਛߩ
BoNT ߅ࠃ߮ HA ߩ৻ߟߢࠆ HA-70 ߪ⩶߇↥↢ߔࠆ࠻ࡊࠪࡦ᭽ࡊࡠ࠹ࠕ㧔Trypsin-like
protease; TLP㧕ߦࠃߞߡಽሶౝㇱߩ․⇣⊛ㇱ߇ಾᢿߐࠇࠆߎߣ߇⍮ࠄࠇߡࠆ㧔࿑ 2㧕8), 9), 10)ޕ
ߒ߆ߒޔಽሶౝಾᢿߦࠃࠆ᭴ㅧ߿Ქ⚛ᵴᕈ߳ߩᓇ㗀ߦߟߡߪߛᧂޔਇߥὐ߇ᄙᧄߢߎߘޕ
⎇ⓥߢߪޔHA ᚑಽߩ᭴ㅧ߿․ᕈߦ⌕⋡ߒߡޔD ဳ HA-70 ߣ NTNHA ߩ┙᭴ㅧߩ੍᷹ࠍⴕߞ
ߚޕ
A
C ᧃ┵
B
N ᧃ┵
C
Lc
࿑ 2 (A) C ဳ BoNT ߩ᭴ㅧ (B) TLP ᭴ㅧ
(C) BoNT/TLP ⶄว᭴ㅧ
X ✢⚿᥏᭴ㅧ⸃ᨆߦࠃࠆ NTNHA ߩ┙᭴ㅧߪ߹ߛቯߐࠇߡߥޔߒ߆ߒޕNTNHA ߣ
40%ߩࡎࡕࡠࠫࠍ߽ߟ BoNT ߣ TeNT㧔tetanus neurotoxin㧕ߩ C ᧃ┵㗔ၞߪ᭴ㅧ߇ቯߐࠇ
ߡࠆ 11), 12), 13)⎇ᧄޔߢߎߘޕⓥߢߪޔBoNT ߩ A ဳ᭴ㅧ㧔PDB ID: 3BTA㧕ߣ B ဳ᭴ㅧ㧔PDB
3
− 95 −
ID: 1EPW㧕ߥࠄ߮ߦ TeNT ᭴ㅧ㧔PDB ID: 1DIW㧕ࠍ㍌ဳߣߒޔNTNHA ߩ㧯ᧃ┵㗔ၞߩ᭴ㅧ
੍᷹ࠍⴕߞߚ 14)ޕ࿑ 3 ߪ NTNHA ߣ㍌ဳ㈩ߣߩࡑ࡞࠴ࠕࠗࡔࡦ࠻ߩ⚿ᨐߢࠆޕ࿑ 4(A)ߪ
੍᷹ߐࠇߚ NTNHA ߩ㧯ᧃ┵㗔ၞߩ᭴ㅧߢࠆ㧔Leu837㨪Ser1191㧕ޕ࿑ 4(B)߆ࠄ࿑ 3 ߢߺࠄ
ࠇߚ TeNT ㈩ߩࠡࡖ࠶ࡊ㗔ၞ߇㐳࡞ࡊࠍᒻᚑߒߡࠆߎߣ߇ಽ߆ࠆ⚿ߩߎޕᨐߪ TeNT ߩ
᭴ㅧߦᲧߴ NTNHA ߩ᭴ㅧ߇ࠦࡦࡄࠢ࠻ߢࠆߎߣࠍ␜ໂߒߡࠆޕ
࿑ 3 NTNHA ߣ㍌ဳ᭴ㅧ㑆ߩ㈩ࠕࠗࡔࡦ࠻㧔TeNT ߪ C ᧃ┵㗔ၞ㧕
D ဳ HA-70 ߩࡕ࠺ࡦࠣߦ㑐ߒߡߪޔC ဳ HA-70 ߩ᭴ㅧ㧔PDB ID: 2ZS6㧕ࠍ㍌ဳߣߒߡ↪
ߒߚޕC ဳߣ D ဳߩ HA-70 㑆ߦߪ 90%ߩ㈩ࡎࡕࡠࠫ߇ࠆ߇ޔC ဳ HA-70 ߪ㒰㗔ၞ
㧔deleted retion㧕ࠍࠎߢࠆߚߩߎޔ㒰㗔ၞࠍ߁ߚࠗࡦ࠹ࠣࡦߩㇱಽ᭴ㅧ㧔PDB ID:
3FCS㧕ࠍ㍌ဳߣߒߡ↪ߒߚ 15㧕ޕ࿑ 4 ߪ੍᷹ߐࠇߚ HA-70 ᭴ㅧߢࠆޕ
A
B
࿑ 4 (A) ੍᷹ߐࠇߚ NTNHA ߩ C ᧃ┵᭴ㅧ (B) ㍌ဳ᭴ㅧߣߩ㊀ߨวࠊߖ᭴ㅧ
㤛㧦NTNHA⊕ ޔ㧦BoNT/A ޔ㕍㧦BoNT/Bࠢࡦࡇޔ㧦TeNT
4
− 96 −
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5
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ࠣࡈᦠ឵߃⸒⺆ REGREL ߦ߅ߌࠆ
ਗⴕࠣࡈᠲࠢࠛߩ
ℂᎿቇㇱ ᖱႎ⑼ቇ⑼
ᱞ↰ ᱜਯ
Abstract
ㄭᐕߢߪ⸘▚ߩኻ⽎ߣߒߡߍࠄࠇࠆ㗴ߪᄙ᭽ൻߒߡ߈ߡࠆ㧚ߘࠇࠄߩ㗴ࠍߔ
ࠆߚߩ᭴ᢥ߿ᗧ⺰߽ᄙ᭽ߢࠅ㧘ᣢሽߩࡊࡠࠣࡒࡦࠣ⸒⺆ࠍ↪ߡ㧘⸘▚ࠍⴕ
߁ߎߣ߇࿎㔍ߢࠆ႐ว߇ࠆ㧚ߘߩ㗴ࠍ⸃ߔࠆߚߦߪ㧘᭽ߥޘ㗴㗔ၞߩ᭴ᢥ߿
ᗧ⺰ࠍ࡙ࠩ߇છᗧߦቯ⟵น⢻ߢࠆࠃ߁ߥⅣႺ߇ᦸ߹ࠇࠆ㧚
ᧄ⎇ⓥߢߪ㧘ߘߩࠃ߁ߥ᳢↪ߩࡔ࠲ࡊࡠࠣࡒࡦࠣ߇น⢻ߢࠆⅣႺߩ᭴▽ࠍ⋡⊛ߣߒߡ
ࠆ㧚ᦨೋߩᲑ㓏ߣߒߡ㧘ฦ⒳᭴ᢥ߿ᗧߩ߇น⢻ߥ⸘▚ࡕ࠺࡞ߩ᭴▽ࠍ⋡ᮡߣߔࠆ㧚
ߘߩࠃ߁ߥ⸘▚ࡕ࠺࡞ߣߒߡ㧘ࡈࠢ࠹ࠖࡉߥࠣࡈᦠ឵߃⸒⺆ REGREL(REflective
Graph REwriting Language)ࠍឭ᩺ߔࠆ㧚REGREL ߪ DACTL ࠍၮ␆ߣߒߡቯ⟵ߐࠇߡ
ࠆ߇㧘ࠃࠅജࠍ㜞ࠆߚߩᎿᄦ߇ⴕࠊࠇߡࠆ㧚ߘߒߡ㧘REGREL ࠍ↪ߚᔕ↪ߣߒ
ߡ㧘᳢↪ߩࠣࡈᠲࠢࠛ⸒⺆ࠍቯ⟵ߒߡߘߩ⹏ଔࠍⴕ߁㧚
REGREL ߪࠣࡈᦠ឵߃♽ߢࠅ㧘REGREL ࠣࡈߣ߫ࠇࠆ㧘㗂ὐߣធ⛯ߩਔᣇߦ
ࡌ࡞ࠍᜬߟะࠣࡈࠍᦠ឵߃ኻ⽎ߣߔࠆ㧚REGREL ࠣࡈߩᦠ឵߃ߪᦠ឵߃ⷙೣߦࠃߞߡ
ቯ⟵ߐࠇࠆ㧚ߘߩᦠ឵߃ߩኻ⽎ߪࡄ࠲ࡦࡑ࠶࠴ߢត⚝ߐࠇ㧘ߐࠄߦㅀ⺆ࠍ↪ߡ⛉ࠅㄟ
ߎߣ߇ߢ߈ࠆ㧚߹ߚ㧘ಽ㘃ߣ߫ࠇࠆᯏ᭴ߦࠃߞߡᦠ឵߃ⷙೣߩㆡ↪▸࿐ࠍ㒢ቯߔࠆߎߣ߇
ߢ߈ࠆ㧚
REGREL ߢߪ REGREL ࠣࡈߦࠃߞߡࡊࡠࠣࡓߩ᭴ㅧ߿᭴ᢥࠍߒ㧘ᦠ឵߃ⷙೣߦ
ࠃߞߡߘߩᗧࠍߔࠆ㧚ߘߒߡ㧘ᦠ឵߃ⷙೣ߽߹ߚࠣࡈߢߐࠇߡࠆߚ㧘ᦠ឵
߃ⷙೣࠍᦠ឵߃ࠆ㜞㓏ᦠ឵߃߇น⢻ߣߥߞߡࠆ㧚ߘߩ․ᓽࠍ↪ߒߡ㧘ࡈ࡚ࠢࠪࡦࠍ
ታߒߡࠆ㧚߹ߚ㧘REGREL ߩᦠ឵߃ߪਗⴕ߆ߟ㕖ቯ⊛ߦⴕࠊࠇࠆ㧚ߥ߅ㅙᰴേ߿
ቯᕈേߪߘߩ․ߥ႐วߣߒߡߐࠇࠆ㧚
XQuery ߿ LINQ ߥߤߩᧁ᭴ㅧ߿ࠣࡈᠲࠢࠛߪ㧘ၮᧄ⊛ߦߪㅙᰴ⊛ߦേߔࠆ߽ߩ
ߣߒߡቯ⟵ߐࠇߡࠆ㧚ߘࠇߦኻߒߡ㧘ᔅⷐߥ▎ᚲߢߪ␜ߖߕߣ߽ㅙᰴಣℂ߿หᦼࠍⴕ㧘
ၮᧄ⊛ߦߪਗⴕߦേߔࠆࠢࠛ߇ቯ⟵ߢ߈ࠇ߫㧘ߘࠇࠍ↪ߚ⸘▚ߪਗⴕ⸘▚⾗Ḯࠍല
ߦᵴ↪ߒߚ߽ߩߦߥࠆߣᦼᓙߐࠇࠆ㧚
ታߣߒߡࠣࡈߩᠲࠢࠛ⸒⺆ࠍߍ㧘ಽ㘃ߣ߫ࠇࠆᦠ឵߃ⷙೣߩㆡ↪▸࿐ࠍ㒢ቯ
ߔࠆᯏ᭴ߣ㧘㜞㓏ᦠ឵߃ࠍ↪ࠆߎߣߢ㧘ᚻᬺߢୃᱜߔࠆߎߣߥߊᦠ឵߃ⷙೣߩേࠍ⹏
ଔᚢ⇛ߦ㑐ߒߡ․ᱶൻߔࠆߎߣ߇ߢ߈ߚ㧚߹ߚ㧘ቯ⟵ߒߚࠢࠛߪၮᧄ⊛ߦਗⴕߦേߒ㧘
␜ߖߕߣ߽ᔅⷐߥ▎ᚲߢߪㅙᰴಣℂ߿หᦼࠍⴕ߁ࠃ߁ߦേߐߖࠆߎߣ߇ߢ߈ߚ㧚
− 98 −
505 ߩ․ᓽࠍ⠨ᘦߦࠇߚࡀ࠶࠻ࡢࠢࡕ࠺࡞ߩ⹏ଔ
ℂᎿቇㇱ ᖱႎ⑼ቇ⑼
ᱞ↰ ᱜਯ
Abstract
ㄭᐕ㧘ࠦࡦࡇࡘ࠲ᛛⴚߩ⊒ዷߦࠃߞߡ⤘ᄢߥ࠺࠲ࠍኈᤃߦᛒ߃ࠆࠃ߁ߦߥࠅ㧘ታߩ
ࡀ࠶࠻ࡢࠢߩᢙℂ⸃ᨆ߇ᵴ⊒ߦⴕࠊࠇߡࠆ㧚⚻ᷣߩ‛ᵹࠪࠬ࠹ࡓ߿વᨴ∛ߩવ㧘↢
ᘒ♽ߥߤታ␠ળߩ⽎ߩ⢛ᓟߦߪࡀ࠶࠻ࡢࠢ߇㑐ࠊߞߡ߅ࠅ㧘ࡀ࠶࠻ࡢࠢߩ᭴ㅧ߇
ਈ߃ࠆᓇ㗀ߦߟߡᄙߊߩ⎇ⓥ߇ߐࠇߡࠆ㧚ⶄ㔀ࡀ࠶࠻ࡢࠢߩಽ㊁ߢߪ㧘ታߩࡀ࠶
࠻ࡢࠢߩᕈ⾰ࠍᜬߟࡀ࠶࠻ࡢࠢࡕ࠺࡞߇ឭ᩺ߐࠇ㧘ࡀ࠶࠻ࡢࠢࡕ࠺࡞ࠍ↪ࠆߎߣ
ߦࠃࠅ㧘ಽᨆ߇࿎㔍ߢࠆታߩࡀ࠶࠻ࡢࠢߩ⽎ߦኻߒߡᣂߒ⊒ࠍᓧࠆߎߣ߇ᦼ
ᓙߐࠇߡࠆ㧚
߹ߚ㧘SNS(Social Networking Service)߿ࡉࡠࠣ߇ Web ߩࠦࡒࡘ࠾ࠤ࡚ࠪࡦࠨࡆࠬ
ߣߒߡᵈ⋡ࠍ㓸㧘SNS ࠍኻ⽎ߦߒߚࡀ࠶࠻ࡢࠢಽᨆ߇ⴕࠊࠇ㧘SNS ߩ࡙ࠩ߇ᒻᚑߔ
ࠆࡀ࠶࠻ࡢࠢࠍߔ᭽߇࡞࠺ࡕࠢࡢ࠻࠶ࡀߥޘឭ᩺ߐࠇߡࠆ㧚SNS ߩࡀ࠶࠻ࡢࠢ↢
ᚑේℂߩℂ⸃ߪ㒢ቯ⊛ߢߪࠆ߇ታߩੱ㑆㑐ଥࠍߔࠆน⢻ᕈ߇ࠆ㧚
SNS ߩࡀ࠶࠻ࡢࠢࠍߔࠆࡕ࠺࡞ߪߊߟ߆ឭ᩺ߐࠇߡࠆ߇㧘ᓥ᧪ߩᄙߊߩ
SNS ߪ⚫ޟߣޠ߫ࠇࠆ․ᓽࠍᜬߟ㧚ߒ߆ߒߥ߇ࠄ㧘ㄭᐕ⊓ޟ㍳ߣޠ߫ࠇࠆᒻᘒߩ
SNS ߇Ⴧടߒ㧘ᓥ᧪ߩࡀ࠶࠻ࡢࠢࡕ࠺࡞ߢߪ㧘ߎߩ⊓㍳ SNS ߩࡀ࠶࠻ࡢࠢࠍߔ
ࠆߦߪਇචಽߢࠆน⢻ᕈ߇ࠆ㧚ߘߎߢᧄ⎇ⓥߢߪ㧘ᣢሽߩࡀ࠶࠻ࡢࠢࡕ࠺࡞ߦኻߒߡ
ታߔࠆ⚫㧘⊓㍳ߩ SNS ߩࡀ࠶࠻ࡢࠢࠍᲧセߒ㧘ᣂߚߦ⊓㍳ߩࡀ࠶࠻ࡢࠢࡕ࠺
࡞ࠍߔࠆᣂߒࡕ࠺࡞ࠍឭ᩺ߒ㧘⹏ଔߔࠆ㧚
ౕ⊛ߦߪ⚫ߣ⊓㍳ߩ SNS ࠍ 3 ߟ↪ߡ㧘ឭ᩺ߐࠇߡࠆࡀ࠶࠻ࡢࠢࡕ࠺࡞
㧔CNN㧘CNNR㧘PFT ࡕ࠺࡞㧕ߩᲧセࠍⴕߞߚ㧚ߘߩ⚿ᨐ㧘PFT ࡕ࠺࡞߇৻⇟ SNS ࠍ
ߢ߈ߡࠆࡕ࠺࡞ߣ⚿⺰ߢ߈ߚ㧚ߒ߆ߒ㧘ߤߩࡀ࠶࠻ࡢࠢࡕ࠺࡞ߢ߽㧘SNS ߦᣂⷙෳടߔ
ࠆࡁ࠼ߪᣢሽߩࡁ࠼ߣࡦࠢࠍ⚿߱ߚ㧘⊓㍳ߩ SNS ࠍߔߎߣߪߢ߈ߕ㧘SNS ߩ
․ᓽࠍߔࡀ࠶࠻ ࡢࠢࡕ࠺࡞ߪሽߒߥ߆ߞߚ㧚ߘߎߢ㧘PFT ࡕ࠺࡞ߦᒛࠍട߃ߚࡕ
࠺࡞ࠍឭ᩺ߒ㧘ߎߩឭ᩺ᚻᴺߢ↢ᚑߒߚࡀ࠶࠻ࡢࠢ߇ઁߩࡀ࠶࠻ࡢࠢࡕ࠺࡞ࠃࠅ߽ታ㓙
ߩ⊓㍳ SNS ߩࡀ࠶࠻ࡢࠢߦㄭߎߣࠍ␜ߒߚ㧚
− 99 −
࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢಽᨆߩ
࡙ࠩ߳ߩᓇ㗀ߣవⴕᕈߦࠃࠆ⸃
ℂᎿቇㇱ ᖱႎ⑼ቇ⑼
ᱞ↰ ᱜਯ
Abstract
Web ߦ߅ߌࠆᖱႎ㊂ߪ㧘ㄭᐕߩ Blog ߿ SNS㧘ࡑࠗࠢࡠࡉࡠࠣ╬ߩ CGM(Cunsumer
Generated Media)ߩᶐㅘߦࠃߞߡ߹ߔ߹ߔჇടߒ㧘ߘߩ⤘ᄢߥᖱႎߩਛ߆ࠄ↪ߥᖱႎࠍ
ߟߌߔߎߣ߇᳞ࠄࠇߡࠆ㧚ᣣߺ↢ޘߐࠇࠆᖱႎߩਛߦߪ㧘࡙ࠩߩ༵ᅢ㧘⥝
ߦߟߥ߇ࠆᖱႎ߽ᄙߊ㧘ߎࠇࠄߩಽᨆ߿㧘ද⺞ࡈࠖ࡞࠲ࡦ ࠣ╬ࠍ↪ߚផ⮈ߩᛛⴚߪ
ᣣ⎇ޘⓥ߇ⴕࠊࠇߡࠆ㧚
࠰ࠪࡖ࡞ࡉ࠶ࠢࡑ߽ࠢ㧘࡙ࠩߩ༵ᅢ㧘⥝ࠍᤋߒߚᖱႎࠍᜬߟ Web ࠨࡆࠬߩ৻
ߟߢࠆ㧚࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߢߪ㧘࡙ࠩߪ⥝ࠍᜬߞߚ Web ࡍࠫߦኻߒ㧘࠲ࠣߣ
߫ࠇࠆ⥄↱ߥᢥሼߢߢ߈ࠆಽ㘃ฬࠍਈ߃㧘ࡉ࠶ࠢࡑࠢߣߒߡ㐿㧘ߔࠆߎߣ
߇ߢ߈ࠆ㧚࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߦࡉ࠶ࠢࡑࠢߐࠇߚ Web ࡍࠫߪ㧘࡙ࠩߦࠃߞߡ♖
ᩏߐࠇ㧘ࠄ߆ߩଔ୯߇ࠄࠇߚ Web ࡍࠫߢࠆߣ⠨߃ࠄࠇࠆߚ㧘⋉ߥ Web ࡍ
ࠫߩᖱႎࠍᄙߊᜬߟࠨࡆࠬߣߒߡ↪⠪ࠍ₪ᓧߒߡࠆ㧚߹ߚ⎇ⓥ㗔ၞ߆ࠄ߽㧘หߓℂ↱
ߦട߃㧘࡙ࠩߦ⚌ߠߌࠄࠇߚ༵ᅢ㧘⥝ߩᖱႎࠍᄙߊᜬߟࠨࡆࠬߢࠆߎߣ߆ࠄᵈ⋡ߐ
ࠇ㧘࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢࠍ↪ߒߚ⎇ⓥ߇⋓ࠎߦⴕࠊࠇߡࠆ㧚
ផ⮈ߩᚻᴺߦࠃߊ↪ࠄࠇࠆද⺞ࡈࠖ࡞࠲ࡦࠣߢߪ㧘 ࡙ࠩߩ༵ᅢߦว߁ࠕࠗ࠹ࡓࠍ
ߟߌࠆߚ㧘༵ᅢߩㄭߒ࡙ࠩ߇ᅢࠕࠗ࠹ࡓࠍឭ␜ߔࠆ㧚ߎࠇࠍ࠰ࠪࡖ࡞ࡉ࠶ࠢࡑ
ࠢߦㆡ↪ߔࠆ႐ว㧘࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߦ߅ߌࠆࡉ࠶ࠢࡑࠢ߇㧘ᐢᄢߥ Web ⓨ㑆߆
ࠄ⊒ߒߚ Web ࡍࠫߦଔ୯ࠍ㧘ㆬᛯߔࠆⴕὑߛߣ⸒߃ࠆߚ㧘 ༵ᅢߣߪࡉ࠶ࠢࡑ
ࠢߒߚ Web ࡍࠫߩะߣߥࠆ㧚ߚߛߒ㧘ᄙߊߩ࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߢߪ㧘৻ቯᦼ㑆ߢ
ߩࡉ࠶ࠢࡑࠢᢙߦࠃࠆࡦࠠࡦࠣ╬ߩ◲නߥࡉ࠶ࠢࡑࠢಽᨆࠍឭଏߒ㧘࡙ࠩߩଢᕈ
ࠍ㜞ߡࠆ㧚
ߎ߁ߒߚᖱႎߩឭଏ߇࡙ࠩߩࡉ࠶ࠢࡑࠢߦᓇ㗀ߔࠆߎߣߪචಽߦ⠨߃ࠄࠇࠆ㧚Cho ࠄ
ߩႎ๔ߦࠃࠆߣ㧘ᬌ⚝ࠛࡦࠫࡦߦ߅ߡߦࡦࠠࡦࠣߐࠇࠆࡍࠫߪ㧘ࠃߊੱߩ⋡ߦߟ
ߊߎߣࠍℂ↱ߣߒߡ㧘㜞㗅ࠍᜬߔࠆะߦࠆߣߐࠇߡࠆ㧚ห᭽ߩߎߣ߇࠰ࠪࡖ
࡞ࡉ࠶ࠢࡑࠢߢ߽⊒↢ߔࠆߎߣ߇⠨߃ࠄࠇࠆ㧚ߔߥࠊߜ㧘ࡦࠠࡦࠣߦឭ␜ߐࠇ㧘ࠃߊੱ
ߩ⋡ߦߟߊࠃ߁ߦߥߞߚ Web ࡍࠫ߇ࠃߊࡉ࠶ࠢࡑࠢߐࠇࠆࠃ߁ߦߥࠅ㧘ᄙߊߩࡉ࠶ࠢࡑ
ࠢࠍ㓸ࠆߣ߁᭴ㅧߢࠆ㧚ߘߒߡหᤨߦ㧘ࠃߊ⋡ߦߟߊࡦࠠࡦࠣߦ⊓႐ߔࠆ Web ࡍ
ࠫ߫߆ࠅࠍఝవ⊛ߦࡉ࠶ࠢࡑࠢߔࠆ࡙ࠩߩ⊓႐੍߽ߐࠇࠆ㧚
࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߦද⺞ࡈࠖ࡞࠲ࡦࠣࠍㆡ↪ߔࠆ㓙ߦ↪ߔࠆ༵ᅢᖱႎߪ㧘Web
− 100 −
ⓨ㑆߆ࠄߩ⥝㧘༵ᅢߦ߆ߥߞߚᖱႎߩㆬᛯߢࠅ㧘ផ⮈ߔࠆࠕࠗ࠹ࡓߩᩮߪ㧘ㄭߒ༵
ᅢߩ࡙ࠩߦ♖ᩏߐࠇ㧘ଔ୯ࠍࠄࠇߚ Web ࡍࠫߢࠆߎߣߦࠆ㧚ࡦࠠࡦࠣ߆ࠄㆬ
ᛯߔࠆ࡙ࠩߩⴕേߪ㧘ᔅߕߒ߽ߘࠇߦว⥌ߔࠆ߽ߩߢߪߥߊ㧘หߦᛒࠊࠇࠆߴ߈߽ߩߢ
ߪߥ㧚
ߘߎߢᧄ⎇ⓥߢߪ㧘࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߦ߅ߌࠆ࡙ࠩߩᝄࠆ⥰ߦߟߡ㧘ߘ߁ߒ
ߚࡉ࠶ࠢࡑࠢᢙࡦࠠ ࡦࠣߩࠃ߁ߥ࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢಽᨆߩᓇ㗀ࠍតࠅ㧘߹ߚߘ߁
ߒߚ࡙ࠩߩᓇ㗀ߦኻಣߔࠆߚ㧘ࠃࠅᣧᦼߦࡉ࠶ࠢࡑࠢߔࠆ࡙ࠩࠍఝߦ⹏ଔߔࠆ㧘
వⴕᕈߣ߁ᜰᮡࠍዉߔࠆߎߣࠍⴕߞߚ㧚
ᧄ⎇ⓥߢߪ㧘࠰ࠪࡖ࡞ࡉ࠶ࠢࡑࠢߢߩ࡙ࠩߩᝄࠆ⥰ߦߟߡ㧘࠰ࠪࡖ࡞ࡉ࠶
ࠢࡑࠢߩᜬߟࡉ࠶ࠢࡑࠢಽᨆߦᄢ߈ߥᓇ㗀ࠍฃߌࠆ࡙ࠩߩሽࠍ␜ߒߚ㧚߹ߚ㧘ߘࠇ
߇ද⺞ࡈࠖ࡞࠲ࡦࠣ╬ߩ⸘▚ߦᓇ㗀ߒ߁ࠆߎߣࠍ␜ߒߚ㧚ᓐࠄߪ৻⒳ߩߪߕࠇ୯ߩࠃ߁ߥ
࡙ࠩߢࠆ߇㧘ᓐࠄࠍឃ㒰ߔࠆߚߦߪ㧘࡙ࠩߩࡉ࠶ࠢࡑࠢઙᢙࠍ᧦ઙߣߒߚ႐วߦ
ߪಽᨆኻ⽎ߩ࡙ࠩࠍඨᷫߐߖࠆ⒟ᐲߩ᧦ઙࠍ߅߆ߥߌࠇ߫ߥࠄߕ㧘߹ߚ URL ߦ㑐ߒߡ᧦
ઙࠍਈ߃ࠆ႐ว㧘ᧄ᧪㘃ૃᐲߩ⸘▚ߦ↪ߢࠆߪߕߩᄙߊߩ࡙ࠩߦࡉ࠶ࠢࡑࠢߐࠇߡ
ࠆ URL ⟲ࠍಽᨆ߆ࠄขࠅ㒰߆ߥߌࠇ߫ߥࠄߥ㧚
ߘߩߚᧄ⎇ⓥߢߪផ⮈ᜰᮡߦవⴕᕈߣ߁ࡉ࠶ࠢࡑࠢߩᤨ㑆⊛೨ᓟ㑐ଥߩⷐ⚛ࠍ⚵ߺ
ㄟߎߣࠍឭ᩺ߒߚ㧚ផ⮈ᜰᮡߩ▚ߦవⴕᕈࠍ⚵ߺㄟߎߣߢ㧘వㅀߒߚߪߕࠇ୯ߦߚ
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ߩߣ⠨߃ࠆ㧚
− 101 −
ᬀ‛ߩ∉ᓮߦ߅ߌࠆᖱႎ⚛ሶߣࠪࠣ࠽࡞ࡀ࠶࠻ࡢࠢ
↢⎇ⓥࠣ࡞ࡊ 㧔ℂᎿቇㇱ ᔕ↪↢‛⑼ቇ⑼ ᵤ⎇ⓥቶ㧕
ᵤᐘ(ᢎ)
᧪㗇ቁశ(✚ว⎇ⓥᯏ᭴ࡊࡠࠫࠚࠢ࠻⎇ⓥຬ)
ự↰᥍ᐽ(D2)
Abstract. 21 ♿ߩੱ㘃߇⋥㕙ߔࠆ㊀ⷐߥ⺖㗴ߢࠆ㘩♳ޔⅣႺࠡ࡞ࡀࠛޔ㗴ߩᧄ
⾰ࠍ⠨߃ࠆߣੱޔ㘃ߪߘߩᄙߊࠍᬀ‛ߦଐሽߒߡࠆߎߣߦ᳇ߠߊޕᬀ‛ߪ↢ޔሽߦㆡ
ߐߥᖡⅣႺ߆ࠄ⒖േߔࠆേ‛ߣ⇣ߥࠅޔ⒖േߒߥ߆ࠊࠅߦᩮࠍਅࠈߒߚ႐ᚲߩⅣ
Ⴚߩᄌൻ߿∛ේߥߤߩᄖᢜࠍ⚛ᣧߊᗵ⍮ߒߩࠄ⥄ޔࠍࠅᦧ߃ࠆߎߣߦࠃߞߡㆡ
ᔕߔࠆ⢻ജࠍ⊒㆐ߐߖߡ᧪ߚޕㄭᐕޔฦ↢‛ߩ⸳⸘࿑ߦ⋧ᒰߔࠆࠥࡁࡓᖱႎ߇ᰴ⸃ߣޘ
⺒ߐࠇࠍࠄࠇߘޔᲧセߔࠆߎߣߦࠃࠅ‛↢ޔㅴൻߩㆊ⒟ߢߩฦ↢‛ߩᚢ⇛ࠍ〔ߠߌࠆߎ
ߣ߇ߢ߈ࠆࠃ߁ߦߥߞߡ᧪ߚ⚦ޕ⢩ౝߩᖱႎવ㆐ߦ㑐ਈߔࠆ࿃ሶࠍേᬀ‛㑆ߢᲧセߔࠆ
ߣࠆߥ⇣߇ࠇߙࠇߘޔㆮવሶ⟲ࠍᄙ᭽ൻߐߖߡ᧪ߚߎߣ߇ࠊ߆ࠆޕᄙ᭽ߥⅣႺᄌൻࠍኤ
⍮ߔࠆ⢻ജࠍ⊒㆐ߐߖߡ᧪ߚᬀ‛߇₪ᓧߒߚ⁛⥄ߩᖱႎવ㆐ᯏ᭴ࠍ⎇ⓥߔࠆߎߣߪޔၮ
␆ޔᔕ↪ߩਔ㕙߆ࠄᓟ߹ߔ߹ߔ㊀ⷐᕈࠍჇߔߣ⠨߃ࠄࠇࠆޕ
1. ᖱႎ⚛ሶߣߒߡߩࠞ࡞ࠪ࠙ࡓࠗࠝࡦ
↢‛ߪޔฃኈߣ߫ࠇࠆ࠲ࡦࡄࠢ⾰ߢᄖ⇇ߩᖱႎࠍ⼂ߒ৻ޔㅪߩᖱႎવ㆐ᯏ᭴ࠍߒߡޔㆮ
વሶ⊒ߘߩઁߩ⚦⢩ᯏ⢻ࠍᓮߔࠆޕᖱႎવ㆐ߩㆊ⒟ߢߪ㉄ࡦߩ⾰ࠢࡄࡦ࠲ޔൻ߿࠲ࡦࡄࠢ
⾰ห჻ߩ⋧↪ߣਗࠎߢޔੑᰴࡔ࠶ࡦࠫࡖߣ✚⒓ߐࠇࠆૐಽሶ‛⾰߿ࠗࠝࡦ߇㊀ⷐߥ߈
ࠍᜂ߁ޔ῎ੇޕૐ᷷ߥߤߩⅣႺࠬ࠻ࠬޔធ⸅ೝỗ∛ޔේߩᗵᨴ߿ᓸ↢‛ߩ↢ޔᬀ‛ࡎ࡞ࡕ
ࡦߦኻߔࠆᔕ╵ߥߤޔᄙ⒳ᄙ᭽ߥೝỗߩવ㆐ߦ(ࡦࠝࠗࡓ࠙ࠪ࡞ࠞޔCa2+)߇ᖱႎવ㆐ߦ㑐ਈߔࠆޕ
↢‛߇ߐ߹ߑ߹ߥⅣႺᄌൻࠬ࠻ࠬߥߤࠍ㑐⍮ߒޔㆡᔕߔࠆㆊ⒟ߢߪ⚦ޔ⢩ౝߩ Ca2+Ớᐲ߇ᄌ
േߒ߇ࠇߎޔᖱႎવ㆐ߩᜂᚻ㧔⚛ሶ㧕ߣߒߡᯏ⢻ߔࠆޕ
⚦⢩ౝߩࠆ‛⾰߇ᖱႎࠍᜂ߁ߚߦߪޔᐔᏱᤨߦỚᐲ߇ૐߊߚࠇޔೝỗߦᔕ╵
ߒߡỚᐲ߇㜞߹ࠅޔೝỗ߇વࠊࠆߣౣ߮Ớᐲࠍૐߊߔࠆᔅⷐ߇ࠆ⚦ޕ⢩⾰ Ca2+Ớᐲ ߪޔᐔ
Ᏹᤨߦߪޔ10-100 nM ⒟ᐲߦ⛽ᜬߐࠇࠆ৻ᣇ⚦ޔ⢩ᄖߩỚᐲߪ mM ࡌ࡞ߢ⚦ޔ⢩⤑ౝᄖߦ
ߪᢙਁ⒟ᐲߩ Ca2+ߩỚᐲ൨㈩߇ሽߔࠆޕᬀ‛⚦⢩ߩⓍߩᄢㇱಽࠍභࠆᶧ⢩߿ޔዊ⢩
(ER)╬ߩ⚦⢩ౝዊེቭߪޔౝㇱߦ⚦⢩ᄖߣห᭽ߩࡌ࡞ߩ Ca2+ࠍᜬߜ⚦ޔ⢩ౝ Ca2+⾂⬿ᐶ
ߣߒߡᯏ⢻ߔࠆ✛⪲ޔࠕ࠼ࡦࠦ࠻ࡒߦࠄߐޕౝߦ߽㜞Ớᐲߩ Ca2+߇ሽߔࠆ⚿ߩߘޕᨐޔ
⤑ࠍ⽾ㅢߒߡሽߔࠆ Ca2+࠴ࡖࡀ࡞࠲ࡦࡄࠢ⾰ࠍߒߡ⚦⢩ᄖ߿⚦⢩ౝዊེቭ߆ࠄ⚦⢩⾰ߦ
ࠊߕ߆ߥ㊂ߩ Ca2+߇ᵹߔࠆߛߌߢޔᄢ߈ߥ Ca2+ߩᄌേ߇⺃ዉߐࠇᖱႎ⚛ሶߣߒߡߊߎߣ
ࠍน⢻ߦߒߡࠆޕ
ߘࠇߙࠇߩೝỗߦኻߒߡޔᄙ᭽ߥ Ca2+࠴ࡖࡀ࡞ߩᓮߦࠃࠅ․ޔᓽ⊛ߥᤨ㑆⊛ⓨ
㑆⊛ࡄ࠲ࡦࠍ␜ߔ⚦⢩ౝ Ca2+Ớᐲᄌേ߇ⷰኤߐࠇࠆޔߪߢࠇߘޕCa2+Ớᐲᄌൻߪ⚦ޔ⢩ౝ
ߢߤߩࠃ߁ߦಽߌࠄࠇޔᖱႎ⚛ሶߣߒߡ⼂ߐࠇࠆߩߛࠈ߁߆?
ࡕ࠺࡞ᬀ‛ࠪࡠࠗ࠽࠭࠽߿ࠗࡀߩోࠥࡁࡓᖱႎ߇⸃⺒ߐࠇߚ⚿ᨐޔᬀ‛ߦߪޔ㕖Ᏹߦᄙ
ߊߩ⒳㘃ߩ Ca2+⚿ว࠲ࡦࡄࠢ⾰߇ሽߔࠆߎߣ߇ࠄ߆ߣߥߞߚߩࠄࠇߎޕᄙߊߪ Ca2+Ớᐲ
ᄌൻߩࡦࠨߣߒߡ⚦ޔ⢩⾰ Ca2+Ớᐲᄌൻߩᤨⓨ㑆ࡄ࠲ࡦߩᜬߟᖱႎࠍ⸃⺒ߒޔᖱႎࠍ
ਅᵹߦᱜ⏕ߦવ߃ࠆᯏ⢻ࠍᜂ߁ߣ⠨߃ࠄࠇࠆޕ
2. ᬀ‛ߩ∉ᔕ╵
ᬀ‛ߪ ߪࠆ࠲ࠪࠛޔPAMP/MAMP (pathogen/microbe associated molecular pattern)
ߣ߫ࠇࠆ∛ේߩᚑಽߦኻߔࠆฃኈ࠲ࡦࡄࠢ⾰ࠍᜬߜ∛ޔේᓸ↢‛ߩᗵᨴࠍ⼂ߔࠆ⼂ޕ
− 102 −
ᓟᢙಽએౝߦ⚦ޔ⢩⤑ߩ⤑㔚ᄌൻ⚦ޔ⢩⤑ࠍߒߚ Ca2+ߩᵹ)⚦ޔ⢩ౝᄖߩ pH ᄌൻޔᵴᕈ㉄
⚛ ⒳ (reactive oxygen species:ROS) ߩ ↢ ᚑ ㉄ ࡦ ⾰ ࠢ ࡄ ࡦ ࠲ ޔൻ ㉂ ⚛ MAP ࠠ ࠽
(mitogen-activated protein kinase:MAPK)ߩᵴᕈൻߥߤ৻ㅪߩᖱႎવ㆐ᔕ߇⺃ዉߐࠇࠆ⚳ᦨޕ
⊛ߦ᛫⩶ᕈ‛⾰ߩ⫾Ⓧߥߤ߇ߎࠅޔੑᰴ⊛ߥᗵᨴࠍ㒐ߋߣߦޔᗵᨴㇱዪᚲ⊛ߥ∛ේࠍ
ㅪࠇߦߒߚ⚦⢩ߩ⥄Ვ(ࡊࡠࠣࡓ⚦⢩ᱫ)ࠍᒁ߈ߎߔ(࿑ 1)ޕ
࿑ 1 ∛ේߩᗵᨴߦኻߔࠆᬀ‛ߩ∉ᔕ
ᗵᨴㇱߢ∛ේࠍㅪࠇߦߒߚዪᚲ⊛ߥ⚦⢩ߩ⥄Ვ߇⺃ዉߐࠇࠆߣหᤨߦޔ࿐ߢ᛫⩶ᕈ‛⾰ߩวᚑ
߿⚦ޔ⢩ოߩᒝൻ߇⺃ዉߐࠇ∛ޔේߩోり߳ߩᗵᨴࠍኽߓㄟࠆޕ
ᬀ‛߇⺃ዉߔࠆ∉ᔕߩᄙߊߪ⚦ޔ⢩ౝ߳ߩ Ca2+ߩᵹࠍᛥߔࠆߎߣߦࠃࠅޔ
ᒝߊᛥ߃ࠄࠇࠆ⚦ޕ⢩ᄖ߆ࠄ⚦⢩⾰߳ߩ Ca2+ᵹ߇ޔᬀ‛ߩ㒐ᓮᔕ╵ࠍ⺃ዉߔࠆߚߦ㊀ⷐ
ߥᓎഀࠍᨐߚߔߣ⠨߃ࠄࠇࠆޕ
࿑ 2 ᬀ‛∉ߩࡕ࠺࡞ታ㛎♽ߦᵴ↪ߒߚࠗࡀߩၭ
㙃⚦⢩
(Ꮐ)✛⦡Ⱟశ࠲ࡦࡄࠢ⾰(GFP)ࠍ⊒ߐ
ߖߚࠗࡀၭ㙃⚦⢩ߩⰯశ㗼ᓸ㏜
(ฝ)ၭ㙃ߒޔჇᱺߐߖߡࠆࠗࡀၭ㙃⚦⢩
ታ㓙ߩᬀ‛ߢߪ৻ޔㇱߩ⚦⢩ߩߺߢޔㇱߦࠃࠅ⇣ߥࠆ∉ᔕ╵߇⺃ዉߐࠇࠆߚ
∉ޔᔕ╵ࠍߎߔ⚦⢩ߩਛߢ⺃ዉߐࠇࠆᔕࠍ⚦ߦ⎇ⓥߔࠆߎߣ߇㔍ߒ⑳ߢߎߘޕ
ߚߜߪޔᬀ‛ߩ⚦⢩ࠍၭ㙃ߒޔဋ৻ൻߒߚ⚦⢩♽(࿑ 2)ࠍ↪ߡ∛ޔේ↱᧪ߩࠪࠣ࠽࡞ಽሶ
ࠍਈ߃ߚ㓙ߩᔕࠍ⸃ᨆߔࠆࡕ࠺࡞ታ㛎♽ࠍ㐿⊒ߒޔᬀ‛ߩ∉ࠪࠬ࠹ࡓߩ⸃ࠍㅴߡ
ࠆߩߎޕታ㛎♽ߪ⚦ޔ⢩ߦㆮવሶࠍዉߒߡߘߩᯏ⢻ࠍ⸃ᨆߔࠆߎߣ߽ኈᤃߢޔᗵᨴࠪࠣ࠽
࡞ಽሶࠍ⼂ߔࠆߎߣߦࠃࠅ⺃ޔዉߐࠇࠆࡊࡠࠣࡓ⚦⢩ᱫ߿ޔ᛫⩶ᕈ‛⾰ߩวᚑޔROS ߩ
⊒↢ߥߤޔᬀ‛߇ᒁ߈ߎߔ∉ᔕߩᄙߊࠍ⚦⢩ޔಽሶߩࡌ࡞ߢ⸃ᨆߢ߈ࠆޕ
એਅߦ∛ޔේ↱᧪ߩࠪࠣ࠽࡞ಽሶߦኻߔࠆၭ㙃⚦⢩ߩᔕߩ⸃ᨆ߆ࠄ⸃ߐࠇߚޔ
⚦⢩⾰ Ca2+Ớᐲᄌൻߩᤨⓨ㑆ࡄ࠲ࡦߩᒻᚑߣޔCa2+ߦࠃߞߡᜂࠊࠇߚᖱႎߩ Ca2+ࡦࠨ
ߦࠃࠆ⸃⺒ߦ㊀ὐࠍ⟎ߡޔᬀ‛ߩ∉ᔕ╵ߦ߅ߌࠆ Ca2+ߩᓎഀߦߟߡ⠨ኤߔࠆޕ
3. ⚦⢩ౝࠞ࡞ࠪ࠙ࡓࠗࠝࡦỚᐲᄌൻߩᤨⓨ㑆ࡄ࠲ࡦ
2008 ᐕߦࡁࡌ࡞ൻቇ⾨ࠍฃ⾨ߒߚਅ⣑ඳ჻ߩ⎇ⓥߦࠃࠇ߫߇ࠥࠢࡦࡢࠝޔశࠆߩߪ‛ޔ
ℂ⊛ߥೝỗߦᔕߓߡࠢࠥߩ⚦⢩ౝ Ca2+Ớᐲ߇ߒޔCa2+ᗵฃᕈ⊒శ࠲ࡦࡄࠢ⾰ࠗࠢࠝࡦ߇
శࠍߔߚߢࠆޕฬߥ GFP ߪߣࡦࠝࠢࠗޔද⺞⊛ߦߊⰯశ࠲ࡦࡄࠢ⾰ߢࠆߚ⑳ޕ
− 103 −
ߜߪࠄ߆ࠥࠢޔන㔌ߒߚࠗࠢࠝࡦㆮવሶࠍޔㆮવሶᎿቇ⊛ᚻᴺߦࠃࠅߐ߹ߑ߹ߥᬀ‛⚦⢩ౝ
ߢ⊒ߐߖߡ⚦ޔ⢩ౝߩ Ca2+Ớᐲᄌൻࠍ⸃ᨆߔࠆታ㛎♽ࠍ᭴▽ߒߚ∛ޕේ↱᧪ߩߐ߹ߑ߹ߥࠛ
ࠪ࠲(PAMP/MAMP)ࠍ⼂ߔࠆߣޔᄙ᭽ߥࡄ࠲ࡦߩ⚦⢩⾰ Ca2+Ớᐲᄌൻ߇ᒁ߈ߎߐࠇ
ࠆޕ
࠲ࡃࠦߩၭ㙃⚦⢩ BY-2 ᩣߪ∛ޔේᕈෆ⩶ Phytophthora cryptogea ↱᧪ߩ࠲ࡦࡄࠢ⾰
ࠢࡊ࠻ࠥࠗࡦࠍ㑐⍮ߒߡ⚦ࡓࠣࡠࡊޔ⢩ᱫࠍ߁∉ᔕ╵ࠍ⺃ዉߔࠆࡦࠗࠥ࠻ࡊࠢޕ
ಣℂᓟ⚂ 1 ಽ߆ࠄ৻ޔㆊᕈ߆ߟੑ⋧ᕈߩ⚦⢩⾰ Ca2+Ớᐲᄌൻ߇⺃ዉߐࠇ(࿑ 3 Ꮐ)ߦࠇߘޔᒁ
߈⛯ߡ ROS ߩ↢ᚑ߇ⷰኤߐࠇࠆ⮎ޕℂቇ⊛ߥ⸃ᨆ߆ࠄ⚦ߪ⋡⋧৻ޔ⢩ᄖ߆ࠄߩ Ca2+ᵹޔ
ੑ⋧⋡ߪ⚦⢩ౝ Ca2+⾂⬿ེቭ↱᧪ߩ Ca2+ߣ⠨߃ࠄࠇࠆ৻ޕᣇࡀࠗޔၭ㙃⚦⢩ߦߜ߽ޔ
∛⩶↱᧪ߩࠝࠧ♧ࠛࠪ࠲ࠍᷝടߔࠆߣޔࠅߥ⇣ߪߣࡦࠗࠥ࠻ࡊࠢޔಣℂᓟ⚂ 12 ⑽⒟
ᐲ߆ࠄ৻ޔㆊᕈߩ⚦⢩⾰ Ca2+Ớᐲᄌൻ߇ⷰኤߐࠇࠆ(࿑ 3 ฝ)ޕᗵᨴࠪࠣ࠽࡞ߩ⒳㘃ߦࠃࠅޔ
⚦⢩⾰ Ca2+Ớᐲᄌൻߩᤨⓨ㑆ࡄ࠲ࡦߦᄙ᭽ᕈ߇ࠄࠇࠆߎߣ߆ࠄޔᬀ‛∉ᔕ╵ㆊ⒟ߦ߅
ߌࠆ Ca2+േຬߦߪ⚦ޔ⢩⤑ࠍߒߚ Ca2+ᵹߣ⚦⢩ౝዊེቭ߆ࠄߩ Ca2+ࠍⶄᢙߩ
Ca2+േຬᯏ᭴߇㑐ਈߒⶄޔ㔀ߥᓮࠍฃߌߡࠆߎߣ߇ࠊ߆ࠆޕ
࿑ 3 ࠗࠢࠝࡦࠍ↪ߚᬀ‛⚦⢩ߦ߅ߌࠆ⚦⢩ౝ Ca2+Ớᐲᄌൻߩ᷹ቯ
(Ꮐ)࠲ࡃࠦၭ㙃⚦⢩ BY-2 ߦ߅ߡࡦࠗࠥ࠻ࡊࠢޔಣℂߒߚ㓙ߦ⺃ዉߐࠇࠆ⚦⢩ౝߩ Ca2+Ớᐲᄌൻ (ޕฝ)
ࠗࡀၭ㙃⚦⢩ߦ߅ߡࠍ࠲ࠪࠛ♧ࠧࠝޔಣℂߒߚ㓙ߦ⺃ዉߐࠇࠆ⚦⢩ౝߩ Ca2+Ớᐲᄌൻޕ
4㧚ࠞ࡞ࠪ࠙ࡓࠗࠝࡦࠍャㅍߔࠆ࠴ࡖࡀ࡞࠲ࡦࡄࠢ⾰
⚦⢩⾰ Ca2+Ớᐲᄌൻߩᤨⓨ㑆ࡄ࠲ࡦߪ⚦ޔ⢩⤑߿⚦⢩ౝዊེቭߦሽߔࠆ Ca2+࠴ࡖࡀ࡞࠲ࡦ
ࡄࠢ⾰߇ᵴᕈൻߐࠇޔCa2+ߩỚᐲ(㔚᳇ൻቇࡐ࠹ࡦࠪࡖ࡞)൨㈩ߦᓥߞߡ Ca2+߇⚦⢩⾰ౝߦേຬߐ
ࠇࠆߎߣߦࠃࠅᒻᚑߐࠇࠆޕ
࿑ 4 TPC ဳࠗࠝࡦ࠴ࡖࡀ࡞ࠍߒߚ∉ᔕ╵ߩᓮ
⌀⩶↱᧪ߩ࠲ࡦࡄࠢ⾰ TvX ߦࠃࠅ⺃ዉߐࠇ
ࠆࠗࡀၭ㙃⚦⢩ߩ∉ᔕ╵ߦ߅ߌࠆ OsTPC1 ߩᓎഀޕ
OsTPC1 ᯏ⢻⎕უᩣ(Ostpc1)ߢߪޔTvX ߦࠃࠅ⺃ዉߐࠇ
ࠆࡊࡠࠣࡓ⚦⢩ᱫ߇ޔ㊁↢ဳߦᲧߴߡ㗼⪺ߦᛥߐࠇ
ࠆ ޕ
− 104 −
∉ᔕ╵ߩೋᦼㆊ⒟ߢߪޔCa2+േຬߣਗࠎߢ⚻⚦⢩ߩᵴേ㔚ߣૃߚᵄᒻߩ⤑㔚ߩ
⣕ಽᭂ߇⺃ዉߐࠇࠆߎߣ߆ࠄ⤑ޔ㔚ᄌൻߦࠃࠅᵴᕈ߇ᓮߐࠇࠆ Ca2+࠴ࡖࡀ࡞߇㑐ਈߔࠆน⢻
ᕈ߇ᗐߐࠇࠆޕᬀ‛ߦߪޔേ‛ߩ⚻♽ߥߤߢ㊀ⷐߥᓎഀࠍᨐߚߔ L ဳ㔚ଐሽᕈ Ca2+࠴ࡖ
ࡀ࡞ߣ㘃ૃߩ࠲ࡦࡄࠢ⾰߿ㆮવሶߪሽߒߥߦࠇߎޕኻߒߡ TPC (two-pore channel)ࡈࠔࡒ
㓁ࠗࠝࡦ࠴ࡖࡀ࡞ߪູޔേ‛ߣᐢ▸ߥᬀ‛ߩᣇߦሽߔࠆޕ
࠲ࡃࠦ TPC1 ㆮવሶߩ⊒ࠍᛥߒߚၭ㙃⚦⢩ߢߪ⺃ࡦࠗࠥ࠻ࡊࠢޔዉᕈߩ⚦⢩
2+
⾰ Ca Ớᐲᄌൻ߿ࡊࡠࠣࡓ⚦⢩ᱫ߇㗼⪺ߦᛥߐࠇߡߚ ߩࡀࠗߚ߹ޕTPC1 ㆮવሶ
OsTPC1 ࠍㆊߦ⊒ߐߖߚࠗࡀၭ㙃⚦⢩ߢߪޔᬀ‛∛ේ⩶ Trichoderma viride ↱᧪ߩ xylanase
࠲ࡦࡄࠢ⾰(TvX)ߦኻߔࠆᗵฃᕈ߇㜞߹ࠅ∉ޔᔕ╵߇ଦㅴߐࠇߚ৻ޕᣇࡐࠬࡦ࠻ࡠ࠻ޔ
࠱ࡦߩᝌߦࠃࠅ OsTPC1 ㆮવሶ߇⎕უߐࠇߚᩣߢߪޔㅒߦ TvX ߦኻߔࠆᗵฃᕈ߇㗼⪺ߦૐ
ਅߒ⚦ࡓࠣࡠࡊޔ⢩ᱫߩ⺃ዉ߇ᛥߐࠇߚ(࿑ 4)ޕએ߆ࠄޔTPC ࡈࠔࡒߪᬀ‛ߩ∉
ᔕ╵ߩᓮߦ㑐ਈߔࠆน⢻ᕈ߇⠨߃ࠄࠇࠆޕ
ᬀ‛ߩ∉ᔕ╵ߩ⺃ዉㆊ⒟ߢߪⶄޔᢙ⒳ߩ Ca2+࠴ࡖࡀ࡞ߩද⺞⊛ᓮ߇ᗐቯߐࠇࠆޕ
ࠗࠝࡦ࠴ࡖࡀ࡞ߩᯏ⢻ߩหቯߪޔ㐳⎇ⓥߩᱧผࠍᜬߟേ‛⚦⢩ߢ߽ኈᤃߢߪߥߊޔᣂⷙ࿃
ሶߩหቯߣᯏ⢻⸃ᨆߩദജ߇⛯ߌࠄࠇߡࠆޕᬀ‛ߩ∉ᔕ╵ߦ߅ߡ Ca2+േຬࠍᜂ߁ᧂ⍮
ߩ Ca2+࠴ࡖࡀ࡞ߩหቯߪޔᓟߩ⎇ⓥߩ㊀ⷐߥ⺖㗴ߣߒߡᱷߐࠇߡࠆޕ
5㧚ࠞ࡞ࠪ࠙ࡓࠗࠝࡦ࠴ࡖࡀ࡞ߩᓮᯏ᭴
৻ㆊᕈߩ⚦⢩⾰ Ca2+ Ớᐲᄌൻ߇⺃
ዉߐࠇࠆ႐วߦߪޔCa2+࠴ࡖࡀ࡞߇
ᵴᕈൻߐࠇࠆߣߦߩߘޔᓟ⋥ߜߦ
ਇᵴᕈൻߐࠇࠆᯏ᭴߇ᔅⷐߢࠆޕ
∉ᔕ╵ߦ߅ߡ㊀ⷐߥᓎഀࠍᜂ߁
Ca2+࠴ࡖࡀ࡞ߪߥ߁ࠃߩߤޔᯏ᭴ߦ
ࠃࠅᵴᕈൻਇᵴᕈൻߐࠇࠆߩߛࠈ
߁߆㧫ᦨㄭࡀࠗޔߪߜߚ⑳ޔၭ㙃⚦
⢩ࠍ࠲ࡦࡄࠢ⾰⣕ࡦ㉄ൻ㉂⚛㒖ኂ
ࠞࠢࡦ A ߢ೨ಣℂߔࠆߣޔᗵ
ᨴࠪࠣ࠽࡞⺃ዉᕈ⚦⢩⾰ Ca2+ Ớᐲ
ᄌൻ߇ᒝߊᛥߐࠇࠆߎߣࠍ⊒ߒ
ߚޕᬀ‛∉ᔕ╵ߦ߅ߌࠆ Ca2+ㅘㆊ
ᕈ࠴ࡖࡀ࡞ߩਇᵴᕈൻߦࠢࡄࡦ࠲ޔ
⾰ࡦ㉄ൻᔕ߇㑐ਈߔࠆน⢻ᕈ߇
␜ໂߐࠇࠆߩࠄࠇߎޕಽሶᯏ᭴ߪో
ߊᧂ⍮ߢࠅߪߜߚ⑳ޔޔCa2+
േຬߩᓮߦ㑐ਈߔࠆࡦ㉄ൻ࠲ࡦ
ࡄࠢ⾰ߩหቯࠍ⋡ᜰߒ⎇ޔⓥࠍㅴ
ߡࠆޕ
6㧚Ca2+Ớᐲᄌൻࡄ࠲ࡦߩ⸃⺒
Ca2+࠴ࡖࡀ࡞ࠍߒߡᒻᚑߐࠇࠆ⚦
⢩⾰ Ca2+ Ớᐲᄌൻߩᤨⓨ㑆ࡄ࠲
࿑ 5 ᬀ‛ߦ߅ߌࠆઍ⊛ߥ Ca2+ࡦࠨ࠲ࡦࡄࠢ⾰ߩ᭴ㅧߣ
ᵴᕈൻᯏ᭴ޕ
ࡦߩᜬߟᖱႎߪߦ߁ࠃߩߤޔਅᵹߦ
Ca2+ߣ⚿วߔࠆߎߣߢޔCaM ߪ CaM ଐሽᕈࠠ࠽
વ㆐ߐࠇࠆߩߛࠈ߁߆㧫ᬀ‛ߦߪޔ
(CaMK)߿ઁߩᮡ⊛࠲ࡦࡄࠢ⾰ߣ⚿วߒߘߩᵴᕈࠍ⺞▵ߔࠆ
ࠞ࡞ࡕࠫࡘࡦ(calmodulin:CaM)ޔ
(1) ޕCBL ߪ CIPK ߣ⚿วߒޔᵴᕈൻߔࠆ(2)ޕCDPK ߪߘࠇ⥄
ࠞ࡞ࠪ࠾ࡘࡦ B ᭽࠲ࡦࡄࠢ⾰
りߩ࠲ࡦࡄࠢ⾰ࡦ㉄ൻ㉂⚛ᵴᕈ߇ᵴᕈൻߐࠇࠆ(3)
(calcineurin B-like protein:CBL)ޔ
2+
2+
Ca ଐሽᕈ࠲ࡦࡄࠢ⾰ࡦ㉄ൻ㉂⚛ (Ca -dependent protein kinase:CDPK)ߥߤߩ Ca2+ߣ․⇣
⊛ߦ⚿วߔࠆ㗔ၞ(EF-hand ࠼ࡔࠗࡦ)ࠍᜬߟ࠲ࡦࡄࠢ⾰߇ߘࠇߙࠇᄙ⒳ሽߔࠆޕᵴᕈ㉄⚛ࠍⓍ
− 105 −
ᭂ⊛ߦ↢ᚑߔࠆᬀ‛ߩ㉂⚛߽ Ca2+ߩ⚿วߦࠃࠅ⋥ធᵴᕈൻߐࠇࠆޔߪ⾰ࠢࡄࡦ࠲ߚߒ߁ߎޕCa2+
߇⚿วߔࠆߣ┙ޔ᭴ㅧࠍᄌൻߐߖޔ᭽ߥޘᮡ⊛࠲ࡦࡄࠢ⾰ߩᵴᕈ߿ㆮવሶ⊒ߩᓮࠍⴕ߁ߎ
ߣ߆ࠄޔCa2+ࡦࠨߣߒߡᯏ⢻ಽᜂߒߡࠆߣ⠨߃ࠄࠇࠆ(࿑ 5)ޕ
േ‛ߩࠞ࡞ࠪ࠾ࡘࡦߪޔCa2+ࡦࠨߢࠆ B ࠨࡉ࡙࠾࠶࠻ߣޔ2B ဳ࠲ࡦࡄࠢ⾰
⣕ࡦ㉄ൻ㉂⚛ߢࠆ A ࠨࡉ࡙࠾࠶࠻߆ࠄ᭴ᚑߐࠇߢߤߥ♽∉ޔ㊀ⷐߥᓎഀࠍᨐߚߔࠆޕᬀ‛
ߦߪ ࡦࡘ࠾ࠪ࡞ࠞޔB ߣࠃߊૃߚ CBL ߇ᄙ⒳ሽߔࠆ߇ ࡦࡘ࠾ࠪ࡞ࠞޔA ߩ⋧ห࠲ࡦ
ࡄࠢ⾰߿ 2B ဳ࠲ࡦࡄࠢ⾰⣕ࡦ㉄ൻ㉂⚛ߪߐࠇߡߥޕCBL ߪޔേ‛ߦߪሽߒߥ৻
⟲ߩ࠲ࡦࡄࠢ⾰ࡦ㉄ൻ㉂⚛ CIPK(CBL-interacting protein kinase)ߣ Ca2+ଐሽ⊛ߦ⚿วߒޔᵴ
ᕈൻߔࠆޕ
࿑ 6 ࠗࡀ∉ᔕ╵ࠍᓮߔࠆ Ca2+ᓮဳ࠲ࡦࡄࠢ
⾰ࡦ㉄ൻ㉂⚛ OsCIPK14/15 ࠲ࡦࡄࠢ⾰
(Ꮐ)OsCIPK14/15 ⊒ᛥၭ㙃⚦⢩ߢߪޔ
⌀⩶↱᧪ߩᗵᨴࠪࠣ࠽࡞ಽሶ TvX ߦࠃࠅ⺃ዉߐࠇࠆ
᛫⩶ᕈ‛⾰ߩวᚑ⢻߇ૐਅߔࠆ(ޕฝ) OsCIPK15 ㆊ
⊒ၭ㙃⚦⢩ߢߪޔTvX ⺃ዉᕈߩ᛫⩶ᕈ‛⾰ߩว
ᚑ⢻߇Ⴧᒝߐࠇࠆޕ
ࠪࡠࠗ࠽࠭࠽ߦߪ 10 ⒳ߩ CBL ߣ 25 ⒳ߩ CIPK ߇ሽߒߺ⚵ߩߘޔวࠊߖߦࠃࠅޔ
Ca2+ࠍߒߚᄙ᭽ߥᖱႎવ㆐♽ߦ㑐ਈߔࠆߣ੍ᗐߐࠇߡࠆޕታ㓙்ޔኂޔૐ᷷ޔ῎ੇޔ㜞
Ⴎߦࠃࠅ⺃ዉߐࠇࠆᖱႎવ㆐♽ߦߐ߹ߑ߹ߥ CBL-CIPK ♽߇㊀ⷐߥᓎഀࠍᨐߚߔᦨޕㄭ⑳ߚ
ߜߪޔᬀ‛∉ߩᓮߦ㑐ਈߔࠆ 2 ⒳ߩ CIPK(OsCIPK14,15)ࠍ⊒ߒߚ ߩߎޕCIPK ࠲ࡦࡄࠢ
⾰ ߪ ․ ቯ ߩ CBL ߦ ࠃ ࠅ ࠲ ࡦ ࡄ ࠢ ⾰ ࡦ ㉄ ൻ ᵴ ᕈ ߇ ⺞ ▵ ߐ ࠇ ࠆ ޕRNA ᐓ ᷤ ᴺ ߦ ࠃ ࠅ
OsCIPK14/15 ㆮવሶߩ⊒ࠍᛥߒߚ⚦⢩߿ޔOsCIPK15 ㆮવሶࠍᄢ㊂ߦ⊒ߐߖߚ⚦⢩ߩ⸃
ᨆ߆ࠄޔOsCIPK14,15 ߇ߩࡀࠗޔ᛫⩶ᕈ‛⾰ߩ⫾Ⓧ(࿑ 6)߿⚦ࡓࠣࡠࡊޔ⢩ᱫߥߤߩᄙ᭽ߥ
∉ᔕ╵ߩᓮߦ㊀ⷐߥᓎഀࠍᨐߚߔߎߣ߇ࠄ߆ߣߥߞߚ(࿑ 7) ޕ
࿑ 7 ࠗࡀߩ∉ᔕ╵ߦ߅ߌࠆᖱႎ⚛ሶߣߒߡߩ Ca2+߿ Ca2+ࡦࠨ࠲
ࡦࡄࠢ⾰ߩᓎഀ
7㧚ߔ߮
Ca2+࠴ࡖࡀ࡞ߦࠃࠆ⚦⢩⾰ Ca2+Ớᐲᄌൻࡄ࠲ࡦߩᒻᚑߣޔCa2+ࡦࠨ࠲ࡦࡄࠢ⾰ߦࠃࠆߘߩ
⸃⺒߆ࠄߥࠆ Ca2+ࠪࠣ࠽࡞વ㆐♽ߪޔേᬀ‛ߦㅢߩ᥉ㆉ⊛ߥᖱႎಣℂࠪࠬ࠹ࡓߛ߇ޔ㑐ਈߔࠆ
ಽሶ߿ߘߩᓮᯏ᭴ߦߪޔᄢ߈ߥᄙ᭽ᕈ߇ࠄࠇࠆޕᬀ‛ߩ∉ᔕ╵ߦ߅ߌࠆᖱႎ⚛ሶߣߒߡߩ
Ca2+ߩ㊀ⷐᕈߪߒ߫ߒ߫ᜰ៰ߐࠇߡ᧪ߚ߇ޔ㑐ਈߔࠆಽሶߩታ߿ߩߘޔᓮᯏ᭴ߣᯏ⢻ߩ⸃
− 106 −
ߪᦨߊ߿߁ࠃޔㄭ߇✜┵ޔᓧࠄࠇߟߟࠆޕ
ᬀ‛ࠍℂ⸃ߒޔᬀ‛ߩᜬߟẜ⢻ജࠍ↢߆ߒੱޔ㑆߇ᬀ‛ߣ߁߹ߊઃ߈วߞߡߊߎߣ
ߪޔᓟߩੱ㘃ߦߣߞߡᔅⷐਇนᰳߥ⺖㗴ߢࠆᦨޕㄭޔᬀ‛ߩ∉ജࠍ㜞ࠆߎߣߦࠃࠅޔᲕ
⩶Ვ⯻╬ߩൻቇㄘ⮎ߩ↪ࠍシᷫߔࠆ⹜ߺ߇ᵈ⋡ࠍ㓸ߡࠆޕຠ⒳ᡷ⦟߿ㆮવሶ⚵឵߃ᛛ
ⴚߥߤߦࠃࠆ⠴∛ᕈߩㆮવ⊛Ⴧᒝߣޔᬀ‛ߩ∉ജࠍ㜞ࠆ↢ℂᵴᕈ‛⾰ߩត⚝↪ߩᣇߩ
ࠕࡊࡠ࠴ߪޔᰴઍߩㄘᬺࠍᄢ߈ߊᄌ߃ࠆน⢻ᕈࠍ⒁ߡࠆ⎇ᧄޕⓥߩㆊ⒟ߢ⑳ߚߜߩ⎇ⓥ
ࠣ࡞ࡊ߽ᦨㄭޔᬀ‛ߩ∉ജࠍ㜞ࠆᣂⷙߩ‛⾰ࠍ⊒ߒޔ↪ᯏ᭴ߩ⸃ߣታ↪ൻߩᬌ⸽ࠍ
ㅴߡࠆ⎇ߚߒ߁ߎޕⓥࠍㅴࠆߢߩᩮᐙߪޔᬀ‛߇⁛⥄ߦㅴൻߐߖߡ᧪ߚޔേ‛ߣߪోߊ
⇣ߥࠆ∉ࠪࠬ࠹ࡓߩᖱႎࡀ࠶࠻ࡢࠢࠍಽሶࡌ࡞ߢℂ⸃ߔࠆߎߣߦࠆޕᬀ‛ߩ∉ᔕ╵ߦ
߅ߌࠆᖱႎવ㆐ᯏ᭴ࠍ⎇ⓥߔࠆߎߣߪޔᓟ߹ߔ߹ߔ㊀ⷐᕈࠍჇߔߣ⠨߃ࠄࠇ⎇ޔⓥߩ⊒ዷ߇ᦼ
ᓙߐࠇࠆޕ
㊂ሶ↢ᖱႎ⎇ⓥࡦ࠲ߦ߅ߌࠆ⎇ⓥߩㆊ⒟ߢޔᢙℂ‛ℂቇޔᖱႎ⑼ቇಽ㊁ߩ⎇
ⓥ⠪ߣኻ߿ห⎇ⓥࠍㅴߚޕᬀ‛ߩᖱႎࡀ࠶࠻ࡢࠢࠍ㊂ሶ↢ᖱႎߩⷞὐ߆ࠄᝒ߃ࠆ
⎇ⓥߪߎࠇ߹ߢ߶ߣࠎߤⴕࠊࠇߡ᧪ߥ߆ߞߚߎߣ৻ޔᣇߢᓟቇ㓙⊛⎇ⓥࠍㅴࠆߎߣߢ㘧
べ⊛ߥㅴዷ߇ᦼᓙߢ߈ࠆߎߣ߇ࠄ߆ߣߥߞߚޕ2010 ᐕ 3 ߩ IC-QBIC ߿ 2010 ᐕ 11 ߦ੩
ㇺᄢቇၮ␆‛ℂቇ⎇ⓥᚲߢ㐿߆ࠇߚ⎇ⓥળޟ㊂ሶ⑼ቇߦ߅ߌࠆኻᕈߣࠬࠤ࡞ߢޠଐ㗬ߐ
ࠇߚ⻠Ṷߩᓟߩ㗀ߩᄢ߈ߐߪߚߞ߆ߥߦߢ߹ࠇߎޔᣂߚߥቇ㓙⎇ⓥߩ⪚⧘ࠍ੍ᗵߐߖߚޕ
ߎ߁ߒߚᗧᄖᕈߦንቇߩᣂዷ㐿ߎߘ߇᧲ޔ੩ℂ⑼ᄢቇ㊂ሶ↢ᖱႎ⎇ⓥࡦ࠲ߦ߅ߌ
ࠆ⎇ⓥߩᄢ߈ߥᚑᨐߩ৻ߟ߆߽⍮ࠇߥޕ
− 107 −
⎇ⓥᬺ❣㧔⎇ⓥ⠪㧕 ↢ࠣ࡞ࡊ
ጊ⊓৻㇢
ቇⴚ⺰ᢥ
1.
Khandoker Mohammad Mozaffor Hossain, Mohammad Yamin Ali, and Ichiro Yamato:
Antibody levels against Newcastle Disease Virus in chickens in Rajshahi and
surrounding districts of Bangladesh. International Journal of Biology 2, in press (2010)
䋨䋩
2.
Khandoker Mohammad Mozaffor Hossain, Mohammad Takabbar Hossain, and Ichiro
Yamato: Seroprevalence of Salmonella and Mycoplasma gallisepticum infection in
chickens in Rajshahi and surrounding districts of Bangladesh. International Journal of
Biology, accepted (2010)䋨䋩
3.
Wankee Kim, Ichiro Yamato and Tadashi Ando: Alanine-based Peptides Containing
Polar Residues Presumably Favor ǂ-Helical Structure Entropically. Mol. Simul. In press
(2010)䋨䋩
4.
Kenji Mizutani, Misaki Yamamoto, Ichiro Yamato, Yoshimi Kakinuma, Mikako Shirouzu,
John E. Walker, Shigeyuki Yokoyama, So Iwata and Takeshi Murata: Structure of the
Na+
unbound
rotor
Na+-transporting
5.
ring
modified
with
N,N’-Dicyclohexylcarbodiimide of the
V-ATPase. Proc. Natl. Acad. Sci. USA 䋨ᛩⓂਛ䋩
Shinya Saijo, Satoshi Arai, K. M. Mozaffor Hossain, Ichiro Yamato, Yoshimi Kakinuma,
Yoshiko Ishizuka-Katsura, Takaho Terada, Mikako Shirouzu, Shigeyuki Yokoyama, So
Iwata and Takeshi Murata: Crystal structure of the central axis (NtpD-NtpG) in the
catalytic portion of Enterococcus hirae V-type Na+-ATPase and its interaction with
central stalk subunit NtpC. Proc. Natl. Acad. Sci. USA 䋨Ḱਛ䋩
6.
K. M. Mozaffor Hossain, Satoshi Arai, Shinya Saijo, Yoshimi Kakinuma, Takeshi Murata
and Ichiro Yamato: Expression and purification of the central stalk subunits of
sodium-translocating V-type ATPase from Enterococcus hirae. African J. Biotechnology
䋨ᛩⓂਛ䋩
ችፒ ᥓ
ቇⴚ⺰ᢥ
1.
Study of transcriptional regulatory network BASED on cis module database, S. Akasaka,
T. Urushibara, T. Suzuki and S. Miyazaki, Quantum Bio-Informatics, 27 Ꮞ, 2010 (in
press) 䋨ᩏ⺒䉍䋩
2.
The prediction of botulinum toxin structure based on in silico and in vitro analysis, T.
Suzuki and S. Miyazaki, Quantum Bio-Informatics, 27 Ꮞ, 2011 䋨in press䋩 䋨ᩏ⺒䈅䉍䋩
− 108 −
3.
Sialic acid-dependent binding and transcytosis of serotype D botulinum neurotoxin and
toxin complex in rat intestinal epithelial cells, K. Niwa, T. Yoneyama, H. Ito, M. Taira, T.
Chikai, H. Kouguchi, T. Suzuki, K. Hasegawa, K. Miyata, K. Inui, T. Ikeda, T. Watanabe
and T. Ohyama, Vet. Microbiol., 141, 312-320, 2010
4.
Identification of Metal Transporter Genes Using Coding Theory, H. Sato, Y. Kwon and S.
Miyazaki, JSBi2010, P-18, 2010.
5.
ᖱႎቇ⊛ℂ⺰䉕↪䈇䈢䉟䊮䊐䊦䉣䊮䉱䉡䉟䊦䉴䈱ㅴൻ䊜䉦䊆䉵䊛䈱⸃, ጊᧄ⋿๋, ᰨ ᆻᄢ, ች
ፒ ᥓ, SIG-BIO2010, A4-27, 2010.
6.
ㆮવሶォ౮ᓮ㗔ၞ䉕↪䈚䈢ㆮવሶ⟲䈱ㅢᕈត⚝䈮䈧䈇䈩, ⿒ဈᔒᵤ, ᰨ ᆻᄢ, ችፒ ᥓ,
SIG-BIO2010, A3-20, 2010.
7.
ᖱႎ⑼ቇ⊛ᚻᴺ䈮䉋䉎䉲䉴䉣䊧䊜䊮䊃㈩䈫᭴ㅧ䊄䊜䉟䊮䈱ㅴൻ⸃ᨆ, ᧄㇹᴕᅥ, ችፒ ᥓ,
CBI2010, P5-04, 2010.
8.
Proposal of Pharmacoinformatics in the Post-Genome Era, S. Miyazaki, Center for
Ecclesiastical Services, Rome, Italy, November 7-10, 2010, ISABEL2010
ᱞ↰ᱜਯ
ቇⴚ⺰ᢥ
1.
䉫䊤䊐ᦠ឵䈋⸒⺆ REGREL 䈮䈍䈔䉎ਗⴕ䉫䊤䊐ᠲ䉪䉣䊥䈱䇮᧲㆐゠䊶ᱞ↰ᱜਯ䇮FIT2010
╙䋹࿁ᖱႎ⑼ቇᛛⴚ䊐䉤䊷䊤䊛䇮A-021䇮pp.197-204䇮Sep. 2010 䋨ᩏ⺒ή䋩
2.
DP 䊙䉾䉼䊮䉫䉕↪䈇䈢Ṷᄼ䈱⟎⸃ᨆᚻᴺ䈱ឭ᩺䇮⿒ᧄੳผ䊶ᱞ↰ᱜਯ䇮FIT2010 ╙䋹࿁
ᖱႎ⑼ቇᛛⴚ䊐䉤䊷䊤䊛䇮E-031䇮pp.289-292䇮Sep. 2010 䋨ᩏ⺒ή䋩
3.
䉸䊷䉲䊞䊦䊑䉾䉪䊙䊷䉪ಽᨆ䈱䊡䊷䉱䈻䈱ᓇ㗀䈫వⴕᕈ䈮䉋䉎⸃䇮ᄥ↰㘧㠽䊶᧻Ỉᥓผ䊶ᱞ↰ᱜ
ਯ䇮FIT2010 ╙䋹࿁ᖱႎ⑼ቇᛛⴚ䊐䉤䊷䊤䊛䇮F-001䇮pp.337-342䇮Sep. 2010 䋨ᩏ⺒ή䋩
4.
SNS 䈱․ᓽ䉕⠨ᘦ䈮䉏䈢䊈䉾䊃䊪䊷䉪䊝䊂䊦䈱⹏ଔ䇮Ꮏ⮮ආᶈ䊶᧻Ỉᥓผ䊶ᱞ↰ᱜਯ䇮FIT2010
╙䋹࿁ᖱႎ⑼ቇᛛⴚ䊐䉤䊷䊤䊛䇮F-057䇮pp.525-528䇮Sep. 2010 䋨ᩏ⺒ή䋩
5.
ᗐᓂೞ䈮䈍䈔䉎 6 ゲ䊝䊷䉲䊢䊮䉶䊮䉰䉕↪䈇䈢ಾᚻᴺ䇮ⱃ㑆ብ䊶ᱞ↰ᱜਯ䇮FIT2010 ╙䋹
࿁ᖱႎ⑼ቇᛛⴚ䊐䉤䊷䊤䊛䇮I-007䇮pp.253-258䇮Sep. 2010 䋨ᩏ⺒ή䋩
ᵤᐘ
ቇⴚ⺰ᢥ
1.
Effects of growth phase and cell density on cryptogein-induced programmed cell death in
suspension-cultured tobacco BY-2 cells:development of a model system for 100% efficient
hypersensitive cell death., Sawai, Y., Tamotsu, S., Kuchitsu, K., Sakai, A., Cytologia
75(4) in press. 2010䋨ᩏ⺒䋩
2.
Negative Feedback Regulation of Microbe-Associated Molecular Pattern-Induced
Cytosolic Ca2+ Transients by Protein Phosphorylation., Kurusu, T., Hamada, H.,
− 109 −
Sugiyama, Y., Yagala, T., Kadota, Y., Furuichi, T., Hayashi, T., Umemura, K., Komatsu,
S., Miyao, A., Hirochika, H., Kuchitsu, K. J. Plant Res. in press. 2010䋨ᩏ⺒䋩
3.
Roles of calcineurin B-like protein-interacting protein kinases in innate immunity in
rice., Kurusu, T., Hamada, J., Hamada, H., Hanamata, S., Kuchitsu, K. Plant Signaling
& Behavior 5(8): 1045-1047, 2010䋨ᩏ⺒䋩
4.
Regulation of microbe-associated molecular pattern-induced hypersensitive cell death,
phytoalexin
production
and
defense
gene
expression
by
calcineurin
B-like
protein-interacting protein kinases, OsCIPK14/15, in rice cultured cells.䋬Kurusu, T.,
Hamada, J., Nokajima, H., Kitagawa, Y., Kiyoduka, M., Takahashi, A., Hanamata, S.,
Ohno, R., Hayashi, T., Okada, K., Koga, J., Hirochika, H., Yamane, H., Kuchitsu, K.
Plant Physiol. 153(2): 678-692, 2010䋨ᩏ⺒䋩
5.
Signaling Network of Environmental Sensing and Adaptation in Plants: Key roles of
Calcium Ion., Kurusu, T., Kuchitsu, K., In: QUANTUM BIO-INFORMATICS III From
Quantum Information to Bio-Informatics. edited by Accardi, L., Freudenberg, W., Ohya,
M., World Scientific, in press.
6.
ᬀ‛䈱∉ᓮᯏ᭴-ᖱႎ⚛ሶ䈫䈚䈩䈱䉦䊦䉲䉡䊛䉟䉥䊮䈱ᓎഀ- ᧪㗇 ቁశ䇮ᵤ ᐘ ⑼ቇ䊐
䉤䊷䊤䊛 in press.
ᐕᐲਛߦ ੍ቯߩ⺰ᢥ
1.
Dynamic reorganization and function of the cytoskeleton and vacuoles in defense
responses and programmed cell death. Higaki, T., Kurusu, T., Hasezawa, S., Kuchitsu,
K.
2.
Cryptogein-induced cell cycle arrest at G2 phase is associated with inhibition of
cyclin-dependent kinases due to suppression of expression of cell cycle-related genes and
protein degradation in synchronized tobacco BY-2 cells, Ohno, R., Kadota, Y., Fujii, S.,
Sekine, M., Umeda, M., Kuchitsu, K.
3.
A plasma membrane protein OsMCA1 is involved in regulation of hypoosmotic
stress-induced Ca2+ influx and modulates production of reactive oxygen species in rice
cultured cells., Kurusu, T., Nishikawa, D., Yamazaki, Y., Sakurai, Y., Gotoh, M.,
Nakano, M., Hamada, H., Yamanaka, T., Iida, K., Nakagawa, Y., Shinozaki, K., Iida, H.,
Kuchitsu, K.
4.
Regulation of microbe-associated molecular pattern-induced Ca2+ influx and production
of phytoalexins by a putative voltage-gated cation channel, OsTPC1, in cultured rice
cells., Hamada, H., Nokajima, H., Kiyoduka, M., Kurusu, T., Koyano, T., Sugiyama, Y.,
Okada, K., Koga, J., Yamane, H., Miyao, A., Hirochika, H., Kuchitsu, K.
ᓙ⻠Ṷ
1.
ᬀ‛䈱 Ca2+/ᵴᕈ㉄⚛ᖱႎવ㆐䊈䉾䊃䊪䊷䉪䈫ᗵᨴ㒐ᓮᔕ╵䈱ᓮ ᵤᐘ䋬ጟጊᄢቇ⾗Ḯᬀ
‛⑼ቇ⎇ⓥᚲ䉶䊚䊅䊷䋬ጟጊᄢቇ⾗Ḯᬀ‛⑼ቇ⎇ⓥᚲ(ୖᢝᏒ)䋬2010
2.
ᬀ‛䈱ᵴᕈ㉄⚛⒳-Ca2+䉲䉫䊅䊦䊈䉾䊃䊪䊷䉪䈫⥄ὼ∉䈱ᓮ䋬ᵤᐘ䋬ᣣᧄㄘ⧓ൻቇળਛ
− 110 −
྾࿖ᡰㇱ ╙ 3 ࿁ㄘ⧓ൻቇ䈱ᧂ᧪㐿ᜏ䉶䊚䊅䊷䋬ጟጊᄢቇ ᵤፉ䉨䊞䊮䊌䉴䋬2010
3.
ᬀ‛䈱 Ca2+-ᵴᕈ㉄⚛ᖱႎવ㆐䊈䉾䊃䊪䊷䉪䈫⥄ὼ∉䍃ᒻᘒᒻᚑ䈱ᓮ䋬ᵤᐘ䋬╙ 67 ࿁
Plant Science Seminar䋨ർᄢℂቇ⎇ⓥ⑼䉶䊚䊅䊷䋩䋬ർᶏᄢቇℂቇㇱ䋬2010
4.
Ca2+ -ROS signaling network regulating innate immunity, programmed cell death and
development in plants䋬Kuchitsu, K. International Symposium of Innovative Research
Center for Agricultural Sciences “Multiple Functions of Plant Membrane Protein”䋬᧲ർ
ᄢቇㄘቇㇱ䋬2010
5.
ᬀ‛䈱ᵴᕈ㉄⚛-Ca2+ᖱႎવ㆐䊈䉾䊃䊪䊷䉪䈫⥄ὼ∉䊶ᒻᘒᒻᚑ䈱ᓮ䋬ᵤᐘ䋬╙ 25 ࿁᧲
ർᄢቇ↢⑼ቇ㕍⪲ጊ䉶䊚䊅䊷䋬᧲ർᄢቇℂቇㇱ䋬2010
6.
ᬀ‛䈱↢㒐ᓮ䈫ᵴᕈ㉄⚛-Ca2+䉲䉫䊅䊦䊈䉾䊃䊪䊷䉪䋬ᵤᐘ䋬᧪㗇ቁశ䋬⾐ደ⑲㓉䋬╙ 21 ࿁
ᣣᧄ↢㒐ᓮቇળ䉲䊮䊘䉳䉡䊛䋬บᏒᚢἴᓳ⥝⸥ᔨ㙚䋬2010
7.
ᬀ‛䈱↢ᖱႎ䊈䉾䊃䊪䊷䉪䋬ᵤᐘ䋬QBIC 䉰䊙䊷䉴䉪䊷䊦䋬⺪⸰᧲੩ℂ⑼ᄢቇ䋬2010
8.
ᗵᨴ⼂䈱ೋᦼᔕ䈫䉦䊦䉲䉡䊛Ɇᵴᕈ㉄⚛䉲䉫䊅䊦䋬ᵤᐘ䋬᧪㗇ቁశ䋬ự↰᥍ᐽ䋬⾐ደ⑲
㓉䋬ᣣᧄᬀ‛ቇળ╙ 74 ࿁ᄢળ䋬ਛㇱᄢቇ䋬2010
9.
ᬀ‛䈱↢䈐ᣇ䈫ᖱႎ䊈䉾䊃䊪䊷䉪㩷 ᵤᐘ㩷 ੩ㇺᄢቇၮ␆‛ℂቇ⎇ⓥᚲ⎇ⓥળ䍀㊂ሶ⑼ቇ䈮䈍
䈔䉎ኻᕈ䈫䉴䉬䊷䊦䍁, ੩ㇺᄢቇၮ␆‛ℂቇ⎇ⓥᚲḡᎹ⸥ᔨ㙚, 2010
10. Ca2+-ROS Signaling Network Regulating Innate Immunity, Programmed Cell Death and
Development in Plants., Kuchitsu, K., Plant Biotechnology Center, Ohio State
University, USA, 2010
11. Ca2+-ROS Signaling Network Regulating Innate Immunity, Programmed Cell Death and
Development in Plants., Kuchitsu, K., Department of Biology, Duke University, USA,
2010.
12. ᬀ‛䈱ᗵᨴ㒐ᓮᔕ╵䈫䉥䊷䊃䊐䉜䉳䊷㩷 ᵤᐘ, ⧎ୀ❥, ᧪㗇ቁశ㩷 ᣣᧄᬀ‛↢ℂቇળ䉲䊮䊘
䉳䉡䊛䇸䊔䊷䊦䉕⣕䈑ᆎ䉄䈢ᬀ‛䉥䊷䊃䊐䉜䉳䊷䇹, 2011
⪺ᦠ
1.
ᬀ‛䈱㒐ᓮᔕ╵䈮䈍䈔䉎䉦䊦䉲䉡䊛䉲䉫䊅䊥䊮䉫䋬ự↰᥍ᐽ䋬᧪㗇ቁశ䋬ᵤᐘ䋬ᬀ‛䈱䉲䉫䊅
䊦વ㆐䋭ಽሶ䈫ᔕ╵䋬┙ 䋬✚㗁 238䋬pp.105-112䋬ᩑᧄㄖ↵䋬㜞ጊ⺈ม䋬↰Ⓞ䋬᧻ጟ
ା ✬䋬2010
ᣂ⡞ႎ
1.
䉟䊈䈱∉ᯏ᭴ᓮ䌾䈢䉖䈴䈒⾰䉕⊒䋬ᣣ⚻↥ᬺᣂ⡞䋬ᐔᚑ 22 ᐕ 7 13 ᣣ
2.
䉟䊈䈱∛ේᗵᨴ䉕㒐ᓮ䌾ᣂⷙ䉺䊮䊌䉪⾰䉕⊒䋬⑼ቇᣂ⡞䋬ᐔᚑ 22 ᐕ 6 18 ᣣ
3.
䉟䊈䈱∉ᯏ᭴ᓮ䌾䈢䉖䈴䈒⾰䉕⊒䋬ᣣೀᎿᬺᣂ⡞䋬ᐔᚑ 22 ᐕ 6 11 ᣣ㩷
− 111 −
㩷
㩷
㩷
㊂ሶ⎇ⓥ䉫䊦䊷䊒㩷
㩷
㩷
ጀ⁁વዉߩᩏ࠻ࡦࡀ࡞ಽశߦࠃࠆ⎇ⓥ
㊂ሶࠣ࡞ࡊ 㧔ℂቇㇱ‛ℂቇ⑼ ဈ↰⎇ⓥቶ㧕
ဈ↰⧷㧔ᢎ㧕ޔട⮮ᜏ㧔ഥᢎ㧕
Abstract. ㌃㉄ൻ‛㜞᷷વዉߦ߅ߌࠆਇ⚐‛ߩᓸⷞ⊛ലᨐࠍ⺞ߴࠆߚߦޔZn ࠍਇ
⚐‛ߣߒߡട߃ߚ⹜ᢱߦߟߡߩᭂૐ᷷ᩏ࠻ࡦࡀ࡞ಽశ᷹ቯࠍⴕߞߚޕਇ⚐‛ࠨࠗ࠻
ߩㄭߊߢߪޔਇ⚐‛ࠨࠗ࠻ࠍਛᔃߦ 4 ࿁ኻ⒓ߩࠦࡦ࠳ࠢ࠲ࡦࠬࡇࠢ߇ⷰኤߐࠇߚߎޕ
ߩࠦࡦ࠳ࠢ࠲ࡦࠬߩჇടߪᢙᩰሶ㐳⒟ᐲᐢ߇ߞߡࠆ߇ߩߎޔᐢ߇ࠅᣇ߇ઁߩਇ⚐‛ࠨ
ࠗ࠻ߣߩ⋧ኻ⟎ߦଐሽߔࠆߎߣ߇ࠄ߆ߦߥߞߚߪࠇߎޕන৻ਇ⚐‛⁁ᘒ㑆ߩᐓᷤߣ
ߒߡℂ⸃ߢ߈ࠆޔߚ߹ޕਇ⚐‛ࠨࠗ࠻ߢⷰኤߐࠇࠆࠦࡦ࠳ࠢ࠲ࡦࠬࡇࠢߪޔવዉࠡ
ࡖ࠶ࡊߩᄢ߈ߐ߇ᄢ߈ߣⷰኤߐࠇߥߊߥࠆߣ߁ߎߣ߇ಽ߆ߞߚⷰߪࠇߎޕኤߐࠇߡ
ࠆࠦࡦ࠳ࠢ࠲ࡦࠬࡇࠢߩᚑ࿃ߦߟߡߩ⍮ࠍਈ߃ࠆޕ
1㧚ߪߓߦ
ᭂૐ᷷ᩏ࠻ࡦࡀ࡞ಽశᴺߪޔේሶࡌ࡞ߢውߞߚត㊎ࠍ↪ߡ⹜ޔᢱߩ㕙᭴ㅧ߮⁁ᘒኒᐲ
ࠍ᷹ቯߢ߈ࠆᣇᴺߢࠆࡠࠢࠗࡑޕᵄ╬ࠍઃടߔࠆߎߣߦࠃࠅޔ㊂ሶࠦࡦࡇࡘ࠲߳ߩᔕ↪ࠍ⠨
߃ߚታⓨ㑆ࠬࡇࡦߩⷰኤߩ⹜ߺ߽ⴕࠊࠇߡࠆޕᩏ࠻ࡦࡀ࡞ಽశᴺࠍવዉߦㆡ↪ߒߚ႐วޔ
વዉࠡࡖ࠶ࡊߩሽ᧤⏛ޔ㊂ሶޔ㔚⩄⒎ᐨߥߤߐ߹ߑ߹ߥ⽎ࠍⷰኤน⢻ߢࠆޕવዉߦ
ਇ⚐‛ࠍᷝടߔࠆߣޔવዉォ⒖᷷ᐲߥߤߩᏂⷞ⊛ߥવዉ․ᕈߦᄢ߈ߥᓇ㗀ࠍਈ߃ࠆߎߣ߇⍮
ࠄࠇߡࠆ߇ߩߎޔਇ⚐‛߇ሽߔࠆࠨࠗ࠻ࠍ⋥ធᩏ࠻ࡦࡀ࡞ಽశᴺߢⷰኤߢ߈ࠆߎߣ߇ࠄ
߆ߦߥߞߡ߈ߚޕ㌃㉄ൻ‛㜞᷷વዉߦ߅ߡߪޔਇ⚐‛ࠨࠗ࠻ߢࡠࡃࠗࠕࠬߦ߅ߌࠆࠦ
ࡦ࠳ࠢ࠲ࡦࠬ߇ࠪࡖࡊߥࡇࠢࠍᒻᚑߔࠆࠬࡦ࠲ࠢ࠳ࡦࠦࠆߌ߅ߦࠬࠕࠗࡃࡠޔߚߩߎޕ
ࠍⓨ㑆⊛ߦࡑ࠶ࡇࡦࠣߒߡߌ߫ޔਇ⚐‛ߩ⟎ࠍ⍮ࠆߎߣ߇ߢ߈ࠆޔߚ߹ޕਇ⚐‛ࠨࠗ࠻ߛ
ߌߢߥߊߩߘޔㄭறߩࠨࠗ࠻ߦ 4 ࿁ኻ⒓ߩࠦࡦ࠳ࠢ࠲ࡦࠬࡇ߽ࠢࠆߪࠢࡇߥ߁ࠃߩߎޕ
ਇ⚐‛ߦࠃࠆ߽ߩߢࠆߎߣߪ⏕ߢࠆ߇ߩߘޔᚑ࿃ߦߟߡߪࠄ߆ߦߥߞߡߥࠇߎޕ
߹ߢߦන⚐ߥࠢࡠࡦᢔੂߦࠃࠆ߽ߩޔ⋧ਇ⚐‛ലᨐߦࠃࠆ߽ߩޔㄭ⮮ലᨐߦࠃࠆ߽ߩ╬ߩน
⢻ᕈ߇ᜰ៰ߐࠇߡࠆޕ
2㧚ਇ⚐‛ߩᓸⷞ⊛ᐓᷤലᨐ
Fig.1 ߪ㌃㉄ൻ‛㜞᷷વዉ Bi2Sr2CuO6 ߦ߅ࠆ
ࡠࡃࠗࠕࠬࠦࡦ࠳ࠢ࠲ࡦࠬߩࡑ࠶ࡊߢࠆޕ㧔4.2K㧕
࿑ߩ⦡ߩỚߣߎࠈ߇ࠦࡦ࠳ࠢ࠲ࡦࠬߩ㜞ㇱಽࠍ
ߔⷞޕ㊁ౝߦ 3 ߟߩਇ⚐‛߇ࠄࠇࠆߩ߅ߩ߅ޕ
ߩਇ⚐‛ㄭறߢߪ߽ߣߞ߽ߩࠬࡦ࠲ࠢ࠳ࡦࠦޔ㜞
㧔⦡ߩỚ㧕ਛᔃߩਇ⚐‛ࠨࠗ࠻ߩ߹ࠊࠅߦ 4 ࿁ኻ
⒓ߩࡇࠢ߇ࠄࠇࠆޕਛᄩਅߩ 2 ߟߩਇ⚐‛ࠨࠗ
࠻ߦᵈ⋡ߔࠆߣߩࠄࠇߘޔᄖߦࠦࡦ࠳ࠢ࠲ࡦࠬߩ
Ⴧടߒߡࠆ㗔ၞ߇ᐢ߇ߞߡࠆߎߣ߇ಽ߆ࠆޕ࿑
߆ࠄࠊ߆ࠆࠃ߁ߦߩߎޔᐢ߇ࠅᣇߦ⇣ᣇᕈ߇ሽߔ
ࠆ㧦࿑ߩ❑ᣇะ㧔ઁߩਇ⚐‛߇ሽߒߥᣇะ㧕ߢ
ߪߎߩᐢ߇ࠅߪ⁜ߊޔ࿑ߩᮮᣇะ㧔ઁߩਇ⚐‛߇ሽ
ߔࠆᣇะ㧕ߢߪᐢ㗔ၞߦᐢ߇ߞߡࠆߎߣ߇ࠊ߆
ࠆޔߪࠇߎޕ2 ߟߩਇ⚐‛⁁ᘒ߇ᐓᷤߒߡࠆߚߣ
⠨߃ࠄࠇࠆޕℂ⺰⊛⎇ⓥߦࠃࠆߣޔ2 ߟߩਇ⚐‛ߩ
1
− 113 −
Fig. 1 Bi2Sr2CuO6 ߦ߅ߌࠆ
ࡠࡃࠗࠕࠬࠦࡦ࠳ࠢ࠲ࡦࠬࡑ࠶ࡊ
⋧ኻ⊛ߥ⟎߿〒㔌ߦࠃߞߡޔᄢ߈ߊᐓᷤലᨐ߇⇣ߥࠆߎߣ߇ႎ๔ߐࠇߡࠆߩ߆ߟߊޕਇ⚐
‛ߩ㈩⟎ߦኻߒߡห᭽ߩ᷹ቯࠍⴕޔℂ⺰ߣߩᲧセࠍⴕߞߡ߈ߚߣ⠨߃ߡࠆޕ
3㧚ࠛࡀ࡞ࠡࠡࡖ࠶ࡊߣਇ⚐‛ࡇࠢ
㌃㉄ൻ‛㜞᷷વዉߢߪવዉࠡࡖ࠶ࡊ߇ⓨ㑆⊛ߦᄢ
߈ߊᄌൻߒߡࠆߎߣ߇⍮ࠄࠇߡࠆޕFig.2 ߪ㌃㉄ൻ
‛㜞᷷વዉ Bi2Sr2CaCu2O8 ߩㆊ࠼ࡊ㧔OD㧕߅
ࠃ߮ਇ⿷࠼ࡊ㧔UD㧕ߩ⹜ᢱߦ߅ߌࠆࠡࡖ࠶ࡊߩᄢ߈
ߐߩಽᏓࠍ␜ߒߚ߽ߩߢࠆ⹜ߩࠄࠇߎޕᢱߦਇ⚐‛ߣ
ߒߡ Zn ࠍᷝടߒޔਇ⚐‛ࡇࠢ߇ࠇࠆ႐ᚲࠍ⺞ߴߡ
ߺࠆߣ࿑ߩࠃ߁ߦ UD ߩ⹜ᢱߢߪಾᢿࠛࡀ࡞ࠡ߇ሽ
ߔࠆߎߣ߇ಽ߆ߞߚࠡ࡞ࡀࠛࠆࠅ߹ߟޕ㧔Ǎcut㧕એ
ߩࠡࡖ࠶ࡊ㗔ၞߢߪਇ⚐‛ࡇࠢߪⷰኤߢ߈ߥ߆ߞߚޕ
ߎࠇߪޔਇ⚐‛ࠨࠗ࠻ߢⷰኤߐࠇࠆࠦࡦ࠳ࠢ࠲ࡦࠬࡇ
ࠢߩᚑ࿃߇ࠡࡖ࠶ࡊߩ୯ߦ⋧㑐ߒߡࠆߎߣࠍ␜ߒߡ
ࠆ⚿ߩߎޕᨐߪ߹ߛࠄ߆ߢߥਇ⚐‛ࡇࠢࠍ⠨߃ࠆ
ߢ㊀ⷐߥ⍮ࠍਈ߃ࠆࡗࡖࠠߣ࠭ࠗࠨࡊ࠶ࡖࠡޕ㊂
ߩ⋧㑐߇ታ㛎⊛ߦᜰ៰ߐࠇߡࠆ⚿ߩߎߚߩߎޕᨐߪ
ਇ⚐‛ࡇࠢߩᒻᚑߦߪࠠࡖࡗ㊂ߩᄢዊ߇㊀ⷐߥᓇ㗀
ࠍਈ߃ߡࠆߣ߁ߎߣࠍ␜ໂߒߡࠆߒ߆ߒޕ⠨߃ࠄ
ࠇࠆ㧟ߟߩࠪ࠽ࠝࠍߩ߁ߜ߆ࠄ߭ߣߟࠍ․ቯߔࠆ߹ߢ
ߦߪ⥋ߞߡߥޕ
4㧚߹ߣ
㌃㉄ൻ‛㜞᷷વዉߦ߅ߌࠆਇ⚐‛ߩᓸⷞ⊛ലᨐࠍ⺞
Fig.2 Bi2Sr2CaCu2O8 ߦ߅ߌࠆࠡࡖ
ߴࠆߚߦޔZn ࠍਇ⚐‛ߣߒߡട߃ߚ⹜ᢱߦߟߡߩᭂ
࠶ࡊߩᄢ߈ߐߩಽᏓߣਇ⚐‛ࡇࠢ߇
ⷰኤߐࠇߚࠡࡖ࠶ࡊߩᄢ߈ߐߩಽᏓ
ૐ᷷ᩏ࠻ࡦࡀ࡞ಽశ᷹ቯࠍⴕߞߚࡇࠬࡦ࠲ࠢ࠳ࡦࠦޕ
ࠢߩⷰኤߦࠃࠅޔਇ⚐‛⁁ᘒ㑆ߩᐓᷤലᨐ߇ࠆߎߣ
ࠍ⏕ߒߚޕવዉߦ߅ߡਇ⚐‛ߩ㊂ࠍჇ߿ߒߡߊߣવዉォ⒖ὐ߇ૐਅߔࠆߪࠇߎޕਇ
⚐‛ߦࠃࠆ᭴ㅧߩੂࠇޔᢔੂߦࠃࠆߣ⠨߃ࠄࠇࠆ߇ࠄࠇߎޔએᄖߦ߽ਇ⚐‛⁁ᘒߩᐓᷤലᨐߦࠃ
ࠆᓇ㗀߽ࠆ߆߽ߒࠇߥޕਇ⚐‛ࡇࠢߩᚑ࿃ߣࠊߖߡᓟᬌ⸛ߒߥߌࠇ߫ߥࠄߥ⺖㗴ߢ
ࠆޕ
References
(1) Tadashi Machida, Takuya Kato, Hiroshi Nakamura, Masaki Fujimoto, Takashi Mochiku,
Shuuichi Ooi, Ajay D. Thakur, Hideaki Sakata, and Kazuto Hirata Phys. Rev. B 82
180507 (2010).
(2) Tadashi Machida, Marat B. Gaifullin1, Shuuich Ooi, Takuya Kato, Hideaki Sakata, and
Kazuto Hirata Jpn. J. Appl. Phys., Vol. 49, 116701 (2010).
2
− 114 −
⥄Ꮖᚑ㐳 InAs ㊂ሶ࠼࠶࠻ߢ⚿วߒߚવዉ⏛᧤㊂ሶᐓᷤ⸘
✚ว⎇ⓥᯏ᭴
㜟ᩉ⧷
ᚒ⥄ߪޘᏆᚑ㐳 InAs ㊂ሶ࠼࠶࠻ߢ⚿วߒߚવዉ⏛᧤㊂ሶᐓᷤ⸘(SQUID)ࠍߒߘޔ
ߩേࠍ⏕ߒߚޔߪࠬࠗࡃ࠺ߩߎޕ㊂ሶ࠼࠶࠻ㄭறߦࠥ࠻㔚ᭂࠍ߃ߡ߅ࠅߩߘޔ
ࠥ࠻㔚ᓮߦࠃߞߡޔSߕࠇߚ㔚ᵹ̆⋧㑐ଥߣޔSធวߩേࠍᗧߔࠆߩ⽶ޔ
વዉ㔚ᵹࠍ᷹ⷰߒߚ ߩߎޕSQUID ߪޔ᧪⊛ߦ࠼࠶࠻ਛߩࠬࡇࡦࠍߒߡޔવዉ㊂ሶ
ࡆ࠶࠻ਛߩ㊂ሶᖱႎߣశ♽ߩ㊂ሶᖱႎߣߩ㑆ߩ߿ࠅߣࠅࠍᇦߔࠆ࠺ࡃࠗࠬߣߒߡᦼᓙ
ߐࠇࠆޕ
1㧚⥄Ꮖᚑ㐳 InAs ㊂ሶ࠼࠶࠻ߢ⚿วߒߚવዉ⏛᧤㊂ሶᐓᷤ⸘
㧜ᰴర᭴ㅧࠍᜬߟ㊂ሶ࠼࠶࠻ߪ‛ޔℂ⊛ߦ߽ᔕ↪߽ߦ߹ߑ߹ߐޔዷ㐿ߐࠇߡࠆޕᚒޔߪޘ
⥄Ꮖᚑ㐳㊂ሶ࠼࠶࠻ߢ⚿วߒߚવዉ⏛᧤㊂ሶᐓᷤ⸘(SQUID)ߩ⎇ⓥࠍⴕߞߡࠆޕ࿑㧝ߦ⥄Ꮖ
ᚑ㐳 InAs ㊂ሶ࠼࠶࠻ߩ AFM ߣᮨᑼ࿑ࠍ␜ߔߩߎޕ㊂ሶ࠼࠶࠻ߩᚑ㐳ߦߪ MBE(ಽሶ✢ࠛࡇ
࠲ࠠࠪᴺ)߇↪ࠄࠇߩߘޔᄢ߈ߐߪ 0.1~0.2 Pm ߢࠆ᧲ޕᄢ↢↥⎇ߩᐔᎹ⎇ⓥቶߣߩห⎇
ⓥߢࠅ⎇ߩߎޔⓥቶࠃࠅޔ㊂ሶ࠼࠶࠻⹜ᢱߪଏ⛎ߐࠇࠆޕ
InAs QDs (dotsize:100~200 nm)
AFM image
GaAs (bufferlayer)
200 nm
AlGaAs (barrierlayer)
100 nm
Si-GaAs (bufferlayer)
200 nm
n+ GaAs substrate
࿑㧝㧚⥄Ꮖᚑ㐳 InAs ㊂ሶ࠼࠶࠻ߩ AFM (Ꮐ)ߣᮨᑼ⊛᭴ㅧᢿ㕙࿑ޕ
㧿㧽㨁㧵㧰ߣߪࡦ࠰ࡈ࡚ࠫޔធวࠍવዉ࡞ࡊ᭴ㅧߩߎߣߢ᧤⏛ޔᄌൻߦኻߒߡᦨޔ
߽ᢅᗵߥ⚛ሶߢࠆޕㅢᏱߪߎߩ࡚ࠫࡈ࠰ࡦ⚛ሶߣߒߡޔવዉ/⛘✼⤑/વዉ(SIS)᭴ㅧࠍᜬ
ߟ࠻ࡦࡀ࡞⚛ሶ߇↪ࠄࠇࠆ߇ޔᚒ ߩߎߪޘSIS ⚛ሶߩ߆ࠊࠅߦ㧘InAs ㊂ሶ࠼࠶࠻ࠍ↪ߚ
SQUID ࠍߒߚޕInAs ㊂ሶ࠼࠶࠻ߪ 0.1㨪0.2 Pm ߣ㕖Ᏹߦᓸዊߢߩߘ߽߆ߒޔᚑ㐳ߒߡࠆ
⟎ߪోߊࡦ࠳ࡓߢࠆࠆޔߚߩߘޕ㧞ߟߩ㊂ሶ࠼࠶࠻ࠍㆬ߮ߩߘޔ㧞ߟࠍવዉߢ
ࠆ Al ߩ࡞ࡊߩਛߦ⚿วߔࠆߦߪޔAFM ߣ㔚ሶࡆࡓ㔺శᴺߦࠃࠆ㜞ᐲߩ⟎วࠊߖᛛⴚ߇ᔅ
ⷐߢࠆޕᚒ ߪޘ10nm એਅߩ⟎วࠊߖᐲᛛⴚࠍ㐿⊒ߒޔInAs ㊂ሶ࠼࠶࠻㧞ߦ⚿วߒߚ
SQUID ࠍߔࠆߎߣߦᚑഞߒߚޕ࿑㧞ߪߩߘޔ㔚ሶ㗼ᓸ㏜౮⌀ߢޔ㧞ߟߩ InAs ㊂ሶ࠼࠶࠻ࠍ
ߒߚવዉ࡞ࡊߩߢ߈ߡࠆߎߣ߇ࠊ߆ࠆޕ
2㧚㔚ᵹ̆㔚․ᕈߣ SQUID േ
ߎߩ SQUID ߩ㔚ᵹ̆㔚 (I-V)․ᕈࠍ 30 mK ߩૐ᷷ߢ᷹ቯߒߚߣߎࠈޔ࿑㧟ߦ␜ߔࠃ߁ߥ
⍎ߥવዉ㔚ᵹࠍ᷹ⷰߒߚޔߜࠊߥߔޕInAs ㊂ሶ࠼࠶࠻ࠍߒߡޔવዉ㔚ᵹ߇ᵹࠇߚߎߣࠍᗧ
ߔࠆ⚿ᨐߢࠆޔߒ߆ߒޕવዉ㔚ᵹ߇ߚߛᵹࠇߚߛߌߢߪ SQUID ߣߒߡേߒߚߣߪ⸒߃
1
− 115 −
ߥޕSQUID ߪવዉߩ࡞ࡊߢࠆ߆ࠄߩ᧤⏛ޔ㊂ሶൻ߇ᚑ┙ߒ⏛߇᧤⏛ߊ⽾ࠍࡊ࡞ޔ
᧤㊂ሶ)0 ߏߣߦᦨᄢવዉ㔚ᵹ Ic ߩᝄേ߇ࠇࠆߪߕߢࠆޕ࿑㧠ߪޔIc ߩ⏛႐ଐሽᕈߢ߈ޔ
ࠇߥᝄേ⽎ࠍ␜ߒߡࠆߩߘޔ߽߆ߒޕᦼ߽⸘▚ߣ߈ߜࠎߣߒߚ৻⥌ࠍ␜ߒߚߣߎߩߎޕ
߆ࠄޔInAs ㊂ሶ࠼࠶࠻⚿ว SQUID ߇߈ߜࠎߣേߒߡࠆߎߣ߇⏕ߢ߈ߚߣ⸒߃ࠆޕ
<Junction B>
(Junction B)
Al
Al
(Dot size x:223 nm, y:156 nm)
Al/InAs SAQD /Al
Josephson junction
Loop area:
3.87x3.08 Pm2
<Junction A>
(Junction A)
(Dot size x:200 nm, y:141 nm)
648,'FRXSOHGZLWK,Q$V6$4'!
*We fabricated samples in Nanotechnology Innovation Center of NIMS.
࿑㧞㧚InAs ㊂ሶ࠼࠶࠻⚿ว SQUID ߩ㔚ሶ㗼ᓸ㏜౮⌀
࿑㧟㧚㔚ᵹ̆㔚․ᕈ ࿑㧠㧚Ic ߩ⏛႐ଐሽᕈ
3㧚Sធวォ⒖ߣ⽶ߩવዉ㔚ᵹ
ߎߩ SQUID ߦߪޔ࿑㧞ߦ␜ߔࠃ߁ߦ߇࠻ࠥ࠼ࠗࠨޔߦ߫ߘߋߔߩ࠻࠶࠼ޔ᭴ᚑߐࠇߡࠆޕ
ߎߩࠨࠗ࠼ࠥ࠻ߪߣ࠻࠶࠼ޔવዉ㔚ᭂߣߩ⚿วᐲࠍᓮߔࠆ߈߇ࠆ৻ޕᣇޔSQUID ࠍ
᭴ᚑߒߡࠆၮ᧼ߩⵣ㕙ߦ߽ࡃ࠶ࠢࠥ࠻߇᭴ᚑߐࠇߡߡ࠻࠶࠼ޔߪ࠻ࠥߩߎޔਛߩ㔚ሶᢙ
ࠍᓮߔࠆ߈߇ࠆ࠻࠶࠼ޕਛߩ㔚ሶᢙࠍᄸᢙߦߒߡߣ࠻࠶࠼ߦᦝޔવዉ㔚ᭂߣߩ⚿วᐲ
ࠍᒙߡޔ㔚ᭂߣ࠼࠶࠻ࠍߚ♽ߢޔㄭ⮮ലᨐ߇߅ߎࠄߥࠃ߁ߥ⁁ᘒߦߔࠆߣࠍ࠻࠶࠼ޔ
ߒߚࠢࡄኻߩ࠻ࡦࡀࡦࠣߦ߅ߡޔ⋧߇Sߕࠇߚലᨐ߇⊒↢ߔࠆࠍࠇߎޕㅢᏱߩ 0 ធว
ߦኻߒߡޔSធวߣࠃ߱ޕ࿑㧡ߪޔធว㧝ߦㄭធߒߚࠥ࠻㧝ߦട߃ࠆࠥ࠻㔚ࠍ 0V ߆ࠄ-0.4V
ߦ⽶ߦჇടߐߖߡવዉ㔚ᭂߣ࠼࠶࠻ߣߩ⚿วᐲࠍᒙߊߒ⁁ߩߘޔᘒߢ࠻ࠥࠢ࠶ࡃޔ㔚ࠍᄌ
ൻߐߖߚߣ߈ߩ Ic ߩ⏛႐ଐሽᕈߢ࠻ࠥޔ㧝ߩ㔚߇-0.4V ߢ࠻ࠥࠢ࠶ࡃޔ㔚߇-0.2V ߩᤨޔ
Ic ߣᄖㇱ⏛႐ߣߩ㑐ଥ߇Sߛߌߕࠇߡࠆߎߣࠍ␜ߒߡࠆ ߩߎޔߡߒ߁ߎޕSQUID ߢ߽߈ߜ
ࠎߣSធวォ⒖ߩ߅ߎࠆߎߣ߇ታ⸽ߐࠇߚޕ
ㅢᏱߩ࡚ࠫࡈ࠰ࡦធวߢߪߘߩ㔚ᵹ̆⋧㑐ଥߪ ޔI Ic sin T ߣߥࠆޔߢߎߎޕTߪੑߟߩ
વዉ㔚ᭂ㑆ߩᏂⷞ⊛⋧ᏅߢࠆޕSធวߢߪޔ㔚ᵹp⋧㑐ଥߪ I Ic sin(T S ) sin T ߣߥࠆ
2
− 116 −
ߩߢޔવዉ㔚ᵹߪㅒะ߈ߦߩ⽶ࠅ߹ߟޔવዉ㔚ᵹ߇ᵹࠇࠆߎߣߦߥࠆޕᚒޔߪޘ㧞ߟߩធว
ߦᵹࠇࠆવዉ㔚ᵹࠍ⸃ᨆߔࠆߎߣߦࠃߞߡޔធว㧝߇ p ォ⒖ࠍߎߒߚᤨߩ⽶ޔવዉ㔚ᵹߩ
ᵹࠇࠆߎߣߩ⏕ߦ߽ᚑഞߒߚޕ
ᓟߩߎޔSធวォ⒖ࠍߚ SQUID ․ᕈࠍࠃࠅ⚦ߦ⺞ߴࠆߣߦ ߩߎޔInAs ㊂ሶ࠼࠶࠻
ߦะߒߚࠩࡆࡓࠍᾖߒߡ࠻࠶࠼ޔਛߦࠬࡇࡦࠍബߒޔSធวߣߩ㑐ㅪࠍ⎇ⓥߔ
ࠆ੍ቯߢࠆޔߪࠇߎޕ᧪ߩశ㊂ሶ♽ߣવዉ㊂ሶ♽ߣߩ㑆ߩ㊂ሶᖱႎߩ߿ࠅขࠅߜࠊߥߔޔ
㊂ሶࠗࡦ࠲ࡈࠚࠬߩ⎇ⓥ߳ߣ⊒ዷߔࠆߘߩ৻ᱠߣߥࠆߢࠈ߁ޕ
࿑㧡㧚Sធวォ⒖
ෳ⠨ᢥ₂
(1) S. Kim, R. Ishiguro, M. Kamio, Y. Doda, E. Watanabe, D. Tsuya, K. Shibata, K.
Hirakawa, and H. Takayanagi, to be published in the Proceedings of 30th International
Conference on the Physics of Semiconductors (ICPS 2010), COEX, Seoul, Korea, July
25~30, 2010
(2) S. Kim, R. Ishiguro, M. Kamio, Y. Doda, E. Watanabe, D. Tsuya, K. Shibata, K.
Hirakawa, and H. Takayanagi, submitted to Appl. Phys. Lett. arXiv:1010.5892
3
− 117 −
Model for the occurrence of Fermi pockets without the
pseudogap hypothesis in underdoped cuprate superconductors
潮
上村
秀樹
洸 （東京理科大学特別顧問、総合研究機構客員教授）
（東京工業高等専門学校名誉教授、総合研究機構客員教授）
平成 22 年 12 月 6 日
概 要
Central issues in the electronic structure of underdoped cuprate superconductors are to
clarify the shape of the Fermi surfaces and the origin of a pseudogap. Based on the model
proposed by Kamimura and Suwa which bears important characteristics born from the interplay of Jahn-Teller Physics and Mott Physics, we show that the feature of Fermi surfaces is
the Fermi pockets constructed by doped holes under the coexistence of a metallic state and
of the local antiferromagnetic order. Below Tc the holes on Fermi pockets form Cooper pairs
with d-wave symmetry in the nodal region. In the antinodal region there are no Fermi surfaces. Thus the introduction of a pseudogap is no longer necessary. Calculated angle-resolved
photoemission spectrum below Tc show a coherent peak at the nodal region while the real
transitions of photo-excited electrons from the occupied states below the Fermi level to a free
electron state above the vacuum level appear in the antinodal region. From this feature the
origin of the two distinct gaps in observed ARPES is elucidated. The finite-size-effects of a
spin-correlation length coexisting with a metallic state are discussed. In particular, we discuss a possibility of the spatially inhomogeneous distribution of Fermi-pocket-states and of
large-Fermi-surface-states above Tc . Finally a new phase diagram for underdoped cuprates is
proposed.
1
Introduction
Undoped copper oxide La2 CuO4 is an antiferromagnetic Mott insulator, in which an electron
correlation plays an important role [1]. Thus we may say that undoped cuprates are governed
by Mott physics. In 1986 Bednorz and Müller discovered high temperature superconductivity
in copper oxides by doping hole carriers into La2 CuO4 [2]. Their motivation was that higher Tc
could be achieved for copper oxide materials by combining Jahn-Teller (JT) active Cu ions with the
structural complexity of layer-type perovskite oxides. In order to investigate the mechanism of high
temperature superconductivity, most of models assumed that the doped holes itinerate through the
orbitals extended over a CuO2 plane in the systems consisting of the CuO6 octahedrons elongated
by the JT effect. Those models are called “single-component theory”, because the orbitals of hole
carriers extend only over a CuO2 plane.
In 1989, Kamimura and his coworkers showed by first-principles calculations that the apical oxygen in the CuO6 octahedrons tend to approach towards Cu2+ ions, when Sr2+ ions are substituted
− 118 −
for La3+ ions in La2 CuO4 , in order to gain the attractive electrostatic energy in ionic crystals
such as cuprates [3, 4]. As a result the elongated CuO6 by the JT effect shrink by doping holes.
This deformation against the JT distortion is called “anti-Jahn-Teller effect” [5]. By this effect
the energy separation between the two kinds of orbital states which have been split originally by
the JT effect becomes smaller by doping hole-carriers, where the spatial extension of one kind is
parallel to the CuO2 plane while that of the other kind is perpendicular to it.
By taking account of the anti-Jahn-Teller effect Kamimura and Suwa proposed that one must
consider these two kinds of orbital states equally in forming the metallic state of cuprates, and
they constructed a metallic state coexisting with the local antiferromagnetic (AF) order [6]. This
model is called “Kamimura-Suwa (K-S) model” [7]. Since “anti-Jahn-Teller effect” is a central
issue of the Jahn-Teller physics, we may say that the K-S model bears important characteristics
born from the interplay of Jahn-Teller Physics and Mott Physics. Since these two kinds of orbitals
extend not only over the CuO2 plane but also along the direction perpendicular to it, the K-S
model represents a prototype of “two-component theory”, in the contrast to the single-component
theory.
In this paper we will discuss three important subjects emerged from the interplay of JahnTeller Physics and Mott Physics, based on the K-S model. The first one is concerned with the
shape of Fermi surfaces in underdoped cuprates. The clarification of Fermi surface structures
is nowadays a central issue in underdoped cuprates. There are two views for Fermi surfaces
in cuprates. One view is based on the single-component theory, where a metallic state has a
large Fermi surface (FS) [8, 9, 10]. Since angle-resolved photoemission spectroscopy (ARPES)
experiments did not show the evidence of a large FS, the phenomenological idea of a pseudogap
was introduced [11]. An alternative view is based on the two-component theory developed by
Kamimura and Suwa [6, 12, 13, 7], and they have shown that the coexistence of a metallic state
and local antiferromagnetic(AF) order results in the Fermi pockets constructed by doped holes in
the nodal region. This key-point of the K-S model results in the coexistence of superconductivity
and the local AF order below Tc , as shown by Kamimura and coworkers. [14, 15]. The appearance
of Fermi pockets and small Fermi surfaces in cuprates has been recently reported by various
experimental groups [16, 17, 18, 19, 20, 21].
The second one is concerned with the phenomenological concept of pseudogap and the clarification of observed two-gaps-structure in ARPES spectra in the underdoped regime. Based on the
electronic structure calculated from the K-S model [12, 13], we will show that the angle-resolved
photoemission spectroscopy (ARPES) spectra in cuprates below Tc exhibit a peculiar feature consisting of a coherent peak due to the superconducting density of states at the nodal region and the
real transitions of electrons from the occupied states below the Fermi level to a free-electron state
above the vacuum level in the antinodal region. Since the latter transitions reflect the density of
states (DOS) of the highest energy band for the doped holes, these transitions may appear as a
broad hump with a small peak. In this context it will be concluded that the pseudogap in the
underdoped regime is absent, based on the K-S model.
2
− 119 −
Concerning the ARPES experiments in underdoped cuprates, Tanaka and his coworkers reported
very interesting gap features in the observation of ARPES spectra. Their result exhibits a coherent
peak in the nodal region and a broad hump in the antinodal region in underdoped Bi2212 samples
below Tc [22]. From the quantitative agreement between theory and experiment we will conclude
that the observed broad hump corresponds to the photo-electron excitations from the occupied
states below the Fermi level to the free electron state above the vacuum level and thus the idea of
a pseudogap in underdoped cuprates is no longer necessary.
The third subject is the finite size effect of a metallic state in the K-S model on the spin-electronic
structures of underdoped cuprates. In connection with the observed finite size of a spin-correlated
region of the AF order [23, 24], Hamada and his coworkers [25] and Kamimura and Hamada [26]
determined the ground state of the K-S model in a two-dimensional (2D) square lattice system
with 16 (4 × 4) localized spins by the exact diagonalization method. They clarified that, in the
presence of hole-carriers, the localized spins in a spin-correlated region tend to form an AF order
rather than a random spin-singlet state and thus that the hole-carriers can lower the kinetic energy
of itineration in the spin-correlated region by taking the two kinds of orbitals alternately in the
lattice of AF order. This mechanism of lowering the kinetic energy of the hole-carriers has led to
the coexistence of a metallic state and the local AF order. This is the essential key-point of the
K-S model. We call the above-mentioned mechanism of lowering the kinetic energy of a doped
hole “the mechanism of the K-S model”. In this paper we will suggest a possibility that a spatially
inhomogeneous distribution of Fermi-pocket-states and of large-Fermi-surface states may appear
due to the finite size effect when a temperature is higher than Tc . Finally a new phase diagram
for underdoped cuprates is proposed.
The organization of the present paper is the following: It consists of six parts. After Introduction,
at the beginning of the Section II we will first describe the essential features of the K-S model which
bears important characteristics born from the interplay of Jahn-Teller Physics and Mott Physics.
Then we will show from the calculated many-body-effects included energy bands that the key
features of the Fermi surfaces in underdoped cuprates are Fermi pockets. Further we will show
that the “Fermi arcs” observed in ARPES is not a portion of a large Fermi surface, but it should
be one of the edges of Fermi pockets in the nodal region. In Section III, on the basis of the manybody-effects included energy bands obtained from the K-S model we will predict the key features
of ARPES spectra and clarify the origin of the two-gap scenario proposed from the experimental
results by Tanaka et al [22]. In Section IV we will discuss the finite size effects on the Fermi surfaces
in cuprates, because a metallic state in the K-S model coexists with local antiferromagnetic (AF)
order constructed from the localized spins whose spin-correlation length is finite. In connection
with the finite size effects we will discuss a possibility of spatially inhomogeneous distribution of
Fermi-pocket-states and of large-Fermi-surface states. Taking account of the finite size effect, a
new phase diagram is proposed in Section V. Section VI is devoted to conclusion and concluding
remarks.
3
− 120 −
2
Electronic structure of a metallic state in underdoped
LSCO described by the K-S model
In this section we will first describe the electronic structure of a metallic state in underdoped
LSCO calculated by Kamimura and Suwa[6], emphasizing the important roles due to the interplay
of JahnTeller physics and Mott physics.
Figure 1 shows the energy-level landscape starting from the orbitally doubly-degenerate eg and
triply-degenerate t2g states of a Cu2+ ion in a CuO6 octahedron with octahedral symmetry embedded in La2 CuO4 at the left column. By the JT effect the Cu eg orbital state splits into a1g and b1g
orbital states, which form antibonding and bonding molecular orbitals of A1g and B1g symmetry
with the molecular orbitals constructed from the in-plane oxygen pσ and apical oxygen pz orbitals
in a CuO6 octahedron with tetragonal symmetry, respectively. These molecular orbitals are denoted by a∗1g , a1g , b∗1g and b1g , as shown at the middle column, where the asterisk * represents
the antibonding orbital. In an undoped case, 7 electrons occupy these molecular orbitals, so that
the highest occupied b∗1g state is half filled, resulting in an S = 1/2 state, where a b∗1g antibonding
orbital has mainly Cu dx2 −y2 character. Following Mott physics, we introduce the Hubbard U
interaction (U = 10eV) as a strong electron-correlation effect. Then the half-filled b∗1g state splits
into the lower and upper Hubbard bands denoted by L.H. and U.H. in the figure. The localized
electrons in the L.H. band give rise to the localized spins around the Cu sites. These localized
spins form the antiferromagnetic (AF) order by the superexchange interaction via intervening O2−
ions in undoped La2 CuO4 .
2.1
Anti-Jahn-Teller Effect
When Sr2+ ions are substituted for La3+ ions in LSCO, one may think intuitively that apical
oxygen (O2− ion) in the CuO6 octahedrons tend to approach toward central Cu2+ ions in order to
gain the attractive electrostatic energy. Theoretically it was shown by the first-principles variational
calculations of the spin-density-functional approach [3, 4] that the optimized distance between
apical O and Cu in La2−x Srx CuO4 (LSCO) which minimizes the total energy of LSCO decreases
with increasing Sr concentration. As a result the elongated CuO6 octahedrons by the Jahn-Teller
(JT) interactions shrink by doping holes. This shrinking effect against the Jahn-Teller distortion
is the “anti-Jahn-Teller effect” [5], as already explained in Introduction.
By this anti-Jahn-Teller effect, the energy separation between the two kinds of orbital states
which have been split originally by the JT effect becomes smaller. These two states are the a1g
anti-bonding orbital state |a∗1g and the b1g bonding orbital state |b1g shown in the third column
from the left in Fig. 1. Here the a∗1g anti-bonding orbital state is constructed by Cu dz 2 orbital
and six surrounding oxygen p orbitals including apical O pz -orbitals while |b1g orbital consists
of four in-plane O pσ orbitals with a small Cu dx2 −y2 component parallel to a CuO2 plane. The
spatial extensions of a∗1g and b1g orbitals are schematically shown in Fig. 2.
4
− 121 −
図 1: Energy-level landscape due to the interplay of Jahn-teller physics and Mott physics, starting
from the orbitally doubly-degenerate eg and triply-degenerate t2g orbitals of a Cu2+ ion in a
regular CuO6 octahedron with octahedral symmetry. The third column from the left shows the
splitting of Cu eg and t2g orbitals by the JT effect. The middle column shows the antibonding and
bonding molecular orbitals of A1g and B1g symmetry formed by Cu a1g and b1g orbitals and the inplane oxygen pσ and apical oxygen pz orbitals in an elongated CuO6 octahedron with tetragonal
symmetry. In the undoped case, these molecular orbitals accommodate 7 electrons. The right
column shows the splitting of the highest half-filled b∗1g state into L.H. and U.H. bands by the
Hubbard U interaction. The spins of localized holes in the L.H. band at the neighboring A and B
sites form an AF order. The energy in this figure is taken for electron-energy but not hole-energy
図 2: Spatial extension of |a∗1g antibonding orbital and |b1g bonding orbital in a CuO2 layer
5
− 122 −
2.2
The important role of two kinds of many-electron states
Kamimura and Eto calculated the lowest state energies of the many-electron states in which one
hole is added into a CuO6 octahedron embedded in LSCO [27, 28]. This means that an electron
is taken out from the system shown in Fig. 1. Hereafter we consider a hole as a doped carrier
instead of the electron picture. In this case, there appear two kinds of many-particle states called
multiplets. One case is that a dopant hole occupies an antibonding a∗1g orbital. In this case its
spin becomes parallel to a localized spin in the b∗1g orbital to get a spin-triplet state, because the
Hund’s coupling makes the spins of holes in a∗1g and b∗1g orbitals with different symmetry parallel.
As a result the energy of the a∗1g orbital state for a single hole decreases by the Hund’s coupling
energy of 0.5 eV. Thus this spin-triplet multiplet is called the “Hund’s coupling triplet” denoted
by 3 B1g , which is schematically shown in Fig. 3a.
The other case is that a dopant hole occupies a bonding b1g orbital, and its spin becomes
antiparallel to the localized spin in the antibonding b∗1g orbital, since the dopant and localized
holes occupy the orbitals of the same symmetry. As a result the energy of b1g orbital state for a
single hole decreases by the spin-singlet exchange interaction of 3.0 eV. This spin-singlet multiplet
may correspond to the “Zhang-Rice singlet” in the t-J model [29], and it is denoted by 1 A1g .
Zhang-Rice singlet state is schematically shown in Fig. 3b.
The energy separation between the two kinds of orbital states, the a∗1g anti-bonding orbital state
and the b1g bonding orbital state, becomes smaller by the anti-Jahn-Teller effect when the hole
concentration increases in the underdoped regime. Thus the Hund’s coupling triplet state and the
Zhang-Rice singlet state also appear at nearly the same energy in the underdoped region.
By the first-principles cluster calculations which take into account the Madelung potential due
to all the ions surrounding a CuO6 cluster in LSCO and also the anti-Jahn-Teller effect, Kamimura
and Eto showed that the lowest-state energies of these two multiplets are nearly equal when both
the Hund’s coupling exchange interaction and the spin-singlet exchange interaction are included
[27, 28]. Consequently the energy difference between the highest occupied orbital states a∗1g in
3
B1g multiplet and b1g in 1 A1g multiplet becomes only 0.1eV for the optimum doping (x = 0.15)
in La2−x Srx CuO4 , as will be shown in the subsection D.
2.3
The explanation of the K-S model by the picture of a “two-story
house model”
By using the results of the first-principles cluster calculations for the lowest state of CuO6
octahedron in LSCO by Kamimura and Eto mentioned in a previous subsection and assuming that
the localized spins form an AF order by the superexchange interaction via intervening oxygen ions,
Kamimura and Suwa [6] constructed a metallic state of LSCO for its underdoped regime. Since
this model has already been explained in detail in refs. 6 and 7, here we explain the key-features
of the model in a heuristic way using the picture of a two-story house model shown in Fig.4. In
this figure the first story of a Cu house with (yellow) roof is occupied by the Cu localized spins,
6
− 123 −
図 3: Schematic pictures of Hund’s coupling triplet 3 B1g (a) and Zhang-Rice singlet 1 A1g (b). The
long and short arrows represent the spins of localized and doped holes, respectively.
which form the AF order in the spin-correlated region by the superexchange interaction.
The second story in a Cu house consists of two floors due to the anti-JT effect, lower a∗1g floor and
upper b1g floor. The second story between neighboring Cu houses are connected by oxygen rooms
with (blue-color) roof, reflecting the hybridization of Cu d and O p orbitals. In the second story a
hole-carrier with up spin enters into the a∗1g floor at the left-hand Cu house due to Hund’s coupling
with Cu localized up-spin in the first story (Hund’s coupling triplet), as shown in the extreme left
column of the figure. By the transfer interaction marked by a long (red-color) arrow in the figure,
the hole is transferred into the b1g floor at the neighboring Cu house (the second from the left)
through the oxygen room, where the hole with up spin forms a spin-singlet state with a localized
down spin at the second Cu house from the left (Zhang-Rice singlet). The key feature of the K-S
model is that the hole-carriers in the underdoped regime of LSCO form a metallic state, by taking
the Hund’s coupling triplet and the Zhang-Rice singlet alternately in the presence of the local AF
order without destroying the AF order, as shown in the figure. Since the second story consists
of the two floors of different symmetry, the two-story house model represents the two-component
theory. As seen in Fig.4, the characteristic feature of the K-S model is the coexistence of the AF
order and a normal, metallic (or a superconducting) state in the underdoped regime. This feature
in the K-S model (two-component theory) is different from that of the single-component theory.
As seen in Fig.4, the wavefunctions of a hole-carrier with up and down spins have the following
phase relation:
Ψk↓ (r) = exp(ik · a)Ψk↑ (r).
(1)
Kamimura et al have shown that this unique phase relation leads to the d-wave superconductivity
7
− 124 −
図 4: Explanation of the K-S model by the picture of a two-story house model
[14, 15].
2.4
Effective Hamiltonian for the K-S model
The following effective Hamiltonian is introduced in order to describe the “K-S model” following
Kamimura and Suwa [6] (see also ref. [7]). It consists of four parts: The one-electron Hamiltonian
Hsing for a∗1g and b1g orbital states, the transfer interaction between neighboring CuO6 octahedrons
Htr , the superexchange interaction between the Cu dx2 −y2 localized spins HAF , and the exchange
interactions between the spins of dopant holes and of dx2 −y2 localized holes within the same CuO6
octahedron Hex . Thus we have
H = Hsing + Htr + HAF + Hex
†
εm Cimσ
Cimσ
=
i,m,σ
†
tmn Cimσ
Cjnσ + h.c.
+
i,j,m,n,σ
+J
Si · Sj +
i,j
Km si,m · Si ,
(2)
i,m
where εm (m = a∗1g or b1g ) represents the one-electron energy of the a∗1g and b1g orbital states,
†
and Cimσ are the creation and annihilation operators of a dopant hole with spin σ in the
Cimσ
i-th CuO6 octahedron, respectively, tmn the transfer integrals of a dopant hole between m-type
and n-type orbitals of neighboring CuO6 octahedrons, J the superexchange interaction between
the spins Si and Sj of dx2 −y2 localized holes in the b∗1g orbital at the nearest neighbor Cu sites
i and j (J > 0 for AF interaction), and Km the exchange integrals for the exchange interactions
between the spin of a dopant hole sim and the dx2 −y2 localized spin Si in the i-th CuO6 octahedron.
8
− 125 −
There are two exchange constants, Ka∗1g and Kb1g , for the Hund’s coupling triplet and Zhang-Rice
singlet, respectively, where Ka∗1g < 0 and Kb1g > 0. The appearance of the two kinds of exchange
interactions in the fourth term is due to the interplay of Mott physics and Jahn-Teller physics.
This is the key-feature of the K-S model.
The electron-electron interactions between doped hole-carriers are very weak due to two reasons:
One is a the low concentration of hole-carriers in the underdoped regime and the other is that the
wave functions of hole-carriers with up and down spins in a CuO6 octahedron occupy a∗1g and b1g
orbitals, respectively, as seen in equation (1) and Fig. 4. By these reasons we have negelected the
electron-electron interactions between doped holes in the effective Hamiltonian (2).
By replacing the localized spins Si ’s in Hex by their average value S in the mean-field sense,
we can calculate the change of the total energy upon moving of a hole from an a∗1g orbital state
in the Hund’s coupling spin-triplet at Cu site i to an empty b1g orbital state in the Zhang-Rice
spin-singlet at the neighbouring Cu site j. At the first step the hole moves from Cu site i to
infinity. The change of the total energy in the mean field approximation is equal to εa∗1g + 14 Ka∗1g .
At the second step the hole moves from infinity to an empty b1g orbital state at Cu site j to form
the Zhang-Rice singlet. The change of the total energy in the second step is equal to εb1g − 34 Kb1g .
As a result the change of the total energy by the transfer of the hole from the occupied a∗1g orbital
state at Cu site i to the empty b1g orbital state at Cu site j is
1
3
eff
εaeff
= εa∗1g + Ka∗1g − εb1g + Kb1g , .
(3)
∗ − εb
1g
1g
4
4
Here εaeff
and εbeff
represents the effective one electron energy of a∗1g and b1g orbital states
∗
1g
1g
eff
including the exchange interaction term Hex , respectively. Thus the energy difference (εaeff
∗ −ε
b1g )
1g
corresponds to the energy difference between a∗1g and b1g floors in the second story in Fig. 4.
Now let us estimate the energy difference, εaeff
- εbeff
, by using the values of the parameters in
∗
1g
1g
the effective Hamiltonian (2). The values of the parameters in equation (2) have been determined
in the case of LSCO in ref. [6] (see also ref. [7]). Those are: J = 0.1, Ka∗1g = −2.0, Kb1g = 4.0,
ta∗1g a∗1g = 0.2, tb1g b1g = 0.4, ta∗1g b1g = ta∗1g a∗1g tb1g b1g ∼ 0.28, εa∗1g = 0, εb1g = 2.6 in units of eV,
where the values of Ka∗1g and Kb1g are taken from the first principles cluster calculations for a CuO6
octahedron in LSCO [27, 28], and the values of tmn ’s are obtained from band structure calculations
[3, 4]. The energy difference of the one-electron energies between a∗1g and b1g orbital states in a
CuO6 octahedron for a certain value of x has been determined so as to reproduce the difference of
the lowest state energies between the Hund’s coupling spin-triplet state and the Zhang-Rice spinsinglet state for the same value of x in LSCO calculated by the Multi-Configuration Self-Consistent
Field (MCSCF) cluster calculations which includes the anti-JT-effect [6].
- εbeff
is 0.1 eV for the case of the optimum doping (x = 0.15).
Thus the calculated value of εaeff
∗
1g
1g
Then, by introducing the transfer interaction of ta∗1g b1g = 0.28, a coherent metallic state in the
normal phase is obtained under the coexistence with the local AF order for the underdoped regime.
This situation is schematically shown in Fig. 5, According to Kamimura and Eto [27, 28], the ground
state energy of the Hund’s coupling spin-triplet is nearly equal to that of the Zhang-Rice spinsinglet state for any value of x in the underdoped regime, so that the energy difference between
9
− 126 −
図 5: Energy diagrams of a∗1g orbital state and the b1g orbital state at the neighbouring Cu sites i
and j before and after the transfer of a hole from Cu site i to j, where the localized spins at i and
j sites are also shown. Energy in this figure is taken for the hole-energy.
the a∗1g orbital state and the b1g orbital state at the neighbouring Cu sites i and j, εaeff
- εbeff
, is
∗
1g
1g
almost zero in the underdoped regime.
2.5
Features of the many-body-effects included energy bands and Fermi
surfaces of underdoped LSCO coexisting with the AF order
In a previous subsection we have shown that the effective Hamiltonian (2) for the K-S model
can lead to a unique metallic state in the normal phase which results in of the coexistence of
a superconducting state and AF order below Tc . In 1994 Kamimura and Ushio calculated the
energy bands and Fermi surfaces of underdoped LSCO in the normal phase, based on the effective
Hamiltonian (2), by treating the fourth term Hex in the effective Hamiltonian (2) by the mean-field
approximation, that is by replacing the localized spins Si ’s by their average value S [12, 13]. Thus
the effect of the localized spin system was dealt with as an effective magnetic field acting on the
hole carriers. As a result they could separate the localized hole-spin system in the AF order and
the hole-carrier system from each other, and calculated the “one-electron type” energy band for a
carrier system assuming a periodic AF order. Here “one-electron type” is meant by the inclusion
of many-body-effects in the energy bands. That is, the exchange interactions between carrier’s and
localized spins are included by the mean field approximation.
In Fig. 6, the calculated many-body-effect included energy band structure for up-spin (or downspin) doped holes for LSCO is shown for various values of wave-vector k and symmetry points in
the antiferromagnetic (AF) Brillouin zone, where the AF Brillouin zone is adopted because of the
coexistence of a metallic state and the AF order, and it is shown at the left side of the figure. Here
10
− 127 −
図 6: The many-body-effect included band-structure for up-spin (or down spin) dopant holes. The
highest occupied band is marked by the #1 band (right) and the AF-Brillouin zone (left). The
∆-point corresponds to (π/2a, π/2a, 0), while the G1 -point corresponds to (π/a, 0, 0).
one should note that the energy in this figure is taken for electron-energy but not hole-energy.
Further the Hubbard bands for localized b∗1g holes which contribute to the local AF order are
separated and do not appear in this figure.
In the undoped La2 CuO4 , all the energy bands in Fig. 6 are occupied fully by electrons so that
La2 CuO4 is an antiferromagnetic Mott insulator, consistent with experimental results. In this
respect the present effective energy band structure is completely different from the ordinary LDA
energy bands [30, 31]. When Sr are doped, holes begin to occupy the top of the highest band in
Fig. 6 marked by #1 at ∆ point which corresponds to (π/2a, π/2a, 0) in the AF Brillouin zone.
At the onset concentration of superconductivity, the Fermi level is located just below the top of
the #1 band at ∆, which is a little higher than that of the G1 point. Here the G1 point in the AF
Brillouin zone lies at (π/a, 0, 0), and corresponds to a saddle point of the van Hove singularity.
Based on the calculated band structure shown in Fig. 6, Kamimura and Ushio [12, 13] calculated
the Fermi surfaces for the underdoped regime of LSCO. In Fig. 7 the Fermi surface structure of
hole-carriers calculated for x = 0.15 is shown as an example, where the Fermi surface (FS) consists
of four Fermi pockets of extremely flat tubes around ∆ point, (π/2a, π/2a, 0), and the other three
equivalent points (the nodal region) in the momentum space. The total volume of the four Fermi
pockets is proportional to the concentration of the doped hole-carriers. Thus the feature of Fermi
pockets constructed from the doped holes shown in Fig. 7 is consistent with Luttinger’s theorem
in the presence of AF order [32].
In 1996 to 1997 Mason et al [23] and Yamada et al [24] reported independently the magnetic
coherence effects on the metallic and superconducting states in underdoped LSCO by neutron
inelastic scattering measurements. Since then a number of papers suggesting the coexistence of
local AF order and superconductivity in cuprates by neutron and NMR experiments have been
published [33, 34, 35, 36, 37, 38].
The Fermi surface structure in Fig. 7 is completely different from that of the single components
theory, in which a Fermi surface is large. Recently Meng and his coworkers reported the existence
of the Fermi pocket structure in the ARPES measurements of underdoped Bi2 Sr2−x Lax CuO6+δ
11
− 128 −
図 7: The Fermi surface of hole-carriers for x = 0.15 calculated for the #1 band. Here two
kinds of Brillouin zones are also shown. One at the outermost part is the ordinary Brillouin zone
corresponding to an ordinary unit cell consisting of a single CuO6 octahedron and the inner part is
the Brillouin zone for the AF unit cell in LSCO. Here the kx axis is taken along ΓG1 , corresponding
to the x-axis (the Cu–O–Cu direction) in a real space.
(La-Bi2201) [16]. Their results are the clear experimental evidence for our prediction of the Fermipocket-structure for underdoped LSCO in 1994 [12].
It is interesting to see the effect of the interplay of Mott physics and Jahn-Teller physics, the
fourth term Hex in the effective Hamiltonian (2), on the Fermi-pockets as follows: By averaging the
character of wavefunction for each wave-vector k over the Fermi surface for x = 0.1 in LSCO, we
have obtained the mixing ratio of characters of the Hund’s coupling triplet state to the Zhang-Rice
singlet state in the metallic state at the Fermi level, which is found to be 1 to 5. Thus the existence
ratio of the Hund’s coupling triplet state in the metallic state at the Fermi level is only 20 %. This
value is close to the observed intensity ratio of the E c to the E ⊥ c polarization of x-ray at the
Cu L3 satellite peak for x = 0.1 in LSCO in the polarized x-ray absorption spectra reported by
Chen et al [39].
In 1997 Anisimov, Ezhov and Rice calculated the energy band structure of the ordered alloy
La2 Li0.5 Cu0.5 O4 by the LDA+U method [40], and they showed that a fairly modest reduction of
the apical Cu-O bond length is sufficient to stabilize the Hund’s coupling spin triplet state with
dopant holes in both b1g and a∗1g orbitals. Their calculated result supports the K-S model.
12
− 129 −
3
The characteristic features of ARPES spectra obtained
from the K-S model and the conclusion of the absence of
pseudogap
Recently considerable attention has been paid to the phenomenological idea of a pseudogap.
When a portion of the Fermi surface in cuprates were not seen in the ARPES experiments, an idea
of pseudogap was proposed as a kind of gap to truncate the Fermi surface in the single particle
spectrum [8, 9]. The disconnected segments of the Fermi surface are called “Fermi arc” [9, 19, 11].
Further ARPES experiments reported that such pseudogap develops below a temperature called
T ∗ which depends on the hole concentration x in the underdoped regime of cuprates, and thus we
write T ∗ (x) hereafter. The T ∗ (x) decreases with increasing the hole concentration x and disappears
at a certain concentration xo in the overdoped region [10]. In this section, on the basis of the K-S
model we clarify the origins of the pseudogap and of T ∗ (x).
3.1
Features of the calculated ARPES spectra, clarification of the observed two-gap scenario, and the absence of pseudogap
Following the K-S model, the Fermi surface in the underdoped regime is the Fermi pockets in
the nodal region as shown in Fig. 7. Here we project the 3D picture of the Fermi pockets in Fig. 7
on the kx -ky plane in the momentum space. The projected 2D picture of the four Fermi pockets
around ∆ point, (π/2a, π/2a), and the other three equivalent points in the momentum space is
shown on the antiferromagnetic Brillouin zone in Fig. 8(a). Below Tc the hole carriers in these
Fermi pockets form Cooper pairs, contributing to the formation of a superconducting state, and a
superconducting gap appears across the Fermi level. This feature is consistent with the Uemura
plot [41]. In Fig. 8(b) the d-wave node below Tc predicted by the K-S model [14, 15] is schematically
shown as dots, and the d-wave superconducting density of states (DOS) is schematically shown in
Fig. 8(c). Here one should note that AF order still coexists with a superconducting state below
Tc so that we can use the same antiferromagnetic Brillouin zone, as seen in Fig. 8(b).
On the other hand, in the antinodal region the states occupied by electrons which do not participate in the formation of superconductivity still exist below Tc . As an example of such states,
a state A is exemplified in Fig. 8(b), and the state corresponding to A above Tc is also shown in
Fig. 8(a). Then the real transitions of electrons from the occupied states, say the state A, below
the Fermi level εF in the #1 energy band in Fig. 6 to a free-electron state above the vacuum level
occur by photo-excitation both above and below Tc around the G1 -point, (π/a, 0, 0), and other
equivalent points in the momentum space. These transitions appear in the antinodal region in the
momentum space. Such transitions are illustrated in Fig. 9, where the density of states (DOS) ρ(ε)
calculated for the #1 energy band by Ushio and Kamimura [13] is shown. Since the intensity of
such photo-excited transitions is proportional to the density of occupied states at A below εF , we
can predict that such photo-excited transitions appear as a broad hump with a peak in ARPES
13
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図 8: The 2D projected Fermi pockets in the nodal region above Tc (a), their change to the d-wave
nodes below Tc (b) and the d-wave superconducting density of states (c.)
spectra, reflecting the density of occupied states at A below εF in the shaded area of Fig. 9.
Based on the situation shown in Fig. 8(a) to Fig. 8(c) and Fig. 9, we can predict the features
of ARPES spectra below Tc theoretically in the following way: A coherent peak due to the superconducting density of states appears at the nodal region around ∆ point while a broad hump with
a peak reflecting the DOS at the initial state in the photo-excited transition in Fig. 9 appears in
a wide region centered at the antinodal point G1 in the momentum space. The latter corresponds
to the real transitions of electrons from the occupied states below the Fermi level εF to a freeelectron state above the vacuum level, with the intensity proportional to the density of states at
the initial state of the transition, ρ(ε), below the Fermi level εF shown in Fig. 9, where εF varies
with the hole-concentration x, so that we write εF (x) hereafter. Thus we designate a broad hump
as “antinodal transitions”. This theoretical ARPES spectra are very similar to those reported by
Tanaka et al. for Bi2212 [22], where the experimental results revealed two distinct energy gaps
exhibiting different doping dependence.
From ARPES experiments for the “antinodal transitions” one can measure the energy difference
between the energy ε(A) of state A and the Fermi energy εF , |ε(A) − εF (x)|, shown in Fig. 9. Since
the G1 point at the edge of the antiferromagnetic BZ corresponds to a saddle-point singularity, we
may expect the appearance of a peak in the broad hump for the transition from the G1 point. In
this context we have calculated the doping dependence of the energy difference |ε(G1 ) − εF (x)| for
LSCO. Here we designate |ε(G1 ) − εF (x)| as “the antinodal transition energy”. The Fermi energy
in the undoped case, εF (0) ≡ εF (x = 0), lies at the right edge of ρ(ε) (DOS) in Fig. 9, and thus
the antinodal transition energy decreases with increasing x. Since a shape of DOS for the highest
conduction band does not depend on cuprate materials, we compare in Fig. 10a the calculated
doping dependence of the antinodal transition energy for LSCO (solid lines) with the experimental
results of “antinodal gap” for Bi2212 in ref. [22] which are shown as dots in the figure. As seen in
14
− 131 −
図 9: Schematic picture illustrating the real transitions of photo-excited electrons from state A
below the Fermi level εF to a free-electron state above the vacuum level. This is the origin of a
broad hump in the ARPES spectra, reflecting the density of occupied states shaded in this figure.
The broad hump appears in the antinodal region around the G1 -point in the momentum space.
図 10: Calculated ARPES spectra for LSCO and comparison with experimental results of Bi2212.
a, Calculated energy difference |ε(G1 )−εF (x)| as a function of the hole concentration x at T = 0K.
Experimental results by Tanaka et al [22] are shown by dots. b, Calculated T ∗ (x) is shown by a
solid line. c, Calculated temperature dependence of the antinodal transition energy for x = 0.05,
0.1, and 0.15. Experimental data by Norman et al [44] (triangle) and Lee et al [45] (square) are
shown.
15
− 132 −
Fig. 10a, the agreement between theory and experiment is remarkably good. Here we would like
to emphasize that there are no disposal parameters in the present case. From this quantitative
agreement we can conclude that among the observed two gaps below Tc , a gap associated with
antinodal regime corresponds to the real transitions of electrons from the occupied states below
the Fermi level to a free-electron state above the vacuum level while the other gap associated with
the near-nodal regime corresponds to the superconducting gap created on the Fermi pockets. From
the marvelous agreement between the present theory and experimental results by Tanaka and his
coworkers, we can conclude that the pseudogap is absent in cuprates.
Recently Yang et al [46] have suggested from their ARPES experiments on Bi2212 that the
opening of a symmetric gap related to superconductivity occurs only in the antinodal region and
that the pseudogap reflects the formation of preformed pairs, contrary to the ARPES experimental
results reported by Tanaka et al [22]. In the present theory we have clearly shown in Fig. 8
and Fig. 10 that, in the ARPES experiments a peak related to superconductivity appears only
in the nodal region and the spectra in the antinodal region correspond to the photoexcitations
from the occupied states below εF to a free-electron state above the vacuum. If the antinodal
region in ref. [46] means the region around the G1 point in the present paper, their suggestion is
different from our theoretical results. Finally we should remark that any proposed theory must
explain both the doping and temperature dependences of ARPES spectra in the underdoped regime
consistently. From this standpoint we will investigate the temperature dependence of the ARPES
spectra theoretically in the next subsection.
3.2
Physical meaning of T ∗ (x) and the temperature dependence of ARPES
spectra
For the purpose of calculating the temperature dependence of the “antinodal transition energy”,
first of all we would like to clarify the physical meaning of T ∗ (x). When a hole concentration x is
fixed at a certain value in the underdoped region and temperature increases beyond Tc , at a certain
temperature in the normal phase the local AF order constructed by superexchange interaction in
a CuO2 plane becomes destroyed thermally and thus the coexistence of a metallic state with the
Fermi pockets and of the local AF order will disappeargradually. As a result the K-S model will
not hold at that temperature in the underdoped regime. This temperature is defined as T ∗ (x).
Thus the phase of the Fermi pockets coexisting with the local AF order in the K-S model holds
only below T ∗ (x). We designate the phase of the Fermi pockets as the “small Fermi-surface (FS)”
phase. Hereafter it is abbreviated as the SF-phase. When temperature increases beyond T ∗ (x), a
system changes from the SF-phase to the electronic phase consisting of a large FS without the AF
order. We call this phase the large FS-phase, abbreviated as the LF-phase.
In this context we may consider that the T ∗ (x) represents a crossover from the SF-phase to the
LF- phase. Now let us introduce a quantity defining the difference between the free energies of the
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− 133 −
LF- and SF- phases:
∆F (T, x) ≡ FLF (T, x) − FSF (T, x),
(4)
where FLF (T, x) and FSF (T, x) are the free energies of LF- and SF- phases, respectively. Here the
free energy F (T, x) is defined as
F (T, x) = E(T, x) − T S(T, x),
(5)
where E(T, x) and S(T, x) are the internal energy and entropy of each phase, respectively. These
quantities are calculated from
∞
ερ(ε)f (ε, µ(x))dε,
(6)
E(T, x) =
−∞
and
S(T, x) = −kB
∞
[f (ε, µ(x)) ln f (ε, µ(x))
−∞
+ {1 − f (ε, µ(x))} ln {1 − f (ε, µ(x))}]
ρ(ε) dε ,
(7)
where µ(x) is a chemical potential of each phase and f (ε, µ(x)) the Fermi distribution function at
energy ε and chemical potential µ(x) . Then T ∗ (x) is defined by the equation
∆F (T ∗ (x), x) = 0.
(8)
Kamimura, Hamada and Ushio calculated the electronic entropies for the SF- and LF- phases of
LSCO [7]. According to their results the difference of electronic entropy between the SF- and LFphases increases with increasing the hole-concentration x in the underdoped regime. Using this
result we find that T ∗ (x) calculated from equation (8) decreases with increasing x, as shown in
Fig. 10b. In this calculation we have introduced two parameters T ∗ (x = 0.05) and xo , where xo
means the critical concentration that satisfies T ∗ (xo )=0, and T ∗ (x = 0.05) represents a quantity
related to the energy difference between the phase of doped AF insulator and the LF phase at
the onset concentration of the metal-insulator transition (x = 0.05). In Fig. 10b we have chosen
T ∗ (x = 0.05) to be 300K [47].
On the other hand, the physical meaning of T ∗ (xo ) = 0 can be explained in the following way:
When the hole concentration exceeds the optimum doping (x = 0.15) for LSCO and enters a slightly
overdoped region, the local AF order via superexchange interaction in a CuO2 plane begins to be
destroyed by an excess of hole-carriers. Thus the K-S model does not hold at a certain concentration
xo in the overdoped region, and hence the small FS in the K-S model changes to a large FS. Thus
T ∗ (x) vanishes at xo . From the analysis of various experimental results we choose xo = 0.17 for
LSCO [48, 49]. From this explanation we can say that the area below T ∗ (x) in the underdoped
regime represents the region in which the normal, metallic phase above Tc and the superconducting
phase below Tc coexist with the local AF order.
17
− 134 −
In this circumstance it is clear that the “antinodal transition energy” defined by |ε(G1 ) − εF (x)|
appears at temperatures below T ∗ (x) and vanishes at T ∗ (x). Now we calculate the temperature
dependence of the antinodal transition energy by using equations (4), (5) , (6) and (7). The calculated results for three concentrations x = 0.05, 0.10 and 0.15 of LSCO in the underdoped to
optimum-doped region are shown in Fig. 10c as a function of temperature, where the parameter
T ∗ (0.05) = 300K is used. As seen in the figure, the antinodal transition energy increases slightly
with temperature up to T ∗ (x) and vanished sharply at T ∗ (x). These calculated results are compared with experimental results of underdoped sample of Bi2212 in refs. [44] and [45], which are
indicated by triangle and square, respectively, in Fig. 10c. As seen in the figure, the agreement
between theory and experiment is remarkably good.
4
Spatially inhomogeneous distribution of Fermi-pocket-states
and of large-Fermi-surface-states due to the finite size effects
4.1
Finite size effects of a metallic state on Fermi surfaces
According to the results of neutron inelastic scattering experiments by Mason [23] and Yamada
[24], the AF spin-correlation length λs in the underdoped region of LSCO is finite. In the underdoped regime of LSCO it increases as the Sr concentration increases from x = 0.05 in LSCO,
the onset of superconductivity, and reaches a value of about 50Å or more at the optimum doping
(x = 0.15). In this subsection we discuss the effects of the finite size of the AF spin-correlation
length on the structure of Fermi pockets shown in Fig. 7. According to the K-S model in Fig. 4,
in the spin-correlated region a doped hole in the underdoped regime of LSCO can itinerate coherently by taking the a∗1g and b1g orbitals alternately in the presence of the local AF order without
destroying the AF order.
In the case of a finite spin-correlated region, one may think that there exist the frustrated
spins on the boundary between the spin-correlated region of the AF order and the region of the
“resonating valence bond” (RVB) state proposed by Anderson [1]. Here the frustrated spins mean
that the localized spins on the boundary are not in the AF order, but directed parallel to each
other. Suppose that one of frustrated spins of a parallel direction on the boundary has changed
its direction from parallel to antiparallel by the fluctuation effect in the 2D Heisenberg AF spin
system during the time of τs defined by τs ≡ /J, where J is the superexchange interaction (∼0.1
eV). In the time of τs , on the other hand, the hole-carriers at the Fermi level can move with the
Fermi velocity inside the spin-correlated region of the AF order. The traveling time of a doped
hole at the Fermi level over an area of the spin-correlation length is given by τF ≡ λs /vF , where
vF is the Fermi velocity of a doped hole at the Fermi level. In the case of underdoped LSCO,
τs is 6 × 10−15 sec. Since vF is estimated to be 2.4 × 104 m/sec from the dispersion of the #1
band in Fig. 6, τF is 2 × 10−13 sec for the underdoped region of x = 0.10 to x = 0.15 in LSCO,
18
− 135 −
図 11: A special case in which a doped hole reaches site 1 on the boundary and at the same time
one of the frustrated spins with up direction at site 2 on the boundary changes its direction from
up to down. (a) The doped hole with up spin reached site 1 occupies the a∗1g orbital, where the
localized spin at site 2 changes its direction from up to down. Then a doped hole in the a∗1g orbital
at site 1 moves to the b1g orbital at site 2 on the boundary and it forms Zhang-Rice singlet with
the localized down spin at site 2. (b) The Zhang-Rice singlet moves to site 3 in the region of the
RVB state like a quasi-particle in the t-J model.
where for the spin-correlation length λs at x = 0.15, we have chosen 50Å. Thus τF becomes much
longer than τs . As a result the frustrated spins on the boundary change their directions from
parallel to antiparallel before a hole-carrier in the spin-correlated region of the AF order reaches
the boundary. Thus a metallic state for a doped hole becomes much wider than the observed
spin-correlation length by its passing through the boundary with the mechanism of the K-S model.
4.2
Occurrence of the AF order in a metallic state to lower the kinetic
energy of doped holes
Concerning a finite system of cuprates, Hamada and coworkers [25] and Kamimura and Hamada
[26] tried to determine the ground state of the effective Hamiltonian (2) for the K-S model by
19
− 136 −
carrying out the exact diagonalization of the Hamiltonian (2) using the Lanczos method for a
two-dimensional (2D) square lattice system with 16 (4 × 4 ) localized spins with the presence of
one and two doped holes, respectively. As a result they clarified that, in the presence of holecarriers, the localized spins in a spin-correlated region tend to form an AF order rather than a
random spin-singlet state and thus that the hole-carriers can lower the kinetic energy by itinerating
in the lattice of AF order with the mechanism of the K-S model. In this way they have proved
that the coexistence of the metallic state and AF order makes the kinetic energy of hole-carriers
lower so that such coexistence state corresponds to the lowest energy state in the K-S model.
Thus generally, a hole-carrier in the spin-correlated region of the AF order can propagate through
the boundary of the spin-correlated region with the mechanism of the K-S model and hence a
region of a metallic state coexisting with the AF order becomes much wider than the observed
spin-correlated region. In fact, Kamimura and coworkers estimeted the length of a metallic region
at the optimum doping of LSCO to be about 300 Åfrom the Tc value at the optimum doping
[15]. Recently an idea similar to ours with regard to the lowering of the kinetic energy has been
proposed by Wrobel and coworkers who have shown that the lowering of the kinetic energy is the
driving mechanism to give rise to superconductivity [42, 43].
As a special case of destroying the coexistence of a metallic state and the local AF order on the
boundary, let us consider a case shown in Fig. 11, where a doped hole with up-spin reaches site 1
on a boundary when one of the frustrated spins with up direction at site 2 is about to change its
direction from up to down. Here the localized spins inside the boundary form the AF order while
those outside the boundary take the RVB state, as shown in Fig. 11, where the RVB state consists
of the nearest neighbor spin-singlet pairs (S = 0), allowing the singlet pairs to move. Thus the
doped hole with up spin at site 1 can move to site 2 by occupying the b1g orbital. As a result it
forms the Zhang-Rice singlet with a localized down spin at site 2. In this case the doped hole with
up spin in the Zhang-Rice singlet can not move to a∗1g orbital state at site 3 because the localized
spins at site 3 take the RVB state. Thus the coherent motion of a hole-carrier by taking the a∗1g
and b1g orbitals in the presence of the local AF order without destroying the AF order stops at
this moment.
Instead of it the Zhang-Rice singlet moves from site 2 to site 3 and to further sites in the region
of the RVB states like a quasi-particle in the t-J model [29], until a certain region of the RVB state
changes to the lower energy state of the local AF order to create the coexistence state of a metallic
state and the local AF order again. Thus we can say that an spatially inhomogeneous distribution
of Fermi-pocket-states and of large-Fermi-surface-states appears as the finite size effect when a
temperature is higher than Tc , and such distribution may change with time.
Our prediction with regard to the coexistence of Fermi-pocket-states and of large-Fermi-surfacestates above Tc is consistent with the experimental results by Meng et al [16] who recently reported
the coexistence of Fermi pockets and a large Fermi surface in Bi2 Sr2−x Lax CuO6+δ (La-Bi2201)
which has a similar structure to LSCO with regard to the array of CuO6 octahedrons. However,
when temperature becomes below Tc , the regions of the coexistence of a superconducting state and
20
− 137 −
図 12: A new phase diagram for LSCO
the local AF order becomes dominant, because the energy of such coexistence state is lower than
that of the hole-carriers in the RVB state, according to Hamada and his coworkers [25].
5
A new phase diagram calculated for underdoped cuprates
based on the K-S model
From the calculated results shown in Fig. 10a to c, we can construct the T vs x phase diagram
for LSCO by choosing T ∗ (x) as a phase boundary, as shown in Fig. 12. In this phase diagram we
have clarified the physical meaning of each area: There is no longer a pseudogap phase. Below
T ∗ (x) in the underdoped region the SF phase constructed from the Fermi pockets appears under
the coexistence of the local AF order. However, owing to various effects such as the mixing effects of
the SF and LF phases due to temperature, hole-concentration and the finite size effect of a metallic
state, each new phase is not sharply defined. For example, we have pointed out a possibility of the
spatially inhomogeneous distribution of Fermi-pocket-states and of large-Fermi-surface-states above
Tc . In such a way the LF-phase may be mixed into the SF-phase spatially and/or thermally even
below T ∗ (x). Thus, when temperature increases at a certain hole concentration in the underdoped
region, the shape of a Fermi pocket changes to a large Fermi surface gradually with increasing
temperature. This fact can explain the strange temperature-evolution of a Fermi arc observed
by Norman et al [9] and Kanigel et al [10] without introducing the pseudogap. Further, in the
superconducting phase below Tc indicated by the green color in Fig. 12, an s-wave component of
superconductivity originated from the LF-phase may be mixed into the d-wave superconductivity.
Such mixing effect was experimentally reported by Müller [50]. In this context we conclude that
T ∗ (x) represents a crossover from the SF-phase to the LF-phase rather than a phase boundary.
Further we can predict that the spin susceptibility shows the 2D-like antiferromagnetic features
21
− 138 −
mainly below T ∗ (x) while the Pauli-like temperature-dependent behavior above T ∗ (x). We find
that this prediction is also consistent with the experimental results for LSCO [49, 48]. In this
context it should be emphasized that the K-S model is shown to explain successfully not only
the ARPES experimental results [16, 22, 44, 45] but also a number of other experimental results
such as NMR results showing the coexistence of a superconducting state and antiferromagnetic
order [38], polarized X-ray absorption spectra [39, 51], site-specific X-ray absorption spectroscopy
[52], anomalous electronic entropy [53, 7], d-wave superconductivity [54, 55], etc., without introducing adjustable parameters. Theoretically the K-S model is also supported by LDA + U band
calculations [40], as already mentioned in Section IIE.
6
Conclusion and concluding remarks
In this paper we have shown on the basis of the K-S model how the interplay of Mott physics
and Jahn-Teller physics plays an important role in determining the superconducting as well as
the metallic state of underdoped cuprates. It was pointed out for underdoped cuprates that Mott
physics gives rises to the existence of local antiferromagnetic order due to the localized spins while
that the anti-Jahn-Teller effect as a central issue of Jahn-Teller physics creates the existence of two
kinds of orbitals parallel and perpendicular to a CuO2 plane which are energetically nearby. The KS model which bears important characteristics born from the interplay of Jahn-Teller Physics and
Mott Physics has led to the coexistence of the local AF order and a metallic states above Tc . This
coexistence has resulted in the occurrence of Fermi pockets. Further below Tc the superconductivity
and antiferromagnetism coexist, leading to the appearance of d-wave superconductivity even in the
phonon-involved mechanism, as was shown by Kamimura, Matsuno, Suwa and Ushio [14].
In connection with the interplay of Jahn-Teller physics and Mott physics, the following important
results have been obtained in this paper: It has been clarified on the basis of the K-S model that
a concept of pseudogap discussed theoretically [56, 57, 42] and reported by ARPES, STM and
tunneling experiments below T ∗ (x) in underdoped cuprates [11, 10, 58] is not necessary. We have
shown that the strange phenomena observed in the antinodal region are explained by the real
transitions of photo-excited electrons from the occupied states in the highest conduction band in
the antinodal region to a free-electron state above the vacuum level. In this context we conclude
that the concept of the pseudogap in the underdoped cuprates is no longer necessary. Further the
physical meaning of the T ∗ is not related to the pseudogap but it represents a crossover line from
the phase consisting mainly of the Fermi pockets in the normal state to the phase consisting of a
large Fermi surface.
Finally several remarks are made on the small Fermi surface and the shadow bands in the
underdoped regime of cuprates. In 1996 Wen and Lee developed a slave-boson theory for the t-J
model at finite doping, and showed that Fermi pockets at low doping continuously evolved into
the large Fermi surface at high doping concentrations [59]. Although their theoretical model is
different from the K-S model, it is interesting to find that they obtained a similar result to the
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− 139 −
prediction by the K-S model in 1994 with regard to the change from small FS to a large FS with
increasing the hole concentration. Recently a proposal was made to reconcile the experimental
result of the coexistence of antiferromagnetism and superconductivity [60]. Further, in relation to
the small Fermi surface the idea of a shadow Fermi surface was proposed as a replica of the main
Fermi surface transferred by Q = (π/a, π/a) by Kampf and Schrieffer theoretically [61] and then by
Aebi et al experimentally [62]. Checking the idea of the shadow FS experimentally, the observation
of shadow bands in the ARPES spectra have been reported [63, 64, 65, 66]. Responding to the
problems of shadow FS and shadow bands from the standpoint of the K-S model it should be
emphasized that the Fermi pockets in the metallic state calculated from the K-S model have been
derived by the two-component theory as the result of the interplay of Jahn-Teller physics and Mott
physics and thus the origin of Fermi pockets is different from that of a single-component theory.
Therefore, the Fermi pockets shown in Fig. 7 are neither the shadow Fermi surface nor related
to the shadow bands. Thus we conclude that the Fermi pockets in the present paper belong to a
new category of a small Fermi surface derived from the interplay of Jahn-Teller physics and Mott
physics.
Acknowledgments
We would like to thank Dr. Wei-Shen Lee, Prof. Atsushi Fujimori and Prof. Tomohiko Saitoh for
their valuable discussion on experimental results and Dr. Jaw-Shen Tsai for valuable comments
on the present work. This work was supported by Quantum Bio-Informatics Center in Tokyo
University of Science.
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26
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㕖ᓥ᧪ဳવዉߩવዉᯏ᭴ߦ㑐ߔࠆ⎇ⓥ
㊂ሶࠣ࡞ࡊ 㧔ℂቇㇱ╙৻ㇱ ᔕ↪‛ℂቇ⑼ ችᎹ⎇ⓥቶ㧕
ችᎹት㧔ಎᢎ㧕
Abstract. 㕖ᓥ᧪ဳવዉߩ᧚ᢱวᚑ߮ߘߩャㅍ․ᕈߦ㑐ߔࠆ⎇ⓥ࡞ࡀࡦ࠻߮ࠃ߅ޔ
ಽశ⎇ⓥ߇ⴕࠊࠇߚޕ㕖ᓥ᧪ဳߣߒߡᧄᐕᐲߩᚒ⎇ߩޘⓥኻ⽎ߦߒߚ᧚ᢱߪޔ㌃㉄ൻ‛
વዉޔ㋕♽વዉ㧔1111 ဳ ߮ࠃ߅ޔ11 ဳ㧕ߢࠆᧄޕᐕᐲߪᤓᐕᐲ߹ߢߩ⎇ⓥ
ࠍߐࠄߦㅴࠆߚߦ㌃㉄ൻ‛ߦኻߒߡߪ㔚ሶ࠼ࡊဳߦ߅ߌࠆવዉߣᒝ⏛ᕈߩ
ሽ⋧ߦߟߡߩ⎇ⓥ㧔න⚿᥏⢒ᚑ߆ࠄ࠻ࡦࡀ࡞ಽశ᷹ቯ㧕ߦဳࡊ࠼࡞ࡎߚ߹ޔኻߒ
ߡߪޔTl ♽વዉዉߩ 2 ጀ♽߮ 3 ጀ♽ߦߟߡߩ࠻ࡦࡀ࡞⎇ⓥ ߚ߹ޔBi2212 ♽ߦኻ
ߒߡߪኻᒻᚑ⋧↪ߦ㑐ߔࠆ⎇ⓥࠍ࠻ࡦࡀ࡞࠺࠲⸃ᨆߦࠃࠅⴕ⏛ޔ᳇⊛ߥ⋧
↪߇㊀ⷐߥᓎഀࠍߒߡࠆߎߣࠍࠄ߆ߦߒߚߦࠄߐޕ㋕♽વዉߦኻߒߡߪޔ11 ♽
ߦߟߡߪޔᱜᣇ⋧ߣᣇ⋧ߩ㔚ሶ⁁ᘒߦ㑐ߔࠆ⎇ⓥ⚛࡛࠙ߩߘߦࠄߐޔ㔓࿐᳇ਛߩࠕ
࠾࡞ലᨐߦ㑐ߔࠆ⎇ⓥߩߘߚ߹ޔᵷ↢‛⾰ߦ㑐ߔࠆ⎇ⓥࠍⴕߞߚ ߦࠄߐޕ1111 ♽ߦኻ
ߒߡߪ࡞ࡀࡦ࠻ޔಽశ⎇ⓥࠍⴕࠍ♽࠼ࡦࡃ࠴࡞ࡑߪߡ߅ߦ♽ߩߎޔᤋߒߡ 2 ⒳㘃
ߩવዉࠡࡖ࠶ࡊ߇ሽߔࠆߎߣࠍࠄ߆ߦߒ߽࠭ࠗࠨࡊ࠶ࡖࠡߩߘߦࠄߐޔࠄ߆ߦ
ߒߚޕ
1㧚ߪߓߦ
㊂ሶࠦࡦࡇࡘ࠲ߢߪޔ㊂ሶࡆ࠶࠻ߦࠃࠅ 1 ࡆ࠶࠻ߦߟ߈ 0 ߣ 1 ߩ୯ࠍછᗧߩഀวߢ㊀ߨวࠊߖ
ߡᜬߔࠆߎߣ߇น⢻ߢࠆߚޔ㨚㊂ሶࡆ࠶࠻ߢࠇ߫ 2n ߩ⁁ᘒࠍหᤨߦ⸘▚ߢ߈ࠆߚߦޔ
ߩᦨㅦߩࠬࡄࠦࡦࡇࡘ࠲ߢ߽⸃ߌߥࠃ߁ߥ⸘▚ߢ߽ޔ㕖Ᏹߦ⍴ᤨ㑆ߢߎߥߔߎߣ
߇น⢻ߦߥࠆߩߘޔߢߎߎޕၮᧄ⚛ሶߢࠆ㊂ሶࡆ࠶࠻ߪޔၮᧄ⊛ߦߪߤߩࠃ߁ߥ㊂ሶੑḰ♽
ߢ߽ታߢ߈ࠆߚߦޔવዉ⚛ሶߢ߽㊂ሶࡆ࠶࠻⎇ⓥߪ⋓ࠎߦⴕࠊࠇߡࠆߩߎޕ᭽ߥ㊂ሶࡆ
࠶࠻⎇ⓥߢߪߢߎߘޔ↪ߔࠆવዉ᧚ᢱߩ‛ᕈ⹏ଔ߮᧚ᢱ㐿⊒⎇ⓥ߽㕖Ᏹߦ㊀ⷐߦߥࠆߘޕ
ࠇࠁ߃ޔᚒߪޘሽߔࠆવዉ᧚ᢱߩਛߢ㜞વዉ⥃⇇ᐲ TC ࠍ㌃㉄ൻ‛વዉߣ㋕♽વ
ዉߩ‛ᕈ⎇ⓥࠍⴕߞߚޕ㌃㉄ൻ‛㜞᷷વዉߢߪޔ㧔c-1㧕ࠢࡄኻࠍᒻᚑߔࠆߚߩ㔚ሶ
㑆ᒁജ⋧↪ߩḮߦㄼࠆ⎇ⓥߣߒߡࠗࡁࠗޔᎿ⑼ᄢቇߩࠣ࡞ࡊߣߩห⎇ⓥߢ Bi2212
ߩࡉࠗࠢធวᴺߦࠃࠆ࠻ࡦࡀ࡞વዉᐲ࠺࠲⸃ᨆ(ޔc-2)㌃㉄ൻ‛♽ߩጀᢙଐሽᕈࠍࠄ߆ߦߔ
ࠆ⋡⊛ߢ Tl12(n-1)♽㧔n=2,3㧕ߩ࠻ࡦࡀ࡞ಽశ⎇ⓥ߅ࠃ߮ޔn=2 ߦኻߒߡߪ࠼ࡇࡦࠣଐሽᕈࠍ
⺞ߴ(ޔc-3)㔚ሶ̆ࡎ࡞㧔㕖㧕ኻ⒓ᕈࠍ⺞ߴࠆ⋡⊛ߢޔ㔚ሶ࠼ࡊ♽ߩන⚿᥏⢒ᚑ߮ߘߩ࠻ࡦ
ࡀ࡞ಽశ⎇ⓥࠍⴕߞߚߦࠄࠇߎޕઃߌട߃ߡޔ㋕♽વዉߢߪ(F-1)TC ߇Ყセ⊛㜞 1111 ♽
ߩ࠻ࡦࡀ࡞વዉᐲߩ࠼ࡇࡦࠣଐሽᕈߦ㑐ߔࠆ⎇ⓥ(ޔF-2)᭴ㅧ߇ᦨ߽ࠪࡦࡊ࡞ߢࠆ 11 ♽ߩ࡛
࠙⚛㔓࿐᳇ࠕ࠾࡞ലᨐ߮ߎߩ♽ߩᱜᣇ⋧ߣᣇ⋧ߩ㔚ሶ⁁ᘒߦ㑐ߔࠆ⎇ⓥࠍⴕߞߚޕ
2㧚⚿ᨐ
(c-1) Eliashberg Analysis of Tunneling Experiments㧦Support for the Pairing Glue Hypothesis
ߎߩ⎇ⓥߪࠗࡁࠗޔᎿ⑼ᄢቇߣߩห⎇ⓥߢࠅޔᐢ doping 㗔ၞߦࠊߚࠆ Bi2212 න⚿᥏
ߩ SIS ࡉࠗࠢធวᴺߦࠃࠆ࠻ࡦࡀ࡞࠺࠲ࠍ modified Eliashberg ⸃ᨆᴺࠍ↪ߡኻᒻᚑ⋧
↪ߩḮࠍ⺞ߴࠆ⎇ⓥ߇ⴕࠊࠇߚޕ㕖ኻⷺࡍࠕࡦࠣ⥄Ꮖࠛࡀ࡞ࠡߩ⯯ㇱ Im)(Z)ߣᚒߩޘ
ታ㛎࠺࠲ࠍౣߔࠆߚߦߞߚ⋧↪ߩࠬࡍࠢ࠻࡞㑐ᢙD2F(Z)߇Ყセߐࠇޔ⍦⋫߇ߥߎ
ߣࠍ␜ߔߎߣ߇ߢ߈ߚᦨߚ߹ޕㆡ࠼ࡊ Bi2212 ߩ࠻ࡦࡀ࡞વዉᐲ⚿ᨐ߳ߩ㩖㨲㨹㩍㨲㩧㩂㩨ߦࠃࠅ᳞
ࠄࠇߚኻⷺ⥄Ꮖࠛࡀ࡞ࠡߩታㇱ Re6(Z)߇ ARPES ߆ࠄⓍ߽ࠄࠇߚ⥄Ꮖࠛࡀ࡞ࠡߣ߆ߥࠅ㘃
ૃߒߚ߽ߩߦߥࠆߎߣࠍࠄ߆ߦߒߚ⸃ߩࠄࠇߎޔߡߒߘޕᨆࠃࠅ߇߉ࠄំࡦࡇࠬޔኻᒻᚑ⋧
↪ߦ㊀ⷐߥᓎഀࠍᨐߚߒߡࠆߎߣࠍ␜ߔߎߣ߇ߢ߈ߚ⚿ߩࠄࠇߎޕᨐߪ Phys.Rev.Lett.߳ᛩ
Ⓜ੍ቯߢࠆޕ
1
− 144 −
(c-2) ㌃㉄ൻ‛♽ߩવዉࠡࡖ࠶ࡊ'ߩጀᢙଐሽᕈ㧦Tl12(n-1)♽㧔n=2,3㧕ߩ࠻ࡦࡀ࡞ಽశ⎇ⓥ
Tl ♽㌃㉄ൻ‛વዉߪޔCuO2 㕙ߩᐔမᕈࠁ߃ߦᧄ⾰⊛ߥ CuO2 㕙ߩᕈ⾰ࠍ⺞ߴࠆߢℂᗐ⊛
ߥ ‛ ⾰ ߢ ࠆ ޔ ߢ ߎ ߘ ޕᚒ ߪ ޘTlBa2CuO4.5+G(Tl1201) ޔTlBa2CaCu2O6.5+G (Tl1212) ޔ
TlBa2Ca2Cu3O8.5+G (Tl1223)(㜞วᚑ)ޔTlBa2Ca3Cu4O10.5+G(Tl1234)ߩ᧚ᢱวᚑߒࡀࡦ࠻ߩߘޔ
࡞વዉᐲߩ࠼ࡇࡦࠣଐሽᕈޔ᷷ᐲଐሽᕈࠍ⺞ߴߚ⚿ߩߘޕᨐ࡞ࡀࡦ࠻ޔવዉᐲߪ Tl1212ޔTl1223ޔ
Tl1234 ߣ߽ߦࠡࡖ࠶ࡊߩᄖߦ dip ᭴ㅧࠍߒޔdip ߪዋߥߊߣ߽ࡎ࡞࠼ࡊဳ㌃㉄ൻ‛વ
ዉߦㅢߩ․ᓽߢࠆߎߣ ߚ߹ޔTl1223ޔTl1234 ߢߪ⚿᥏ቇ⊛ߦ㕖╬ଔߥ Inner Plane(IP)
ߣ Outer Plane(OP)ߦ࿃ߔࠆ 2 ⒳㘃ߩᄢ߈ߐߩࠡࡖ࠶ࡊ߇ࠆߎߣࠍߒߚ৻ߩߘޕᣇߢޔ
2 ⒳㘃ߩ㕖╬ଔߥ CuO2 㕙ࠍߒߡߥ Tl1212 ߦ߅ߡߪ⸘⛔ߩࡊ࠶ࡖࠡޔಽᏓߪ࡞ࠣࡦࠪޔ
ࠡࡖ࠶ࡊߩ᭽⋧ࠍ␜ߒߡࠆߎߣࠍࠄ߆ߦߒߚߩࠄࠇߎޕᚑᨐߪޔn>3 ߩᄙጀ♽㌃㉄ൻ‛ߦ߅
ߌࠆ࡙࠾ࠢߥᝄࠆ⥰ߢࠆߎߣࠍࠄ߆ߦߒߡ߅ࠅޔᓟࡑ࡞࠴ࡃࡦ࠼ߣ㜞᷷વዉߩ㑐ଥ
ࠍࠄ߆ߦߔࠆߎߣߦࠃࠅࠆߥࠄߐޔℂ⸃߇ㅴߣᦼᓙߐࠇࠆߩࠄࠇߎޕᚑᨐߪ Physica C ߦႎ
๔ߐࠇߚޕ
Tl1212(opt-109K)
Tl1223 (nearly opt-126K)
Tl1234(TC~120K)
OP
Probability [%]
P ro b a rb ility [% ]
P ro b a rb ility [% ]
OP
IP
OP
' [meV]
IP
OP
IP
' [meV]
' [meV]
Fig.4 Tl12(n-1)n ߩવዉࠡࡖ࠶ࡊߩ⛔⸘ಽᏓߩጀᢙଐሽᕈ
2
− 145 −
(c-3) 㔚ሶ࠼ࡊဳ㌃㉄ൻ‛વዉߩන⚿᥏⢒ᚑ߮ߘߩ࠻ࡦࡀ࡞ಽశ⎇ⓥ
ࡎ࡞࠼ࡊဳ㌃㉄ൻ‛ߦ߅ߡߪޔ⒎ᐨࡄࡔ࠲߇ d ᵄኻ⒓ᕈߢࠆߎߣޔવዉࠡࡖ࠶ࡊ
ߩ⇣Ᏹߥ࠼ࡇࡦࠣଐሽᕈࡊ࠼࠳ࡦࠕޔ㗔ၞߢߩ⇼ࠡࡖ࠶ࡊߥߤႎ๔ߐࠇߡࠆ߇ޔ㔚ሶ࠼
ࡊဳ㌃㉄ൻ‛ߩႎ๔ߪࡎ࡞࠼ࡊဳߣᲧセߔࠆߣዋߥߊޔㅀߩὐߦ߅ߡ߹ߛ⺰ਛߢ
ࠅ߹ߔޔߢߎߘޕ㔚ሶ࠼ࡊဳ㌃㉄ൻ‛વዉߩ FZ ᴺߦࠃࠆන⚿᥏⢒ᚑ߮ߘߩ࠻ࡦࡀ࡞ಽ
శ⎇ⓥࠍⴕߞߚޕPr1-xLaCexCuO4㧔PLCCO㧕ߩන⚿᥏⹜ᢱߦኻߒߡ࠻ࡦࡀ࡞ಽశ᷹ቯࠍⴕߞߡ
߈ߡࠆ⚿ߩߘޕᨐࠢࡇࠬࡦ࠲ࠢ࠳ࡦࠦޔ⟎ߢߚવዉࠡࡖ࠶ࡊࠨࠗ࠭ VP ߩ࠼ࡇࡦࠣ
ଐሽᕈߪ TC ߦ⋧㑐ߒߡࠆߎߣ߇ಽ߆ߞߚޔߚ߹ޕᒝ⏛ᕈߣવዉߩሽ߇␜ໂߐࠇߡࠆ
ࠕࡦ࠳࠼ࡊ㗔ၞߩ⹜ᢱ Pr0.91LaCe0.09CuO4 ߩ࠻ࡦࡀ࡞ࠬࡍࠢ࠻࡞ߦ߅ߡࠬࡦ࠲ࠢ࠳ࡦࠦޔ
ࡇࠢ߆ࠄ⚂ 20mV ઃㄭߦ TN ߦ㑐ㅪߔࠆน⢻ᕈ߇ࠆᡆࠡࡖ࠶ࡊߩࠃ߁ߥ᭴ㅧࠍ᷹ⷰߦᚑഞߒ
ߚᧄޔߢߎߘޕᐕᐲߪ RE1-85Ce0-15CuO4(RE:Nd, Pr)ߩන⚿᥏ࠍ TSFZ ᴺߢ⢒ᚑߦᚑഞߒࠗࡐޔ
ࡦ࠻ធวߦࠃࠅ࠻ࡦࡀ࡞ಽశ᷹ቯࠍⴕߞߚ࡞ࡀࡦ࠻ޕಽశߩ⚿ᨐߪ߹ߛ੍⊛ߥᚑᨐߒ߆ᓧࠄࠇ
ߡߥ߇ߣ♽ࡊ࠼࡞ࡎޔห᭽ߦࡁ࠳࡞ࠡࡖ࠶ࡊࠍߒߡࠆน⢻ᕈ߇㜞ߎߣࠍ␜ߔ⚿
ᨐ߇ᓧࠄࠇߡࠆ ߚ߹ޕPLCCO ߩ᷷ᐲଐሽᕈߩᚑᨐߪ PhysicaC ߦႎ๔ߐࠇߚޕ
(F-1) TC ߇Ყセ⊛㜞 1111 ♽ߩ࠻ࡦࡀ࡞વዉᐲߩ࠼ࡇࡦࠣଐሽᕈߦ㑐ߔࠆ⎇ⓥ
ᤓᐕᐲߦᒁ߈⛯߈ޔ㋕♽વዉ NdFeAsO1-y㧔Nd1111㧕
ߩ࠻ࡦࡀ࡞ಽశ⎇ⓥ߅ࠃ߮ Nd ࠍ La ߦ⟎߈឵߃ߚ⹜ᢱߦ
߅ߌࠆ࠻ࡦࡀ࡞᷹ቯ߽ⴕߞߚ⚿ߩߘޕᨐޔᤓᐕᐲႎ๔ߒߚ
ࠃ߁ߦ ߩߎޔ1111 ♽ߦ߅ߡߪޔዋߥߊߣ߽ 2 ⒳㘃ߩ
વዉࠡࡖ࠶ࡊ߇ሽߔࠆߎߣߪ㑆㆑߇ߥߐߘ߁ߢࠆߎ
ߣࠍࠄ߆ߦߒߚޔߦࠄߐޕዊߐߥࠡࡖ࠶ࡊߩࠨࠗ࠭ߪ
BCS ߢ੍ᦼߐࠇࠆࠨࠗ࠭એਅߢࠅޔ㋕♽વዉߦ߅
ߡߪᄢ߈ߥࠡࡖ࠶ࡊ߇વዉ TC ࠍᡰ㈩ߒߡࠆน⢻ᕈ߇
㜞ߣᕁࠊࠇࠆᚑᨐ߇ᓧࠄࠇߚޕ
(F-2) ᭴ㅧ߇ᦨ߽ࠪࡦࡊ࡞ߢࠆ 11 ♽ߩ࡛࠙⚛㔓࿐᳇ࠕ
࠾࡞ലᨐ
Fig.5 ࡛࠙⚛㔓࿐᳇ࠕ࠾࡞ߐࠇߚ
㋕♽વዉߩਛߢੑᰴరጀ⁁᭴ㅧࠍߔࠆવዉߣߒߡ
♽ߩᛶ᛫₸ߩ᷷ᐲଐሽᕈ
FeSe(TC=8K) ߇⊒ߐࠇߚޕFeTe ߪ FeSe ߣห᭽ߩ⚿᥏
᭴ㅧ(ᱜᣇ᥏, ⓨ㑆⟲ P4/nmm) ࠍߒߡࠆ߇ޔવዉࠍ␜ߐߥ ߒ߆ߒޕTe ߩ৻ㇱࠍ S ߿
Se ߢ⟎឵ߒߡߊߣ⏛ޔ᳇ォ⒖߇ᛥߐࠇޔવዉ߇⊒ߔࠆߎߣ߇⏕ߐࠇߡࠆߩߎޕ
FeTe0.8S0.2 ߪ⹜ޔᢱࠍߒߚ⋥ᓟߪવዉࠍ␜ߐߥ߇ᤨޔ㑆߇⚻ㆊߔࠆߣવዉ߇⊒ߒޔ
ᤨ㑆ߩ⚻ㆊߦᓥ TC ߇ᓢߦޘߔࠆߎߣ߿ޔ᳓߿߅ḡ߿߅㈬ߦᶐߔߣવዉߦߥࠆߎߣ߽⍮
ࠄࠇߡࠆ ߚ߹ޕFeTe0.8S0.2 ߦ㉄⚛ࠕ࠾࡞ࠍᣉߔߣવዉ߇⊒ߒޔFeSe ߣหߓࠃ߁ߥᛶ᛫
₸ߩ᷷ᐲଐሽᕈࠍߣࠆߎߣ߇ႎ๔ߐࠇߡࠆ߇⚛㉄ࠅࠃࠇߎޕਛߦࠅㄟߎߣߢ⚿᥏᭴ㅧ߇߭
ߕߺޔવዉ߇⊒↢ߔࠆߣ⠨߃ࠄࠇࠆޔ߫ࠄߥࠆߔߣߛ߁ߘߒ߽ޕේሶ߇ጀ㑆ߦࠅㄟߎߣߢޔ
߭ߕߺ߇↢ߓવዉ߇⊒↢ߔࠆߪߕߢࠆߒ߆ߒޕ㔚᳇ൻቇ⊛ߥᣇᴺߢ FeSe ࠴࠙ࡓࠗࡦ࠲
࡚ࠞࠪࡦࠍᣉߒߚ߽ߩߪવዉォ⒖᷷ᐲ߇ߔࠆߣ߁ႎ๔ߪߐࠇߡߥޕ㓁ࠗࠝࡦߢ
ࠆ Li ߢߪᄌൻ߇ߥߩߦኻߒࡑࠗ࠽ࠬ㧞ଔࠍߣࠆ㒶ࠗࠝࡦߩ㉄⚛߇߹ࠇࠆߣ⊛ߥᄌൻ߇
ࠆߎߣ߆ࠄޔหߓ㒶ࠗࠝࡦߢࠆ࡛࠙⚛ࠍ߹ߖࠆߣߤࠎߥലᨐ߇ࠆ߆ߣ⠨߃ታ㛎ࠍⴕߞߚޕ
ߘߒߡᚒޔߪޘ࿕⋧ᔕᴺߢߐࠇߚ FeTe0.8S0.2 ߦ࡛࠙⚛㔓࿐᳇ߢߩࠕ࠾࡞ലᨐࠍᣉߒߚ
IxFeTe0.8S0.2 (x ߪ࡛࠙⚛ߣ FeTe0.8S0.2 ߩࡕ࡞Ყ)ޕas-grown ߢ FeTe0.8S0.2 ࠍߒޔⓨ᳇ਛߦ
⟎ߒߡ⚻ㆊᣣᢙ߇ 27 ᣣ⋡ߩ߽ߩߪ TzeroC =4K ߢߞߚ߇ޔIxFeTe0.8S0.2 ࠍߒߚ⋥ᓟߦ᷹
ቯߒߚ߽ߩߪ TzeroC =8K ࠍ␜ߒߡ߅ࠅޔTC ߇ะߔࠆߎߣ߇ࠊ߆ߞߚޔߚ߹ޕIxFeTe0.8S0.2 ߣ
㉄⚛ࠕ࠾࡞ࠍᣉߒߚ FeTe0.8S0.2 ߪਔᣇߣ߽ TzeroC =8K ߹ߢߒߡ߅ࠅૃߚࠃ߁ߥᛶ᛫₸ߩ
᷷ᐲଐሽᕈߩᝄࠆ⥰ࠍ␜ߔߎߣࠍࠄ߆ߦߒߚޕ
3
− 146 −
References
(1) Eliashberg Analysis of Tunneling Experiments: Support for the Pairing Glue
Hypothesis, O. Ahmadi, L. Coffey, J.F. Zasadzinski, N. Miyakawa, L. Ozyuzer, to be
submitted to Phys. Rev. Lett.
(2) Electron correlation in the FeSe superconductor studied by bulk-sensitive
photoemission spectroscopy, A. Yamasaki, Y. Matsui, S. Imada, K. Takase, H. Azuma, T.
Muro, Y. Kato, A. Higashiya, A. Sekiyama, S. Suga, M. Yabashi, K. Tamasaku, T.
Ishikawa, K. Terashima, H. Kobori, A. Sugimura, N. Umeyama, H. Sato, Y. Hara, N.
Miyakawa,
S.
I.
Ikeda,
Phys.
Rev.
B
82,
184511
(2010),
DOI:
10.1103/PhysRevB.82.184511
(3) Tunneling spectroscopy of an optimally-doped TlBa2CaCu2O6.5+G with TC~109K, S.
Kawashima, T. Inose, S. Mikusu, K. Tokiwa, T. Watanabe and N. Miyakawa, Physica C
(2010), doi:10.1016/j.physc.2010.02.010
(4) Temperature dependence of tunneling conductance on an overdoped Pr0.82LaCe0.18CuO4
with TC~16K, M. Minematsu, M. Fujita, K. Yamada, N. Miyakawa, Physica C (2009),
doi:10.1016/j.physc.2009.12.011
(5) Superconductivity on FeSe synthesized by various sintering temperature, N. Umeyama,
K. Takase, S. Hara, S. Horiguchi, A. Tominaga, H. Sato, Y. Hara, N. Miyakawa, S.I.
Ikeda, Physica C (2010),doi:10.1016/j.physc.2010.05.241
(6) Structural and physical properties of FeSe crystals fabricated by the chemical vapor
transport method, Y. Hara, K. Takase, A. Yamasaki, H. Sato, N. Miyakawa, N.
Umeyama, S.I. Ikeda, Physica C (2010), doi:10.1016/j.physc.2010.02.021
(7) PROBING THE SUPERCONDUCTING GAP FROM TUNNELING CONDUCTANCE
ON NdFeAsO0.7 WITH TC=51K, N. MIYAKAWAA, M. MINEMATSUA, S. KAWASHIMAA,
K. OGATAA, K. MIYAZAWAB,C, H. KITOC, P. M. SHIRAGEC, H. EISAKIC, A. IYOC,
Journal of Superconductivity and Novel Magnetism 23 (2010) pp.575-578, DOI
10.1007/s10948-010-0689-9
4
− 147 −
ᩰሶ᭴ㅧࠍઃടߒߚવዉ-ඨዉ-વዉធวߦ߅ߌࠆ
ࠕࡦ࠼ࠛࡈࡐࡠࡦߩャㅍ․ᕈ
㩷 ㊂ሶ䉫䊦䊷䊒㩷 䋨ℂቇㇱ╙৻ㇱᔕ↪‛ℂቇ⑼㩷 㜟ᩉ⎇ⓥቶ䋩㩷
੫ᄥ㇢㧔ഥᢎ㧕
Abstract. ᩰሶ᭴ㅧ䉕ઃട䈚䈢વዉ-ඨዉ-વዉ(S-Sm-S)ធว䈱ャㅍ․ᕈ䉕⺞
䈼䈢䇯․ቯ䈱ᵄᢙ 1.77 GHz 䈱䊤䉳䉥ᵄᾖਅ䈮䈍䈇䈩䇮ᓸಽᛶ᛫䈲㔚䈱㑐ᢙ䈫䈚䈩ᝄേ
䈜䉎䇯䊤䉳䉥ᵄ(RF ᵄ)䈮䉋䈦䈩㑆ធ⊛䈮ബ䈘䉏䈢㐳ᵄ㐳㖸㗀䊐䉤䊉䊮䈫વዉḰ☸ሶ䈫䈱ᒝ
䈇⚿ว䈮䉋䈦䈩ᣂ䈚䈇ⶄว☸ሶ䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䈏ᒻᚑ䈘䉏䈢䈫⠨䈋䉌䉏䉎䇯
1㧚ߪߓߦ
વዉ-ඨዉ-વዉ(S-Sm-S)ធว䈮䈍䈇䈩䈲䇮વዉ㔚ᵹ䈏ᵹ䉏䈭䈒䈩䉅ᄙ㊀䉝䊮䊄䊧䊷䉣䊐
䈭䈬ᄙ᭽䈭⽎䈏⎇ⓥ䈘䉏䈩䈐䈩䈇䉎䇯䈖䈖䈪ᚒ䇱䈲㐳ᵄ㐳㖸㗀䊐䉤䊉䊮䈫વዉḰ☸ሶ䈫䈱ᒝ䈇
⚿ว䈮䉋䈦䈩ᒻᚑ䈘䉏䈢ᣂ䈚䈇ⶄว☸ሶ䇮䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䈮䈧䈇䈩ႎ๔䈜䉎䇯䈖䉏䉁䈪䊘䊷䊤
䊨䊮䈲䇮㖸㗀䊐䉤䊉䊮䈏䉎ᄌᒻ䊘䊁䊮䉲䊞䊦䈱䉋䈉䈭㔚ሶ-ᩰሶ⋧↪䇮䉅䈚䈒䈲䉟䉥䊮⚿᥏ਛ䈮䈍䈔
䉎䉪䊷䊨䊮⋧↪䈱䉋䈉䈮䇮Ᏹવዉ㊄ዻਛ䉅䈚䈒䈲⺃㔚ਛ䈪䈱䉂⍮䉌䉏䈩䈇䈢䇯৻ᣇવዉਛ䈪
䈲䇮䊐䉤䊉䊮䉕䈚䈢㔚ሶ㑆ᒁജ⋧↪䈏䉪䊷䊌䊷ኻᒻᚑ䈱ⷐ࿃䈫䈚䈩ฬ䈪䈅䉎䈔䉏䈬䉅䇮䉪䊷䊌䊷
ኻએᄖ䈱䉨䊞䊥䉝䈏䊐䉤䊉䊮䈫⚿ว䈜䉎䈫䈲⠨䈋䉌䉏䈩䈖䈭䈎䈦䈢䇯
2㧚ታ㛎
࿑ 1 ᩰሶ᭴ㅧࠍઃടߒߚવዉ-ඨዉ-વዉធวߩᮨᑼ⊛ᢿ㕙࿑
࿑ 1 䈮ᩰሶ᭴ㅧ䉕ઃട䈚䈢 S-Sm-S ធว䈱ᮨᑼ⊛ᢿ㕙䉕␜䈜䇯S-Sm-S ធว䉕᭴ᚑ䈜䉎 2
䈧䈱䉝䊦䊚䊆䉡䊛(Al)વዉ㔚ᭂ䈫 n ဳ䉧䊥䉡䊛⎆⚛(n-GaAs)ጀ䈱㑆䈮 GaAs/GaAsNSe ᩰሶ᭴ㅧ䈏
䉁䉏䈩䈇䉎(1)䇯Al 㔚ᭂ䉩䊞䉾䊒䈱㐳䈘 L 䈫ធว w 䈲䈠䉏䈡䉏 0.5 / 1.0 Ǎm 䈫 0.6Ǎm 䈪䈅䉎䇯ቶ᷷
䈮䈍䈔䉎 n- GaAs ጀ䈱㔚ሶኒᐲ䈫㔚ሶ⒖േᐲ䈲䈠䉏䈡䉏 5㬍1018 cm-3 䈫 1000 cm2/Vs 䈪䈅䉎䇯Ꮧ㉼಄
ಓᯏ᷷ᐲ 45 mK 䈮䈍䈔䉎 n- GaAs ጀ䈱ᐔဋ⥄↱ⴕ⒟ l 䈍䉋䈶ᾲ⊛䉮䊍䊷䊧䊮䉴㐳 N 䈲䈠䉏䈡䉏 0.1
Ǎm 䈍䉋䈶 0.3 Ǎm 䈫Ⓧ䉅䉌䉏䇮ᓥ䈦䈩 n-GaAs ጀ䈪䈱Ḱ☸ሶ䉻䉟䊅䊚䉪䉴䈲Ḱ䊋䊥䉴䊁䉞䉾䉪䈫⸒䈋䉎䇯
䈖䉏䈮ኻ䈚䈩ᩰሶ᭴ㅧ䈮䈍䈔䉎㔚ሶ䈱ᐔဋ⥄↱ⴕ⒟䈲䉌䈎䈮ᩰሶ᭴ㅧ䈱ฦጀ䈱ෘ䉂 1-2 nm ⒟
ᐲ䈪䈅䉍䇮Ḱ☸ሶ䉻䉟䊅䊚䉪䉴䈲ᢔ⊛䈪䈅䉎䈫⸒䈋䉎䇯
Ꮧ㉼಄ಓᯏ䉕↪䈇䈩ૐ᷷ 45 mK 䈮䈍䈇䈩䈖䈱ធว䈱ᓸಽᛶ᛫․ᕈ䉕㔚䈱㑐ᢙ䈫䈚䈩᷹ቯ
䈚䈢䇯᷹ቯ䈮䈲䊨䉾䉪䉟䊮䉝䊮䊒䉕↪䈇䇮䊦䊷䊒䉝䊮䊁䊅䉕↪䈇䈩䊤䉳䉥ᵄ(RF ᵄ)䈱ᾖ䉕ⴕ䈦䈢䇯䊤䉳
䉥ᵄജᵄᢙ䈶ജ䊌䊪䊷䈲 RF ⊒↢ེ䈪ᓮ䈚䈢䈏䇮䊦䊷䊒䉝䊮䊁䊅䈮䈍䈔䉎䉟䊮䊏䊷䉻䊮䉴ਇ
ᢛว䈱䈢䉄䈮䇮ታ㓙䈱 RF ᵄᾖ䊌䊪䊷䈲ജ䊌䊪䊷䈱 1 %⒟ᐲ䈫ផቯ䈘䉏䉎䇯
1
− 148 −
࿑ 2 ធวߩᓸಽᛶ᛫․ᕈ
3㧚⚿ᨐߣ⼏⺰
࿑ 2 䈮 L=0.5 Ǎm 䈱⹜ᢱ䈮䈍䈔䉎Ꮧ㉼಄ಓᯏ᷷ᐲ 45 mK 䈪䈱ᓸಽᛶ᛫․ᕈ䉕㔚䈱㑐ᢙ䈫䈚䈩␜䈜䇯
RF ᵄ䉕ᾖ䈚䈩䈇䈭䈇ᤨ䈱ᓸಽᛶ᛫䈲 Vpeak=110V 䈮䈍䈇䈩䊏䊷䉪䉕␜䈜䇯䈖䈱 Vpeak 䈱୯䈲䊋䊦䉪
Al 䈱વዉ䉩䊞䉾䊒⋧ᒰ㔚䈱ੑ 2ƦAl/e = 340 V 䈮Ყセ䈜䉎䈫䈎䈭䉍ዊ䈘䈇䈏䇮䈖䈱㔚䉕Ⴚ䈮Ḱ☸
ሶ䉻䉟䊅䊚䉪䉴䈏ᄢ䈐䈒ᄌൻ䈚䈩䈇䉎䈖䈫䈮ᵈ⋡䈚䈩䈖䈱୯䉕વዉ䉩䊞䉾䊒䉣䊈䊦䉩䊷Ʀeff /e 䈫䈭䈜䇯
ᵄᢙ 1.77 GHz 䈱 RF ᵄ䉕ᾖ䈜䉎䈫 Vpeak=110V ㄭற䈱㔚䈮䈍䈇䈩ᓸಽᛶ᛫䈲㔚䈱㑐ᢙ䈫䈚
䈩ᝄേ䈜䉎䇯ᝄേ㔚㑆㓒䈲 25V 䈎䉌 15V 䉁䈪䇮㔚䈱⛘ኻ୯䈫䈮න⺞䈮ᷫዋ䈜䉎䇯․䈮ᦨዊ䈱ᝄ
േ㔚㑆㓒 15V 䈲ᾖ RF ᵄ䈱శሶ䉣䊈䊦䉩䊷䈱 2 䈮৻⥌䈚䈩䈇䉎䇯
ᩰሶ᭴ㅧ䉕ᜬ䈢䈭䈇⹜ᢱ䈪䈲 RF ᵄᾖ䈮䉋䉎ᓸಽᛶ᛫䈱ᝄേ䈲᷹ⷰ䈘䉏䈭䈇䇯ㅒ䈮ᩰ
ሶ᭴ㅧ䉕ᜬ䈧⹜ᢱ䈪䈲䇮વዉ㔚ᭂ᧚⾰(Al / Nb)䇮Al 㔚ᭂ䉩䊞䉾䊒䈱㐳䈘 L 䈫ធว w 䈮䈾䈫䉖䈬䉋䉌
䈝৻ቯ୯ 1.7-1.8 GHz 䈱 RF ᵄᾖ䈮䉋䈦䈩ᓸಽᛶ᛫䈱ᝄേ䈏᷹ⷰ䈘䉏䉎(2)䇯ᚒ䇱䈲ᩰሶ᭴ㅧ䈮↱
᧪䈚䈢䊐䉤䊉䊮䊝䊷䊄䈏 RF ᵄ䈮䉋䈦䈩ബ䈘䉏䇮ធว䈱ャㅍ․ᕈ䈮ᓇ㗀䉕ਈ䈋䈩䈇䉎䈫⠨䈋䉎䇯
࿑ 3 䈮ᓸಽᛶ᛫ᝄേ䈱ᝄേ䊏䊷䉪⟎䉕ᾖ RF ᵄ䈱ᵄᢙ䇮䊌䊪䊷䈱㑐ᢙ䈫䈚䈩␜䈜䇯ᾖ
ᵄᢙ䉕䇸ᝄ䇹ᵄᢙ 1.77 GHz 䈎䉌ᄌൻ䈘䈞䉎䈫䇮᷹ⷰ䈘䉏䉎䊏䊷䉪䈲䈍䈇䈱㑆㓒䉕䈾䈿৻ቯ䈮
䈤䈭䈏䉌ో䈫䈚䈩ૐ㔚䈮Ⓩ䉇䈎䈮䉲䊐䊃䈜䉎䇯᷹ⷰ䈘䉏䉎䊏䊷䉪䈱ᧄᢙ䈲䇮ᾖᵄᢙ䈏䇸
ᝄ䇹ᵄᢙ 1.77 GHz 䈎䉌䈝䉏䉎䈫ᷫዋ䈚䇮20 MHz 䈝䉏䉎䈫ᝄേ⊛䈭ᝄ⥰䈇䈲ቢో䈮ᶖᄬ䈜䉎䇯᷹ⷰ䈘
䉏䉎䊏䊷䉪䈱ᧄᢙ䈲䇮ᾖ䊌䊪䊷䈮䉋䈦䈩䉅ᄌൻ䈜䉎䇯ᾖ䊌䊪䊷䉕Ⴧᄢ䈜䉎䈫ૐ㔚䊶㜞㔚㗔ၞ
䈮ᣂ䈚䈇䊏䊷䉪䈏䈜䉎䇯
࿑ 3 ᾖࠫࠝᵄߦࠃࠆࡇࠢ㔚⟎ߩᄌൻ(ޕa)ᵄᢙଐሽᕈ (b)ࡄࡢଐሽᕈ
2
− 149 −
䉁䈝ೋ䉄䈮㑐ଥ䈜䉎䊐䉤䊉䊮䊝䊷䊄䈮䈧䈇䈩⠨ኤ䈜䉎䇯RF ᵄ䈪ബ䈪䈐䉎䉋䈉䈭ૐ䉣䊈䊦䉩䊷
䈱䊐䉤䊉䊮䈲㖸㗀䊝䊷䊄䈮䈍䈇䈩䈱䉂ሽ䈜䉎䇯䊋䊦䉪 GaAs 䈮䈍䈔䉎㖸㗀䊐䉤䊉䊮䈱㖸ㅦ((001)ᣇะ䈮
ኻ䈚䈩❑ᵄ䈪 4500 m/s䇮ᮮᵄ䈪 2800 m/s)䉕⠨ᘦ䈜䉎䈫(3) 䇮ᵄᢙ 1.77 GHz 䈮⋧ᒰ䈜䉎ᵄ㐳䈲❑ᵄ
䈪 1.6Ǎm䇮ᮮᵄ䈪 2.6Ǎm 䈫Ⓧ䉅䉌䉏䉎䇯䈖䉏䈲ធว䈮䈍䈔䉎 Al વዉ㔚ᭂ䈱䉩䊞䉾䊒㐳䈘 L 䈫ห⒟ᐲ
䈪䈅䉍䇮RF ᵄ䈮䉋䈦䈩ធวਛ䈱 n-GaAs ጀ䈮㖸㗀䊐䉤䊉䊮䈱ቯᵄ䈏ബ䈘䉏䈩䈇䉎䈫⠨䈋䉌䉏䉎䇯䈚䈎
䈚䈭䈏䉌䇮㖸㗀䊐䉤䊉䊮䊝䊷䊄䈮䉋䉎ᝄേ䈲䇮⚿᥏ਛ䈮㔚᳇ᭂሶ䊝䊷䊜䊮䊃䉕⺃䈚䈭䈇䈱䈪䇮㔚⏛ᵄ䈫
䈱⚿ว䈏ᒙ䈒䇮ᓥ䈦䈩 RF ᵄ䈱శሶ䉣䊈䊦䉩䊷䉕㖸㗀䊐䉤䊉䊮䊝䊷䊄䈮વ㆐䈜䉎䈮䈲䈱䊐䉤䊉䊮䊝䊷䊄
䈱ሽ䈏ᔅⷐ䈪䈅䉎䇯䈖䈱䊐䉤䊉䊮䊝䊷䊄䈲 RF ᵴᕈ䈎䈧ᩰሶ᭴ㅧ䈫ኒធ䈮㑐ଥ䈚䈩䈇䈭䈔䉏䈳䈭䉌䈭
䈇䇯ᦨജ䈲ᩰሶ᭴ㅧౝ䈱 GaAs/GaAsNSe ⇇㕙䈮ᴪ䈦䈩䉎⇇㕙䊝䊷䊄䈪䈅䉎䇯⇇㕙䈪⺖䈞䉌
䉏䉎ᒢᕈႺ⇇᧦ઙ䈱䈢䉄䈮⇇㕙䊝䊷䊄䈲৻⥸䈮㕖✢ဳ䈭ಽᢔ㑐ଥ䉕ᜬ䈧䈔䉏䈬䉅䇮ૐ䉣䊈䊦䉩䊷㐳ᵄ
㐳ᭂ㒢䈮䈍䈔䉎㖸ㅦ䈲䊋䊦䉪 GaAs ጀ䈱ᮮᵄ䈱㖸ㅦ䉋䉍䉒䈝䈎䈮ዊ䈘䈇䇯⇇㕙䊝䊷䊄䈲䊋䊦䉪 GaAs ጀ
䈱㖸㗀䊐䉤䊉䊮䊝䊷䊄䈫䉣䊈䊦䉩䊷䈏ㄭ䈒ᡆ⋧ᢛว᧦ઙ䉅㐳〒㔌䈮ᷰ䈦䈩ḩ䈢䈜䈖䈫䈏น⢻䈭䈢䉄䇮ᒝ
䈒⚿ว䈪䈐䉎(4-6)䇯
⚿ᨐ䈫䈚䈩䇮ᩰሶ᭴ㅧઃടធว䈱 n-GaAs ጀ䈮䈍䈇䈩䈲䇮ᩰሶ/n-GaAs ⇇㕙ㄭற䈮વዉ
Ḱ☸ሶ䈫㐳ᵄ㐳㖸㗀䊐䉤䊉䊮䈏䈮ቯᵄ䈫䈚䈩 ሽ䈜䉎䈖䈫䈮䈭䉎䇯ᰴర㐽䈛ㄟ䉄ലᨐ䈮䉋䈦䈩ᒝ䈇
Ḱ☸ሶ-䊐䉤䊉䊮⋧↪䈏䈐䇮ᣂ䈚䈇ⶄว☸ሶ䈪䈅䉎䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䈏ᒻᚑ䈘䉏䉎䇯
․ቯ䈱㖸㗀䊐䉤䊉䊮䈫ᒝ䈒⚿ว䈚䈢৻ᰴర⊛䈭વዉḰ☸ሶ䈱♽䉕⠨䈋䉎䇯◲න䈱䈢䉄䈮䉷䊨
㔚⁁ᘒ䉕ቯ䈚䇮વዉḰ☸ሶ䈱䉣䊈䊦䉩䊷Ḱ䈏චಽ䈮㔌ᢔ⊛䈪䈅䉎䈫䈭䈚䈩䇮ᢿᾲㄭૃ䈱ਅ
䈪⚿ว♽䈱䊊䊚䊦䊃䊆䉝䊮䉕៨േ䈮䉋䈦䈩⸃䈒䈫䇮⚿ว♽䈱࿕䉣䊈䊦䉩䊷䈫䈚䈩ᰴ䉕ᓧ䉎䇯
En
H0 V1
2
1·
§
=: ¨ n ¸
2 =:
2¹
©
(1)
䈖䈖䈮㱑0 䈲⚿ว䈱䈭䈇႐ว䈱વዉḰ☸ሶ䉣䊈䊦䉩䊷䇮ŧƺ 䈲㖸㗀䊐䉤䊉䊮䈱䉣䊈䊦䉩䊷䇮䈠
䈚䈩䇴V1䇵䈲⚿ว䉣䊈䊦䉩䊷䈱ᦼᓙ୯䈪䈅䉎䇯㊂ሶᢙ n 䈲 1 䈧䈱વዉḰ☸ሶ䈏䊘䊷䊤䊨䊮䈫䈭䈦䈢㓙り
䈮䉁䈫䈦䈩䈇䉎䊐䉤䊉䊮䈱ᢙ䈫⸃㉼䈪䈐䉎䇯࿑ 4 䈮䉷䊨㔚⁁ᘒ䈮䈍䈔䉎䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮Ḱ䈱
ᮨᑼ࿑䉕䇮㖸㗀䊐䉤䊉䊮䈱ᗵ䈛䉎ᢿᾲ䊘䊁䊮䉲䊞䊦䈫䈮␜䈜䇯㒢䈱⚿ว䉣䊈䊦䉩䊷䈮䉋䈦䈩ᢿᾲ䊘䊁
䊮䉲䊞䊦䈲㒢䈱䊐䉤䊉䊮ᐳᮡ Q 䈮䈍䈇䈩ᦨዊ୯䉕ข䉍䇮䈠䈱୯䈲䈱વዉḰ☸ሶ䉣䊈䊦䉩䊷㱑0 䉋䉍
䉅䇴V1䇵2/2 ŧƺ 䈣䈔ዊ䈘䈇䇯䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䈱䉣䊈䊦䉩䊷Ḱ㑆㓒䈲៨േ䈱ㄭૃ▸࿐ౝ䈪㖸
㗀䊐䉤䊉䊮䈱䉣䊈䊦䉩䊷ŧƺ 䈮╬䈚䈇
㔚⁁ᘒ䈮䈍䈔䉎ャㅍ․ᕈ䉕䇮ᐔⴧ⁁ᘒ䈱䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮Ḱ䈪⸥ㅀ䈜䉎䈖䈫䈲
৻⥸䈮䈲ᱜ䈚䈒䈭䈇䇯䈚䈎䈚ㄭૃ⊛䈮䈲ਔ㔚ᭂ㑆䈱㔚Ꮕ䈏䊐䉢䊦䊚Ḱ䈎䉌᷹ቯ䈚䈢Ḱ䈱ੑ 2En
䉅䈚䈒䈲䈠䈱ᢛᢙಽ䈱৻䈮ኻᔕ䈜䉎ᤨ䈮ᓸಽᛶ᛫䈮ᄙ㊀䉝䊮䊄䊧䊷䉣䊐䈮䈉᭴ㅧ䈏䉏䉎䇯䈖䈱
䇸㡆䇹㔚᧦ઙ䉕ḩ䈢䈜㔚㑆㓒䈲 2 ŧƺ 䈮ኻᔕ䈚䈩䈍䉍䇮᷹ቯ䊂䊷䉺䈫৻⥌䈚䈩䈇䉎䇯
࿑ 4 ࠕࡦ࠼ࠛࡈࡐࡠࡦḰߩᔨ࿑
3
− 150 −
4㧚ߔ߮ߣᓟߩ੍ቯ
ᩰሶ᭴ㅧ䉕ઃട䈚䈢વዉ-ඨዉ-વዉធว䈱ャㅍ․ᕈ䉕⺞䈼䇮1.77 GHz 䈫䈇䈉․ቯ䈱䊤䉳
䉥ᵄᾖਅ䈮䈍䈇䈩ᓸಽᛶ᛫䈏㔚䈱㑐ᢙ䈫䈚䈩ᝄേ䈜䉎䈖䈫䉕⊒䈚䈢䇯ᚒ䇱䈲䇮䊤䉳䉥ᵄ䈮䉋䈦䈩
㑆ធ⊛䈮ബ䈘䉏䈢㐳ᵄ㐳㖸㗀䊐䉤䊉䊮䈫વዉḰ☸ሶ䈫䈱ᒝ䈇⚿ว䈮䉋䈦䈩ᝄേ䉕ቯ㊂⊛䈮⺑䈚䇮
ᒝ䈇⚿ว䈮䉋䈦䈩ᣂ䈚䈇ⶄว☸ሶ䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䈏ᒻᚑ䈘䉏䈩䈇䉎䈫ឭ᩺䈚䈢䇯
ᓟ䈲વዉ㔚ᭂ䈮䊆䉥䊑䋨Nb䋩䉕↪䈇䈢⹜ᢱ䈱⚿ᨐ䈮䈧䈇䈩ቯ㊂⊛䈭⸃ᨆ䉕ㅴ䉄䉎䈫䈮䇮
㖸ᵄ⚛ሶ䈮䉋䈦䈩㐳ᵄ㐳㖸㗀䊐䉤䊉䊮䉕⋥ធബ䈚䈩䉝䊮䊄䊧䊷䉣䊐䊶䊘䊷䊤䊨䊮䉕᷹ⷰ䈜䉎ታ㛎䈱Ḱ
䉕ㅴ䉄䉎䇯
5㧚⻢ㄉ
ᧄ⎇ⓥߪ⑼ޔቇᛛⴚᝄ⥝ᯏ᭴ᚢ⇛⊛ഃㅧ⎇ⓥផㅴᬺ(JST-CREST)ߩࡊࡠࠫࠚࠢ࠻(⎇ⓥ⺖㗴ฬ㧦
ޟવዉࡈࠜ࠻࠾ࠢࠬߩഃᚑߣߘߩᔕ↪⎇ޠⓥઍ⠪㧦ർᶏᄢቇ ᧃቬᐞᄦᢎ)ߣߒߡㆀⴕߐ
ࠇߡࠆ⹜ޕᢱߩᓸ⚦ടᎿߩᛛⴚᡰេߦኻߒߡ᧚⾰‛ޔᢱ⎇ⓥᯏ᭴࠽ࡁ࠹ࠢࡁࡠࠫⲢวࡦ
࠲ߩᵤ⼱ᄢ᮸ޔᷰㄝ⧷৻㇢ਔඳ჻ߦᗵ⻢ߔࠆޕ
References
(1) K. Uesugi and I. Suemune, Appl. Phys. Lett. 79, 3284, 2001.
(2) R. Inoue, H. Takanayagi, M. Jo, T. Akazaki, and I. Suemune, J. Phys. Conf. Series 109,
012033, 2008.
(3) D. Stauch and B. Dorner, J. Phys. Condens. Matter 2, 1457, 1990.
(4) K. Hess, Appl. Phys. Lett. 35, 484, 1979.
(5) S. Yu, K. W. Kim, M. A. Stroscio, G. J. Iafrate, and A. Ballato, Phys. Rev. B 50, 1733,
1994.
(6) V. G. Litovchenko, D. V. Korbutyak, S. Krylyuk, H. T. Grahn and K. H. Ploog, Phys. Rev.
B 55, 10621, 1997.
4
− 151 −
⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ࠍ↪ߚ SQUID
㊂ሶࠣ࡞ࡊ 㧔ℂቇㇱᔕ↪‛ℂቇ⑼ 㜟ᩉ⎇ⓥቶ㧕
⍹㤥 ੫テ㧔ഥᢎ㧕
Abstract. ⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻↪ߚવዉ㊂ሶ࠼࠶࠻વዉធวࠍવዉ࡞
ࡊߦ㧞㊂ሶ࠼࠶࠻ SQUID ࠍߒ․ߩߘޔᕈߩ⹏ଔࠍⴕߞߚޕ㊂ሶ࠼࠶࠻ߩࠛ
ࡀ࡞ࠡḰࠍࡃ࠶ࠢࠥ࠻ߣࠨࠗ࠼ࠥ࠻㔚ߦࠃࠅᓮߒޔ㊂ሶ࠼࠶࠻ធวࠍવዉ
ߩ⋧߇ធว㑆ߢᄌൻߒߥ㧜ធว߆ࠄޔǸߩᄌൻࠍ߁Ǹធว߳ߩォ⒖ࠍ᷹ⷰߒߚޕ
ߪߓߦ
㊂ሶ࠼࠶࠻ࠍવዉߢࠎߛ㊂ሶ࠼࠶࠻વዉធวߪ࠽ࡁ࠴ࡘࡉ߿࠽ࡁࡢࠗࡗࠍ↪ߚ
SQUID ߿⥄ޔᏆᒻᚑ InAs ㊂ሶ࠼࠶࠻ធวࠍ↪ߡޔ㊂ሶ࠼࠶࠻ߦࠇࠆㄭ⮮ലᨐߣવዉߣߩ
┹ว߿ 0 ធว߆ࠄǸធว߳ߩォ⒖ߥߤ♖ജ⊛ߥ⎇ⓥ߇ⴕࠊࠇߡࠆ[1,2,3]⎇ᧄޕⓥߢߪ⥄Ꮖᒻᚑ
㊂ሶ࠼࠶࠻ࠍធวߦ↪ߚ SQUID ࠍߒޔ㊂ሶ࠼࠶࠻ SQUID ߩ․ᕈࠍ⹏ଔߒߚ⥄ޕᏆᒻᚑ
㊂ሶ࠼࠶࠻ߪ⌀ߦࡠᰴర᭴ㅧࠍᜬߞߡࠆߚߩߎޔ㊂ሶ࠼࠶࠻ࠍធวߦ↪ߚ SQUID ߪ࠽
ࡁ࠴ࡘࡉ߿࠽ࡁࡢࠗࡗߦࠃࠆ SQUID ࠃࠅ߽᧪ߩ㓸Ⓧൻࠬࠗࡃ࠺ޔൻߥߤߦὐ߇ࠆߣ
⠨߃ࠄࠇߡࠆߩߎޕ㊂ሶ࠼࠶࠻ SQUID ߪࠥ࠻㔚ߦࠃࠅᓮߦࠃߞߡធว㑆ߩቯߥ⋧
Ꮕ߇㧜߆ࠄǸߦᄌൻߔࠆߎߣ߇⏕ߐࠇ࠻ࠥߩߎޔᓮน⢻ߥ 0-Ǹធวォ⒖ࠍ↪ߒߚ㊂ሶࡆ
࠶࠻߽ᦼᓙߐࠇࠆߩߎߦࠄߐޕ㊂ሶ࠼࠶࠻ធวߦశࠍᒰߡࠆߎߣߢߩߎޔ㊂ሶ࠼࠶࠻ SQUID ߪޔ
વዉ㊂ሶࡆ࠶࠻ߣశߦࠃࠆ㊂ሶࡆ࠶࠻㑆ࠍࠬࡇࡦ㊂ሶࡆ࠶࠻ߢ⚿วߒߚߩ㊂ሶࠗࡦ࠲ࡈࠚ
ࠬߣߒߡേ߇ᦼᓙߐࠇࠆޕ
2㧚ታ㛎⚿ᨐ
⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ၮ᧼ࠍ↪ޔၮ᧼ߦࡦ࠳ࡓߦ㈩ߒߚ㊂ሶ࠼࠶࠻ߦ㔚ሶࡆࡓឬ
↹ᴺߦࠃࠅ⁓ߞߡવዉ㔚ᭂࠍઃߌߚޕ㊂ሶ࠼࠶࠻ߩࠨࠗ࠭ߪ 200nm એਅߢࠅߩߎޔ᭽ߥࠨࠗ
࠭ߩ߽ߩߦ⁓ߞߡ㔚ᭂࠍઃߌࠆߎߣߪ࿎㔍ߣ⠨߃ࠄࠇߡߚ߇ޔቯ⊛ߦߔࠆᛛⴚࠍ⏕┙ߒ
ߚޕ
࿑㧝ߪ㊂ሶ࠼࠶࠻ SQUID ߩ SEM ߢ⿒ਣߢ⸥
ߒߚㇱಽ߇વዉ/㊂ሶ࠼࠶࠻/વዉធวߢࠆޕ
વዉ㔚ᭂߣߒߡߪෘߐ 100nm ߩࠕ࡞ࡒ࠾࠙ࡓ
ࠍ↪ޔ㊂ሶ࠼࠶࠻ߣߩࠦࡦ࠲ࠢ࠻ᕈߩะߩߚ
ߦߩߚߦ࠴࠲ࡦࠍ 5nm ⫳⌕ߒߡࠆ⥄ޕᏆ
ᒻᚑ InAs ㊂ሶ࠼࠶࠻ߩࠨࠗ࠭ߪ⋥ᓘ 100㨪
200nm ߩ߽ߩࠍ↪ߡࠆߚ߹ޕធวㄭறߦߪ
ߘࠇߙࠇࠨࠗ࠼ࠥ࠻㔚ᭂ߇ขࠅઃߌߡࠅߘޔ
ࠇߙࠇߩធวߦኻߒߡ⁛┙ߦࠥ࠻ᠲ߇น⢻
ߢߣߊߦࠨࠗ࠼ࠥ࠻ߪ࠼࠶࠻ߣવዉ㔚ᭂߣ
ߩ⚿วࠍᓮߔࠆߎߣ߇ಽ߆ߞߚߚ߹ޕ㧞ߟߩធ
ว߇หᤨߦᓮߐࠇࠆࡃ࠶ࠢࠥ࠻㔚ᭂߪ․ߦ
㊂ሶ࠼࠶࠻ౝߩ㔚⩄⁁ᘒࠍᄌ⺞ߔࠆޕ
ߎߩ㊂ሶ࠼࠶࠻ SQUID ߪᏗ㉼಄ಓᯏߦࠃࠅ
࿑㧝㧚㊂ሶ࠼࠶࠻ SQUID ߩ SEM
30mK ⒟ᐲ߹ߢ಄ළߒߘߩ․ᕈࠍ⹏ଔߒߚ⸘ޕ
᷹ߦߪ㜞ᵄૐ᷷ࡈࠖ࡞࠲ࠍ↪ࠆߎߣߢ
વዉ㔚ᵹߩ᷹ⷰ߇น⢻ߦߥߞߚޕ
1
− 152 −
࿑ 2 ߪ㊂ሶ࠼࠶࠻ SQUID ߩ 2 ߟߩធวߘࠇߙࠇߩ⥃⇇㔚ᵹߩࠥ࠻ଐሽᕈࠍ␜ߒߡࠆޕ
SQUID ోߩ⥃⇇㔚ᵹߩ⏛႐ᄌൻߩ⸃ᨆߦࠃࠅ 2 ߟߩធวߘࠇߙࠇߩ⥃⇇㔚ᵹߦಽ㔌ߔࠆߎߣ
ߦᚑഞߒߚ࠻ࠥ࠼ࠗࠨޕ㔚ߣࡃ࠶ࠢࠥ࠻㔚ߩਔᣇࠍᄌ߃ࠆߣ(c)ߩ৻ㇱߢធว 1 ߩ⥃⇇㔚
ᵹ߇ᐭߦߥࠆㇱಽ߇ࠄࠇࠆߪࠇߎޕធว 1 ߩ㊂ሶ࠼࠶࠻ធว߇ 0 ߆ࠄǸធวߦォ⒖ߒߚߎߣ
ࠍ␜ߒߡࠆߩ࠻ࠥࠢ࠶ࡃޕᄌ⺞ߩߺߢ 0-Ǹォ⒖߇᷹ⷰߐࠇߥ߆ߞߚߎߣߪߎߩ 0-Ǹធวォ⒖
ߪ㊂ሶ࠼࠶࠻ߣવዉ㔚ᭂߣߩ⚿วࠛࡀ࡞ࠡࠍᄌൻߐߖߚߎߣߢታߒߚߣ߃ࠆޕ
࿑ 2. ⥃⇇㔚ᵹߩࠥ࠻ଐሽᕈޔធว㧝ߦ߅ߌࠆ⥃⇇㔚ᵹ߇(㨏)ߦ߅ߡ⽶ߩ୯ߣߥࠆㇱಽ߇
ࠅǸធวߣߥߞߚߎߣࠍ␜ߒߡࠆ 4ޕ
3. ߔ߮
⥄Ꮖᒻᚑ㊂ሶ࠼࠶࠻ࠍ↪ߚ㊂ሶ࠼࠶࠻ SQUID ߩၮᧄ⊛ߥᕈ⾰ࠍࠄ߆ߦߒߩߎޔ㊂ሶ࠼࠶
࠻ធว߽ 0-Ǹォ⒖ࠍߎߔߎߣࠍࠄ߆ߦߒߚ․ߩߎޕᕈࠍ↪ࠆߎߣߢ㊂ሶ࠼࠶࠻ SQUID ߇
శߣવዉ㊂ሶࡆ࠶࠻㑆ߩ㊂ሶࠗࡦ࠲ࡈࠚࠬߣߒߡേߢ߈ࠆߎߣ߇ᦼᓙߐࠇࠆޕ
References
(1) J. -P. Cleuziou et al., Nature Nanotech. 1, 53, (2006)
(2) J. A. Van
Dam et al., Nature 442, 667 (2006)
(3) Y Kanai et al J. Phys.: Conf. Ser. 150 022032 (2009)
(4) S. Kim et al. arXiv:1010.5892v1 [cond-mat.mes-hall]
2
− 153 −
⎇ⓥᬺ❣㧔⎇ⓥ⠪㧕 ㊂ሶࠣ࡞ࡊ
ဈ↰⧷
ቇⴚ⺰ᢥ㧔ᩏ⺒㧕
1. Infrared reflectivity of the phonon spectra in multiferroic TbMnO3 (ᩏ⺒ࠅ)
PHYSICAL REVIEW B 82, 144309 (2010) R. Schleck, R. L. Moreira, H. Sakata,
and R. P. S. M. Lobo
2.
Development of Near-Field Microwave Microscope with the Functionality of
Scanning Tunneling Spectroscopy (ᩏ⺒ࠅ) Jpn. J. Appl. Phys., Vol. 49, 116701
(2010) Tadashi Machida, Marat B. Gaifullin1, Shuuich Ooi, Takuya Kato, Hideaki
Sakata, and Kazuto Hirata
3.
Magnetoelectric excitations in multiferroic TbMnO3 by Raman scattering (ᩏ⺒
ࠅ) PHYSICAL REVIEW B 81 054428 (2010) P. Rovillain, M. Cazayous, Y. Gallais,
A. Sacuto, and M-A. Measson, H. Sakata
4.
Disappearance of zinc impurity resonance in large-gap regions of
Bi2Sr2CaCu2O8+Ǭ probed by scanning tunneling spectroscopy (ᩏ⺒ࠅ) Phys.
Rev. B 82 180507 (2010) Tadashi Machida, Takuya Kato, Hiroshi Nakamura,
Masaki Fujimoto, Takashi Mochiku, Shuuichi Ooi, Ajay D. Thakur, Hideaki
Sakata, and Kazuto Hirata
㜟ᩉ⧷
ቇⴚ⺰ᢥ
1. Transport characteristics of a superconductor-based LED, R. Inoue, H. Takayanagi,
T. Akazaki, K. Tanaka and I. Suemune,Supercond. Sci. Technol. 23, 2010,
034025-034028, 2010 㧔ᩏ⺒㧕.
2. Superconducting Transport in an LED with Nb Electrodes, H. Takayanagi, R. Inoue, T. Akazaki, K.
Tanaka and I. Suemune Physica C 470, 814-817. 2010 (ᩏ⺒).
3. A Cooper-Pair Light-Emitting Diode: Temperature Dependence of Both Quantum
Efficiency and Radiative Recombination Lifetime, I. Suemune, Y. Hayashi, S.
Kuramitsu, K. Tanaka, T. Akazaki, H. Sasakura, R. Inoue, H. Takayanagi, Y. Asano,
E. Hanamura, S. Odashima, and H. Kumano Appl. Phys. Express 3, 054001-1
~054001-3 2010 (ᩏ⺒).
4. Evaluation of spin polarization in p-In0.96Mn0.04As using Andreev reflection
spectroscopy, T. Akazaki, T. Yokoyama, Y. Tanaka, H. Munekata and H.
Takayanagi,J. of Physics:Conference Series 234 , 042001, 2010 㧔ᩏ⺒㧕.
− 154 −
ᓙ⻠Ṷ
1. SQUID coupled with self-assembled InAs Quantum Dot, H. Takayanagi, Plasma
2010 ᒄ೨ 2010.
2.
Transport Characteristics of a Sperconductor-Based LED, H. Takayanagi,
Workshop one the Physics of Micro and Nano Scale Systems, Ystad, Sweden, 2010
3.
Spin Detection by applying the Inverse Proximity Effect on p –InMnAs / n –InAs / Nb junction,
H. Takayanagi, SM2010 Superconductivity and Magnetism: Paestum, Italy, 2010.
4.
Transport properties of a superconductor-semiconductor junction with superlattice
structure,
H.
Takayanagi,
ESF-NES
WORKSHOP
2010
Nanoscale
Superconductivity, Fluxonics and Plasmonics Crete, Greece, 2010.
5.
Transport of a superconducting LED and Andreev Polaron, H. Takayanagi,
ICAUM2010, Cheju, Korea, 2010.
ᵲ
ቇⴚ⺰ᢥ㧔ᩏ⺒㧕
1. 1. Model for the occurrence of Fermi pockets without the pseudogap hypothesisi in
the underdoped cuprates superconductors – Interplay of Jahn-Teller physics and
Mott physics – 㩷 H. Kamimura and H. Ushio (submitted to Phys. Rev. B 2010)
䋨ᩏ⺒䋩 (Condmat䋺arXiv: 1006.0586)
2.
Coherent proton-induced conduction in the excited state of hydrogen-bonded
materials exhibiting a metal-insulator transition H. Kamimura, S. Ikehata, Y.
Yoshida and Y. Matsuo
To be published in the 30th international Conference on
Physics of Semiconductors, AIP ConferenceProceedings (New York, USA, 2010)
3.
On the Interplay of Jahn-Teller physics and Mott physics in the mechanism of high
Tc superconductivity H. Ushio, S. Matsuno and H. Kamimura To be published in
Vibronic Interactions and the Jahn-Teller Effect, Progress of Theoretical Physics
and Chemistry, Springer, Proceedings of the XXth International Conference on the
Jahn-Teller Effect䋨Springer, Heidelberg, 2010䋩 䋨ᩏ⺒䋩
4.
On the mechanism of d-wave high Tc superconductivity by the interplay of
Jahn-Teller physics and Mott physics H. Ushio, S. Matsuno and H. Kamimura
Quantuem Bio-Informatics IV, eds., L. Accardi, W.Freudenberg and M. Ohya,
(World Scientific, Singapore, 2010) 䋨ᩏ⺒䋩
ᓙ⻠Ṷ
1. NIMS Computational Materials Science (CMS) ࡒ࠽ Roles of Mott Physics in
Semiconductor Impurity Bands and Copper-Oxide Superconductors ᵲޔ2010
ᐕ 6 9 ᣣ᧚⾰‛ޔᢱ⎇ⓥᯏ᭴ 㧔ߟߊ߫㧕
2.
╳ᵄᄢቇ⸘▚⑼ቇ⎇ⓥࡦ࠲ ⻠Ṷ Model for the occurrence of Fermi pockets
without the pseudogap hypothesisi in the underdoped cuprates superconductors ̄
− 155 −
Interplay of Jahn-Teller physics and Mott physics ̄ ᵲޔ2010 ᐕ 6 23 ᣣ╳ޔ
ᵄᄢቇ⸘▚⑼ቇ⎇ⓥࡦ࠲
3.
HTSC one-day symposium, Dept of Physics, University of Tokyo Model for the
occurrence of Fermi pockets without the pseudogap hypothesisi in the underdoped
cuprates superconductors – Interplay of Jahn-Teller physics and Mott physics –
ᵲ ޔ2010 ᐕ 7 3 ᣣ᧲ޔ੩ᄢቇℂቇㇱ㧝ภ㙚 ዊᩊࡎ࡞
4.
Coherent proton-induced conduction in the excited state of hydrogen-bonded
materials exhibiting a metal-insulator transition ᵲ 2010 ᐕ 7 26 ᣣ ޔ30th
International Conference on Physics of Semiconductors International Convention
Center, Seoul, Korea (৻⥸ߩญ㗡⻠Ṷ)
1.
Plenary talk entitled “From Impurity Bands to High Temperature Superconductors”
ᵲ 2009 ᐕ 11 30 ᣣ International Symposium on Advanced Nanodevices
and Nanotechnology, Kaanapari, Maui, Hawaii, USA (November 29-December 4,
2009)
․⻠Ṷ
㪈㪅 ᚭጊ䉥䊷䊒䊮䉦䊧䉾䉳㩷㩷 ㊂ሶ‛ℂ䈫ᣣᏱ↢ᵴ䊶␠ળ䈫䈱㑐䉒䉍㩷 㩷 ᵲ㩷 䋨ౝ㑑ᐭ䊶ᦨవ┵⎇
ⓥ㐿⊒䊒䊨䉫䊤䊛䊶㊂ሶᖱႎᓮ䊒䊨䉳䉢䉪䊃䊶䉝䊄䊋䉟䉱䊷䋩㪉㪇㪈㪈 ᐕ 㪈 㪉㪐 ᣣ䋨䋩ඦᓟ 㪉 ᤨ
㵪㪋 ᤨ䇮᧲੩ㇺ┙ᚭጊ㜞ᩞ䋨䉴䊷䊌䊷䉰䉟䉣䊮䉴䊊䉟䉴䉪䊷䊦䋩ᄢળ⼏ቶ㩷
ችᎹት
ቇⴚ⺰ᢥ
1. Electron correlation in the FeSe superconductor studied by bulk-sensitive
photoemission spectroscopy, A. Yamasaki, Y. Matsui, S. Imada, K. Takase, H.
Azuma, T. Muro, Y. Kato, A. Higashiya, A. Sekiyama, S. Suga, M. Yabashi, K.
Tamasaku, T. Ishikawa, K. Terashima, H. Kobori, A. Sugimura, N. Umeyama, H.
Sato, Y. Hara, N. Miyakawa, S. I. Ikeda, Phys. Rev. B 82, 184511 (2010), DOI:
10.1103/PhysRevB.82.184511㧔ᩏ⺒㧕
2.
Superconductivity on FeSe synthesized by various sintering temperature, N.
Umeyama, K. Takase, S. Hara, S. Horiguchi, A. Tominaga, H. Sato, Y. Hara, N.
Miyakawa, S.I. Ikeda, Physica C (2010),doi:10.1016/j.physc.2010.05.241㧔ᩏ⺒㧕
3.
Structural and physical properties of FeSe crystals fabricated by the chemical
vapor transport method, Y. Hara, K. Takase, A. Yamasaki, H. Sato, N. Miyakawa,
N. Umeyama, S.I. Ikeda, Physica C (2010), doi:10.1016/j.physc.2010.02.021㧔ᩏ⺒㧕
4.
Eliashberg Analysis of Tunneling Experiments: Support for the Pairing Glue
Hypothesis, O. Ahmadi, L. Coffey, J.F. Zasadzinski, N. Miyakawa, L. Ozyuzer, to be
submitted to Phys. Rev. Lett.
− 156 −
੫ᄥ㇢
ቇⴚ⺰ᢥ
1. Transport Characteristics of a superconductor-based LED, R. Inoue, H.
Takayanagi, T. Akazaki, K. Tanaka and I. Suemune, Superconductor Sci. Tech. vol.
23 no. 3, 034025, 2010㧔ᩏ⺒㧕
2.
Tranport properties of Andreev polarons in superconductor-semiconductorsuperconductor junction with superlattice structure R. Inoue, H. Takayanagi, M.
Jo, T. Akazaki and I. Suemune, (ᛩⓂਛ)
࿖㓙ቇળ
1. Transport Characteristics of superconductor-based light-emitting diode structure,
R. Inoue, H. Takayanagi, T. Akazaki, K. Tanaka and I. Suemune, International
Symposium on Quantum Nanostructures and Spin-related Phenomena
(QNSP2010), March 9-11, 2010, Komaba.
2.
Transport properties of a superconductor-semiconductor junction with superlattice
structure, R. Inoue, H. Takayanagi, M. Jo, T. Akazaki and I. Suemune,
International Conference on the Physics of Semiconductors (ICPS 2010) July 25-30,
2010, Seoul.
ട⮮ᜏ
ቇⴚ⺰ᢥ
1. Scanning tunneling microscopy and spectroscopy on iron chalcogenide
superconductor Fe1+DžSe1-xTex, T. Kato, Y. Mizuguchi, H. Nakamura, T. Machida, Y.
Takano, H. Sakata, J. Supercond. Nov. Magn., accepted. 䋨ᩏ⺒䋩
2.
Impurity-related local density-of-states modulation in Bi2Sr2Ca(Cu1-xZnx)2O8+Dž
probed by scanning tunneling spectroscopy, T. Machida, T. Kato, H. Nakamura, M.
Fujimoto, T. Mochiku, S. Ooi, H. Sakata, K. Hirata, J. Supercond. Nov. Magn.,
accepted. 䋨ᩏ⺒䋩
3.
Effect of Zn, Ni and Fe impurities on Bi2Sr1.6La0.4CuO6+Dž, H. Nakamura, H.
Funahashi, M. Fujimoto, M. Iguchi, T. Yamasaki, T. Machida, T. Kato, H. Sakata,
J. Supercond. Nov. Magn., accepted. 䋨ᩏ⺒䋩
4.
Quasiparticle density of states in cuprate superconductor Bi2Sr2-xLaxCuO6+Dž in a
magnetic field studied by scanning tunneling spectroscopy, T. Kato, H. Nakamura,
M. Fujimoto, H. Funahashi, T. Machida, H. Sakata, S. Nakao, T. Hasegawa,
Physica C, in press. 䋨ᩏ⺒䋩
5.
Scanning tunneling spectroscopy on Bi2SrCaCuO6+Dž, H. Sakata, T. Sakuyama, T.
Kato, Physica C, in press. 䋨ᩏ⺒䋩
6.
Microwave responses on locally modified Bi2Sr2CaCu2O8+Dž by near-field microwave
microscope, T. Machida, S. Ooi, M. Tachiki, T. Mochiku, T. Kato, H. Sakata, K.
− 157 −
Hirata, Physica C, in press. 䋨ᩏ⺒䋩
7.
Disappearance of zinc impurity resonance in large-gap regions of Bi2Sr2CaCu2O8+Dž
probed by scanning tunneling spectroscopy, T. Machida, T. Kato, H. Nakamura, M.
Fujimoto, T. Mochiku, S. Ooi, H. Sakata, K. Hirata, Phys. Rev. B 82, 180507(R)
(2010) 䋨ᩏ⺒䋩
8.
Development of near-field microwave microscope with the functionality of scanning
tunneling spectroscopy, T. Machida, M.B. Gaifullin, S. Ooi, T. Kato, H. Sakata, K.
Hirata, Jpn. J. Appl. Phys. 49, 116701 (2010) 䋨ᩏ⺒䋩
⍹㤥੫テ
ቇળ⊒
1. ⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ࠍធวߦ↪ߚ SQUID ߩ⎇ⓥ II ⍹㤥੫テ㧘የలᒎ㧘㊄
㞲⟤㧘ᷰㄝ⧷৻㇢㧘ᵤ⼱ᄢ᮸㧘ᩊ↰ᙗᴦ㧘ᐔᎹ৻ᒾ㧘㜞ᩉ⧷ ᣣᧄ‛ℂቇળ 2010 ᐕ
⑺ቄᄢળ 2010 ᐕ 9 24 ᣣ
2.
ᒝ⚿ว⁁ᘒߦ߅ߌࠆ⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ߣવዉߩャㅍ․ᕈ የలᒎ㧘㊄
㞲⟤㧘⍹㤥੫テ㧘ᷰㄝ⧷৻㇢㧘ᵤ⼱ᄢ᮸㧘ᩊ↰ᙗᴦ㧘ᐔᎹ৻ᒾ㧘㜞ᩉ⧷ ╙ 71 ࿁ᔕ
↪‛ℂቇળቇⴚ⻠Ṷળ 2010 ᐕ 9 14 ᣣ
3.
Side-gate controlled electrical properties of superconducting quantum interference
device coupled with self-assembled InAs quantum dot S. Kim, R. Ishiguro, M.
Kamio, Y. Doda, E. Watanabe, D. Tsuya, K.Shibata, K. Hirakawa, and H.
Takayanagi [30th International Conference on the Physics of Semiconductors
(ICPS 2010) 2010 ᐕ 7 27 ᣣ
4.
SQUID coupled with self-assembled InAs Quantum Dot H.Takayanagi, S. Kim, R.
Ishiguro, D. Tsuya, K. Shibata and K. Hirakawa
Plasma 2010 2010 ᐕ 4 30
ᣣ
5.
⥄Ꮖᚑ㐳 InAs ㊂ሶ࠼࠶࠻ࠍធวߦ↪ߚ SQUID ߩ⎇ⓥ ⍹㤥੫テ㧘ᷓᎹዏ⟵㧘የ
లᒎ㧘ਛፉ⠍㧘᳇⼱ථ㧘㊄㞲⟤㧘ᷰㄝ⧷৻㇢㧘ᵤ⼱ᄢ᮸㧘ᩊ↰ᙗᴦ㧘ᐔᎹ৻ᒾ, 㜞ᩉ⧷
ᣣᧄ‛ℂቇળ ╙ 65 ࿁ᐕᰴᄢળ 2010 ᐕ 3 21 ᣣ
6.
㗼ᓸ⊒శࠬࡍࠢ࠻࡞ࠍ↪ߚ GaAs ࡋ࠹ࡠធว߳ߩ㓸᧤ࠗࠝࡦࡆࡓᾖߦࠃࠆ៊்
ߩ⹏ଔ ባᚭਯ㧘㊁ᄥ㇢㧘᧻ᧄືᦶ㧘ᨰ⼱⡡㧘⍹㤥੫テ㧘㜞ᩉ⧷
2010 ᐕᤐ
ቄ╙ 57 ࿁ᔕ↪‛ℂቇ㑐ଥㅪว⻠Ṷળ 2010 ᐕ 3 19 ᣣ
7.
Nb nanoSQUID ߩ Ic ૐᷫߦะߌߡߩࠕࡊࡠ࠴ ᧻ᧄືᦶޔᨰ⼱⟤ޔᩊ↰⡸ޔ⍹㤥
੫テޔጊญޔ㜞ᩉ⧷ޔ㊁ᄥ㇢ޔᨰ⼱⡡ 2010 ᐕᤐቄ ╙ 57 ࿁ ᔕ↪‛ℂቇ㑐
ଥㅪว⻠Ṷળ 2010 ᐕ 3 19 ᣣ
8.
⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ࠍ↪ߚ SQUID ߦ߅ߌࠆᄙ㊀ࠕࡦ࠼ࠛࡈ የల
ᒎ㧘ਛፉ⠍㧘᳇⼱ථ㧘ᷓᎹዏ⟵㧘㊄㞲⟤㧘⍹㤥੫テ㧘ᷰㄝ⧷৻㇢㧘ᵤ⼱ᄢ᮸㧘ᩊ↰ᙗ
ᴦ㧘ᐔᎹ৻ᒾ㧘㜞ᩉ⧷ 2010 ᐕᤐቄ╙ 57 ࿁ᔕ↪‛ℂቇ㑐ଥㅪว⻠Ṷળ 2010 ᐕ 3
− 158 −
18 ᣣ
9.
⥄Ꮖᒻᚑ InAs ㊂ሶ࠼࠶࠻ߣ⚿วߒߚ SQUID ߩࠥ࠻ᓮ ㊄㞲⟤㧘⍹㤥੫テ㧘ၴ↰
ᵏผ㧘ᷰㄝ⧷৻㇢㧘ᵤ⼱ᄢ᮸㧘ᩊ↰ᙗᴦ㧘ᐔᎹ৻ᒾ㧘㜞ᩉ⧷ 2010 ᐕᤐቄ╙ 57 ࿁
ᔕ↪‛ℂቇ㑐ଥㅪว⻠Ṷળ
2010 ᐕ 3 18 ᣣ
10. Quantum Dot Superconducting Quantum Interference Devices with self-assembled
InAs R. Ishiguro, S. Kim, E. Watanabe, D. Tsuya, K. Shibata, K. Hirakawa, H.
Takayanagi
APS March Meeting 2010 ᐕ 3 18 ᣣ
11. Gate control of SQUID coupled with self-assembled InAs Quantum Dot S. Kim, R.
Ishiguro, Y. Doda, E. Watanabe, D. Tsuya, K. Shibata, K. Hirakawa, H. Takayanagi
The MANA International Symposium 2010
2010 ᐕ 3 5 ᣣ
12. Multiple Andreev Reflection In SQUID with self-assembled InAs Quantum Dot R.
Ishiguro, N. Fukagawa, M. Kamio, S. Nakajima, S. Kitani, S. Kim, E. Watanabe, D.
Tsuya, K. Shibata, K. Hirakawa, H. Takayanagi The MANA International
Symposium 2010 2010 ᐕ 3 5 ᣣ
⺰ᢥ
1. pi-junction transition in InAs self-assembled quantum dot coupled with SQUID S.
Kim, R. Ishiguro, M. Kamio, Y. Doda, E. Watanabe, D. Tsuya, K. Shibata, H.
Hirakawa, and H. Takayanagi 㧔ᛩⓂਛ㧕
− 159 −
㩷
㩷
㩷
ઃ㩷 ㍳㩷
㩷
䊶㪨㪙㪠㪚 䉰䊙䊷䉴䉪䊷䊦㩷
䊶㪨㪙㪠㪚 䉶䊚䊅䊷㩷
㩷
2010 ᐕ QBIC ࠨࡑࠬࠢ࡞ႎ๔
ᣣ⒟㧦8 20 ᣣ㧔㊄㧕13㧦00㨪8 25 ᣣ㧔᳓㧕12㧦00
႐ᚲ㧦⺪⸰᧲੩ℂ⑼ᄢቇ
ෳട⠪㧦ቇ↢ 34 ฬ߅ࠃ߮ᧄቇᢎຬ 19 ฬਗ߮ߦࠥࠬ࠻⻠Ṷ⠪ 2 ฬޔว⸘ 55 ฬ
⋡⊛㧦ᢙℂ↢ޔޔ㊂ሶߩ 3 ࠣ࡞ࡊߩᨒࠍ߃ߡޔQBIC ߩ⋡ᮡࠍᔨ㗡ߦޔઍ⑼ቇߩၮ
␆߆ߟᧄ⾰ߦ㑐ࠊࠆࠕࠗ࠺ࠕ߿㗴ὐߥߤࠍ⼏⺰ߒߩߘޔℂ⸃ࠍᷓޔᓟߩ⎇ⓥᵴേߩ
น⢻ᕈࠍᬌ⸛ߔࠆޕ
ീᒝળߩㅴᣇ㧦
ඦ೨㧦ၮᧄࠢ࠴ࡖߦࠃࠆ㗴ឭଏޔ߮㗴ឭ
ඦᓟ㧦ࡈ࠺ࠖࠬࠞ࠶࡚ࠪࡦࡦ࡚ࠪ࠶ࠞࠬࠖ࠺࡞ࡀࡄޔ
ീᒝળߩਥߥౝኈ㧦㧔ᷝઃ⾗ᢱ㧦ീᒝળᤨ㑆ഀࠍෳᾖ㧕
٧ㆡᔕജቇߩືℂ㧦⥄ὼߪޔᚒߩ߳ࠇߘ߇ޘ㑐ࠊࠅᣇ㧔ⷰᣇ㧕ߦḰߓߡߩߘޔᖱࠍᄌ߃
ࠆ⁁ߚߒ߁ߘޔ߫ࠇߔߣޕᴫߦㆡᔕߒߚജቇࠍ᭴▽ߔࠆᔅⷐ߇ࠆߩߘޕ᭴▽ߣߪ㧚㧚㧚
㧔㗴ឭଏ⠪㧦ᄢ⍫㓷ೣ㧕
٧ࡒࠢࡠࡑࠢࡠኻᕈ㧦
(ࡠࠢࡒޟᓸⷞ⊛)ߣࡑࠢࡠ(Ꮒⷞ⊛)ޔޠ
ޟฎౖߣ㊂ሶ߁ߣޠੑ⚵ߩ
ኻᔨ߇ߦߤ߁㑐ࠊࠅ߁ߤࠍࠇߘޔℂ⸃ߔࠆߎߣ߇ߢ߈ࠆ߆㧚㧚㧚
㧔㗴ឭଏ⠪㧦ዊ᎑ᴰ㧔੩ㇺᄢቇᢙℂ⸃ᨆ⎇ⓥᚲ㧕㧕
٧⥄ὼℂ⸃ߦ߅ߌࠆ⏕₸ജቇߩน⢻ᕈ㧦ࡀ࡞࠰ࡦᵹߩ⥄ὼ⸃㉼㧔ࡒࠢࡠ☸ሶߩࡉ࠙ࡦㆇ
േ߆ࠄޔᵄേജቇࠍዉߊ㧕ߪޔㆡᔕജቇ߿ࡒࠢࡠࡑࠢࡠኻᕈߣᨐߚߒߡᢛวߔࠆ߆㧚㧚㧚
㧔㗴ឭଏ⠪㧦᧻ጟ㓉ᔒޔጊ⊓৻㇢㧕
٧㊂ሶ‛ℂߩᦨవ┵㧦㔚ሶࠍ↪ߡੑ㊀ࠬ࠶࠻ߩታ㛎ࠍⴕ߁ߎߣ߇ߢ߈ࠆߣ߁ޕᨐߚ
ߒߡᚒޔߪޘ㔚ሶ 1 ߿శሶ 1 ࠍߤߎ߹ߢᓮߢ߈ߡࠆߩ߆㧫 ታ㛎࠺࠲ߣ㊂ሶജ
ቇߩၮ␆ߣߩᢛวᕈ߿ታ㛎ᚻᴺߩᦨవ┵ߩ⁁ߣߪ㧚㧚㧚
㧔㗴ឭଏ⠪㧦ਛࡁ㓳ੱ㧔NTT ‛ᕈၮ␆⎇㧕
٧ᬀ‛ߩ∉ࠪࠬ࠹ࡓߣᖱႎવ㆐㧦ᬀ‛ߪᄙ᭽ߥⅣႺᄌൻࠍኤ⍮ߔࠆ⢻ജࠍ⊒㆐ߐߖࠆߎ
ߣߢߣࡓ࠹ࠬࠪ∉ߩ⥄⁛ޔᖱႎવ㆐ᯏ᭴ࠍ₪ᓧߔࠆߦ⥋ߞߡࠆߩߘޕᖱႎવ㆐ߩ⚵
ߺߦὶὐࠍᒰߡࠆߎߣߢ‛↢ࠆߊߡ߃ޔㅴൻߩ৻㕙ߣߪ㧚㧚㧚
㧔㗴ឭଏ⠪㧦ᵤᐘ㧕
ീᒝળߩᚑᨐ㧦ฦಽ㊁ߩၮ␆ߦ㑐ߔࠆᔨ߿⠨߃ᣇࠍਛᔃߣߒߚ⻠⟵ࠍߒߡޔෳട⠪ߩ
㑆ߢᵴ⊒ߥᗧ឵߇ߥߐࠇߚ⚿ߩߘޕᨐޔᢙℂ↢ޔޔ㊂ሶߣ߁ಽ㊁ߩᨒࠍ߃ߚ
ㅢߩⷞὐߩ₪ᓧߣࠄ߆ߎߘޔฦಽ㊁ߩ⁛⥄ᕈߦ┙ߜᚯࠆߣ߁ޟㇱಽߣో߽ߢߣޠߴ
ࠆߴ߈ᣇᴺ⺰ߩน⢻ᕈࠍޔෳട⠪ฦ⥄߇ᗵߓࠆߎߣ߇ߢ߈ߚߣᕁ߃ࠆޕ
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− 162 −
㪈㪎㪑㪊㪇
㪈㪎㪑㪇㪇
㪈㪍㪑㪊㪇
㪈㪋㪑㪊㪇
㪈㪌㪑㪇㪇
㪈㪊㪑㪇㪇
㪈㪉㪑㪇㪇
㪈㪈㪑㪊㪇
㪈㪇㪑㪇㪇
㩿㪝㫉㫀㪀
㪏㪉㪈ᣣ
㩿㪪㪸㫋㪀
㩿㪪㫌㫅㪀
㪨㫌㪸㫅㫋㫌㫄㩷㪤㪼㪺㪿㪸㫅㫀㪺㫊㩷㫀㫅
㪣㪸㪹㫆㫉㪸㫋㫆㫉㫐㩷䋨ਛ䊉వ↢
䃩
䋩
㪏㪉㪊ᣣ
㩿㪤㫆㫅㪀
䊂䉞䉴䉦䉾䉲䊢䊮㪠㪠
䃩䋺㪥㪫㪫‛ᕈ⑼ቇၮ␆⎇ⓥᚲ
䋳ᰴరశᩰሶ䈮䊃䊤䉾䊒䈘䉏
䈢න৻ේሶ䋨⿒Ⴆ䋩
㪏㪉㪋ᣣ
㩿㪫㫌㪼㪀
㪏㪉㪌ᣣ
䊂䉞䉴䉦䉾䉲䊢䊮㪠㪠㪠
㪤㫀㪺㫉㫆㪤㪸㪺㫉㫆㩷㪛㫌㪸㫋㫀㫃㫐㩷㪠㪠㪠
㩿ዊ᎑వ↢䋩
㪘㪻㪸㫇㫋㫀㫍㪼㩷㪛㫐㫅㪸㫄㫀㪺㫊㩷㪠㪠
㪥㪼㫃㫊㫆㫅㩷㪛㫐㫅㪸㫄㫀㪺㫊
䋨ᄢ⍫వ↢䇮ጊవ↢䇮
䋨᧻ጟవ↢䇮ᚭᎹవ↢䇮 ᵻ㊁㪧㪛䋩
䊂䉞䉴䉦䉾䉲䊢䊮㪠㪭
ጊ⊓వ↢䇮ᑝ↰㪧㪛䋩
䊤䊮䉼䊑䊧䊷䉪
㪘㫇㫇㫃㫀㪺㪸㫋㫀㫆㫅㩷㫆㪽㩷㪘㪻㪸㫇㫋㫀㫍㪼
㪛㫐㫅㪸㫄㫀㪺㫊㩷㫋㫆㩷㪣㫀㪽㪼㩷㪪㪺㫀㪼㫅㪺㪼
䋨⮮వ↢䇮ේవ↢䋩
㪤㫀㪺㫉㫆㪤㪸㪺㫉㫆㩷㪛㫌㪸㫋㫀㫃㫐䇭㪠㪠
䋨ዊ᎑వ↢䋩
㪏㪉㪉ᣣ
ળᦼ䋺㪉㪇㪈㪇ᐕ㪏㪉㪇ᣣ䌾㪉㪌ᣣ䇭䇭ᣈ䋺⺪⸰᧲੩ℂ⑼ᄢቇ
㪣㫀㪽㪼㩷㪪㪺㫀㪼㫅㪺㪼㩷㪠㪠
䊌䊈䊦䊂䉞䉴䉦䉾䉲䊢䊮㪄
䋨ችፒవ↢䇮㋈ᧁవ↢䇮 㪨㫌㪸㫅㫋㫌㫄㩷㪤㪼㪺㪿㪸㫅㫀㪺㫊㩷㫀㫅
䊐䊥䊷䊂䉞䉴䉦䉾䉲䊢䊮
ᰨవ↢䋩
㪣㪸㪹㫆㫉㪸㫋㫆㫉㫐
↢‛කቇᢥ₂䈎䉌䈱∔ᖚ
䋨ဈ↰వ↢䇮ችᎹవ↢䇮
㑐ㅪㆮવሶត⚝ᚻᴺ䈱
⋓᳗వ↢䇮ᷰㆻవ↢䋩
ឭ᩺䋨ᰨᆻᄢ䋩
䊂䉞䉴䉦䉾䉲䊢䊮㪠
㪣㫀㪽㪼㩷㪪㪺㫀㪼㫅㪺㪼㩷㪠
䋨ᵤవ↢䋩
䃨䋺੩ㇺᄢቇᢙℂ⸃ᨆ⎇ⓥᚲ
㪤㫀㪺㫉㫆㪤㪸㪺㫉㫆㩷㪛㫌㪸㫋㫀㫃㫐㩷㪠
䋨ዊ᎑వ↢䃨䋩
㪘㪻㪸㫇㫋㫀㫍㪼㩷㪛㫐㫅㪸㫄㫀㪺㫊
䋨ᄢ⍫వ↢䋩
㪏㪉㪇ᣣ
㪉㪇㪈㪇ᐕ䇭㪨㪙㪠㪚䉰䊙䊷䉴䉪䊷䊦䇭ീᒝળᤨ㑆ഀ
㩿㪮㪼㪻㪀
2010 ᐕᐲ QBIC ࡒ࠽ⷐ
ᣣᤨ㧦10 13 ᣣ㧘20 ᣣ㧘27 ᣣ㧘15㧦00㨪16㧦00
႐ᚲ㧦⸘▚⑼ቇ⎇ⓥࡦ࠲㧠F ᄢળ⼏ቶ
⻠Ꮷ: Roman V. Belavkin; School of Engineering and Information Sciences, Middlesex
University, London NW4 4BT, UK
╙ 1 ࿁ 㗴: The effect of information constraints on decision-making and economic
behaviour
Abstract:
Economic theory is based on the idea of rational agents acting according to their
preferences.
Mathematically, this is represented by maximisation of utility or the
expected utility function, if choices are made under uncertainty.
Originally developed
in game theory, this formalism has become dominant in operations research, machine
learning and even used in cognitive science to model human behaviour.
There is,
however, a number of paradoxical counter-examples in behavioural economists causing
a degree of scepticism about the correctness of the expected utility model of human
decision-making.
I will try to convince you that many of these paradoxes can be
avoided if the problem is treated from a learning theory point of view, where
information constraints are explicitly taken into account.
I will use methods of
functional and convex analyses to demonstrate geometric interpretation of the solution
of an abstract optimal learning problem, and demonstrate how this solution explains
the mismatch between the normative and behavioural theories of decision-making.
╙ 2 ࿁ 㗴: Geometric optimisation of learning and utility of information
Abstract:
I will describe a formalism to represent information states of a classical or quantum
learning system.
These states will correspond to points on a statistical manifold --- a
subset of a vector space endowed with an information topology. An optimal learning
system is represented by points that are solutions to the information utility problem.
Geometric and functional analyses become useful tools for interpretation of the results,
which have close relation to statistical physics as well as adaptive dynamics.
I will
present several results and examples, and then discuss a range of current and potential
applications to economics, machine learning and evolutionary systems.
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╙ 3 ࿁ 㗴: Evolution as an information dynamic system
Abstract:
I will speak about a new research project called `Evolution as an information dynamic
system', which involves collaboration between four universities in the United Kingdom.
This is a three year project between 2010--2012, and its aim is to develop new
understanding of information dynamics in evolution and biology.
In particular, we are
going to derive new optimality conditions for some evolutionary operators, such as
mutation and recombination. Evolutionary states will be represented by probability
measures on the space of genetic sequences, and different operators produce different
evolution of the states. We define the optimality conditions for evolution based on the
maximisation of utility (or fitness) of information principle. The optimal evolution in
this sense achieves the shortest `information distance', and it can be different from an
evolution optimal in another sense, such as the shortest convergence time.
We argue
that the former achieves a better adaptation of organisms living in a dynamic
environment.
I will present several early results related to the optimisation of
mutation rate parameter.
I will review these results in the light of the classical
theories of adaptation (e.g. Fisher's geometric model) and error threshold.
Then I will
outline some future theoretical and experimental work of the project.
-----------------------------------------------------------------------------------------------------------------------ᣣᤨ㧦10 18 ᣣ㧔㧕13:00㨪15:00
႐ᚲ㧦⸘▚⑼ቇࡦ࠲4 㓏ળ⼏ቶ㧘
1) 13:00㨪14:00
⻠Ṷ㗴⋡㧦Dynamics and entanglement of exciton states in clusters of resonantly
interacting fluorescent particles.
⻠Ꮷ㧦 Dr. Irina Basieva (Prokhorov General Physics Institute, Russian Academy of
Sciences, Russia)
The presented theoretical consideration of the exciton states control via laser field can
be applied to any equivalent and equidistant two-level systems: quantum dots, dye
molecules, rare-earth ions, or something else.
Specifically, the parameters relevant to rare-earth ions are used. Rare-earth ions are
promising as quantum computer hardware due to the fact that electron transitions of
interest take place in the internal 4f electronic shell relatively isolated from the
environment.
− 164 −
In clusters of rare-earth ions energy levels split due to interion interaction, for
example, one exciton state does not belong to any specific ion, it is a state of the whole
cluster. It means that one photon is not localized on a specific ion. To describe quantum
evolution of energy states we used half-spin formalism.
We suggest a method from the theory of differential equations to solve the
Shroedinger, or master, equation with a small parameter. The parameter can be laser
intensity, if it is weak comparing with the ion-ion interaction, or, in the case of master
equation, small dephasing parameter.
The method presented proved to be much more
useful and adequate to dealing with these problems than the perturbation theory. For
example, it allows us to predict optimal laser pulse intensities and duration needed to
prepare maximally entangled states, or perform some quantum gate operations.
2) 14:00㨪15:00
⻠Ṷ㗴⋡㧦Exploring the analogy between classical and quantum signals
⻠ Ꮷ 㧦 Prof.
Andrei
Khrennikov,
(Applied
Mathematics,International
Center
Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science,
Linnaeus University)
Recently it was shown that them a indistinguishing features of quantum mechanics
(QM) can be re-produced by a model based on classical random felds, the so-called
prequantum classical statistical feld theory (PCSFT). This model provides a possibility
to represent averages of quantum observables, including correlations of observables on
subsystems of a composite system (e.g., entangled systems) , as averages with respect to
fluctuations of classical (Gaussian) random felds. We consider some con-sequences of
the PCSFT for quantum information theory. They are based on our previous
observation that classical Gaussian channels (important in classical signal theory) can
be represented as quantum channels. Now we show that quantum channels can be
represented as classical linear transforms of classical Gaussian signals.
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C M Y K
CORE
平成 年度 東京理科大学
22
CORE
CORE
CORE
文部科学省私立大学学術研究高度化推進事業
ハイテク・リサーチ・センター整備事業
平成22年度 研究成果報告
量子論から見る情報と生命の研究
量子論から見る
情報と生命の研究
研究成果報告
平成
年３月
23
108330_QBIC研究報告書.indd 1
平成23年3月
東京理科大学総合研究機構
量子生命情報研究センター
10/12/14 18:35
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