第 30 卷第 5 期 光　子　学　报　V o l. 30 No . 5
2001 年 5 月 ACT A
PHOT ONICA SINICA M ay
2001 QUANTUM STATISTIC PROPERTIES OF
SUPERPOSITION OF THE EXCITED TWO-MODE
*
SQUEEZED VACUUM STATES
H uang Chunqing 1, Jiang Junqin 2
1 D ep artment of P hy sics, Foshan U niver sity , F os han 528000, China
2 D ep artment of Phy sics, Guang dong I nstitute of Education, Guangz hou 510303, China
Abstract　T he superposition st ate fro m t he excited t w o -mode squeezed v acuum stat es is
co nstr ucted , and it s noncl assical pro pert ies are st udied . Under cer tain co ndit ions , as t he
phase diff er ence changes , the mean phot on num ber o f t he superposit ion st at e ex hibit s o scillat ing phenomenon of collapses and rev iv als in a manner sim ilar t o t hat of Rabi oscillatio n, sub -Poissonian phot on st at ist ical character and phase squeezing of t he f ield in t he
superposit ion st at es are st ronger t han ones in the single ex cit ed t w o -mo de squeezed vacuum stat e .
Keywords　Superposit ion of t he ex cit ed t w o -mo de squeezed v acuum st at es ; P hot o n number oscillating ; Phase squeezing ; Sub-P oissonian phot on st at ist ical charact er
0　Introduction
It is w ell -know t hat t he t wo -mode squeezed
vacuum st at e ( T M SVS ) is a highly correlat ed
nonclassical st at e. T M SVS has been real ized in
t he labo rato ry 1 and w as used as an entangled Einstein-P odolsky -Rosen st at e in a t elepo rt ation
2
exper im ent . But the pho to n st at ist ics is alw ay s
super -P oissonian in t he T MSVS . Recent ly w e
hav e int ro duced t he ex cit ed t wo -mode squeezed
m
m
〉( nam ely , pho to n-added
vacuum st ate a b
3
t w o -mode squeezed vacuum st ate ) via repeat ed
applicat ion of t he pho to n creat ion oper at or on
t he T M SVS , and st udied it s nonclassical pro perties, our r esult s sho w t hat fo r t he excited t w omode squeezed vacuum st ate, t he phase squeezing ex ist s and t he st atist ics is sub-P oissonian under cert ain condit ions.
T he super position stat es ( Schro¨ ding er cat
st at es) can exhibit int erest ing nonclassical pro per ties, and have recent ly been w idely st udied.
Janszky , Buzek and Schleich st udied the superposit ion st at es f rom t he sing le-mode coherent
st at es; Chai applied t his met ho d t o t he t w o -m ode
*
4
case , const ruct ed the superpo sit io n st at es fro m
t he tw o -m ode coherent st at es , and discussed
t heir nonclassical proper ties such as t he t w o mo de squeezing , t he violat ion of t he Cauchy Schw art z inequalit y and t he sub -Poissonian pho5
t on st at ist ics ; L u and Guo st udied t he noncl assical propert ies of superposit io n st at es f rom t he
excited single -mo de coherent st at es .
In t his paper , w e apply t his method t o t he
excited tw o -mode case , const ruct t he superposi〉 fr om t he ex cit ed t w o -m ode
t io n st at es
m
m
〉
, and investisqueezed v acuum st ates a b
gat e t heir nonclassical pro pert ies, oscillat io ns of
t he mean phot on number , the phase squeezing
and t he sub-P oissonian phot on stat ist ical charact er of t he field.
1　Construction of the superposition
states
In this number -st at e representat ion , t he
6
T MSVS can be ex pr essed as
T he pr oject suppor ted by Guangdong Pr ov incial N at ur al Science F oundation ( 990212) of China
R eceived dat e: 2000-11-07
光　子　学　报　30 卷
524
〉
= S 0a, 0b 〉
∞
= 1
(- i
)n , 〉
coshr n = 0 e t anhr n n
( 1)
w here
*
S= ex p( ab- a b )
( 2)
i
here a and b are annihilat ion oper at o rs, = r e is
a complex squeeze param et er.
T he ex cit ed t w o-mo de squeezed vacuum
3
st at e is defined by
m
m
; m, m〉
= A ma b
〉
∞
Am
i
n
= coshr
( - e t anhr )
n= 0
・[ ( m + n) ! / n! ] m+ n, m + n〉 ( 3)
Now , w e co nst ruct t he superposit ion st at es
fr om t he ex cit ed t w o-mo de squeezed vacuum
st at es
i
*
〉
= B m ( ; m, m 〉
+ e
; m, m 〉
)
F ig . 1　N a v er sus
for =
/ 2, r = 0. 5, m= 0, 1, 3
∞
Cm
n ( m + n) !
( - t anhr )
co shr n = 0
n!
in
i( - n )
・( e + e
) m+ n , m+ n 〉
( 4)
;
,
w here is a relativ e phase bet w een
m m〉
*
; m , m 〉, the phase dif f erence bet w een ;
and
*
,
〉
; m , m 〉is
= 2 , w hen = 0 t he
m m and
〉reduces t o ; m , m 〉
. C m is no rmalizast at e
tio n const ant
=
∞
- 2
2
C m = ( 2/ cosh r )
( t anhr ) 2n
n= 0
・( n! )
- 2
2
[ ( m+ n) ! ] [ 1+ co s( 2n - ) ] ( 5)
2 Nonclassical properties of the
superposition states
First , w e calculat e t he mean pho to n num ber
N.
Fr om Eq . ( 4) , the pro babilit y o f f inding m +
n phot ons in t he f iel d mode a( b ) is
∞
2Cm2
[ ( m+ n) ! ] 2
m
+
n
=
( t anhr ) 2n
P
2
co sh r n = 0
( n! ) 2
・[ 1+ co s( 2n - ) ]
( 6)
T he m ean phot on num ber N a( = N b ) is given by
∞
N = 〈
a a〉
=
a
P m+ n ( m+ n)
( 7)
n= 0
In Fig . 1, t he mean pho to n num ber N a is
show n as a f unct ion of f or = / 2, r= 0. 5, m =
0, 1, 3.
In Fig . 2, t he mean pho to n num ber N a is
show n as a f unct ion of f or = / 2, r= 1. 5, m =
3. F or t he l ar ger r , as changes, t he mean
F ig . 2　N a v ersus
for =
/ 2, r = 1. 5, m = 3
phot on
number
N a ex hibit s o scill at ing
pheno meno n o f collapses and revivals in a m anner similar t o t hat of Rabi o scil lation.
Next , w e st udied the phase squeezing de7
fined by Lo udon and Knight . T he t w o -m ode
quadrat ure operat or s are ginen by
X 1 = ( a+ a + b+ b ) /
8
( 8)
X 2 = ( a- a + b- b ) / i
8
T he phase squeezing is said t o ex ist w henev er X 2j < 1/ 4( j = 1, 2) .
Fo r t he st ate
〉
, w e can easily pr ove t hat
〈
a〉= 〈
b〉= 〈a 〉
= 〈b 〉= 〈
a2 〉= 〈
b2 〉
= 〈a 2 〉=
〈
b 2〉= 〈ab 〉= 〈
b a〉= 0, 〈a b 〉= 〈b a 〉=
〈
ab〉= 〈
ba〉
, such t hat
2
2
2
- 〈
X 1= 〈
X 1〉
X 1〉
- 1
- 1
- 1
= 4 + 2 〈
a a〉
+ 2 〈
ab〉
∞
2
2
N
a
C
m
1
2n+ 1 [ ( m + n) ! ]
= 4+ 2 2
( t anhr )
2
cosh r n = 0
( n! )
( n+ m + 1) 2
×
[ cos + cos( 2n + - ) ]
( 9)
n+ 1
2
In F ig . 3, X 1 is show n as a f unct io n o f f or
= 0. 9 , r = 0. 5, m = 1. As can be seen, under
〉excert ain co ndit ions , t he field in t he st ate
hibit s st ro ng er phase squeezing t han t he f ield in
t he st ate ; m , m 〉( corresponding to = 0) .
5 期　Huang Chunqing , et al. Q uantum statist ic pr oper ties of superposit ion
o f the excited tw o -mo de squeezed v acuum stat es
F ig . 3　X 21 ver sus
fo r = 0. 9 , r = 0. 5, m = 1
Finally, w e discuss t he st at ist ical pr opert ies
of phot ons in t he field m ode a( b) by calculat ing
t he M andel Qa ( Q b = Q a) paramet er , w hich is de8
fined by
2
2
( a a) 2 〉
- 〈a a 〉
] /〈
( 10)
Qa = [ 〈
a a〉
w here
∞
2
〈
( a a) 〉
=
P m + n ( m+ n)
2
( 11)
n= 0
T he phot on st at ist ics is sub-P oissonian
w henev er Qa < 1.
In F ig. 4, Qa is sho w n as a funct ion of f or
= 0. 9 , r = 0. 5, m= 0, 1, 6. As can be seen, under
cert ain condit ions, t he sub-Poissonian phot on
st at ist ical char act er o f t he f iel d in t he st ate
〉
is also st rong er t han o ne in t he st at e ; m , m〉
Fig . 4　Q a ver sus
525
fo r = 0. 9 , r = 0. 5, m= 0, 1, 6
( corr esponding t o = 0) .
3　Conclusion
We have const ruct ed t he super position
st at es f rom t he ex cit ed t w o -mode squeezed vacuum st at es , and have st udied t heir nonclassical
pr opert ies. Our results show t hat under cert ain
co nditions, t he mean phot on number of t he superposit ion st ate exhibit s o scil lating pheno menon of coll apses and revivals in a manner
sim ilar to that of Rabi oscillat ion, sub-P oissonian
phot on stat ist ical char act er and phase squeezing
of the field in t he superpo sit ion st at es are
st ro ng er t han o nes in the single ex cit ed t w omo de squeezed vacuum st at e.
References
1　Slusher R E, Hollber g L W, Yur ke B, M er tz J C , V alley J F. Obser vation of squeezed states g enerated by four -wa ve
mix ing in a n o pt ical cav ity . P hys Rev Let t, 1985, 55( 9) : 2409～2412
2　Furusaw a A , S r ensen J L , Br aunstein S L , Fuchs C A , K imble H J , P olzik E S . U nco nditional quant um telepo r tatio n .
Science, 1998, 282( 6) : 706～709
3　Jiang Junqin, Huang Chunqing and L u Hong . N onclassical pr oper ties of the ex cited two -mo de squeezed vacuum stat e.
A cta P hoto nica Sinica ( China) , 2000, 29( 11) : 989～992
4　Chai Chinlin . T w o -mode nonclassical st ate via superpositio ns of t wo -mo de co her ent states . P hys Rev A , 1992, 46
( 11) : 7187～7191
5　L u Hong , G uo Guangcan. N o nclassical pr operties o f states gener ated by the superposit ion o f ex cit ed coher ent st ates.
A cta P hysica Sinica ( China ) , 1999, 48( 9) : 1644～1649
6　L u Hong . Pho ton statistics of photo n -added and photo n ～ subtr act ed tw o -mo de squeezed v acuum state . Chinese P hys
L ett , 1999, 16( 9) : 646～647
7　L o udon R , K night P L . Squeezed lig ht. J M o d O pt, 1987, 34( 6/ 7) : 709～759
8　M andel L . Sub -Po issonian pho to n stat istics in r eso nance fluo r escence. O pt Let t , 1979, 4( 1) : 205～208
光　子　学　报　30 卷
526
叠加激发双模压缩真空态的量子统计特性
黄纯青
( 佛山大学物理系, 佛山 528000)
江俊勤
( 广东教育学院物理系, 广州 510303)
收稿日期: 2000-11-07
摘　要　从激发双模压缩真空态 a
m
b
m
〉
出发构造了叠加态
〉
, 研究了
〉
的量子统计特
性 . 结果表明: 在一定的条件下, 随着相位差的变化, 叠加态 〉的平均光子数出现类似于
m
m
〉
相比, 在叠加态 〉
Rabi 振荡的崩塌与复原现象, 而且与单个激发双模压缩真空态 a b
中光场的相位压缩和亚泊松光子统计特性都得到了加强 .
关键词　叠加激发双模压缩真空态; 光子数振荡; 相位压缩; 亚泊松光子统计特性
Huang Chunqing w as born in 1963. He g raduated f rom Delpart ment of
Physics , South China Norm al U niversit y , guang zhou , China and r eceived a M .
Edu. degr ee in 1991, Now he is a visit ing scholar in Depar tm ent of P hy sics,
Zhongshan U niversity , Guangzhou, China. He has published more t han t en papers in t he academ ic journals at ho me and abr oad. H is m ajor resear ch is in
phy sics o f part icle and quant um o pt ics .