FULL P APERS
Journal of the Physical Society of Japan 81 (2012) 044301
DOI: 10.1143/JPSJ.81.044301
High Sensitivity Measurement and Accurate Analysis of the Vibrational
Spectroscopy Near the ð6S1=2 þ 6P3=2 Þ Dissociation limit for 1g State of Cs2
Yuqing LI, Jie MA, Jizhou WU, Yichi ZHANG, Gang CHEN, Yanting ZHAO,
Lirong WANG, Liantuan XIAO, and Suotang JIA
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Laser Spectroscopy Laboratory,
Shanxi University, Taiyuan 030006, P. R. China
(Received December 5, 2011; accepted January 26, 2012; published online March 21, 2012)
The photoassociation spectroscopic data of the long-range molecular 1g state of Cs2 in the asymptotic range red
detuned ½5:02; 2:45 cm1 to the ð6S1=2 þ 6P3=2 Þ dissociation limit detected using photon counting technique are
presented and 18 new high-lying vibrational energy levels are obtained. Fitting experimental data to the improved
LeRoy–Bernstein (LRB) formula, we have obtained the long-range molecular coefficient c3 with a high precision and
also demonstrated the necessity of the improved LRB formula for more precise c3 coefficient by analyzing the residual
from fitting procedure.
KEYWORDS: photoassociation, photon counting, the long-range molecular coefficient, residual
1.
Introduction
The high-resolution molecular photoassociation spectroscopy (PAS), especially for the part concentrating on
dissociation limit, has been showing a great deal of interest
as a particular testbed for the ultra-high resolution measurement, studying long-range molecular dipole–dipole interaction1–3) and providing excite energy levels data for closed
channel in Optical Feshbach Resonance.4,5) Besides, it also
provides a particular opportunity to obtain some valuable
information about the long-range molecular excited state,6,7)
investigate atom-molecule colliding8,9) and determine the
long-range molecular coefficient and therefore the molecular
potential energy curves.10) Due to atomic resonance arises
near dissociation limit, all of cold trapped atoms almost
vanish, and so the high-lying vibrational levels data are very
troublesome, where no experimental result is reported except
our group gave spectral data11) for the long-range molecular
state 0u þ of Cs2 with red detuning range ½6:2; 2:5 cm1 below the ð6S1=2 þ 6P3=2 Þ dissociation limit so far.
For the long-range Cs2 exited state, since the PA
experiment is reported firstly by Pillet et al. in 1998,12)
there are many experiments performed to obtain more broad
and richer energy levels data. For example, Pichler et al.
provided the rich experimental data with red detuning range
½49:5; 6:15 cm1 below the 6S1=2 þ 6P3=2 dissociation
limit in 2004.13) In 2009, our group extended spectroscopy range to a larger red detuning with range ½68:12;
7:40 cm1 based on the technique of the modulated
trap-loss fluorescence spectroscopy.14)
In this paper, we use photon counting technique for PAS
spectroscopy just as the way,11) instead of the detection
technique of the conventional ion-spectroscopy or resonant
fluorescence, where the resonant interaction between atoms
and laser increases and suppresses the formation of highlying vibrational serried levels close to the dissociation limit.
Though a small part of molecule are produced, they are very
hard to detect due to spectroscopic sensitivity. For ionization
spectroscopy, the molecule produced in magneto-optical trap
E-mail: [email protected]
(MOT) will be destroyed and ionized to obtain molecular
ion signal and this goes against next experiment. And the
resonant fluorescence by directly monitoring fluorescence of
the trapped cold atoms is also not suitable because stray
noises nearly submerge useful signal in the part of PA
spectroscopy near the dissociation limit. The spectral data
of the high-lying vibrational energy levels of the longrange molecular state 1g of Cs2 with red detuning
½5:02; 2:45 cm1 below the ð6S1=2 þ 6P3=2 Þ dissociation
limit have been presented by using the high sensitive photon
counting technique.
For the long-range part of the excited diatomic molecular
potential expressed as V ðRÞ D þ cn =Rn þ cm =Rm þ ðn > 2; m > nÞ, of interesting is the asymptotic term
D cn =Rn mainly determines the asymptotic behavior of
alkali-metal dimers. Focusing on the cesium atom, the
quantitative analysis of the PAS data is often performed
by using the well-known LeRoy–Bernstein (LRB) formula
to obtain cn coefficient determining the leading long-range
interaction term (cn =R3 ) for an asymptotic Rn potential
of diatomic molecules with n ¼ 3. For more precision, the
improved LRB formula proposed by Comparat through
taking into account the multipole expansion terms is
employed to obtain more precision long-range molecular
coefficient by considering two additional terms.15) Comparing with the theoretical long-range molecular coefficient, the
result obtained by using the improved LRB formula to fit
shows a remarkable improvement. In addition, we have
demonstrated the requirement of the improved LRB formula
by analyzing residual from fitting procedure.
2.
Experiment and Spectral Data
The typical MOT technique11,14) is employed to prepare
cold atomic sample. Consequently the cold caesium atoms
107 are produced in the quartz vacuum chamber with a
background pressure 1 107 Pa. The temperature of the
sample is estimated to be 200 K measured by the method
of time of flight. The PA is performed by a widely tunable
Ti:sapphire laser pumped by the Verdi-10 with a line-width
less than 100 kHz and output power 600 mW. The PA laser
is focused on the sample of cold atoms with a diameter of
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#2012 The Physical Society of Japan
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FULL PAPERS
Y. LI et al.
Fig. 1. Experimental setup for photoassociation of the cold atoms sample
trapped in MOT and detection by using photon counting technology.
500 m resulting in an available intensities 300 W/cm2
shown in Fig. 1. The frequency of PA laser is measured by a
wavelength meter with an accuracy 0:002 cm1 .
In the PA process, a pair of cold ground cesium atoms
absorbs resonantly a photon provided by PA light and forms
the electronic rovibrational level of an artificial long-range
molecular excited state:
Csð6S1=2 , F ¼ 4Þ þ Csð6S1=2 , F ¼ 4Þ
Fig. 2. Nonlinear fitting for long-range state 1g of Cs2 by using wellknown LRB formula (LRB), the first improved LRB formula (F-LRB) and
the second improved LRB formula (S-LRB).
Table I. The experimental data of bound energy of high-lying vibrational
levels for the 1g long-range state of Cs2 below the 6S1=2 þ 6P3=2
dissociation limit.
þ hðv0 L Þ ! Cs2 ½u,g ð6S1=2 + 6P3=2 Þ; v; J;
where v0 is the resonant frequency of Cs atomic
6S1=2 ðF ¼ 4Þ ! 6P3=2 ðF ¼ 5Þ hyperfine transition corresponding to dissociation limit 11732:176 cm1 , L is the
red detuning of the PA laser and v is the frequency of
resonant transition from atomic ground state to the excited
molecular state. The spontaneous emission of excited
photoassociated molecules leads back mostly to dissociation
into two free atoms with a relative kinetic energy larger than
that in the initial cold ground state. Thus, the pairs of heated
atoms generally escape from MOT, and the analysis of the
trap-loss is a convenient way to detect PA. As a kind of
convenient and sensitive detecting technique, the trap-loss
in PA experiments with cold cesium atoms have permitted
the spectroscopy of the three attractive Hund’s c case states
1g, 0u þ and 0g converging to the ð6S1=2 þ 6P3=2 Þ dissociation limit. In this section, we report high sensitivity PAS
data close to the 6S1=2 þ 6P3=2 dissociation limit for 1g
state by using photon counting not directly monitoring the
resonant fluorescence.
The fluorescence from the trapped atoms is collected by a
convex lens and is detected by Si-avalanche photodiode just
as in Fig. 1. The output pulses from the photodiode are
amplified and discriminated with a controllable threshold by
a fast discriminator (ORTEC 9302). Choosing an appropriate threshold voltage, we can obtain the optimal SNR as
Fig. 2.11) The optimal SNR 26 has been achieved as the
threshold level is 0.63 V and a universal counter (Stanford
Research SR620) is used to record the discriminated pulses.
Finally, the high resolution PAS data in the asymptotic range
½5:02; 2:45 cm1 close to the 6S1=2 þ 6P3=2 dissociation
limit for 1g state is shown in Table I.
3.
vD v
"V
(cm1 )
134
2.453
143
3.6333
135
2.5651
144
3.795
136
2.6858
145
3.9845
137
2.7983
146
4.1589
138
139
2.9321
3.0794
147
148
4.3206
4.4747
140
3.2132
149
4.6362
141
3.3528
150
4.814
142
3.4808
151
5.0155
vD v
"V
(cm1 )
Reexamination of the Improved LRB Formula
The PAS data directly reveals bound energy levels of the
excited molecular state, which is particularly essential for
determination of the long-range molecular coefficients cn
by fitting experimental spectral data to the well-known LRB
formula proposed by LeRoy and Bernstein in 197016) and
computing the asymptotic potential curve of the long-range
molecular state. Here we remain only the main asymptotic
term cn =Rn determines the asymptotic behavior of the
alkali-metal dimmer in the long-range molecular potential
expressed as V ðRÞ D þ cn =Rn þ cm =Rm þ ðn > 2;
m > nÞ. The LRB formula is a analytic formula for the
eigen energies of a potential V ðRÞ has an asymptotic form as
cn =Rn . For more precision cn coefficient, we have recalled
and reexamined the derivation of the LRB formula and
obtained the improved LRB formula by considering the two
additional high order terms in the multipole expansion
process.15,17)
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#2012 The Physical Society of Japan
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The starting point of LRB formula used to analyze PAS
data is the application of Bohr–Sommerfeld rule (1) for a
bound vibrational molecular level v with the energy E, u is
the reduced mass, and R and Rþ are the inner and outer
classical turning points of the vibrational motion, solving the
equation V ðRÞ ¼ E:
pffiffiffiffiffi Z R
þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2u
1
E V ðRÞ dR ¼ v þ
:
ð1Þ
h R
2
In order to the simplification of the computing, a new
defined integral is introduced as
pffiffiffiffiffi Z Rþ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2u
¼
E V ðRÞ dR:
ð2Þ
h
R
The integral about formula (2) depends strongly on E and
V ðRÞ, and the differentiation of ,
pffiffiffiffiffi Z R
þ
2u
d
dv
dR
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
¼
¼
ð3Þ
h R
dE dE
E V ðRÞ
which represents the level density, is dominated by its value
in the range defined by V ðRÞ ¼ E, and the asymptotic
potential form is V ðRÞ ¼ D cn =Rn .
The equation links the molecular vibrational constant v
and bound energy " ¼ D E can be deduced from
pffiffiffiffiffi Z R
þ
2u
d
dR
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinffi ;
¼
ð4Þ
d"
h 0
" þ cn =R
where the extension of the integral for the short value of R
is allowed to the limit R ¼ 0 according to appropriate
approximation17) and the Rþ is defined as the method.15)
By integration of the level density and the high order terms
are neglected in the multipole expansion through remaining
only main part, The LRB law is given as formula (5):
ðn2Þ=2n
"
;
ð5Þ
vD v ¼
En
2
32n=ðn2Þ
1
1
þ
6 pffiffiffi
n 7
6 h ðn 2Þ 7
En ¼ 6 pffiffiffiffiffi 1=n
:
ð6Þ
7
4 2u cn
1 1 5
þ
2 n
The above established result gives binding energy " of a
vibrational level versus the quantum vibrational number v by
(5), vD is a constant related to dissociation limit, En is a
parameter connected to the reduced mass u, the long-range
molecular coefficient cn and is the gamma function.
The improved LRB formula is proposed firstly by
Comparat through taking into account the multipole
expansion terms and the non-asymptotic part of the potential
curve.15) After Jelassi et al. show that the well-known LRB
formula can be corrected by three additional terms, with the
first one varying as ", the second one as "2 , and the third as
"7=6 by a slight different theoretical deducing.17) However,
there is few report for application of the improved LRB
formula to fit spectral data. So far, Jelassi et al. employed
the improved LRB formula including only one additional
term varying as bound energy linearly on PA spectral data of
87
Rb2 for the long-range molecular ð5S1=2 þ 5P1=2 Þ0g state,
and the result of fitting presents a very good agreement
between experimental data and theory.18) In 2009, our group
also applied the well-known LRB formula (5) to the
Y. LI et al.
experimental deeply bound 0u þ and 1g levels14) and the
improved LRB formula containing only linear term to
spectral data of the 0u þ state near threshold11) converging to
the ð6S1=2 þ 6P3=2 Þ dissociation limit of Cs2 .
For the high precision long-range molecular cn coefficient,
the well-known LRB formula has to be corrected by
remaining two additional high order terms as (8), with the
first one varying as ", the second one as "2 . Here we employ
two additional modified terms not the third one given by
Jelassi,17) because the power value of energy from the third
modified term is given in the middle between the first one
and the second. And this can reflect the each contribution
of the first and second additional term via shielding the
influence from the third modified term. In order to
comparing, we apply the well-known LRB formula (5) and
the improved LRB formula (7) and (8) to fit our
experimental spectral data,
vD v ¼ Hn1 "ðn2Þ=2n þ ";
vD v ¼
Hn1 "ðn2Þ=2n
ð7Þ
þ " þ " ;
2
ð8Þ
Hn1
is a fitting parameter including the long-range
where
coefficient cn ,
1 1
þ
ðn2Þ=2n pffiffiffiffiffi
2 n
2uðcn Þ1=n
1
; ð9Þ
¼ pffiffiffi
Hn1 ¼
En
1
h ðn 2Þ
1þ
n
and is the first order correction term coefficient and is
the second order correction term coefficient, which can be
obtained as parameters in a nonlinear fit procedure. Only
considering the main term (dominant contribution), eqs. (7)
and (8) become the well-known LRB formula (5).
4.
Analysis of the Spectral Data
The experimental spectral data near the ð6S1=2 þ 6P3=2 Þ
dissociation limit for excited 1g state of Cs2 detected by
using photon counting technique are fitted to the improved
LRB formula (7) and (8) and well-known LRB formula (5)
as n ¼ 3, and this nonlinear fitting procedure is shown in
Fig. 2. Here we regard the improved LRB formula (7) and
(8) as the first improved LRB formula (F-LRB) and the
second improved LRB formula (S-LRB) respectively. The
long-range molecular c3 coefficient can be deduced from
fitting parameter H31 comes from the nonlinear fitting
procedure, and the corresponding values of c3 are 16.6050,
16.6197, and 17.4887 shown in Table II by using LRB (1),
F-LRB (7) and S-LRB (8) to fit, respectively. Comparing to
the theoretical value 17:46,19) the error is 4.90% from the
non-linear fit by using well-known LRB (5). The result
obtained by using F-LRB (7) improves little with the error
4:81%, but a more precise c3 coefficient is deduced from
the parameter of non-linear fitting with the error reduced
to 0.16% by applying S-LRB (8) to the spectral data. Thus
it is necessary for the improved LRB formula [namely
S-LRB(8)] to use to fit the PAS data and derive more
precision long-range molecular coefficient.
For reanalysis of the improved LRB formula including
two additional high order expansion terms, we have also
appraised the significance of the improved LRB formula by
analyzing the residual in the nonlinear fitting procedure of
PAS data. And Pruvost et al. firstly used the residual from
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J. Phys. Soc. Jpn. 81 (2012) 044301
Y. LI et al.
Table II. Values (a.u.) obtained from present result and the previous
theoretical work.
Year
c3 for 1g
LRB formula
This work
2011
17.4887
Second improved
This work
2011
16.6197
First improved
This work
2011
16.6050
Well-known
Theoretical18)
2011
17.46
—
fitting process to analyze the degree of agreement between
theory and experimental data.20) The residual indicates
deviation of theoretical formula from experimental data and
is also a kind of error reflecting the degree of the agreement
between theory and experiment, and therefore the analysis of
experimental data can be only implemented through nonlinear fit where the approximation is looked for with small
residual. Of course, the residual can be also used to
determine whether the improved LRB formula is more
suitable to deal with PAS data. In Fig. 3, the ordinate shows
residual from spectral data and theoretical value, and
independent is wave number indicates the molecular bound
energy or the detuning from dissociation limit. The scattered
points from residual plot provide some crucial information,
for example, the values of residual present increasing or
decreasing trend as the variation of independent, which
indicates the errors from fitting formula and spectral data
present increasing or decreasing tend with independent, and
this tells us fitting formula given is not suitable and includes
non-appropriate approximation generally. On the whole, the
values of residual present increasing trend with the bound
energy in Figs. 3(a) and 3(b). Otherwise the values of
residual have no apparent change with independent and
show discrete points irregularly, which indicates the
theoretical formula is appropriate to deal with experimental
data just like Fig. 3(c). And the amplitudes of residual from
fitting procedure by using S-LRB formula are least, which
shows the least error can be obtained by choosing the S-LRB
formula.
In this section, the obtained residual plots show the
deviation from experimental data and theoretical formula
corresponding to the LRB, F-LRB, and S-LRB respectively
in Figs. 3(a), 3(b), and 3(c). At the same time, the obtained
the long-range molecular c3 coefficients are also shown in
Table II. Comparing the three residual plots and the three
values of the long-range molecular c3 coefficient, we can
know the F-LRB is more appropriate to fit spectral data
rather than LRB formula, and S-LRB formula is the best to
fit PAS data. Thus LRB formula is necessary to improve by
remaining the first and the second order expansion terms in
term of theoretical analysis and fitting for experimental data,
especially for calculating the high precision long-range
molecular coefficient and the real experimental potential
curve.
5.
Conclusion
We presented the PAS data for the long-range molecular
state 1g of Cs2 in the asymptotic range with red detuning
½5:02; 2:45 cm1 below the ð6S1=2 þ 6P3=2 Þ dissociation
limit and 18 new high-lying vibrational levels are detected
by using photon counting technique. The experimental data
are analyzed by fitting to the improved LRB formula,
Fig. 3. (a) The residual from fitting procedure by using well-known LRB
formula, (b) the residual by using the first improved LRB formula and
(c) the residual by using the second improved LRB formula.
and the high precision long-range molecular coefficient is
obtained to be 17.4887 and the error is only 0.16%
comparing to theoretical value.19) We have also demonstrated the improved LRB formula including two additional
high order terms is more appropriate for analyzing PAS
data and obtaining more precision molecular coefficient by
analyzing residual from fitting procedure. Based on this high
sensitive experimental result detected by using photon
counting and theoretical analysis by employing the improved LRB formula, the more precision c3 coefficient is
obtained from the non-linear fitting of the experimental data
for potential curve and actual long-range dipole–dipole
interaction showing real physical image.
Acknowledgments
This work was supported by the 973 Program (grant
2012CB921603), the 863 Program (grant 2009AA01Z319),
the National Natural Science Foundation of China (NSFC)
(grants 61008012, 11074154, 10934004, 60978018,
60978001, and 60808009), the NSFC Project for Excellent
Research Team (grant 60821004), Shanxi Natural Science
Foundation (grant 2009011059-2), and the New Teacher
Fund of the Ministry of Education of China (grant
20101401120004).
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