Density correlations in ultracold atomic Fermi gases
W. Belzig
University of Konstanz, Department of Physics, D-78457 Konstanz, Germany
C. Schroll and C. Bruder
Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 共Received 15 March 2007; published 12 June 2007兲
We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate共BEC兲limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular con- densate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.
DOI:10.1103/PhysRevA.75.063611 PACS number共s兲: 03.75.Ss, 03.75.Hh, 05.30.Fk
I. INTRODUCTION
Following the successful creation of Bose-Einstein con- densates共BECs兲 in ultracold atomic clouds关1兴, recently ul- tracold fermionic clouds have been produced关2–6兴. This has attracted a lot of attention both theoretically and experimen- tally, especially due to the ability to tune the mutual interac- tion between atoms via a Fano-Feshbach resonance. The unique opportunity to study the crossover from weak attrac- tive to strong attractive interactions in one and the same sys- tem makes this interesting from a fundamental many-body point of view.
Theoretically, fermionic systems with weak attractive in- teraction are superfluids and as such described by the Bardeen-Cooper-Schrieffer 共BCS兲 theory 关7兴. This theory can also describe the limit of stronger attractive interaction 关8–10兴, in which a BEC of molecules is formed. In the cross- over regime, the long-range nature of the interaction makes the BCS theory less accurate关11兴.
Recently, several experiments studied the strongly inter- acting BEC-BCS crossover regime using spin mixtures of ultracold fermionic gases共see关12兴for a recent review兲. Mea- surements of the interaction energy of an ultracold fermionic gas near a Feshbach resonance were made, studying the im- pact of the interaction on the time-of-flight expansion关13兴.
Experimental investigation of collective excitations showed a strong dependence on the coupling strength关14,15兴. More- over, the condensation 关16,17兴 and the spatial correlations 关18兴 of the fermionic atom pairs were observed in the full crossover regime. The pairing gap was measured directly via a spectroscopic technique in the whole crossover region关19兴.
Remarkably, the gap values are in relatively good agreement with the BCS model in the whole region.
The use of noise correlations to probe the many-body states of ultracold atoms was proposed in 关20兴 共see also 关21兴兲. Correlation measurements can be applied to detect phase coherence in mesoscopic superpositions关22兴. The den- sity and spin structure factor for the BEC-BCS transition was calculated关23兴. Interferometric measurement schemes of the
spatial pairing order have been proposed based on the atom- counting statistics in the output channels 关24兴. Pairing fluc- tuations of trapped Fermi gases have been studied in 关25兴.
The number statistics of Fermi and Bose gases has also been investigated in关26,27兴. In recent experiments关28兴the spatial structure of an atomic cloud has been directly observed 共without the expansion used in most other experiments兲. This makes it possible to determine the density fluctuations either by repeating the experiment many times or by taking densi- ties at different positions in a homogeneous system to extract the statistics. Atomic shot noise has been experimentally in- vestigated both in bosonic and fermionic systems关29–35兴.
In this article we propose to use the full counting statistics of density fluctuations as a tool to gain access to the many- body nature of the ground state of a fermionic cloud in the BEC-BCS crossover regime共for other work on full counting statistics in ultracold atomic gases see 关36兴 on the experi- mental and关37–40兴on the theoretical side兲. Our main result is a general expression for the particle number statistics of the mean-field BCS wave function. In the limiting cases, the statistics allows a straightforward interpretation. Deep in the molecular BEC limit the statistics is Poissonian—i.e., that of independent pairs of atoms. In the opposite limit, on the BCS side of the crossover, the fluctuations are strongly suppressed and reflect the particle-hole symmetry. The statistics in the crossover regime differs strongly from both the BCS and BEC limits and will be discussed based on several numerical results.
II. COUNTING STATISTICS OF DENSITY FLUCTUATIONS We consider an atomic cloud with spatial distributionn共x兲 and divide the system into bins; see Fig.1. We are interested in the probabilityP共N兲=具␦共N−Nˆ兲典to findNatoms in a “bin”
of the system; see Fig.1. Here,Nˆ=兰Vbinnˆ共x兲is the bin num- ber operator, Vbin is the bin volume, and nˆ共x兲 denotes the density operator. The bin volume is assumed to be much 1050-2947/2007/75共6兲/063611共5兲 063611-1 Konstanzer Online-Publikations-System (KOPS) ©2007 The American Physical Society
URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3229/
smaller than the volume of the full system, but still much larger than the mean interparticle distance cubed. Hence, a single bin can be considered as a grand-canonical system with the surrounding atomic cloud serving as the reservoir.
In practice, it is more convenient to study the Fourier transform ofP共N兲, which is related to the cumulant generat- ing function共CGF兲S共兲 via
e−S共兲=
兺
N eiNP共N兲=具eiNˆ典. 共1兲The cumulants Cn are defined in a standard way as S共兲= −兺nCn共i兲n/n! and can be used to characterize the full counting statistics共FCS兲. We recall thatC1=N¯ is the average number of atoms in a bin and C2=具共N−N¯兲2典 measures the width of the number distribution. The third cumulant is pro- portional to the skewness and, therefore, a measure of the asymmetry of the distribution function. As we will see later, the third cumulant in a fermionic system is related to devia- tions from particle-hole symmetry.
III. NONINTERACTING FERMIONS AND BOSONS We start by recalling some properties of noninteracting Fermi and Bose gases at a given temperatureT. Here, indi- vidual atoms are independent and we can obtain the statistics as
S共兲= ⫿
兺
k ln关1 ±f±共k兲共ei− 1兲兴, 共2兲where f±共k兲=兵exp关共⑀k−兲/kBT兴± 1其−1 is the Fermi 共Bose兲 occupation function. The chemical potential is determined by the average atom number—i.e.,C1=N¯=兺kf±. Obviously, the number statistics of fermions and bosons differs drastically in the degenerate regime. In the nondegenerate regimef±is small and we find the Poissonian statistics of classical par- ticlesS共兲= −N¯关exp共i兲− 1兴, for both fermions and bosons.
For degenerate fermionskBTⰆ⑀F the statistics is
S共兲= −iN¯−共DkBT/4⑀F兲N¯2. 共3兲 Here and in the following, S共兲 is defined in the interval 关−,兴 and extended periodically. Thus, particle-number fluctuations are suppressed by T/⑀F in comparison to the classical case关41兴. Remarkably, the statistics is Gaussian and consequently all cumulantsCnfor n艌3 vanish. This behav- ior resembles fermions in a one-dimensional共1D兲wire关42兴 and can be interpreted as a consequence of antibunching.
Note that the Gaussian nature of the statistics is a conse- quence of particle-hole symmetry and is therefore strictly limited to the degenerate regime. Higher-order corrections in kBT/⑀Fwill introduce deviations from the Gaussian limit and lead, e.g., to the appearance of higher-order odd cumulants, which are directly related to deviations from the perfect particle-hole symmetry.
In contrast to that, we obtain quite a different behavior for free bosons. Approaching the degeneracy temperature TCBEC
= 2ប2n2/3/m共3 / 2兲2/3 关43兴 from above, the fluctuations are enhanced due to the large factorf−共k兲共f−共k兲+ 1兲. In the con- densed regime, the occupation of the ground state becomes macroscopically large and the grand-canonical approach is no longer valid关44,45兴. This can be seen if we take the limit T= 0 of Eq.共2兲, leading toS共兲= ln关1 −N¯共ei− 1兲兴, which cor- responds to a negative binomial distribution, and the fluctua- tions therefore diverge according toCn⬃N¯n. As in the grand- canonical ensemble the chemical potential is= 0 below the critical temperatureT⬍TCBEC, an arbitrarily large number of bosons can be transferred from the reservoir into the bin, leading to unphysically large fluctuations. In order to find the correct fluctuations in this case, we calculate the FCS from Eq. 共1兲 explicitly. We divide the boson operator intoak=bk
+ck, where bk=
冕
Vbind3reikr⌿共r兲, ck=
冕
V⬘d3reikr⌿共r兲, 共4兲 whereV⬘is the volumeVwithout the volumeVbinof the bin.We consider a fully condensed state of Ntot noninteracting bosons兩典=共a0†兲Ntot兩vac典=共b0†+c0†兲Ntot兩vac典. Using Eq. 共1兲for the bin number operator Nˆ=兺kbk†bk we obtain a binomial statistics for a noninteracting bosonic gas in a bin atT= 0:
S共兲= −Ntotln
冉
1 +VVbin共ei− 1兲冊
. 共5兲For bin volumes Vbin/VⰆ1 the particle-number statistics becomes Poissonian—i.e., S共兲⬇−N¯共ei− 1兲, where N¯=NtotVbin/V.
IV. BCS GROUND STATE
We now turn to a Fermi gas with an attractive interaction parameterized by a scattering lengtha. The Hamiltonian for the full system is number conserving and as such shows no number fluctuations at all. Here we consider a bin—i.e. a small subsystem—which we assume to be described by the
n(x)
... i−1 i i+1 ...
bins X
P(N)
N N
C2 (a)
(b)
FIG. 1. 共Color online兲 共a兲 Sketch of a typical atomic number density. The measured observable is the atom numberNgiven by n共x兲integrated over the bin volume共the bins are indicated by the dashed lines兲.共b兲Histogram of probabilities to findNparticles in a bin.
non-number-conserving BCS Hamiltonian关7兴. A straightfor- ward ansatz for a non-number-conserving many-body state, which takes the pairing interaction into account, is the BCS wave function, which will be used in the following and is known to correctly describe both the BCS and BEC limits 关8–10兴. We will later show that the approach also reproduces the correct counting statistics in the two limits, and we there- fore prefer to use this transparent, almost analytical approach instead of more complex canonical approaches. The BCS wave function is given by
兩BCS典=
兿
k 共uk+vkck†↑c−k† ↓兲兩0典. 共6兲The variational procedure yields vk2= 1 −uk2=关1 −共⑀k
−兲/Ek兴/ 2, where Ek2=共⑀k−兲2+⌬2 is the energy of quasi- particle excitations. The order parameter⌬and the chemical potential are fixed by the self-consistency equations
⌬= −
兺
k ukvk, N¯ = 2兺
k vk2, 共7兲whereis the BCS coupling constant. After renormalization of the coupling constant and considering only the low- energy limit, the gap equation can be related to the two- particle scattering amplitude关9,46兴.
The product form of the BCS wave function greatly sim- plifies the calculation of the statistics, since differentkstates can be treated separately. For a single pair of states 共k↑, −k↓兲 the sum over all possible configurations can be easily performed:
e−Sk共兲=具BCS兩ei共nˆk↑+nˆ−k↓兲兩BCS典=uk2+vk2e2i. 共8兲 The sum of all states yields the result
S共兲= −
兺
k ln关1 +vk2共e2i− 1兲兴. 共9兲This is one of the main results of our paper. It is valid in two and three dimensions; the dimension will only enter into the density of states when transforming the sum overk into an energy integration via the standard expression ND
=mD/2共2⑀兲D/2−1/ 2D−1បDforD= 2 , 3. It should be noted that in a strictly two-dimensional system the low-energy scatter- ing amplitude vanishes⬃−1 / ln⑀ and consequently the gap equation共7兲shows a logarithmic divergence for⑀→0. How- ever, for the more realistic situation of a quasi-two- dimensional cloud共i.e., a three-dimensional trapped atomic cloud strongly confined in one dimension兲, this singularity is eliminated. Results derived for the strictly two-dimensional situation are still valid for the quasi-two-dimensional case, however with the chemical potentialshifted by the ground- state energy.
We now discuss some limiting cases in which analytical expressions can be obtained. On the BEC side, ⬍0 and
⌬Ⰶ兩兩 leads to vk2Ⰶ1 for all energies and allows one to expand the logarithm in Eq.共9兲. The result is
S共兲= −N¯
2共e2i− 1兲, 共10兲
which corresponds to a Poissonian number statistics ofpairs of atoms. This supports the picture of strongly bound pairs in a coherent state. Note that the factor of 2 in the exponent leads to exponentially growing cumulants in the fermion number—viz., Cn= 2nN¯/ 2. We therefore expect strong fluc- tuations. Remarkably, the result 共10兲 is in accordance with the number statistics of condensed bosons in a bin with vol- ume much less than the total volume; see the expression given after Eq.共5兲, which was derived using the canonical formalism. Since we are counting single fermions instead of bosons in the present case, the counting fieldis replaced by 2 and there is a prefactor 1 / 2. This agreement is quite remarkable, since the starting point of our approach is the grand-canonical formalism. It is an indication that using the BCS wave function to describe the number fluctuations of a small subsystem works surprisingly well.
On the BCS side the situation is quite different. Here
=⑀FⰇ⌬and we obtain
S共兲= −iN¯ −N¯ D ⌬ 4⑀F
关兩cos共兲兩− 1兴. 共11兲
We observe that the first term is dominant but contributes only to the first cumulant. The fluctuations come from the second term in Eq.共11兲which is smaller by a factor of⌬/⑀F. Furthermore, similar to the degenerate Fermi gas, the odd cumulantsCnforn艌3 vanish, which is again a consequence of particle-hole symmetry.
Due to the constant density of states in 2D, we can obtain an analytical expression for the CGF for arbitraryand⌬, which reads
S共兲= −N¯ ⌬
⑀F
冋
cos共兲atan冉
2⌬⑀Fei冊
− atan冉
2⌬⑀F冊 册
−N¯ 2
⑀F
ln关1 +v02共e2i− 1兲兴. 共12兲
Herev02=共1 +/
冑
2+⌬2兲/ 2 is the BCS coherence function for k= 0. While we do not have an analytical expression in 3D, we expect a similar behavior as in 2D. This will be corroborated later by comparing the numerically obtained cumulants in 3D to the 2D case.In Fig.2we show how the density noise changes through the BEC-BCS crossover. Going from the BEC to the BCS regime strongly suppresses the fluctuations, in agreement with our previous discussion. The inset shows the fluctua- tions normalized to⌬/⑀F; they approach a constant value in the BCS limit. Another important piece of information gained from a noise measurement is the order parameter, which can be extracted from a measurement of C2 in the BCS limit, Eq.共11兲, as⌬/⑀F= 4C2/N¯ D. Figure3shows the third cumulant. The global behavior is rather similar to the second cumulant; i.e.,C3is strongly reduced going from the BEC limit to the BCS limit. However, the behavior of C3 normalized to⌬/⑀Fshown in the inset is qualitatively differ- ent from that shown in the inset of Fig.2sinceC3vanishes also in the BCS limit. Note also thatC3vanishes faster in the 2D case than in the 3D case. To understand the behavior of the third cumulant in more detail we recall thatC3is related
to the skewness of the distribution; i.e.,C3 is a measure of the difference between positive and negative fluctuations. To see this, we note that an elementary binomial event has the property ln兵1 +vk2关exp共i兲− 1兴其=i+ ln兵1 +uk2关exp共−i兲− 1兴其. Using this property and the particle-hole symmetry ofvk2 in the BCS limit, we can rewrite the CGF in the BCS regime as iN¯+N0兰d⑀ln兵1 +vk2uk2关cos共2兲− 1兴其. Here we have used the fact that the density of states close to⑀Fcan be approximated by a constantN0. The second term is even inand therefore only contributes to even cumulants, whereas all odd cumu- lants for n艌3 vanish. The difference between 2D and 3D seen in Fig.3is caused by the共small兲energy dependence of the 3D density of states, which is absent in 2D.
V. FINITE TEMPERATURES
We would now like to discuss the effect of finite tempera- tures in a qualitative way. The excitations in the BCS theory are fermionic quasiparticles. This is a good approximation on the BCS side of the transition. The fluctuations共measured by the second cumulant C2兲 will be reduced with increasing temperature, since they approach the linearly increasingC2 of the free Fermi gas that is lower at the critical temperature thanC2atT= 0,C2free共TCBCS兲⬍C2共0兲. At even higher tempera- turesT艌TF,C2 reaches the classical Poisson value N¯. The situation on the BEC side of the crossover will be very dif- ferent. According to our result the statistics is a Poissonian distribution of molecules and, hence,C2 is doubled in com- parison to the classical value for the atomic gas关47兴. Upon increasing the temperature, the main effect on the fluctua-
tions will be a breaking up of molecules, which will reduce the second cumulantC2above a dissociation temperature to finally reach the value for the classical gas. Thus, we expect quite a different temperature dependence of the number fluc- tuations on the BEC or BCS side of the transition.
VI. CONCLUSION
In conclusion, we have calculated the full counting statis- tics of number densities in an ultracold fermionic atomic cloud with attractive interactions. The number statistics in the vicinity of the BEC-BCS crossover displays interesting features which reveal the nature of the many-body ground state. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We have also discussed the BEC- BCS crossover of the third cumulant and the temperature dependence of the second cumulant. These quantities can be accessed experimentally and provide additional information on the many-body ground state in the crossover regime. The concept of counting statistics in ultracold gases opens inter- esting possibilities to study the interplay between coherence and correlation in quantum many-body systems.
ACKNOWLEDGMENTS
We would like to thank A. Lamacraft for discussions. This work was financially supported by the Swiss NSF, the NCCR Nanoscience, and the European Science Foundation共QUDE- DIS network兲. This research was supported in part by the National Science Foundation under Grant No. NSF PHY05- 51164.
关1兴M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 共1995兲; K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 共1995兲.
关2兴B. DeMarco and D. S. Jin, Science 285, 1703共1999兲.
关3兴A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Par- tridge, and R. G. Hulet, Science 291, 2570共2001兲.
关4兴F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bour- del, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. 87, 080403共2001兲.
关5兴S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E. Thomas,
-2 0 2 4
-(ξ−ξµ=0) 0
1 2 3 4
C3/N -4 -2 0-(ξ−ξ 2 4
µ=0) 0
1 2 3 C3εF/∆N
2D 3D
BEC BCS
FIG. 3. 共Color online兲Third cumulantC3as a function ofas defined in Fig.2. The dotted line corresponds to =0. The inset showsC3normalized to⌬/⑀F.
-2 0 2 4
-(ξ−ξµ=0) 0
0.5 1 1.5 2
C2/N -4 -2 0-(ξ−ξ 2 4
µ=0) 0
1 2 3 C2εF/∆N
2D 3D
BEC BCS
FIG. 2. 共Color online兲 Second cumulantC2 as a function of
= 1 /kFa. For the strongly confined, quasi-two-dimensional system 共2D兲, the Fermi vector can be approximated by the inverse of the ground-state size:kF=/ᐉ0. The dotted line corresponds to =0. The inset showsC2normalized to⌬/⑀F.
Phys. Rev. Lett. 88, 120405共2002兲.
关6兴K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Science 298, 2179共2002兲.
关7兴J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.108, 1175共1957兲.
关8兴D. M. Eagles, Phys. Rev. 186, 456共1969兲.
关9兴A. J. Leggett, inModern Trends in the Theory of Condensed Matter, edited by A. Pedalski and J. Przystawa共Springer, Ber- lin, 1980兲.
关10兴P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 共1985兲.
关11兴S. Simonucci, P. Pieri, and G. C. Strinati, Europhys. Lett. 69, 713共2005兲.
关12兴R. Grimm, e-print arXiv:cond-mat/0703091.
关13兴T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhães, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, Phys. Rev. Lett. 91, 020402共2003兲.
关14兴M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.
H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 203201 共2004兲.
关15兴J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E.
Thomas, Phys. Rev. Lett. 92, 150402共2004兲.
关16兴C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403共2004兲.
关17兴M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. 92, 120403 共2004兲.
关18兴M. Greiner, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 92, 150405共2004兲.
关19兴C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J.
H. Denschlag, and R. Grimm, Science 305, 1128共2004兲. 关20兴E. Altman, E. Demler, and M. D. Lukin, Phys. Rev. A 70,
013603共2004兲.
关21兴A. Lamacraft, Phys. Rev. A 73, 011602共R兲 共2006兲.
关22兴R. Bach and K. Rzazewski, Phys. Rev. Lett. 92, 200401 共2004兲.
关23兴H. P. Büchler, P. Zoller, and W. Zwerger, Phys. Rev. Lett. 93, 080401共2004兲.
关24兴I. Carusotto and Y. Castin, Phys. Rev. Lett.94, 223202共2005兲. 关25兴L. Viverit, G. M. Bruun, A. Minguzzi, and R. Fazio, Phys.
Rev. Lett. 93, 110406共2004兲.
关26兴M. Budde and K. Mølmer, Phys. Rev. A 70, 053618共2004兲. 关27兴D. Meiser and P. Meystre, Phys. Rev. Lett. 94, 093001共2005兲.
关28兴M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.
H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 共2004兲.
关29兴S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I.
Bloch, Nature共London兲 434, 481共2005兲.
关30兴M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D.
Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 共2005兲.
关31兴M. Greiner, C. A. Regal, J. T. Stewart, and D. S. Jin, Phys.
Rev. Lett. 94, 110401共2005兲.
关32兴T. Rom, Th. Best, D. van Oosten, U. Schneider, S. Fölling, B.
Paredes, and I. Bloch, Nature共London兲 444, 733共2006兲. 关33兴J. Esteve, J.-B. Trebbia, T. Schumm, A. Aspect, C. I. West-
brook, and I. Bouchoule, Phys. Rev. Lett. 96, 130403共2006兲. 关34兴T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V.
Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D.
Boiron, A. Aspect, and C. I. Westbrook, Nature共London兲 445, 402共2007兲.
关35兴I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys. Rev. Lett.
98, 080404共2007兲.
关36兴A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett.
95, 090404共2005兲.
关37兴A. Lamacraft, e-print arXiv:cond-mat/0512580.
关38兴A. Nunnenkamp, D. Meiser, and P. Meystre, New J. Phys. 8, 88共2006兲.
关39兴A. Kuklov and H. Moritz, e-print arXiv:cond-mat/0609531.
关40兴B. Capogrosso-Sansone, E. Kozik, N. Prokof’ev, and B. Svis- tunov, Phys. Rev. A 75, 013619共2007兲.
关41兴Y. Castin, e-print arXiv:cond-mat/0612613.
关42兴L. S. Levitov, H. Lee, and G. B. Lesovik, J. Math. Phys. 37, 4845共1996兲.
关43兴The expression forTCBECis calculated for a three-dimensional gas confined by rigid walls.is the Riemann zeta function.
关44兴L. D. Landau, and E. M. Lifshitz,Statistical Physics, 3rd ed.
共Butterworth-Heinemann, Oxford, 1984兲, Pt. I, Sec. 113.
关45兴A. J. Leggett,Quantum Liquids共Oxford University Press, Ox- ford, 2006兲.
关46兴M. Randeria, J.-M. Duan, and L. Y. Shieh, Phys. Rev. B 41, 327共1990兲.
关47兴Note that here the fluctuations of atoms are discussed. This should not be confused with the fluctuations of the molecule number.
