Critical temperature of a trapped Bose gas: Mean-field theory and fluctuations

Houbiers, M.; Stoof, H. T. C.; Cornell, Eric A.

doi:10.1103/physreva.56.2041

Cited by 55 publications

(72 citation statements)

References 18 publications

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“…The predicted effect of critical fluctuations [11,12] is to further increase T c . The nonperturbative A2 estimate lies at the boundary of experimental error, but as mentioned earlier this result does not satisfy the validity requirement for this experiment.…”

Section: 060404 (2006) P H Y S I C a L R E V I E W L E T T E R S mentioning

confidence: 99%

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Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

2006

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We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature T c of a harmonically confined Bose gas as reported in Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data. To unequivocally observe beyond mean-field effects, however, the experimental precision must either improve by an order of magnitude, or consider more strongly interacting systems. This is the first application of a classical field method to make quantitative comparison with experiment. The shift in critical temperature T c with interaction strength for the homogeneous Bose gas has been the subject of numerous studies spanning almost 50 years since the first calculations of Lee and Yang [1,2]. While there is a finite shift to the chemical potential in mean-field (MF) theory, the shift of the critical temperature is zero [3]. The leading order effect is due to long-wavelength critical fluctuations and is inherently nonperturbative. Using effective field theory it was determined that the shift is T c =T 0 c can 1=3 , where n is the particle number density, a is the s-wave scattering length, and c is a constant of order unity [4]. Until recently, results for the value of c disagreed by an order of magnitude and even sign, as summarized in Fig. 1

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Section: 060404 (2006) P H Y S I C a L R E V I E W L E T T E R S mentioning

confidence: 99%

“…Repulsive interactions reduce T c , intuitively due to a lowering of the peak density of the gas. Next-order effects due to fluctuations have been estimated in [8,11,12] and in general predict an increase in T c from the first-order result. For a sufficiently wide trap Ref.…”

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confidence: 99%

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

2006

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“…The classical field approximation requires that all momenta involved in the various integrations are small in comparison with Λ ∼ λ −1 or, in other words, that the integrands are negligibly small for momenta k ∼ λ −1 . Only then, for instance, can we use the approximate form of the statistical factors (29). This requires in particular that k ∼ k c ≪ Λ, yielding a/λ ≪ 1.…”

Section: Linear Dependence Of the Density Correctionmentioning

confidence: 99%

“…We focus throughout on a spatially uniform system; a discussion of the transition temperature of a dilute gas in a trap can be found in Refs. [2,29,30]. For experimental data in the 4 He-Vycor system, see [31].…”

Section: Introductionmentioning

confidence: 99%

Bose-Einstein transition in a dilute interacting gas

et al. 2001

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We study the effects of repulsive interactions on the critical density for the Bose-Einstein transition in a homogeneous dilute gas of bosons. First, we point out that the simple mean field approximation produces no change in the critical density, or critical temperature, and discuss the inadequacies of various contradictory results in the literature. Then, both within the frameworks of Ursell operators and of Green's functions, we derive self-consistent equations that include correlations in the system and predict the change of the critical density. We argue that the dominant contribution to this change can be obtained within classical field theory and show that the lowest order correction introduced by interactions is linear in the scattering length, a, with a positive coefficient. Finally, we calculate this coefficient within various approximations, and compare with various recent numerical estimates.

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“…Especially, it is of great importance to know the behavior of the critical temperature in dependence on the interaction strength and the density. The fluctuations beyond the mean field are especially significant since they establish the deviations of the critical temperature from the noninteracting value and might be observable by adiabatically converting down the trapping frequency [3]. The measurement of the critical temperature of a trapped weakly interacting 87 Rb gas differs from the ideal gas value already by two standard deviations [4].…”

Section: Introductionmentioning

confidence: 99%

Phase diagram for interacting Bose gases

2007

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We propose a new form of the inversion method in terms of a selfenergy expansion to access the phase diagram of the Bose-Einstein transition. The dependence of the critical temperature on the interaction parameter is calculated. This is discussed with the help of a new condition for Bose-Einstein condensation in interacting systems which follows from the pole of the T-matrix in the same way as from the divergence of the medium-dependent scattering length. A many-body approximation consisting of screened ladder diagrams is proposed which describes the Monte Carlo data more appropriately. The specific results are that a non-selfconsistent T-matrix leads to a linear coefficient in leading order of 4.7, the screened ladder approximation to 2.3, and the selfconsistent T-matrix due to the effective mass to a coefficient of 1.3 close to the Monte Carlo data.

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Critical temperature of a trapped Bose gas: Mean-field theory and fluctuations

Cited by 55 publications

References 18 publications

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

Bose-Einstein transition in a dilute interacting gas

Phase diagram for interacting Bose gases

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