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arXiv:cond-mat/0406210 · Physical Review A

Coherence properties of the two-dimensional Bose-Einstein condensate

Christopher Gies, David A. W. Hutchinson

Published Oct 5, 2004• 32 citations
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Abstract

We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the two-dimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.

DOI
10.1103/physreva.70.043606

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We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the two-dimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.
We present a Hartree-Fock-Bogoliubov (HFB) theoretical treatment of the two-dimensional trapped Bose gas and indicate how semiclassical approximations to this and other formalisms have lead to confusion. We numerically obtain results for the quantum-mechanical HFB theory within the Popov approximation and show that the presence of the trap stabilizes the condensate against long wavelength fluctuations. These results are used to show where phase fluctuations lead to the formation of a quasicondensate.
We present a detailed theoretical analysis of Bragg spectroscopy from a Bose-Einstein condensate at T=0K.T=0 \mathrm{K}.T=0K. We demonstrate that within the linear-response regime, both a quantum-field-theory treatment and a mean-field Gross-Pitaevskii treatment lead to the same value for the mean evolution of the quasiparticle operators. The observable for Bragg spectroscopy experiments, which is the spectral response function of the momentum transferred to the condensate, can therefore be calculated in a mean-field formalism. We analyze the behavior of this observable by carrying out numerical simulations in axially symmetric three-dimensional cases and in two dimensions. An approximate analytic expression for the observable is obtained and provides a means for identifying the relative importance of three broadening and shift mechanisms (mean field, Doppler, and finite pulse duration) in different regimes. We show that the suppression of scattering at small values of q observed by Stamper-Kurn et al. [Phys. Rev. Lett. 83, 2876 (1999)] is accounted for by the mean-field treatment, and can be interpreted in terms of the interference of the u and vvv quasiparticle amplitudes. We also show that, contrary to the assumptions of previous analyses, there is no regime for trapped condensates for which the spectral response function and the dynamic structure factor are equivalent. Our numerical calculations can also be performed outside the linear-response regime, and show that at large laser intensities a significant decrease in the shift of the spectral response function can occur due to depletion of the initial condensate.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
We perform an ab initio analysis of the temperature dependence of the phase coherence length of finite-temperature, quasi-one-dimensional Bose gases measured in the experiments of Richard et al. [Phys. Rev. Lett. 91, 010405 (2003)] and Hugbart et al. [Eur. Phys. J. D 35, 155 (2005)], finding very good agreement across the entire observed temperature range ($0.8<T/{T}_{\ensuremath{\varphi}}<28$). Our analysis is based on the one-dimensional stochastic Gross-Pitaevskii equation, modified to self-consistently account for transverse, quasi-one-dimensional effects, thus making it a valid model in the regime where $\ensuremath{\mu}$ is approximately equal to a few $\ensuremath{\hbar}{\ensuremath{\omega}}_{\ensuremath{\perp}}$. We also numerically implement an alternative identification of ${T}_{\ensuremath{\varphi}}$, based on direct analysis of the distribution of phases in a stochastic treatment.
We derive and discuss the temperature dependence of the condensate and noncondensate density profile of a Bose-Einstein condensate gas with Feshbach resonance in a parabolic trap. These quantities are calculated self-consistently using the generalized Hartree-Fock-Bogoliubov (HFB) equations within the Bogoliubov approximation. At zero temperature, the HFB equation can be solved by means of a variation method that gives the low-excitation spectrum. Moreover, within the two-body collision theory, we estimate the relationship between the number of atoms in the condensate and the external magnetic field B . The result is in good agreement with data from recent experiments.
We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (NCB) approach and a stochastic Gross-Pitaevskii equation (SGPE) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the NCB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (NCB, canonical; SGPE, grand-canonical), they yield the correct condensate statistics in a large Bose-Einstein condensate (BEC) (strong enough particle interactions). For smaller systems, the SGPE results are prone to anomalously large number fluctuations, well known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is both computationally convenient and of potential use to experimentalists. This also clarifies the link between condensate and quasicondensate in the Popov theory of low-dimensional systems.