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arXiv:cond-mat/9607202

First quantum corrections for a hydrodynamics of a nonideal Bose gas

K. N. Ilinski, A. S. Stepanenko

Published Jul 28, 1996• 0 citations
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Abstract

In the paper we consider a hydrodynamical description of a nonideal Bose gas in one-loop approximation. We calculate an effective action which consists of mean field contributions and first quantum correction. This provides the equations of motion for the density and velocity of the gas where both mean field contributions and fluctuations are presented. To fulfill the calculation we make use of the formalism of functional integrals to map the problem to a problem of quantum gravity to benefit from a well-developed technique in this field. This effective action provides all correlation functions for the system and is a basis for a consideration of dynamics of the gas. Response functions are briefly discussed. Applications to the trapped bosons are reviewed. Together with Ref.\cite{IS} the paper provide complete description of the condensate fraction and the deplition in the case of Bose condensed gases.

DOI
10.48550/arxiv.cond-mat/9607202

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In the paper we consider a hydrodynamical description of a nonideal Bose gas in one-loop approximation. We calculate an effective action which consists of mean field contributions and first quantum correction. This provides the equations of motion for the density and velocity of the gas where both mean field contributions and fluctuations are presented. To fulfill the calculation we make use of the formalism of functional integrals to map the problem to a problem of quantum gravity to benefit from a well-developed technique in this field. This effective action provides all correlation functions for the system and is a basis for a consideration of dynamics of the gas. Response functions are briefly discussed. Applications to the trapped bosons are reviewed. Together with Ref.\cite{IS} the paper provide complete description of the condensate fraction and the deplition in the case of Bose condensed gases.
Self-consistent hydrodynamic one-loop quantum corrections to the Gross-Pitaevskii equation due to the interaction of the condensate with collective excitations are calculated. It is done by making use a formalism of effective action and ζζζ-function regularization for a contribution from Bogoliubov particles in hydrodynamic approximation. It turns out to be possible to reduce the problem to the investigation of a determinant of Laplace operator on curved space where a metric is defined by density and velocity of the condensate. Standard methods of quantum gravity let us get the leading logarithmic contribution of the determinant and corresponding quantum corrections. They describe an additional quantum pressure in the condensate, local heating-cooling and evaporation-condensation effects of the Bose-condensed fraction. The effects of these corrections are studied for the correction from excited states in equilibrium situation. Response functions and form factors are discussed in the same approximation.
This Thesis discusses approaches to calculating static and dynamic correlation functions in onedimensional gases of interacting bosons. The first part of the Thesis deals with the momentummomentum correlation function in the weakly interacting 1D Bose gas. In the regime of phase fluctuating quasicondensate this correlator is found to differ qualitatively from the form predicted by Bogoliubov theory of the true condensate in that correlations between any two values of momentum become finite. Linear hydrodynamics used to calculate the quasicondensate correlation function prove to be adequate. A classical field approximation is used to smoothly interpolate between the quasicondensate and strongly degenerate gas regimes. In the second part the focus is shifted to the dynamical structure factor and it is shown that hydrodynamics is generally inapplicable to calculating the dynamical correlators. The hydrodynamic treatment is enhanced using impurity theory and an exact model-dependent boundary is obtained for the region of applicability of hydrodynamics on the momentum-energy plane. Numerical estimates show that non-hydrodynamic behavior should be observable for the currently available values of experimental parameters.
Using the hydrodynamic theory of superfluids and the Lee-Huang-Yang equation of state for interacting Bose gases, we derive the first correction to the collective frequencies of a trapped gas, due to effects beyond the mean field approximation. The corresponding frequency shift, which is calculated at zero temperature and for large NNN, is compared with other corrections due to finite size, nonlinearity, and temperature. We show that for reasonable choices of the relevant parameters of the system, the non-mean-field correction is the leading contribution and amounts to about 11%1. The role of the deformation of the trap is also discussed.
We investigate the conditions under which quantum fluctuations are relevant for the quantitative interpretation of experiments with ultracold Bose gases. This requires to go beyond the description in terms of the Gross-Pitaevskii and Hartree-Fock-Bogoliubov mean-field theories, which can be obtained as classical (statistical) field-theory approximations of the quantum many-body problem. We employ functional-integral techniques based on the two-particle irreducible (2PI) effective action. The role of quantum fluctuations is studied within the nonperturbative 2PI 1∕N1∕\mathcal{N}1∕N expansion to next-to-leading order. At this accuracy level memory integrals enter the dynamic equations, which differ for quantum and classical statistical descriptions. This can be used to obtain a classicality condition for the many-body dynamics. We exemplify this condition by studying the nonequilibrium evolution of a one-dimensional Bose gas of sodium atoms, and discuss some distinctive properties of quantum versus classical statistical dynamics.
We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.
Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δvδvδv which includes the hydrodynamic damping due to the shear viscosity ηηη and the thermal conductivity κκκ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for ηηη and κκκ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.