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arXiv:cond-mat/9612117

Hydrodynamics of a Bose condensate: beyond the mean field approximation (II)

Kirill Ilinski, A. S. Stepanenko

Published Dec 12, 1996• 2 citations
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Abstract

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 $ζ$-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.

DOI
10.48550/arxiv.cond-mat/9612117

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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.
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.
We account for the interaction of the Bose{condensed fraction with the normal phase in an eective dynamical equation such as the Gross{Pitaevskii equation.We show that the low{energy excitations can be treated as sound waves with speed dependent on the condensate density.This allows us to reduce the problem to the calculation of the determinant of the Laplace operator on a curved space and apply standard methods of quantum gravity to get the leading logarithmic contribution of the determinant.This produces the rst quantum correction due to the noncondensed fraction to the Gross{Pitaevskii equation for the condensate.The correction describes an additional quantum pressure in the condensate and evaporation{condensation eects.
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.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations. We then use the hydrodynamic equations to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional solutions of the time-dependent Gross-Pitaevskii equation. General formulation In the previous chapter we saw that the equilibrium structure of the condensate is described by a time-independent Schrödinger equation with a nonlinear contribution to the potential to take into account interactions between particles.
Abstract Several hydrodynamic models of the atomic Bose–Einstein condensate (BEC) obtained beyond the mean-field approximation are discussed together from a single point of view. All these models are derived from the microscopic quantum description. The derivation is made within the many-particle quantum hydrodynamics method suggested by L Kuz’menkov. The derivation is demonstrated and discussed for the mean-field regime revealing the Gross–Pitaevskii equation as the simplest illustration. It appears in the first order by the interaction radius. Generalization of the hydrodynamic Euler equation obtained in the third order by the interaction radius is discussed. It includes the contribution of the isotropic short-range interaction (SRI) presented by the third space derivative of the square of concentration. The Euler equation also includes the contribution of the anisotropic part of the SRI proportional to the second order spherical function. A systematic account of the quantum fluctuations in terms of the many-particle quantum hydrodynamics method requires the extension of the set of hydrodynamic equations from the couple continuity and Euler equations to the set of four equations which also includes the pressure evolution equation and the evolution equation for the third rank tensor of the pressure flux. The pressure evolution equation contains no interaction contribution in the first order by the interaction radius. The source of the quantum fluctuations is in the interaction caused term existing in the third rank tensor evolution equation which is obtained in the first order by the interaction radius. The presented models are considered for the single-component BECs.
In the mean-field approximation, a trapped Bose-Einstein condensate at zero temperature is described by the Gross-Pitaevskii equation for the condensate or, equivalently, by the hydrodynamic equations for the number density and current density. These equations receive corrections from quantum field fluctuations around the mean field. We calculate the semiclassical corrections to these equations for a general time-independent state of the condensate, extending previous work to include vortex states as well as the ground state. In the Thomas-Fermi limit, the semiclassical corrections can be taken into account by adding a local correction term to the Gross-Pitaevskii equation. At second order in the Thomas-Fermi expansion, the semiclassical corrections can be taken into account by adding local correction terms to the hydrodynamic equations.