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arXiv:1206.0145 · Cold atoms

A Dynamical Self-Consistent Finite-Temperature Kinetic Theory: The ZNG Scheme

A. Joy Allen, Carlo F. Barenghi, Nick P. Proukakis, E. Zaremba

Published Mar 18, 2013• 4 citations
arXivHTMLPDF

Abstract

We review a self-consistent scheme for modelling trapped weakly-interacting quantum gases at temperatures where the condensate coexists with a significant thermal cloud. This method has been applied to atomic gases by Zaremba, Nikuni, and Griffin, and is often referred to as ZNG. It describes both mean-field-dominated and hydrodynamic regimes, except at very low temperatures or in the regime of large fluctuations. Condensate dynamics are described by a dissipative Gross-Pitaevskii equation (or the corresponding quantum hydrodynamic equation with a source term), while the non-condensate evolution is represented by a quantum Boltzmann equation, which additionally includes collisional processes which transfer atoms between these two subsystems. In the mean-field-dominated regime collisions are treated perturbatively and the full distribution function is needed to describe the thermal cloud, while in the hydrodynamic regime the system is parametrised in terms of a set of local variables. Applications to finite temperature induced damping of collective modes and vortices in the mean-field-dominated regime are presented.

DOI
10.1142/9781848168121_0005

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We review a self-consistent scheme for modelling trapped weakly-interacting quantum gases at temperatures where the condensate coexists with a significant thermal cloud. This method has been applied to atomic gases by Zaremba, Nikuni, and Griffin, and is often referred to as ZNG. It describes both mean-field-dominated and hydrodynamic regimes, except at very low temperatures or in the regime of large fluctuations. Condensate dynamics are described by a dissipative Gross-Pitaevskii equation (or the corresponding quantum hydrodynamic equation with a source term), while the non-condensate evolution is represented by a quantum Boltzmann equation, which additionally includes collisional processes which transfer atoms between these two subsystems. In the mean-field-dominated regime collisions are treated perturbatively and the full distribution function is needed to describe the thermal cloud, while in the hydrodynamic regime the system is parametrised in terms of a set of local variables. Applications to finite temperature induced damping of collective modes and vortices in the mean-field-dominated regime are presented.
In Chapters 11–13, we gave a detailed discussion of the dynamics of a trapped Bose gas at finite temperatures in a region where the collisions described by the C 12 and C 22 terms in the kinetic equation (3.42) do not play the central role. In this “collisionless” region, the dominant interaction effects are associated with the self-consistent fields which both the condensate and noncondensate atoms feel. Thus the dynamics can be understood to a first approximation by neglecting the C 12 and C 22 collision integrals in the kinetic equation and, at the next stage, treating them as a weak perturbation on the collisionless dynamics. In the rest of this book (Chapters 15–19), we turn to the study of the coupled ZNG equations in the opposite limit, where the C 12 and C 22 collision integrals completely determine the dynamics of the thermal cloud. Specifically, the collisions lead to the thermal cloud being in local hydrodynamic equilibrium, and hence this regime is described by the equations of collisional hydrodynamics. Its characteristic feature is that the nonequilibrium behaviour of the thermal cloud atoms can be completely described in terms of a few differential equations involving coarse-grained variables that are dependent on position and time, analogous to the condensate variables n c ( r , t ) and v c ( r , t ). In the present chapter, assuming that the thermal cloud distribution function f ( p, r , t ) is given by the Bose distribution describing partial local equilibrium, (15.16), we show how the ZNG coupled equations lead precisely to Landau's two-fluid equations, reviewed in Chapter 14. This equivalence is not obvious, mainly because Landau's equations are expressed in terms of thermodynamic variables, which are not used in a more microscopic analysis such as that used in the ZNG approach.
We formulate a generalized self-consistent stochastic quantum kinetic theory for finite-temperature ultracold Bose gases interacting via a generic long-range interaction, applicable to a broad range of systems, by means of Keldysh non-equilibrium field theory: such model is explicitly cast in the context of dipolar atomic gases, and is also shown to encompass established stochastic and kinetic treatments for ultracold atomic gases with local interactions as special cases. The condensate and low-lying modes are collectively described by a stochastic Gross-Pitaevskii equation with two collisional terms and their corresponding stochastic noise terms, with thermal particles dynamically modelled through a self-consistently coupled Quantum Boltzmann equation and dipolar interactions included by means of a coupled Poisson-like equation. Additional use of Bogoliubov-de Gennes analysis generating the Lee-Huang-Yang correction term relevant in the T=0T=0T=0 quantum-fluctuation-dominated regime, allows us to postulate the extension of such model offering a plausible scheme for interpolating between quantum-dominated and thermal-dominated fluctuation regimes, the consistency of which remains to be tested against experimental observations.
In this chapter we describe the numerical methods that can be used to solve the ZNG equations in the context of a dynamical simulation. These equations consist of a generalized GP equation (3.21) for the condensate and a Boltzmann equation (3.42) for the thermal component. The fact that the two equations are coupled makes their numerical solution more complex than when either is considered on its own. Indeed, the distinct quantum and classical aspects of the problem require specifically tailored numerical methods. Although most of these methods are well established and described elsewhere (Taha and Ablowitz, 1984; Sanz-Serna and Calvo, 1994), we provide in this chapter a detailed pedagogical discussion that will serve as a guide to those interested in carrying out such calculations for trapped Bose gases. This chapter is based on the papers of Jackson and Zaremba (2002a,b). There are two main parts to the numerical problem. The first is developing a method for solving the time-dependent GP equation for an arbitrary three-dimensional geometry. This we take up in Section 11.1. Second, a method is needed for solving the Boltzmann equation that accounts for the dynamics of the thermal component. Here one must deal both with the Hamiltonian dynamics of the thermal atoms, as they move in the self-consistent mean field of the condensate and thermal cloud, and with the collisions that take place between the thermal atoms themselves (the C 22 collisions) and between the thermal atoms and the condensate (the C 12 collisions). The methods used to account for these two distinct collisional processes are taken up in Section 11.3. As we shall see, collisions play an important role and cannot be neglected even when the dynamical behaviour is dominated by mean-field interactions.
In this paper, we develop a numerical procedure for investigating the dynamics of trapped Bose gases based on the ZGN theory. The dynamical equations used consist of a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud. The former is solved using a fast Fourier transform split-operator technique while the Boltzmann equation is treated by means of N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. This scheme has been applied to a model equilibration problem, dipole oscillations in isotropic traps and scissors modes in anisotropic traps. In the case of the latter, the frequencies and damping rates of the condensate mode have been extracted from the simulations for a wide range of temperatures. Good agreement with recent experiments has been found.
We consider a model of a dilute Bose-Einstein condensed gas at finite temperatures, where the condensate coexists in a trap with a cloud of thermal excitations. Within the Zaremba, Nikuni, and Griffin formalism, the dynamics of the condensate is described by a generalized Gross-Pitaevskii equation, while the thermal cloud is represented by a semiclassical kinetic equation. Our numerical approach simulates the kinetic equation using a cloud of representative test particles, while collisions are treated by means of a Monte Carlo sampling technique. A full three-dimensional split-operator fast Fourier transform method is used to evolve the condensate wave function. We give details regarding the numerical methods used and discuss simulations carried out to test the accuracy of the numerics. We use this scheme to simulate the monopole mode in a spherical trap. The dynamical coupling between the condensate and thermal cloud is responsible for frequency shifts and damping of the condensate collective mode. We compare our results to previous theoretical approaches, not only to confirm the reliability of our numerical scheme, but also to check the validity of approximations which have been used in the past.
Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive kinetic equations for the non-condensate atoms at finite temperatures which include the effect of binary collisions between atoms. The effect of collisions is included using the second-order self-energy given by the Beliaev (gapless) approximation. We limit our discussion to finite temperatures where we can use the single-particle Hartree-Fock spectrum for the excited atoms. In this limit, we can neglect the off-diagonal propagators (g~12\tilde{g}_{12}g~​12​ and g~21\tilde{g}_{21}g~​21​). As expected, this leads to the kinetic equations and collision integrals used in recent work by Zaremba, Nikuni, and Griffin (ZNG) [1]. We also derive a consistent equation of motion for the condensate wavefunction, involving a finite-temperature generalization of the well-known Gross-Pitaevskii equation which includes a dissipative term, as well as the mean field of the non-condensate.