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arXiv:0907.5061

Derivation of non-Markoffian transport equations for trapped cold atoms in nonequilibrium thermal field theory

Yusuke Nakamura, Tomoka Sunaga, M. Mine, Masahiko Okumura, Y. Yamanaka

Published Jul 29, 2009• 0 citations
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Abstract

The non-Markoffian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal filed theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markoffian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markoffian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.

DOI
10.48550/arxiv.0907.5061

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The non-Markoffian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal filed theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markoffian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markoffian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.
In recent experiments of Bose-Einstein condensation of a magnetically trapped dilute alkaline atomic gas, the nonequilibrium process of thermalization into the condensed state has been observed in a reasonably long time scale. This presents a new challenge to develop a theory to treat strongly space-time dependent nonequilibrium processes. In this paper, we present a nonequilibrium formulation of Bose-Einstein condensed states in the field theoretical framework of thermo-field dynamics. By use of an effective Hamiltonian method, a quasi-particle field equation is derived. This equation contains information regarding the renormalized energy, decay width and time-variation of particle distributions. It is shown that, in the present formulation, the strong space-time dependence can be treated, even including quantum corrections, as the use of the quasi-particle field equation enables us to factorize double-time functions of the self-energy.
We study the thermal processes for the system of cold neutral atomic gas in the framework of nonequilibrium Thermo Field Dynamics (TFD) which is a real-time canonical formalism of quantum field theory for thermal situations [1]. The system is suitable for testing quantum many-body theories in nonequilibrium situations because temporal changes in thermal processes are so slow that various nonequilibrium phenomena can be observed experimentally. In nonequilibrium TFD, the quantum transport equation, which is the equation for a time-dependent number distribution introduced as an unknown parameter, is derived from the self-consistent renormalization condition on the self-energy [2]. The correction on the excitation energy is important both physically and in the renormalization of quantum field theory. Recently we have proposed the new self-consistent renormalization method in nonequilibrium TFD, in which the on-shell self-energy in time-dependent case is introduced as a generalization of the ordinary on-shell self-energy in stationary case and the renormalization conditions on it yields the quantum transport equation and the diagonal correction of the quasiparticle energy in a unified manner [2]. In this work, the renormalization condition is extended to the imaginary and off-diagonal parts of the self-energy. In our previous work [3], we applied nonequilibrium TFD to the one dimensional system of confined cold neutral atomic Bose gas. There the nonequilibrium situation was set up by a sudden displacement of the confining potential and a sudden switch-on of the interparticle interaction. We investigated the thermal process for the system by solving the quantum transport equation, but neglecting the correction of the quasiparticle energy. Then it was found that in some particular parameter region it gives rise to an unphysical result, namely the negative number distribution. Here we take account of the correction of the quasiparticle energy and solve the quantum transport equation again. The correction affects drastically the thermal relaxation process and resolves the difficulty in some parameter region, as will be shown.
The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.
We derive a nonequilibrium finite-temperature kinetic theory for a binary mixture of two interacting atomic Bose-Einstein condensates and use it to explore the degree of hydrodynamicity attainable in realistic experimental geometries. Based on the standard separation-of-time-scales argument of kinetic theory, the dynamics of the condensates of the multicomponent system are shown to be described by dissipative Gross-Pitaevskii equations self-consistently coupled to corresponding quantum Boltzmann equations for the noncondensate atoms: On top of the usual mean-field contributions, our scheme identifies a total of eight distinct collisional processes, whose dynamical interplay is expected to be responsible for the system's equilibration. In order to provide their first characterization, we perform a detailed numerical analysis of the role of trap frequency and geometry on collisional rates for experimentally accessible mixtures of $^{87}\mathrm{Rb}\ensuremath{-}^{41}\mathrm{K}$ and $^{87}\mathrm{Rb}\ensuremath{-}^{85}\mathrm{Rb}$, discussing the extent to which the system may approach the hydrodynamic regime with regard to some of those processes as a guide for future experimental investigations of ultracold Bose gas mixtures.
In Chapter 15, we showed that in the limit of short collision times the coupled equations of motion for the condensate and noncondensate atoms lead to Landau's non-dissipative two-fluid hydrodynamics. However the approach used in Chapter 15 was not based on a small expansion parameter, in contrast with the more systematic Chapman–Enskog procedure used to derive hydrodynamic damping in the kinetic theory of classical gases. In the present chapter, we generalize the procedure of Chapter 15 to trapped Bose-condensed gases, in order to derive two-fluid hydrodynamic equations that include dissipation due to transport processes. We solve the kinetic equation by expanding the nonequilibrium single-particle distribution function f ( p, r , t ) around the distribution function f (0) ( p, r , t ) that describes complete local equilibrium between the condensate and the noncondensate components. All hydrodynamic damping effects are included by taking into account deviations from the local equilibrium distribution function f (0) . Our discussion for a trapped Bose gas is a natural extension of the pioneering work of Kirkpatrick and Dorfman (1983, 1985a) for a uniform Bose-condensed gas. This chapter is mainly based on their work as well as on Nikuni and Griffin (2001a,b). We will prove that, with appropriate definitions of various thermodynamic variables, our two-fluid hydrodynamic equations including damping have precisely the structure of those first derived by Landau and Khalatnikov for superfluid He. In particular, the damping associated with the collisional exchange of atoms between the condensate and noncondensate components, which is discussed at length in Chapter 15, is now expressed in terms of frequency-dependent second viscosity coefficients. This special type of damping is a characteristic signature of a dilute Bose superfluid and exists in addition to the hydrodynamic damping associated with the shear viscosity and thermal conductivity of the normal fluid.