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Asia Pacific Physics Conference (APPC)

Numerical Analysis of Quantum Transport Equation for Bose Gas in One Dimensional Optical Lattice

Yukiro Kuwahara, Yusuke Nakamura, Y. Yamanaka

Published Mar 19, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.7566/jpscp.1.012101

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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 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.
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.
Making use of the thermo field dynamics (TFD) we formulate a calculable method for time-dependent nonequilibrium systems in a time representation (t-representation) rather than in the k0-Fourier representation. The corrected one-body propagator in the t-representation has the form of B−1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). The number parameter in B here is the observed number (the Heisenberg number) with a fluctuation. With the usual definition of the on-shell self-energy a self-consistent renormalization condition leads to a kinetic equation for the number parameter. This equation turns out to be the Boltzmann equation, from which the entropy law follows.
We analyze in detail the heating of bosonic atoms in an optical lattice due to incoherent scattering of light from the lasers forming the lattice. Because atoms scattered into higher bands do not thermalize on the time scale of typical experiments, this process cannot be described by the total energy increase in the system alone (which is determined by single-particle effects). The heating instead involves an important interplay between the atomic physics of the heating process and the many-body physics of the state. We characterize the effects on many-body states for various system parameters, where we observe important differences in the heating for strongly and weakly interacting regimes, as well as a strong dependence on the sign of the laser detuning from the excited atomic state. We compute heating rates and changes to characteristic correlation functions based on both perturbation-theory calculations and a time-dependent calculation of the dissipative many-body dynamics. The latter is made possible for one-dimensional systems by combining time-dependent density-matrix-renormalization-group methods with quantum trajectory techniques.
This study aims to construct a theoretical formulation of a nonequilibrium process for a system of Bose--Einstein condensate associated with a spontaneous symmetry breakdown. For this, Thermo Field Dynamics is used. We then describe the most general forms of a 4x4 transformation and an unperturbed Hamiltonian. After calculating the 4×44\times 44×4 self-energy and identifying its on-shell, we impose the renormalization condition in which the entire on-shell self-energy should vanish. This condition provides a sufficient number of independent equations to determine all of the parameters in an unperturbed Hamiltonian, among which the quantum transport equations for normal and anomalous number distributions are included.