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Progress of Theoretical Physics

Nonequilibrium Formulation in Bose-Einstein Condensed States

H. Matsumoto, S. Sakamoto

Published Apr 1, 2001• 10 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.105.573

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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.
In the development of Thermo Field Dynamics (the quantum field theory at finite temperature and density), Prof. Mancini acted as one of the originators and played the important role for its birth through the work of superconductivity. Recent development of the Bose‐Einstein condensation in the dilute alkaline atomic gas enables us to investigate deeply the quantum coherence in bose and fermi particle systems, and gives us a plenty of experimental data even for the problem in nonequilibrium formation of the condensate. There the relaxation time is reasonably slow to make the laboratory observation possible. In this talk, we will show that Thermo Field Dynamics gives us a reasonable approximation scheme to describe time‐dependent ordered states with spatial inhomogeniety by use of a single‐time quasi‐particle field equation, which contains the information of the spatial and temporal dependent order parameter, particle energy and particle distribution.
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So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms.
In the Bose-Einstein condensation of a dilute alkaline atomic gas trapped in a magnetic potential, one has to deal with a strongly space- (and time-) dependent order parameter, which violates the phase symmetry. In this case, certain relations are required among propagators and vertices, known as Ward-Takahashi identities. In this paper, we show that a systematic self-consistent perturbation scheme can be developed even with a space-time order parameter in such a way as to satisfy the requirement of phase symmetry. The key procedure is that the information concerning the full order parameter, which is self-consistently determined, is included in the unperturbed Hamiltonian.
A dynamical many-body theory is presented which systematically extends beyond mean-field and perturbative quantum-field theoretical procedures. It allows us to study the dynamics of strongly interacting quantum-degenerate atomic gases. The nonperturbative approximation scheme is based on a systematic expansion of the two-particle irreducible effective action in powers of the inverse number of field components. This yields dynamic equations which contain direct scattering, memory, and ``off-shell'' effects that are not captured by the Gross-Pitaevskii equation. This is relevant to account for the dynamics of, e.g., strongly interacting quantum gases atoms near a scattering resonance, or of one-dimensional Bose gases in the Tonks-Girardeau regime. We apply the theory to a homogeneous ultracold Bose gas in one spatial dimension. Considering the time evolution of an initial state far from equilibrium we show that it quickly evolves to a nonequilibrium quasistationary state and discuss the possibility to attribute an effective temperature to it. The approach to thermal equilibrium is found to be extremely slow.