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arXiv:0902.4792 · Journal of the Physical Society of Japan

Population Dynamics of a Spin-1 Bose Gas above the Bose–Einstein Transition Temperature

Yuki Endo, Tetsuro Nikuni

Published Aug 28, 2009• 2 citations
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Abstract

We study population dynamics of a trapped spin-1 Bose gas above the Bose-Einstein transition temperature. Starting from the semiclassical kinetic equation for a spin-1 gas, we derive a coupled rate equations for the populations of internal states. Solving the rate equations, we discuss the dynamical evolution of spin populations. We also estimate the characteristic timescale in which the system reaches equilibrium. Finally, we briefly discuss how the presence of the condensate will affect the population dynamics.

DOI
10.1143/jpsj.78.094005

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We study population dynamics of a trapped spin-1 Bose gas above the Bose-Einstein transition temperature. Starting from the semiclassical kinetic equation for a spin-1 gas, we derive a coupled rate equations for the populations of internal states. Solving the rate equations, we discuss the dynamical evolution of spin populations. We also estimate the characteristic timescale in which the system reaches equilibrium. Finally, we briefly discuss how the presence of the condensate will affect the population dynamics.
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We derive a kinetic theory for a spin-1/2 Bose-condensed gas of two-level atoms at finite temperatures. The condensate dynamics is described by a generalized Gross–Pitaevskii equation for the two-component spinor order parameter, which includes the interaction with the uncondensed fraction. The noncondensate atoms are described by a quantum kinetic equation, which is a generalization of the spin kinetic equation for spin-polarized quantum gases to include couplings to the condensate degree of freedom. The kinetic equation is used to derive hydrodynamic equations for the noncondensate spin density. The condensate and noncondensate spins are coupled directly through the exchange mean field. Collisions between the condensate and noncondensate atoms give rise to an additional contribution to the spin diffusion relaxation rate. In addition, they give rise to mutual relaxation of the condensate and noncondensate due to lack of local equilibrium between the two components.
We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived based on a semiclassical approximation to the single-particle-energy spectrum and density of states and are compared with exact results obtained by explicitly summing discrete energy levels for a small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to finite spin-orbit coupling. For a large coupling strength and a finite number of particles NNN, the critical temperature scales as N2/5{N}^{2/5}N2/5 and N2/3{N}^{2/3}N2/3 in three and two dimensions, respectively, contrasted to the predictions of N1/3{N}^{1/3}N1/3 and N1/2{N}^{1/2}N1/2 in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.
Starting from first principles of the statistical mechanics we construct\na dynamics of the condensed Bose gas at the nonzero\ntemperatures. In particular, we derive an equation of motion for\nthe condensate wavefunction and a quantum kinetic equation for\nthe distribution function for the excited atoms. Obtained generalized\nGross-Pitaevskii equation for condensate include effect of\ncollisions with thermal cloud atoms (non-condensate). The Boltzmann\nquantum kinetic equation for non-condensate was obtained\nusing Zubarev’s method of nonequilibrium statistical operator.\nTherefore was obtained a complete dynamics of the condensed\nBose gas at the nonzero temperatures.
We discuss the collective modes of a trapped Bose gas in the hydrodynamic regime where atomic collisions ensure local thermal equilibrium for the distribution function. Starting from the conservation laws, in the linearized limit we derive a closed equation for the velocity fluctuations in a trapped Bose gas above the Bose-Einstein transition temperature. Explicit solutions for a parabolic trap are given. We find that the surface modes above the transition have the same dispersion relation as the one recently obtained by Stringari for the oscillations of the condensate at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 within the Thomas-Fermi approximation. Results are also given for the monopole ``breathing'' mode as well as for the m=0m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0m=0 excitations which result from the coupling of the monopole and quadrupole modes in an anisotropic parabolic well.
The dynamics of Bose-Einstein condensation in a three-dimensional harmonic trap is studied explicitly including the Bogoliubov approximation for temperatures below the critical one. To model the evolution towards equilibrium at each cooling step, we derive quantum kinetic equations that describe the dynamics of the gas for temperatures above and below the transition temperature. These equations, valid in the Born and Markov approximations, consider the essential role of the chemical potential as the main parameter that signals the transition. The kinetic equation that describes the growth of the condensate below the transition temperature is derived within the Bogoliubov approximation. To illustrate our results we propose an energetic cooling protocol and simulate the whole sequence of the formation of a condensate.