Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9701008 · Physical Review A

Quantum kinetic theory. II. Simulation of the quantum Boltzmann master equation

Dieter Jaksch, Crispin W. Gardiner, P. Zoller

Published Jul 1, 1997• 128 citations
arXivHTMLPDF

Abstract

We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.

DOI
10.1103/physreva.56.575

Related papers

Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential

Crispin W. Gardiner, P. Zoller

A quantum kinetic master equation (QKME) for bosonic atoms is formulated. It is a quantum stochastic equation for the kinetics of a dilute quantum Bose gas, and describes the behavior and formation of Bose condensation. The key assumption in deriving the QKME is a Markov approximation for the atomic collision terms. In the present paper the basic structure of the theory is developed, and approximations are stated and justified to delineate the region of validity of the theory. Limiting cases of the QKME include the quantum Boltzmann master equation and the Uehling-Uhlenbeck equation, as well as an equation analogous to the Gross-Pitaevskii equation.

arXiv:quant-ph/9611043•Physical Review A•Apr 1, 1997•163 citations

Quantum kinetic theory for a condensed bosonic gas

R. Walser, Jonathan H. Williams, J. Cooper +1 more

We present a kinetic theory for Bose-Einstein condensation of a weakly interacting atomic gas in a trap. Starting from first principles, we establish a Markovian kinetic description for the evolution towards equilibrium. In particular, we obtain a set of self-consistent master equations for mean fields, normal densities, and anomalous fluctuations. These kinetic equations generalize the Gross-Pitaevskii mean-field equations, and merge them consistently with a quantum-Boltzmann-equation approach.

arXiv:cond-mat/9810150•Physical Review A•May 1, 1999•97 citations

Quantum kinetic theory. III. Quantum kinetic master equation for strongly condensed trapped systems

Crispin W. Gardiner, P. Zoller

We extend quantum kinetic theory to deal with a strongly Bose-condensed atomic vapor in a trap. The method assumes that the majority of the vapor is not condensed, and acts as a bath of heat and atoms for the condensate. The condensate is described by the particle-number-conserving Bogoliubov method developed by one of the authors. We derive equations which describe the fluctuations of particle number and phase, and the growth of the Bose-Einstein condensate. The equilibrium state of the condensate is a mixture of states with different numbers of particles and quasiparticles. It is not a quantum superposition of states with different numbers of particles---nevertheless, the stationary state exhibits the property of off-diagonal long-range order, to the extent that this concept makes sense in a tightly trapped condensate.

arXiv:cond-mat/9712002•Physical Review A•Jul 1, 1998•108 citations

Quantum kinetic theory. IV. Intensity and amplitude fluctuations of a Bose-Einstein condensate at finite temperature including trap loss

Dieter Jaksch, Crispin W. Gardiner, Klaus M. Gheri +1 more

We use the quantum kinetic theory to calculate the steady state and fluctuations of a trapped Bose-Einstein condensate at a finite temperature. The system is divided in a condensate and a noncondensate part. A quantum-mechanical description based on the number-conserving Bogoliubov method is used for describing the condensate part. The noncondensed particles are treated as a classical gas in thermal equilibrium with temperature $T$ and chemical potential $\ensuremath{\mu}$. We find a master equation for the reduced density operator of the Bose-Einstein condensate, calculate the steady state of the system, and investigate the effect of one-, two-, and three-particle losses on the condensate. Using linearized Ito equations, we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. A Lorentzian line shape is found for the intensity correlation function that is characterized by a time constant ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$ derived in the paper. For the amplitude correlation function, we find ballistic behavior for time differences smaller than ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$, and diffusive behavior for larger time differences.

arXiv:cond-mat/9712206•Physical Review A•Aug 1, 1998•74 citations

Equivalence of kinetic theories of Bose-Einstein condensation

J. Wachter, R. Walser, J. Cooper +1 more

We discuss the equivalence of two nonequilibrium kinetic theories that describe the evolution of a dilute, Bose-Einstein condensed atomic gas in a harmonic trap. The second-order kinetic equations of Walser et al. [Phys. Rev. A 63, 013607 (2001)] reduce to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the low- and high-temperature limits, respectively. These kinetic equations thus describe the system in equilibrium (finite temperature) as well as in nonequilibrium (real time). We have found this theory to be equivalent to the nonequilibrium Green's function approach originally proposed by Kadanoff and Baym and more recently applied to inhomogeneous trapped systems by Imamovi\ifmmode \acute{c}\else \'{c}\fi{}-Tomasovi\ifmmode \acute{c}\else \'{c}\fi{} and Griffin [in Progress in Nonequilibrium Green's Functions, edited by M. Bonitz (World Scientific, Singapore, 2000), p. 404].

arXiv:cond-mat/0105181•Physical Review A•Oct 12, 2001•33 citations

QUANTUM FLUCTUATION DYNAMICS DURING THE TRANSITION OF A MESOSCOPIC BOSONIC GAS INTO A BOSE–EINSTEIN CONDENSATE

ALEXEJ SCHELLE

The condensate number distribution during the transition of a dilute, weakly interacting gas of N = 200 bosonic atoms into a Bose–Einstein condensate is modeled within number conserving master equation theory of Bose–Einstein condensation. Initial strong quantum fluctuations occuring during the exponential cycle of condensate growth reduce in a subsequent saturation stage, before the Bose gas finally relaxes towards the Gibbs–Boltzmann equilibrium.

arXiv:1110.6646•Fluctuation and Noise Letters•Nov 20, 2012•1 citation
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
A quantum kinetic master equation (QKME) for bosonic atoms is formulated. It is a quantum stochastic equation for the kinetics of a dilute quantum Bose gas, and describes the behavior and formation of Bose condensation. The key assumption in deriving the QKME is a Markov approximation for the atomic collision terms. In the present paper the basic structure of the theory is developed, and approximations are stated and justified to delineate the region of validity of the theory. Limiting cases of the QKME include the quantum Boltzmann master equation and the Uehling-Uhlenbeck equation, as well as an equation analogous to the Gross-Pitaevskii equation.
We extend quantum kinetic theory to deal with a strongly Bose-condensed atomic vapor in a trap. The method assumes that the majority of the vapor is not condensed, and acts as a bath of heat and atoms for the condensate. The condensate is described by the particle-number-conserving Bogoliubov method developed by one of the authors. We derive equations which describe the fluctuations of particle number and phase, and the growth of the Bose-Einstein condensate. The equilibrium state of the condensate is a mixture of states with different numbers of particles and quasiparticles. It is not a quantum superposition of states with different numbers of particles---nevertheless, the stationary state exhibits the property of off-diagonal long-range order, to the extent that this concept makes sense in a tightly trapped condensate.
We use the quantum kinetic theory to calculate the steady state and fluctuations of a trapped Bose-Einstein condensate at a finite temperature. The system is divided in a condensate and a noncondensate part. A quantum-mechanical description based on the number-conserving Bogoliubov method is used for describing the condensate part. The noncondensed particles are treated as a classical gas in thermal equilibrium with temperature TTT and chemical potential $\ensuremath{\mu}$. We find a master equation for the reduced density operator of the Bose-Einstein condensate, calculate the steady state of the system, and investigate the effect of one-, two-, and three-particle losses on the condensate. Using linearized Ito equations, we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. A Lorentzian line shape is found for the intensity correlation function that is characterized by a time constant ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$ derived in the paper. For the amplitude correlation function, we find ballistic behavior for time differences smaller than ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$, and diffusive behavior for larger time differences.
We discuss the equivalence of two nonequilibrium kinetic theories that describe the evolution of a dilute, Bose-Einstein condensed atomic gas in a harmonic trap. The second-order kinetic equations of Walser et al. [Phys. Rev. A 63, 013607 (2001)] reduce to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the low- and high-temperature limits, respectively. These kinetic equations thus describe the system in equilibrium (finite temperature) as well as in nonequilibrium (real time). We have found this theory to be equivalent to the nonequilibrium Green's function approach originally proposed by Kadanoff and Baym and more recently applied to inhomogeneous trapped systems by Imamovi\ifmmode \acute{c}\else \'{c}\fi{}-Tomasovi\ifmmode \acute{c}\else \'{c}\fi{} and Griffin [in Progress in Nonequilibrium Green's Functions, edited by M. Bonitz (World Scientific, Singapore, 2000), p. 404].