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arXiv:cond-mat/0101210 · Physical Review Letters

Classical-Field Method for Time Dependent Bose-Einstein Condensed Gases

Alice Sinatra, Carlos Lobo, Yvan Castin

Published Nov 2, 2001• 160 citations
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Abstract

We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas.

DOI
10.1103/physrevlett.87.210404

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We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas.
We study the truncated Wigner method applied to a weakly interacting spinless Bose-condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ( r ) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross–Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work (Sinatra et al 2000 J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three-dimensional spatially homogeneous Bose-condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev–Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross–Pitaevskii equation, thermalizes to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy max of the Bogoliubov modes in the simulation does not exceed a few k B T .
We demonstrate that the time-dependent projected Gross-Pitaevskii equation (GPE) derived earlier [M. J. Davis, R. J. Ballagh, and K. Burnett, J. Phys. B 34, 4487 (2001)] can represent the highly occupied modes of a homogeneous, partially-condensed Bose gas. Contrary to the often held belief that the GPE is valid only at zero temperature, we find that this equation will evolve randomized initial wave functions to a state describing thermal equilibrium. In the case of small interaction strengths or low temperatures, our numerical results can be compared to the predictions of Bogoliubov theory and its perturbative extensions. This demonstrates the validity of the GPE in these limits and allows us to assign a temperature to the simulations unambiguously. However, the GPE method is nonperturbative, and we believe it can be used to describe the thermal properties of a Bose gas even when Bogoliubov theory fails. We suggest a different technique to measure the temperature of our simulations in these circumstances. Using this approach we determine the dependence of the condensate fraction and specific heat on temperature for several interaction strengths, and observe the appearance of vortex networks. Interesting behavior near the critical point is observed and discussed.
We introduce a time-dependent projected Gross-Pitaevskii equation to describe a partially condensed homogeneous Bose gas, and find that this equation will evolve randomized initial wave functions to equilibrium. We compare our numerical data to the predictions of a gapless, second order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33, 3847 (2000)], and find that we can determine a temperature when the theory is valid. As the Gross-Pitaevskii equation is nonperturbative, we expect that it can describe the correct thermal behavior of a Bose gas as long as all relevant modes are highly occupied. Our method could be applied to other boson fields.
This thesis is comprised of four separate parts: in the first we calculate the second and third virial coefficients of 3He and 4He at low temperatures with an empirical interatomic potential by Janzen and Aziz and using the Path Integral Monte Carlo (PIMC) technique. We show that, for the calculation of the second coefficient, the method is successful whereas for the third the particular implementation that we chose (free particle sampling) is not sufficient to produce reliable results at low temperatures. In the second part we consider the Josephson Effect between two spatially separated Bose-Einstein condensates of atoms each of which can be in two hyperfine states. We derive simple equations of motion for this system closely analogous to the Bloch equations. We also map the dynamics of the system onto those of a classical particle in a well. We find novel density and spin modes of oscillation and new stable equilibrium points of the motion. Finally we analyze the oscillation modes in the spin-1 (F = 1) case. In the third part of the thesis we propose a powerful method to study the time evolution of Bose condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate an ensemble of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one dimensional Bose gas. The fourth part deals with inhomogeneous superconductivity in the presence of magnetic fields that couple only to the spin and not to the orbital motion. They induce a splitting of the Fermi surfaces of up and down spins. We start by considering a one dimensional system and explicitly write down the wavefunction that pairs states with unequal energies due to the splitting. Next we consider an extension to three dimensions and work out explicitly certain aspects of the crossover from one to two dimensions. We then discuss the relationship of this state with those of Fulde, Ferrell, Larkin and Ovchinnikov. Finally, we consider an extension to the case of a paired state which has a gap with a spherically symmetric spatial dependence.
We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for NNN-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.
We consider the evolution of a gas of N bosons in the three-dimensional Gross–Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of order 1/N ). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as N \to \infty , in the L^{2} (\mathbb{R}^{3N}) -norm. To achieve this goal, we let the Bose–Einstein condensate evolve according to a time-dependent Gross–Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross–Pitaevskii dynamics.