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Physical Review

Order Parameter, Mean-Field Theory, and the Ideal Bose Gas

Michael Schick C, Paul R. Zilsel

Published Dec 5, 1969• 4 citations
Publisher

Abstract

A formalism due to Langer which permits the derivation of the Gross-Pitaevskii (mean-field) theory of many-boson systems is applied to the ideal Bose gas, and the results obtained are compared with those of an exact treatment. With the exact solution known, it is shown that the order parameter is nonvanishing below the Bose condensation temperature. However, this information cannot be obtained from the mean-field theory alone, which indicates that no Gross-Pitaevskii equation can be derived for this system.

DOI
10.1103/physrev.188.522

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•Jan 1, 2017•3 citations
A formalism due to Langer which permits the derivation of the Gross-Pitaevskii (mean-field) theory of many-boson systems is applied to the ideal Bose gas, and the results obtained are compared with those of an exact treatment. With the exact solution known, it is shown that the order parameter is nonvanishing below the Bose condensation temperature. However, this information cannot be obtained from the mean-field theory alone, which indicates that no Gross-Pitaevskii equation can be derived for this system.
We investigate the properties of the one-dimensional Bose gas at zero temperature, for which exact results exist for some model systems. We treat the interactions between particles in the gas with an approximate form of the many-body T-matrix, and find a form of Gross-Pitaevskii equation valid in 1D. The results presented agree with the exact models in both the weakly and strongly interacting limits, and interpolate smoothly between them. We also investigate the use of mean-field treatments of trapped BECs to describe the 1D system in the strongly interacting limit. We find that the use of the many-body T-matrix to describe interactions leads to qualitative agreement with the exact models for some physical quantities. We indicate how the standard mean-field treatments need to be modified to extend and improve the agreement.
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 chapter introduces the order parameter and the Gross--Pitaevskii equation. Although a Bose-Einstein C condensate (BEC) is a quantum many-body system which that can be described fully only using the many-body wavefunction, many of its simplest spatial dynamics can be understood using the order parameter and its dynamics. We first describe how the order parameter can be formally defined, and how this can be considered an effective spatial wavefunction of the BEC. We then derive the time-dependent evolution of the order parameter via the Gross--Pitaevskii equation, then and study various solutions of it. This includes plane-wave solutions, an infinite potential well solutions, and excited state solutions such as vortices and solitons. After introducing key concepts such as the Thomas--Fermi approximation and the healing length, we also discuss how the Gross--Pitaevskii equation can be cast in the form of hydrodynamic equations.
Summary form only given, as follows. The physical basis of the theory being used to describe Bose-Einstein condensed (BEC) gases will be described. The simplest and indeed most successful approach uses mean field theories of the phenomena. In the case of the lowest temperatures reached, this theory reduces to the study of the Gross-Pitaevskii nonlinear Schrodinger equation. I will describe the extensions of mean field theory to finite temperatures and show how dynamics and damping can be included. I will address the links to superfluidity and time-evolving order parameters in general.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding lll particles out of their total number N not in the condensate is exponentially small in the number lll. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.