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Low Temperature Physics

Nonstationary equation for the one-particle wave function of the Bose–Einstein condensate

Viktor Borisovich Bobrov, S. A. Trigger, Anatolij G. Zagorodny

Published Apr 1, 2021• 1 citation
Publisher

Abstract

Based on the self-consistent Hartree–Fock approximation, the nonstationary equation is obtained for the one-particle wave function describing the Bose–Einstein condensate in a rarefied gas of spin-zero bosons. A rarefied gas of bosons is exposed to the static external field, which ensures its finite ground state. The derived equation allows one to correctly determine the ground state energy in the stationary case.

DOI
10.1063/10.0003747

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Based on the self-consistent Hartree–Fock approximation, the nonstationary equation is obtained for the one-particle wave function describing the Bose–Einstein condensate in a rarefied gas of spin-zero bosons. A rarefied gas of bosons is exposed to the static external field, which ensures its finite ground state. The derived equation allows one to correctly determine the ground state energy in the stationary case.
The ground-state energy of a finite inhomogeneous system of bosons located in a scalar external field was found within the framework of the self-consistent Hartree-Fock approximation, on the basis of the representation of the second quantization without using the formalism of anomalous averages. The ground-state wave function corresponds to the stationary Gross-Pitaevskii equation for the Bose-Einstein condensate wave function. It was shown that the ground-state energy can be found using the energy determined by the stationary Gross-Pitaevskii equation only for a system that satisfies the thermodynamic limit.
Within the self-consistent Hartree–Fock approximation, an explicit in this approximation expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose–Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross–Pitaevski approach, is found. The obtained form of kinetic energy is valid beyond the Hartree–Fock approximation and can be applied for arbitrary strong interparticle interaction.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4πħ 2 a / m . In coordinate space this corresponds to a contact interaction U 0 δ( r – r ′), where r and r ′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.
A large system of N N integer-spin atoms, called Bosons, manifests one of the most coherent macroscopic quantum states known to date, the “Bose-Einstein condensate”, at extremely low temperatures. As N → ∞ N\to \infty , this system is usually described by a mean-field limit: a single-particle wave function, the condensate wave function , that satisfies a nonlinear Schrödinger-type equation. In this expository paper, we review kinetic aspects of the mean-field Boson evolution. Furthermore, we discuss recent advances in the rigorous study of second-order corrections to this mean-field limit. These corrections originate from the quantum-kinetic mechanism of pair excitation , which lies at the core of pioneering works in theoretical physics including ideas of Bogoliubov, Lee, Huang, Yang and Wu. In the course of our exposition, we revisit the formalism of Fock space, which is indispensable for the analysis of pair excitation.