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Physical Review A

Bogoliubov-Zubarev Description of a Weakly Interacting Bose Gas

Joseph P. Straley

Published Jul 1, 1972• 13 citations
Publisher

Abstract

Bogoliubov-Zubarev's exact transformation, specific to a weakly interacting Bose gas having a condensate, yields a Hamiltonian describing a gas of interacting excitations. The non-Hermiticity of the Hamiltonian arises from the nonorthogonality of the physical states; it has some significant consequences. The excitation spectrum is gapless in every approximation; the lowest-order correction is evaluated numerically and shown to be small. It is concluded that the approach presents a prospective method for a quantitative description of the weakly interacting Bose gas.

DOI
10.1103/physreva.6.498

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Bogoliubov-Zubarev's exact transformation, specific to a weakly interacting Bose gas having a condensate, yields a Hamiltonian describing a gas of interacting excitations. The non-Hermiticity of the Hamiltonian arises from the nonorthogonality of the physical states; it has some significant consequences. The excitation spectrum is gapless in every approximation; the lowest-order correction is evaluated numerically and shown to be small. It is concluded that the approach presents a prospective method for a quantitative description of the weakly interacting Bose gas.
As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian H^=∑k≠0H^k\hat{H}=\sum_{\bf k\neq 0}\hat{H}_{\bf k}H^=∑k=0​H^k​ is found by diagonalizing each of the Hamiltonians H^k\hat{H}_{\bf k}H^k​ corresponding to a given momentum mode k{\bf k}k independently of the Hamiltonians H^k′(≠k)\hat{H}_{\bf k'(\neq k)}H^k′(=k)​ of the remaining modes. We argue that this way of diagonalizing H^\hat{H}H^ may not be adequate, since the Hilbert spaces where the single-mode Hamiltonians H^k\hat{H}_{\bf k}H^k​ are diagonalized are not disjoint, but have the k=0{\bf k}=0k=0 in common. A number-conserving generalization of Bogoliubov's method is presented where the total Hamiltonian H^\hat{H}H^ is diagonalized directly. When this is done, the spectrum of excitations changes from a gapless one, as predicted by Bogoliubov's method, to one which has a finite gap in the k→0k\to 0k→0 limit.
Abstract We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c -number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.
The effective interaction energy of quasiparticles in the Bose fluid, which is described by ”ladder state diagrams” in the Bogoliubov approximation, was studied for a model potential. It is shown that in the Bogoliubov approximation, the interaction of excitations in liquid He can not be described even qualitatively. It means that a model of the weakly interacting dilute Bose gas fails in describing the interaction between excitation in Bose-condensed liquid, although this model qualitatively incorporates superfluidity in itself.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.