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Bulletin of the Lebedev Physics Institute

On the collective excitations and the existence of the gap in the degenerate Bose gas spectrum

Viktor Borisovich Bobrov, С. А. Тригер

Published May 1, 2010• 1 citation
Publisher

Abstract

Themodel of degenerate weakly nonideal Bose gas is considered without the assumption on the C -number nature of creation and annihilation operators of particles in the zero-momentum state. It was shown that the “density-density” Green’s function pole defines the Bogolyubov spectrum of phonon-roton-type collective excitations. The single-particle excitation spectrum is characterized by the gap whose value is defined by the particle density in the “condensate”.

DOI
10.3103/s1068335610050064

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Themodel of degenerate weakly nonideal Bose gas is considered without the assumption on the C -number nature of creation and annihilation operators of particles in the zero-momentum state. It was shown that the “density-density” Green’s function pole defines the Bogolyubov spectrum of phonon-roton-type collective excitations. The single-particle excitation spectrum is characterized by the gap whose value is defined by the particle density in the “condensate”.
Model of the degenerated weakly non-ideal Bose gas is considered without suggestion on C-number representation of the creation and annihilation operators with zero momentum. The "density-density" correlation function and the one-particle Green function are calculated on basis of the suggestion about C-number representation of the operator of particle density in the Bose condensate. It is shown, that the pole in the "density-density" Green function determines the Bogolyubov's spectrum of the collective excitations, which have the phonon-roton form. At the same time the spectrum of the one-particle excitations possesses a gap, whose value is connected with the density of particles in the Bose condensate.
Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. It is shown that the pole of the "density-density" Green's function exactly coincides with the Bogolybov's phonon-roton spectrum of excitations. At the same time, a gap exists in the one-particle excitation spectrum . This gap is related to the density of particles in the "condensate".
Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. Instead of this assumption, we use the assumption on the C-number nature of the density operator N 0 = a † 0 a 0 with zero momentum. It is shown that the pole of the "density-density" Green function (DDGF), as well as the pole of the single-particle Green function (SPGF), exactly coincide with the Bogoliubov phonon-roton spectrum of excitations for both assumptions. This spectrum, as is known confirmed by many neutron and x-ray experimental measurements of the dynamic structure factor in He II, is straightly related to the DDGF. At the same time, we show that in the other case under consideration, when neither N 0 nor a † 0 and a 0 are C-numbers, a gap can exist in SPGF. This gap in SPGF excitations is straightly related to the density of particles in the "condensate". Therefore, the spectra of excitations for the DDGF and SPGF in the last case under consideration are different, in contrast to the Bogoliubov-type theory where these spectra are identical.
Based on the results of statistical quantum electrodynamics, it is shown that the Coulomb interaction potential of charged particles has no Fourier components at a zero wave vector. This result provides for the possibility of using the grand canonical ensemble to describe the Coulomb system, with independent descriptions of different varieties of charged particles. Based on this, we established that there could be an energy gap in the single-particle excitation spectrum at low pulses, given the presence of Bose-Einstein condensate in the Coulomb system, which does not contradict the existence of collective excitations, characterized by the phonon-roton spectrum.
Abstract In the work the problem of phenomenon of superfluidity is considered within the framework of Bose-Einshtein condensation, theoretically. We discuss the main features of the excitation spectrum in a weakly interacting Bose gas. Our analysis will show that the healing length, which determines the length scale below which collective phenomena are physically more relevant than free-particle-like ones. Since the weakly interacting gas of Bose particles has been mapped onto a gas of noninteracting quasiparticles, we are able to analyze the thermodynamics of the system and find how temperature affect the depletion of the condensate.
The density of states of a Bose-condensed gas confined in a harmonic trap is investigated. The predictions of Bogoliubov theory are compared with those of Hartree-Fock theory and of the hydrodynamic model. We show that the Hartree-Fock scheme provides an excellent description of the excitation spectrum in a wide range of energy, revealing a major role played by single-particle excitations in these confined systems. The crossover from the hydrodynamic regime, holding at low energies, to the independent-particle regime is explicitly explored by studying the frequency of the surface mode as a function of their angular momentum. The applicability of the semiclassical approximation for the excited states is also discussed. We show that the semiclassical approach provides simple and accurate formulas for the density of states and the quantum depletion of the condensate.