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Communications in Theoretical Physics

Excitation Spectra of λ ( ϕϕ † ) 2 Relativistic Bose condensate

Yang Zhang

Published Jun 15, 1996• 1 citation
Publisher

Abstract

Using the Bogolubov transformation method, we calculate excitation energy spectra for a charged relativistic Bose–Einstein condensate with λ ( ϕϕ † ) 2 interactions. There are two distinct spectra. In the long-wavelength limit one of the spectra approaches to the phonon spectrum for an uncharged nonrelativistic boson condensate, and the other to the nonrelativistic particle energy spectrum. The ground state is explicitly constructed as a coherent state which contains zero-momentum bosons and pairs of quanta with nonzero opposite momenta.

DOI
10.1088/0253-6102/25/4/409

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•Jan 1, 1996•0 citations
Using the Bogolubov transformation method, we calculate excitation energy spectra for a charged relativistic Bose–Einstein condensate with λ ( ϕϕ † ) 2 interactions. There are two distinct spectra. In the long-wavelength limit one of the spectra approaches to the phonon spectrum for an uncharged nonrelativistic boson condensate, and the other to the nonrelativistic particle energy spectrum. The ground state is explicitly constructed as a coherent state which contains zero-momentum bosons and pairs of quanta with nonzero opposite momenta.
We study an interacting spinless Bose-Einstein condensate to clarify theoretically whether the spectra of its quasiparticles (one-particle excitations) and collective modes (two-particle excitations) are identical, as concluded by Gavoret and Nozières [Ann. Phys. 28, 349 (1964)]. We derive analytic expressions for their first and second moments so as to extend the Bijl-Feynman formula for the peak of the collective-mode spectrum to its width (inverse lifetime) and also to the one-particle channel. The obtained formulas indicate that the width of the collective-mode spectrum manifestly vanishes in the long-wavelength limit, whereas that of the quasiparticle spectrum apparently remains finite. We also evaluate the peaks and widths of the two spectra numerically for a model interaction potential in terms of the Jastrow wave function optimized by a variational method. It is thereby found that the width of the quasiparticle spectrum increases towards a constant as the wavenumber decreases. This marked difference in the spectral widths implies that the two spectra are distinct. In particular, the lifetime of the quasiparticles remains finite even in the long-wavelength limit.
The elementary excitation spectrum of a two-component Bose—Einstein condensate in different hyperfine states is obtained by Green's function method. It is found to have two branches. In the long wave-length limit, the two branches of the excitation spectrum are reduced to one phonon excitation and one single-particle excitation. The single-particle one has an energy gap. When the energy gap exists, we study the Landau critical velocity and the depletion of the condensate. With the obtained Green's functions, we calculate the structure factor of a two-component condensate. It is found that the static structure factor comprises only the branch of the phonon excitation and the single-particle excitation makes no contribution to the structure factor.
The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, ``antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ``pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.
We study the spectrum of rotating Bose-Einstein condensates in the limit of weak repulsive interactions and present analytical and numerical results for energies and wave functions. While the low-lying states are of collective nature, the high-lying states are dominated by single-particle excitations and can be accurately approximated by simple polynomial expressions. In the limit that the number of particles is large compared to the number of excited quanta, the single-particle states become excellent approximations to the eigenstates, and a rather simple ordering scheme is obtained.
The excitation spectrum of the interacting Bose gas has two branches. The first branch was dominated in the low momentum area and high density of the ground state particles and the second one was dominated in the high momentum area and was existed in the density of the ground state particles which lower than the existed density of the ground state particles of the first branch. The functional integrals method was used. The Hamiltonian was introduced in form of the combination of field operators which was divided into two parts; ground state particles and exicted particles were described respectively the action and partition function were calculated and were introduced partition function poles which were defined as excitation spectrum. According to the definition of excitation energy of ground was equal to zero, parameters were calculated and devided both forms of Bose excitation spectrums in phase space. The interaction between boson particles in the system was replaced by our ansatz interaction. Its behaviour was like the Lennard-Jones potential. The ansatz interaction was mentioned in configuration space was varied with [e -alphar/(betar)2 - e -alphar/(betar)] by assigning α, β as unity with the inversion unit of length.