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Progress of Theoretical Physics

Coherent State Combined with Mixed State for an Imperfect Boson System. II: -- Static Solution and Elementary Excitation --

A. Kuriyama

Published Jun 1, 2005• 0 citations
Publisher

Abstract

The thermal equilibrium state and elementary modes of excitation are investigated for a spinless boson system with a contact interaction. In a homogeneous infinite system, the elementary mode associated with the bose condensation with 0 momentum is automatically damped. In a semi-infinite cubic system, there appear bound states in the semi-infinite direction for a free end condition on the finite side and then two-dimensional modes with a phonon-like spectrum.

DOI
10.1143/ptp.113.1239

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The thermal equilibrium state and elementary modes of excitation are investigated for a spinless boson system with a contact interaction. In a homogeneous infinite system, the elementary mode associated with the bose condensation with 0 momentum is automatically damped. In a semi-infinite cubic system, there appear bound states in the semi-infinite direction for a free end condition on the finite side and then two-dimensional modes with a phonon-like spectrum.
A pure squeezed coherent state is amalgamated with a mixed state for the purpose of describing bose condensation in an imperfect spinless boson system at non-zero temperature. Three types of variables are introduced: One describes the bose condensation, one describes the quantum fluctuations of a condensed pair induced by the interaction, and one describes the thermal properties. We define an entropy in association with the amalgamated state. Then, the thermal equilibrium state is defined through a variational principle with a fixed boson number and entropy.
An imperfect spinless bose system is studied with use of the mixed state representation, which was originally developed to study highly excited nuclear states. With this representation, the degrees of freedom are classified into two types: One describes the thermal behaviour of the non-condensed boson and the other mainly describes the dynamical aspects of the condensed boson. The entropy of the system is specified by only the former type of degrees of freedom. Comparison is made with the conventional results for a dilute bose gas with an interaction of the hard sphere type.
We have developed a microscopic theory of superfluidity in a two-dimensional, slightly nonideal Bose gas. The theory is rooted in the fact that a number of bosons are concentrated in the small momentum region at low temperatures when the interaction is weak, in spite of the fact that there is no Bose condensate in the two-dimensional case when T≠0. This fact lets us reduce the description of the system from one employing creation ak+ and annihilation ak operators for bosons to one based on combinations of these operators ρk and φk(ρk = S−1Σqaq+aq+k, φk· = −(i/ns) Σ(kq/k2 + l)aq+aq+k, where n is the boson density, and S is the area of the system). The description in terms of ρk and φk is simple because the hamiltonian is quadratic with respect to ρk and φk. The operators ρk and φk can also be used to express the occupation number Nk = ak+ak and current operators. We find the elementary excitation spectrum (which is identical to the Bogolyubov spectrum) and the momentum distribution of the bosons. We calculate the normal density due to thermal oscillations in the Bose gas and to interactions with impurity centers in the substrate.
Abstract We find stationary excited states of a one-dimensional system of N spinless point bosons with repulsive interaction and zero boundary conditions by numerically solving the time-independent Gross–Pitaevskii equation. The solutions are compared with the exact ones found in the Bethe-ansatz approach. We show that the jth stationary excited state of a nonuniform condensate of atoms corresponds to a Bethe-ansatz solution with the quantum numbers n 1 = n 2 = … = n N = j + 1 . On the other hand, such values of n 1 , … , n N correspond to a condensate of N elementary excitations (the Bogoliubov quasiparticles) with the quasimomentum ℏ π j / L , where L is the system size. Thus, each stationary excited state of the condensate is ‘doubly coherent’, since it corresponds simultaneously to a condensate of N atoms and a condensate of N elementary excitations. We find the energy E and the particle density profile ρ ( x ) for such states. The possibility of experimental production of these states is also discussed.