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Progress of Theoretical Physics

Coherent State Combined with Mixed State for an Imperfect Boson System. I: -- Thermalization of Pure Squeezed Coherent State --

A. Kuriyama

Published May 1, 2005• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.113.993

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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 apply the mixed state representation to the description of the thermal properties of the pairing model. We derive an extended form of the boson realization of the quasi-spin operators in a classical image. We introduce a density operator and entropy combined with the mixed state representation. Then we define a thermal equilibrium state with use of the variation principle. In this representation, the seniority number is specified by a distribution of the Fermi type as a function of temperature. We study the pairing collective modes that induce no change in the entropy.
The coherent-state representation of the density operator for an ideal Bose gas in a thermal equilibrium is introduced. It is shown that a pure coherent state can be an eigenstate of the Hamiltonian of the non-interacting boson system when the symmetry breaking term is added to it. However, at the limit of the vanishing “classical field” only zero-energy states can be a coherent state. It is also proved that the pure coherent state is the lowest energy state among all the zero-momentum states. The temperature dependence of the coherent state is then discussed. The diminution of the coherent state can clearly be seen as the temperature rises from zero to the critical temperature.
The pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.
The nature of the Einstein-Bose condensation in an imperfect boson gas is discussed. It is argued that no matter how low the temperature, provided it is not zero, macroscopic second-sound waves should be weakly excited as thermal fluctuations about equilibrium. This implies that the single-particle momentum distribution must be smeared so that the state of zero momentum cannot contain a finite fraction of the particles. A microscopic mechanism which independently leads to a momentum spread is discussed. It is suggested that the actual Einstein-Bose condensation of an imperfect gas takes place in energy rather than momentum but that even in terms of this variable it becomes perfectly sharp only at zero temperature. This hypothesis is formulated analytically in terms of correlation functions of the second-quantized boson field.