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Progress of Theoretical Physics

On the Coherent and the Squeezed State in the su(2)-Boson Model

M. Yamamura, A. Kuriyama, Y. Tsue

Published Oct 1, 1992• 1 citation
Publisher

Abstract

With the use of the coherent and the squeezed state combined with canonicity condition in the time-dependent Hartree-Fock theory, quantal fluctuation appearing in the su(2)-boson model, which is well known as Schwinger boson representation, is investigated. Especially, the attention is focussed on the least quantal effect: 1) the minimal uncertainty and 2) the quantal fluctuation energy. The Lipkin model, an example obeying the su(2)-algebra, is adopted for demonstrating the basic idea.

DOI
10.1143/ptp/88.4.719

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With the use of the coherent and the squeezed state combined with canonicity condition in the time-dependent Hartree-Fock theory, quantal fluctuation appearing in the su(2)-boson model, which is well known as Schwinger boson representation, is investigated. Especially, the attention is focussed on the least quantal effect: 1) the minimal uncertainty and 2) the quantal fluctuation energy. The Lipkin model, an example obeying the su(2)-algebra, is adopted for demonstrating the basic idea.
Various classical counterparts to the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal systems is the use of various trial states in addition to the su(2) ⊗ su(2)-coherent state, which may be a natural selection for the two-level pairing model governed by the su(2) ⊗ su(2)-algebra. It is pointed out that a spurious behavior such as a sharp phase transition can be avoided by using the other states including the su(2) ⊗ su(1,1)- and su(1,1) ⊗ su(1,1)-coherent states. This contrasts with the situation regarding the usual Hartree-Fock-Bogoliubov and quasi-particle random phase approximations applied to the original fermion system, which yield a sharp phase transition.
Various works performed by the present authors in the 1990s are reviewed. The topics discussed in this paper are mainly related to the time-evolution of the coherent and the squeezed states of the systems obeying the su(2)- and the su(1,1)-algebra. The formulations are based on the basic idea of the boson mapping and the TDHF theory in canonical form. Under the time-dependent variational procedure for trial states appropriately chosen, the time-evolution of the systems under investigation is described. Further, it is shown that this method enables us to obtain the classical counterparts of the original quantal systems.
With the aim of investigating dynamics of systems governed by su (1, 1)-algebra, a possible state which can give us a classical counterpart for the su (1, 1)-spin system properly is examined and constructed explicitly. By the use of canonicity conditions developed in the time-dependent Hartree-Fock theory, the c -number analogue of the Holstein-Primakoff boson representation is obtained for the su (1, 1)-algebra. It is noted that it is important to introduce a character of two-mode coherent state in order to present a suitable classical counterpart for the su (1, 1)-spin. Furthermore, it is shown that the spreading of uncertainty relations appears inevitably, which suggests that a certain fluctuation can be introduced.
With the aim of investigating dynamics of systems governed by su (1, 1)-algebra, a possible state which can give us a classical counterpart for the su (1, 1)-spin system properly is examined and constructed explicitly. By the use of canonicity conditions developed in the time-dependent Hartree-Fock theory, the c -number analogue of the Holstein-Primakoff boson representation is obtained for the su (1, 1)-algebra. It is noted that it is important to introduce a character of two-mode coherent state in order to present a suitable classical counterpart for the su (1, 1)-spin. Furthermore, it is shown that the spreading of uncertainty relations appears inevitably, which suggests that a certain fluctuation can be introduced.
To go beyond the usual Hartree-Fock approximation, a quasi-spin squeezed state is introduced in the Lipkin model. We define the quasi-spin squeezed state from the su(2)-coherent state in a manner similar to that for the ordinary boson squeezed state. In this paper, we deal only with the case that the parameters included in this state are time-independent. It is then seen that the exact energy eigenvalues of the ground state are reproduced with high accuracy in all regions of the force strength χ by using the quasi-spin squeezed state. Also, it is shown that the quasi-spin squeezed state gives a classical counterpart with a certain correction for the Holstein-Primakoff boson representation of su(2)-spin. We investigate uncertainty relations for su(2)-generators and the meaning of “squeezing” is clarified.