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Progress of Theoretical Physics

Maslov Phase as Geometric Phase in the Time-Dependent Variational Approach with Squeezed Coherent States

Y. Tsue

Published Nov 1, 1992• 24 citations
Publisher

Abstract

Toward the classical description of quantum fluctuations in quantal systems, the Maslov phase occurring in a semi-classical quantization rule is investigated in a framework of the time-dependent variational principle with squeezed coherent states. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to Berry's phases or canonical phases including dynamical effects. The Maslov phase appears as a winding number in this framework. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.

DOI
10.1143/ptp.88.911

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Toward the classical description of quantum fluctuations in quantal systems, the Maslov phase occurring in a semi-classical quantization rule is investigated in a framework of the time-dependent variational principle with squeezed coherent states. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to Berry's phases or canonical phases including dynamical effects. The Maslov phase appears as a winding number in this framework. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.
A general framework for time-dependent variational approach in terms of squeezed coherent states is constructed with the aim of describing quantal systems by means of classical mechanics including higher order quantal effects with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. The Maslov phase occurring in a semi-classical quantization rule is investigated in this framework. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to the Berry phase. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.
To examine dynamical effects on the geometric phase at the semi-classical level, we apply the technique of the time-dependent variational principle to simple models which typically exhibit the geometric phase effects discovered by Berry, namely, the Jaynes-Cummings models and the generalized harmonic oscillator model. Parameters are themselves dynamical variables in our models. Instead of integrating out the “fast” variables at the first stage, we derive the canonical equations of motion both for “fast” and “slow” varibales without the adiabatic assumption. Using special solutions of the canonical equations, we requantize the systems by Einstein's condition, in which Berry's phase appearing as a correction for the Bohr-Sommerfeld condition is modified by the dynamical effects related to the slow degree of freedom.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.
The quantum dynamics of time-dependent Hamiltonians of the kind H(t)=U(t)H(0)U(t)° is studied in the framework of generalized coherent states. It is shown that the quantum and the classical dynamics are isomorphic and that the phase of ‖ψ(t)〉 is simply the classical action. In adiabatic approximation it is easy to extract Berry’s phase from this. A discussion of generalized coherent states (CS’s) necessary to deal with excited states is given; in particular for SU(2), SU(1,1) and E(2) their structure is completely characterized, showing that the deep connection with geometric quantization extends to these generalized CS’s. This technique may be employed to estimate higher-order correction to the adiabatic approximation.
Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere (S2).({\mathcal{S}}^{2}).(S2). An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the $E\ensuremath{\bigotimes}\ensuremath{\varepsilon}$ Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.
In the WKB approximation for coupled channel equations, it has been pointed out by Yabana and Horiuchi that the semi-classical quantization condition should be modified by Berry's phase and another extra phase. Recently Littlejohn and Flynn have rederived them in a quite elegant way. We concentrate on this extra phase in this paper, and derive it for a simple model by means of the path integral method in the generalized coherent state representation. It is shown that it appears as a quantum fluctuation around classical orbits.