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.
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.
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.
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). 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.
We describe the quantum dynamics of the Hubbard model at the semiclassical level, by implementing the time-dependent variational principle (TDVP) procedure on appropriate macroscopic wave functions constructed in terms of SU(2)-coherent states. Within the TDVP procedure, such states turn out to include a time-dependent quantum phase, part of which can be recognized as Berry's phase. We derive two semiclassical model Hamiltonians for describing the dynamics in the paramagnetic, superconducting, antiferromagnetic and charge-density wave phases and solve the corresponding canonical equations of motion in various cases. Noticeably, a vortexlike ground-state phase dynamics is found to take place for U>0 away from half filling. Moreover, it appears that an oscillatorylike ground-state dynamics survives at the Fermi surface at half filling for any U. The low-energy dynamics is also exactly solved by separating fast and slow variables. The role of the time-dependent phase is shown to be particularly interesting in the ordered phases.
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schr\\"odinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions, are cyclic. Criterions for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, can not be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can changes from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.