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arXiv:quant-ph/9907062 · Physical Review A

Periodic Hamiltonian and Berry’s phase in harmonic oscillators

Dae-Yup Song

Published Jan 4, 2000• 16 citations
arXivHTMLPDF

Abstract

For a time-dependent $\ensuremath{\tau}$-periodic harmonic oscillator of two linearly independent homogeneous solutions of the classical equation of motion that are both bounded over time (stable), it is shown that there is a representation of states that are cyclic up to multiplicative constants under $\ensuremath{\tau}$ evolution or $2\ensuremath{\tau}$ evolution depending on the model. The set of the wave functions is complete. Berry's phase, which could depend on the choice of representation, can be defined under the $\ensuremath{\tau}$ or $2\ensuremath{\tau}$ evolution in this representation. If a homogeneous solution diverges as time goes to infinity, it is shown that such a Berry phase cannot be defined in any representation considered. Berry's phase for the driven harmonic oscillator is also considered. For the cases where Berry's phase can be defined, the phase is given in terms of solutions of the classical equation of motion.

DOI
10.1103/physreva.61.024102

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Canadian Journal of Physics•Jun 15, 2021•4 citations
For a time-dependent \ensuremathτ\ensuremath{\tau}\ensuremathτ-periodic harmonic oscillator of two linearly independent homogeneous solutions of the classical equation of motion that are both bounded over time (stable), it is shown that there is a representation of states that are cyclic up to multiplicative constants under \ensuremathτ\ensuremath{\tau}\ensuremathτ evolution or 2\ensuremathτ2\ensuremath{\tau}2\ensuremathτ evolution depending on the model. The set of the wave functions is complete. Berry's phase, which could depend on the choice of representation, can be defined under the \ensuremathτ\ensuremath{\tau}\ensuremathτ or 2\ensuremathτ2\ensuremath{\tau}2\ensuremathτ evolution in this representation. If a homogeneous solution diverges as time goes to infinity, it is shown that such a Berry phase cannot be defined in any representation considered. Berry's phase for the driven harmonic oscillator is also considered. For the cases where Berry's phase can be defined, the phase is given in terms of solutions of the classical equation of motion.
Berry phase of simple harmonic oscillator is considered in a general representation. It is shown that, Berry phase which depends on the choice of representation can be defined under evolution of the half of period of the classical motions, as well as under evolution of the period. The Berry phases do {\em not} depend on the mass or angular frequency of the oscillator. The driven harmonic oscillator is also considered, and the Berry phase is given in terms of Fourier coefficients of the external force and parameters which determine the representation.
Exact Schr\"odinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\ensuremath{\tau}$-periodic Hamiltonian. There may exist $N\ensuremath{\tau}$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.
Shows that (non-adiabatic) Berry phases in time-independent systems arise for fundamentally different reasons than those for time-dependent systems. For time-independent Hamiltonians the Berry phases are seen to only depend on the chosen cyclic initial state, the Hamiltonian merely providing the appropriate period. The author also discusses two time-dependent examples where the time dependence arises from different sources, namely a two-level atom in an intense laser and a harmonic oscillator with a periodic forcing term.
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.
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.
In the dynamic description of physical systems, the two coupled harmonic oscillators’ time-dependent mass, angular frequency, and coupling parameter are recognized as a good working example. We present in this work an analytical treatment with a numerical evaluation of the entanglement and the nonadiabatic Berry phases in the vacuum state. On the basis of an exact resolution of the wave function solution of the time-dependent Schrödinger equation (TDSE) using the Heisenberg picture approach, we derive the wave function of the two coupled harmonic oscillators. At the logarithmic scale, we derive the entanglement entropies and the temperature. We discuss the existence of the cyclical initial state (CIS) based on an instant Hamiltonian and we obtain the corresponding nonadiabatic Berry phases through a period T. Moreover, we extend the result to the case of N coupled harmonic oscillators. We use the numerical calculation to follow the dynamic evolution of the entanglement in comparison to the time dependance of the nonadiabatic Berry phases and the time dependance of the temperature. For two coupled harmonic oscillators with time-independent mass and angular frequency, the nonadiabatic Berry phases present very slight oscillations with the equivalent period as the period of the entanglement. A second model is composed of two coupled harmonic oscillators with angular frequency, which change initially as well as later. Herein, the entanglement and the temperature exhibit the same oscillatory behavior with exponential increase in temperature.