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arXiv:quant-ph/9908005

Berry phase in the simple harmonic oscillator

JeongHyeong Park, Dae-Yup Song

Published Aug 1, 1999• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/9908005

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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.
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.
Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability distribution function (pulsation of the width and oscillation of the center) contribute to the geometric phase which can be given in terms of the parameters of classical solutions. The geometric phase for a general wave function is also given. If a wave function has a parity under the inversion of space coordinate, then the geometric phase can be defined under the evolution of half of the period of classical motions. For the driven case, geometric phases are given in terms of Fourier coefficients of the external force. The oscillator systems whose classical equation of motion is Mathieu's equation are perturbatively studied, and the first term of nonvanishing geometric phase is calculated.
The most general time-dependence Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with a time-dependent mass, a time-dependent friction (or antifriction) 'constant', and a time-dependent spring constant, acted upon by a time-dependent force. The energy, whose time derivative is the power, is in general different from the Hamiltonian. The generalized Berry phase for a given energy eigenstate has a state-dependent part, which vanishes if there is no damping, and an arbitrary state-independent part. If the Hamiltonian is identified as the energy, both the energy eigenvalues and the generalized Berry phase are different. In the adiabatic limit both approaches give the same total phase, which is the sum of the dynamical and Berry phases.
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.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.