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Chinese Physics Letters

Classical analogues of quantum berry's phase

Sun Changpu

Published Nov 1, 1989• 5 citations
Publisher

Abstract

With the classical generalized harmonic oscillator as an example, we show that the analogues of the quantum Berry's phase can appear in the classical mechanic systems with adiabatically-changing parameters, which have the Schrödinger-type evolution equations. The corresponding guage structures are naturally induced in these systems.

DOI
10.1088/0256-307x/6/11/001

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With the classical generalized harmonic oscillator as an example, we show that the analogues of the quantum Berry's phase can appear in the classical mechanic systems with adiabatically-changing parameters, which have the Schrödinger-type evolution equations. The corresponding guage structures are naturally induced in these systems.
Considering that the classical evolution equations are similar to the Schrodinger eqaution, we point out that there is the structure of the Berry's Phase in the solution of a classical evolution eqaution with adiabatic changing parameters. Thus, we can use a classical evolution process to imitate the quantum adiabatic process. As an example, the motion of a charged particle in slowly-changing magnetic field is studied and the corresponding phase factor can be interpreted as a holomony of the fiber bundle on the sphare s~2 in the parameter space.
General time-dependent, bilinear classical Lagrangian mechanics can be interpreted as a charged particle interacting with a gauge scalar and vector potential. For an arbitrary discrete number of degrees of freedom, there are enough symplectic symmetries in classical mechanics to make this system canonically identical to the most general discrete spectrum quantum system. It follows that the general invariance group of the Schr\"odinger system of equations is the symplectic group Sp(2N), not U(N). Several examples show the physical naturalness of the classical coordinates. In the adiabatic limit Berry's phase is identified as the cumulative change of the angle of action-angle coordinates of a normal mode. A generalization of Berry's phase, the time component of the adiabatic connection and its associated electric curvature, is shown to be on an equal footing with the usual Berry phase.
In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.
The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian PTPTPT-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.