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Journal of Physics A Mathematical and General

The geometric phase for chaotic unitary families

Jonathan M. Robbins

Published Feb 21, 1994• 18 citations
Publisher

Abstract

We consider the geometric phase for a family of quantum/classical Hamiltonians in which the effect of changing parameters is simply to induce unitary/canonical transformations. In this case the classical limit of the geometric phase is easily obtained, even when the classical motion is chaotic. The results agree with those previously obtained for general chaotic families, but may be expressed in a simpler form, not involving time integrals of correlation functions. It is also straightforward to establish some results which are problematic in the general case, for example the form of periodic orbit corrections, and the closeness of the classical 2-form. If the parameters are regarded as dynamical variables, evolving slowly so as to maintain adiabaticity, they are subject to geometric magnetism, but not, in contrast to the general case, deterministic friction and Born-Oppenheimer forces. Examples including families of translated and rotated systems are discussed.

DOI
10.1088/0305-4470/27/4/013

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We consider the geometric phase for a family of quantum/classical Hamiltonians in which the effect of changing parameters is simply to induce unitary/canonical transformations. In this case the classical limit of the geometric phase is easily obtained, even when the classical motion is chaotic. The results agree with those previously obtained for general chaotic families, but may be expressed in a simpler form, not involving time integrals of correlation functions. It is also straightforward to establish some results which are problematic in the general case, for example the form of periodic orbit corrections, and the closeness of the classical 2-form. If the parameters are regarded as dynamical variables, evolving slowly so as to maintain adiabaticity, they are subject to geometric magnetism, but not, in contrast to the general case, deterministic friction and Born-Oppenheimer forces. Examples including families of translated and rotated systems are discussed.
Abstract The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies.
Abstract We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.
An isolated classical chaotic system, when driven by the slow change in several parameters, responds with two reaction forces: geometric friction and geometric magnetism. By using the theory of quantum fluctuation relations, we show that this holds true also for open quantum systems and provide explicit expressions for those forces in this case. This extends the concept of Berry curvature to the realm of open quantum systems. We illustrate our findings by calculating the geometric magnetism of a damped charged quantum harmonic oscillator transported along a path in physical space in the presence of a magnetic field and a thermal environment. We find that, in this case, the geometric magnetism is unaffected by the presence of the heat bath.
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.
We study the concepts of adiabatic driving and geometric phases of classical integrable systems under the Koopman-von Neumann formalism. In close relation to what happens to a quantum state, a classical Koopman-von Neumann eigenstate will acquire a geometric phase factor $exp\left\{ iΦ\right\} $ after a closed variation of the parameters λλλ in its associated Hamiltonian. The explicit form of ΦΦΦ is then derived for integrable systems, and its relation with the Hannay angles is shown. Additionally, we use quantum formulas to write a classical adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase.
Abstract Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometrical forms of the interaction, or due to interactions which do not result from conservative forces acting on the system (see Section 1.2). A conscientious reader may point out that all these effects could be entirely explained using the Schrödinger equation only. But the geometrical phase formalism is useful because it sometimes plays a role in fundamental quantum phenomena and it shows that additional phase factors exist which are easily overlooked, making the interpretation of measured effects more mysterious than necessary. Thus, our understand-ing of quantum mechanics has deepened since the existence of geometrical phases was explicitly revealed (Shapere and Wilczek 1989). This happened due to the seminal works of Pancharatnam (1956), Berry (1984), Anandan (1992), and Bhandari (1997). It also shows that a wavefunction often carries much more information than is usually extracted in a standard experiment. A typical example is the assumption of adiabaticity, where the magnetic moment of the neutron produces (inside a slowly varying magnetic field) a dynamical phase shift proportional to the path integral ∫ B ds, but when the direction of the field varies slowly an additional, i.e. geometrical, phase also appears. The back reaction of a quantum measurement can also be described as a kind of an induced topological phase (Aharonov et al. 1998).