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arXiv:quant-ph/9803026 · Physical Review A

Berry’s phase in the presence of a stochastically evolving environment: A geometric mechanism for energy-level broadening

Frank Gaitan

Published Sep 1, 1998• 22 citations
arXivHTMLPDF

Abstract

The generic Berry phase scenario in which a two-level system is coupled to a second system whose dynamical coordinate is slowly varying is generalized to allow for stochastic evolution of the slow system. The stochastic behavior is produced by coupling the slow system to a heat reservoir, which is modeled by a bath of harmonic oscillators initially in equilibrium at temperature $T$, and whose spectral density has a bandwidth that is small compared to the energy-level spacing of the fast system. The well-known energy-level shifts produced by Berry's phase in the fast system, in conjunction with the stochastic motion of the slow system, leads to a broadening of the fast system energy levels. In the limit of strong damping and sufficiently low temperature, we determine the degree of level broadening analytically, and show that the slow system dynamics satisfies a Langevin equation in which Lorentz-like and electriclike forces appear as a consequence of geometrical effects. We also determine the average energy level shift produced in the fast system by this mechanism.

DOI
10.1103/physreva.58.1665

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The generic Berry phase scenario in which a two-level system is coupled to a second system whose dynamical coordinate is slowly varying is generalized to allow for stochastic evolution of the slow system. The stochastic behavior is produced by coupling the slow system to a heat reservoir, which is modeled by a bath of harmonic oscillators initially in equilibrium at temperature TTT, and whose spectral density has a bandwidth that is small compared to the energy-level spacing of the fast system. The well-known energy-level shifts produced by Berry's phase in the fast system, in conjunction with the stochastic motion of the slow system, leads to a broadening of the fast system energy levels. In the limit of strong damping and sufficiently low temperature, we determine the degree of level broadening analytically, and show that the slow system dynamics satisfies a Langevin equation in which Lorentz-like and electriclike forces appear as a consequence of geometrical effects. We also determine the average energy level shift produced in the fast system by this mechanism.
Berry phase was originally defined for systems whose states are separated by finite energy gaps. One might naively expect that a system without a gap cannot have a Berry phase. Despite this we ask whether a Berry phase can be observed in a system which has a continuous spectrum because its coupling to the environment has broadened its energy levels. We find that, contrary to the above naive expectation, there are conditions under which the Berry phase is observable. However it is modified by the presence of the environment and no longer has a simple geometric interpretation. The model system we consider is a spin-half in a slowly rotating magnetic field, with the spin also coupled to a Ohmic environment of harmonic oscillators (spin-boson model). Here we discuss the high-temperature limit of this model. We then interpret our results in terms of a spin under the influence of a classical stochastic field.
Berry's [Proc. R. Soc. London, Ser. A 392, 45 (1984)] geometrical phase is discussed in the context of dissipative evolution of an interacting spin system, governed by the stochastic Liouville equation. An analytical treatment is given for a possible ESR experiment on an interacting electron-nucleus system, modulated by two-site jumps. Geometrical phases are shown to be relevant to systems of this type, when their Hamiltonian changes slowly with time. A method for obtaining higher-order corrections to the adiabatic approximation is demonstrated. It is found that if the jumps are slow relative to the rate of change of the Hamiltonian, their effect reduces to familiar line broadening, and the geometrical phases may be observed experimentally. Equations are also set up for a similar ESR experiment on an electron-nucleus system undergoing isotropic rotational diffusion, and a brief discussion of the equations follows.
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 consider a two-level system coupled to an environment that evolves non-adiabatically. We present a non-perturbative method for determining the persistence amplitude whose phase contains all the corrections to Berry's phase produced by the non-adiabatic motion of the environment. Specifically, it includes the effects of transitions between the two energy levels to all orders in the non-adiabatic coupling. The problem of determining all non-adiabatic corrections is reduced to solving an ordinary differential equation to which numerical methods should provide solutions in a variety of situations. We apply our method to a particular example that can be realized as a magnetic resonance experiment, thus raising the possibility of testing our results in the lab.
We consider the spin 12 model coupled to a slowly varying magnetic field in the presence of a weak damping represented by a Lindblad-form operators. We show that Berry's geometrical phase remains unaltered by the two dissipation mechanism considered. Dissipation effects are twofold: a shrinking in the modulus of the Bloch's vector, which characterizes coherence loss and a time-dependent (dissipation related) precession angle. We show that the line broadening of the Fourier transformation of the components of magnetization is only due to the presence of dissipation.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.