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Physical Review A

Noncyclic geometric phase, coherent states, and the time-dependent variational principle: Application to coupled electron-nuclear dynamics

Erik Sjöqvist, Magnus Hedström

Published Nov 1, 1997• 24 citations
Publisher

Abstract

Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere $({\mathcal{S}}^{2}).$ An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the $E\ensuremath{\bigotimes}\ensuremath{\varepsilon}$ Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.

DOI
10.1103/physreva.56.3417

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Kinematical and dynamical aspects of the noncyclic geometric phase for coherent states are analyzed. It is shown for Glauber and SU(2) coherent states that the geometric phase obeys a geodesic closure rule associated with the coset space of the respective Lie group. In the former case the geodesic is a straight line in phase space, while in the latter the geodesic is a great circle on the Bloch sphere (S2).({\mathcal{S}}^{2}).(S2). An alternative expression for the geometric phase is derived from the time-dependent variational principle by making a specific choice of gauge. The noncyclic geometric phase is numerically calculated in a time-dependent variational treatment of the E\ensuremath⨂\ensuremathεE\ensuremath{\bigotimes}\ensuremath{\varepsilon}E\ensuremath⨂\ensuremathε Jahn-Teller model first introduced by Longuet-Higgins and co-workers. The electronic and nuclear degrees of freedom are parametrized by SU(2) and Glauber coherent states respectively. In particular, the adiabatic limit is studied and shown to yield the anticipated Berry geometric phase.
Using a time-dependent variational method, we study the evolution of nonstationary states in Longuet-Higgin’s model of a Jahn–Teller molecule. Conditions are found for the nuclear motion to be adiabatic. The effects of wave-packet spreading are neglected upon specializing to the case of nearly harmonic motion. It is shown explicitly how the effective vector potential introduced by Mead and Truhlar gives rise to an electronic Berry phase. In a semiclassical approximation sufficient to produce the electronic adiabatic phase anticipated from the result for a given sequence of nuclear configurations, it is demonstrated that the effective vector potential has a negligible effect on the nuclear motion; the effective vector potential, the source of an effective field proportional to ℏ, is seen to affect the nuclear trajectory only in higher order. For the special case of periodic nuclear motion the electronic adiabatic phase is seen as a contribution to the geometric phase attending an arbitrary cyclic evolution. It is demonstrated that a molecular state prepared with identically pseudorotating nuclear wave packets in both electronic levels corresponds, in a weak coupling limit, to a spin 1/2 in a conically varying external field. Geometrical phase differences are shown to make discernible contributions to the frequencies of oscillation of the electronic charge and current densities, which may serve as classical sources for superradiant emission. Our results are shown to be gauge invariant.
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.
The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.
We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.
The theory of quantal revivals is extended to Hamiltonians that are time dependent through the slow noncyclic change of some parameters. It is shown by generalizing the cyclic geometric phase (Berry's phase) to noncyclic Hamiltonians that there is a noncyclic geometric displacement for all quantum revivals. The treatment of the classical mechanical geometric angle is also extended to open paths in parameter space and its semiclassical relation with the noncyclic geometric phase derived. The displacement of the revivals is then shown to be given by the total angle shift on the final torus sum of the dynamical angle and the noncyclic geometric angle.