Non-adiabatic Berry Phase Research Articles

Time evolution of quantum systems with time-dependent Hamiltonian and the invariant Hermitian operator

We study the time evolution of a class of exactly solvable time-dependent quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators with the help of the invariant Hermitian operator. The exact common solutions of the Schrodinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the invariant operator. The adiabatic and non-adiabatic Berry phases are calculated with the exact solutions. Moreover, we derive an explicit time-evolution operator which is used to investigate the time-dependent two-photon squeezing states and SU(2) squeezing states. The squeezing properties of the time-dependent SU(1,1) coherent states are also discussed.

Exact wave functions and nonadiabatic Berry phases of a time-dependent harmonic oscillator.

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.

Exact solution of the linear nonautonomous system with the SU (2) dynamical group: Inversion probability of a neutrino in a time-dependent magnetic field

The exact solution and the invariant Cartan operator of the linear nonautonomous system with the SU(2) dynamical group are obtained with the method of algebraic dynamics. The profound quantum-classical correspondence of the solutions is exhibited clearly: the quantum solutions of the system are determined by that of the corresponding classical equations. The exact solutions are employed to calculate the inversion probability of a neutrino in a time-dependent magnetic field and the particle spin polarization in beam dynamics. The nonadiabatic Berry phase for a periodic system is also calculated.

Nonadiabatic Berry's phase for light propagating in an optical fiber

A Schrodinger-like equation for an approximate description of polarization for a photon propagating in a helix has been obtained. The analytical solution is employed to study the nonadiabatic Berry's phase. A new expression for Berry's phase is given, indicating a close link between the Berry's phase and spin-alignment. Two kinds of experiments are suggested to measure the nonadiabatic Berry's phase.

Gauge transformation approach to the exact solution of a generalized harmonic oscillator

The exact solution and the invariant operator of a generalized harmonic oscillator are obtained by performing three consecutive gauge transformations on the time-dependent Schrodinger equation. In contrast to previous results, they depend only on solutions of a linear differential equation for the associated classical harmonic oscillator. Non-adiabatic and adiabatic Berry phases for the system are calculated on the basis of the exact solution.

Topological aspects of the non-adiabatic Berry phase

The topology of the non-adiabatic parameter space bundle is discussed for evolution of exact cyclic state vectors in Berry's original example of split angular momentum eigenstates. It turns out that the change in topology occurs at a critical frequency. The first Chern number that classifies these bundles is proportional to angular momentum. The non-adiabatic principal bundle over the parameter space is not well-defined at the critical frequency.

Geometric phase, geometric distance and length of the curve in quantum evolution

The geometric phase and the geometric distance function are intimately related via length of the curve (a concept the author introduces) for any parametric evolution of the quantum system. He offers an interpretation of the nonadiabatic Berry phase as the integral over a difference between the differentials of two geometric quantities, which enables him to say that the geometric phase is just (half) the integral of the contracted length of the curve that the system traverses during a cyclic excursion.

Nonadiabatic Berry phase in rotating systems.

The time-dependent Schr\odinger equation for a particle in general rotating systems is solved analytically, and the exact solutions are used to study the nonadiabatic Berry phase. It is shown that the nonadiabatic Berry topological phase appears in general rotating systems for purely mechanical reasons and is observable. An alternative expression for the nonadiabatic Berry phase is given and discussed.

Adiabatic and nonadiabatic Berry phases for two-level atoms.

Adiabatic and nonadiabatic Berry phases for two-level atoms.

Nonadiabatic Berry's phase for a quantum system with a dynamical semisimple Lie group.

The nonadiabatic Berry's phase is investigated for a quantum system with a dynamical semisimple Lie group within the framework of the generalized cranking approach. An expression for nonadiabatic Berry's phase is given, which indicates that the nonadiabatic Berry's phase is related to the expectation value of Cartan operators along the cranking direction in group space, and that it depends on (i) the geometry of the group space, (ii) the time evolution ray generated by the Hamiltonian (i.e., by the dynamics) in some irreducible representation Hilbert space, and (iii) the cranking rate. The expression also provides a simple algorithm for calculating the nonadiabatic Berry's phase. The general formalism is illustrated by examples of SU(2) dynamic group.

Nonadiabatic Berry's phase for a spin particle in a rotating magnetic field.

The time-dependent Schr\odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.

Floquet theory and the non-adiabatic Berry phase

An efficient calculational algorithm is provided for the operator decomposition approach to non-adiabatic Berry phases for systems with periodic Hamiltonians.

Non-adiabatic Berry phase for periodic Hamiltonians

Berry's (1984) formulation of the topological phase for periodic Hamiltonians is extended to the case of non-adiabatic evolution by proper choice of initial states. This restores a full parallel with the non-adiabatic formalism of Aharonov and Anandan (1987) for periodic orbits in a projective Hilbert space. The Berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by Aharonov and Anandan and by Page (1987).