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Physics Research International

Quantum Transitions by Change of Parameters and in Noninertial Motion

Marian Apostol

Published Jan 1, 2008• 0 citations
Publisher

Abstract

As it is well known, a quantum system depending on parameters exhibits the (geometric) Berry phase when parameters are varying in the adiabatic limit. A generalization of the Berry phase is given in the present paper for a nonadiabatic change of parameters, which leads to quantum transitions in the system. This generalization is applied to noninertial motions and it is shown that such motions may induce quantum transitions for a system in an external field governed by Schrodinger′s equation.

DOI
10.1155/2008/570167

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As it is well known, a quantum system depending on parameters exhibits the (geometric) Berry phase when parameters are varying in the adiabatic limit. A generalization of the Berry phase is given in the present paper for a nonadiabatic change of parameters, which leads to quantum transitions in the system. This generalization is applied to noninertial motions and it is shown that such motions may induce quantum transitions for a system in an external field governed by Schrodinger′s equation.
Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.
We investigate an explicitly time-dependent quantum system driven by a secant-pulse external field. By solving the Schr\"odinger equation exactly, we elucidate exotic properties of the system with respect to its dynamical evolution: On the one hand, the system is shown to be essentially nonadiabatic, which prohibits an adiabatic approximation for its dynamics; on the other hand, the loop evolution of the model can induce a geometric phase which, analogous to the Berry phase of the cyclic adiabatic evolution, is in direct proportion to the solid angle subtended by the path of the state vector. Moreover, we extend the model and show that the described properties coincide in a special family of secant-pulse-driven models.
A matrix method is presented for treating the dynamical phases, adiabatic phases and nonadiabatic phases of quantum superposition states. It is effective for any parameter-varying Hamiltonian system. As two examples, the evolution of mass-varying harmonic oscillator and the evolution of coherent states under parameter-varying displaced operator have been studied, Some new phenomena are obtained in the first case and the possible producing of so-called Schrodinger's cat state by geometric phases is pointed out. The quantum state useful for the quantum optical verification of Berry's phase is introduced.