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Physical Review A

Interpretation of geometric phase via geometric distance and length during cyclic evolution

Arun Kumar Pati, Amitabh Joshi

Published Jan 1, 1993• 18 citations
Publisher

Abstract

We cast the nonadiabatic geometric phase in terms of the geometric distance function and the geometric length of the curve for arbitrary cyclic evolution of the quantum states. An interpretation is given to the geometric phase as the value of the integral of the contracted length of the curve along which the system traverses. It is found that for arbitrary cyclic evolution of the quantum states the geometric phase \ensuremath{\beta}(C) acquired by the system cannot be greater than the total length of the curve l(C). We have argued that the geometric phase arises because of the fundamental inequality between the length of the curve and the distance function. Finally, we have illustrated the calculation of the geometric phase based on the geometric distance function and the geometric length of the curve.

DOI
10.1103/physreva.47.98

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We cast the nonadiabatic geometric phase in terms of the geometric distance function and the geometric length of the curve for arbitrary cyclic evolution of the quantum states. An interpretation is given to the geometric phase as the value of the integral of the contracted length of the curve along which the system traverses. It is found that for arbitrary cyclic evolution of the quantum states the geometric phase \ensuremath{\beta}(C) acquired by the system cannot be greater than the total length of the curve l(C). We have argued that the geometric phase arises because of the fundamental inequality between the length of the curve and the distance function. Finally, we have illustrated the calculation of the geometric phase based on the geometric distance function and the geometric length of the curve.
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.
The geometric phase is defined for any arbitrary quantum evolution using a ``reference section'' of the bundle covering the curve in the projective Hilbert space. A canonical one-form is defined whose line integral gives the desired geometric phase. It is manifestly gauge, phase, and reparametrization invariant for all quantum evolutions. A simple proof of the vanishing nature of the geometric phase along the geodesic is given. Also, an elementary proof of the nonadditive nature of the geometric phase is given. In the limit of cyclic evolution of a pure quantum state, this phase reduces to the Aharonov and Anandan phase, precisely. It is observed that in addition to the geometric phase, other geometric structures exist, such as the ``length'' and ``distance'' during any arbitrary quantum evolution. The relations among all of these geometric quantities are pointed out. Finally, two simple examples are studied to illustrate the ideas introduced in this paper.
We calculate the geometric phase associated with the time evolution of the wave function of a Bose-Einstein condensate system in a double-well trap by using a model for tunneling between the wells. For a cyclic evolution, this phase is shown to be half the solid angle subtended by the evolution of a unit vector whose z component and azimuthal angle are given by the population difference and phase difference between the two condensates. For a non-cyclic evolution an additional phase term arises. We show that the geometric phase can also be obtained by mapping the tunneling equations onto the equations os a space curve. The importance of a geometric phase in the context of some recent experiments is pointed out.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of $\ensuremath{\Elzxh},$ and equal to $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle.
We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.