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arXiv:1004.0075 · Journal of Physics A Mathematical and Theoretical

Noncyclic and nonadiabatic geometric phase for counting statistics

Jun Ohkubo, Thomas Eggel

Published Sep 30, 2010• 9 citations
arXivHTMLPDF

Abstract

We propose a general framework of the geometric-phase interpretation for counting statistics. Counting statistics is a scheme to count the number of specific transitions in a stochastic process. The cumulant generating function for the counting statistics can be interpreted as a `phase', and it is generally divided into two parts: the dynamical phase and a remaining one. It has already been shown that for cyclic evolution the remaining phase corresponds to a geometric phase, such as the Berry phase or Aharonov-Anandan phase. We here show that the remaining phase also has an interpretation as a geometric phase even in noncyclic and nonadiabatic evolution.

DOI
10.1088/1751-8113/43/42/425001

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We propose a general framework of the geometric-phase interpretation for counting statistics. Counting statistics is a scheme to count the number of specific transitions in a stochastic process. The cumulant generating function for the counting statistics can be interpreted as a `phase', and it is generally divided into two parts: the dynamical phase and a remaining one. It has already been shown that for cyclic evolution the remaining phase corresponds to a geometric phase, such as the Berry phase or Aharonov-Anandan phase. We here show that the remaining phase also has an interpretation as a geometric phase even in noncyclic and nonadiabatic evolution.
We propose an application of fiber bundles to counting statistics. The framework of the fiber bundles gives a splitting of a cumulant generating function for a current in a stochastic process, i.e., contributions from the dynamical phase and the geometric phase. We will show that the introduced noncyclic geometric phase is related to kinds of excess contributions, which have been investigated a lot in nonequilibrium physics. Using a specific nonequilibrium model, the characteristics of the noncyclic geometric phase are discussed; in particular, we reveal differences between a geometric contribution for the entropy production and the 'excess entropy production' which has been used to discuss the second law of steady-state thermodynamics.
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.
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.
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.