Phase-Space Approach to Berry Phases

We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light into the correspondence between classical and quantum adiabatic phases — both phases are related with the averaging procedure: Hannay’s angle with averaging over the classical torus and Berry’s phase with averaging over the entire classical phase space with respect to the corresponding Wigner function. Generalizations to the non-abelian Wilczek–Zee case and mixed states are also included. Geometric Berry phase [1] and its classical analog, so called Hannay angle, [2] (see also [3]) have found numerous applications in various branches of physics (see e.g. [4] and [5]). Recently, it turned out that adiabatic Berry phase plays important role in quantum computation algorithms as a model of a quantum gate in a quantum computer (see e.g. [6, 7]). In this paper we present a new formula for the Berry phase which is based on the phase space formulation of quantum mechanics. This approach sheds a new light into the correspondence between classical and quantum adiabatic phases. Both Berry’s phase and Hannay’s angles have been introduced in the context of adiabatic evolution in quantum and classical mechanics, respectively. Let us consider for simplicity a classical system with one degree of freedom and let the corresponding phase space be parameterized by canonical coordinates (q, p). Suppose, that a Hamiltonian H(q, p;X) depends on a set of some external parameters X from the parameter space M and that X are changed adiabatically along a circuit C and come back to their initial values, i.e. X(T ) = X(0) for some T > 0. Now, the classical adiabatic theorem [8] states that the system will evolve on the torus defined by the constant value of the action variable I and the angle variable varies according to θ(T ) = ∫ T 0 ω(I;X(t))dt +∆θ(I;C) , (1) where the frequency ω(I,X) = ∂H(I;X)/∂I and the additional shift — Hannay angle ∆θ — is given by the following integral over an arbitrary two-dimensional region Σ in M such that C = ∂Σ ∆θ(I;C) = − ∂ ∂I ∫ ∫ ∂Σ=C F (I;X) , (2) where F c(I;X) denotes the following two-form on M: F (I;X) = 〈 dX p(I;X) ∧ dX q(I;X) 〉 , (3)