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Journal of Physics B Atomic Molecular and Optical Physics

Circuit-dependent adiabatic phase factors from phase integral theory

Peter V. Coveney, Derrick S. F. Crothers, J. H. Macek

Published Apr 14, 1988• 6 citations
Publisher

Abstract

The authors show that circuit-dependent adiabatic phase factors occur naturally in the phase integral theory of atomic collisions, being a physical manifestation of the Stokes phenomenon familiar in asymptotic analysis. This implies a generalisation of Berry's work (1984) on geometric phase factors for situations involving adiabatic parallel transport around closed circuits in the complex plane.

DOI
10.1088/0953-4075/21/7/005

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The authors show that circuit-dependent adiabatic phase factors occur naturally in the phase integral theory of atomic collisions, being a physical manifestation of the Stokes phenomenon familiar in asymptotic analysis. This implies a generalisation of Berry's work (1984) on geometric phase factors for situations involving adiabatic parallel transport around closed circuits in the complex plane.
We consider a two-state model which extends the exponential time-dependent Demkov model of non-adiabatic coupling to complex energies and interactions. Using the method of steepest descent, exact and practicable asymptotic expansions of the Hankel functions H v j (1) (λ), H v j (2) (λ) are obtained when | v j | and | λ | are large and of the same order. The semiclassical scattering two channel matrix within the strong-coupling Demkov model and embracing non-adiabatic parameters is derived. Closed form expressions for the transition probabilities, which incorporate the effects of the classical turning point, are also derived. The matrix elements are shown to reduce correctly in known limiting cases. These include the case of exact symmetric resonance, the Landau-Zener and Callaway-Bartling model and the Stueckelberg noncrossing formula. In the case of a Hamiltonian matrix which is complex Hermitian on the real t axis, Berry has shown that the phenomenon of anholonomy arises as the physical manifestation of a non-integrable real geometric phase in adiabatic parallel transport round a closed circuit in real space. More recently Berry has reinterpreted this theory as arising from transport around a circuit in complex time. The geometric amplitude factor, which is simply the anholonomic phase corresponding to the connection on a complex fibre bundle, occurs naturally in the Demkov model which is an example of a closed system in parameter space driven by a complex Hermitian Hamiltonian. It is shown that the Berry phase phenomenon also involves contributions from both real and complex circuit adiabatic phases.
The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
According to Berry, quantum states of a hamiltonian which varies adiabatically through a circuit C in parameter space may acquire geometrical phase factors exp (iγ(C)) in addition to the normal dynamical phase factors exp ((-i/ħ) ∝ E(t) dt). We present N.M.R. experiments in the rotating frame which bear out these predictions for simple conical circuits, and point out that they are related to familiar behaviour based on the classical Bloch equations and on Haeberlen-Waugh coherent averaging theory. Extensions to coupled spins and electric quadrupolar effects are discussed.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.