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Physical Review A

Nonadiabatic geometric phase in periodic linear systems

Y. C. Ge, Mark Sheard Child

Published Aug 1, 1998• 18 citations
Publisher

Abstract

The method used by Garrison and Wright for the calculation of complex geometrical phases in quantal dissipative systems is extended to all time periodic linear systems, including classical linear systems with nonunitary time evolution. Each eigenvector $|{x}_{i}(t)〉$ of the period map modified by a complex scaling exponent ${\ensuremath{\theta}}_{i}(t)$ has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the $|{x}_{i}(t)〉$ commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.

DOI
10.1103/physreva.58.872

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The method used by Garrison and Wright for the calculation of complex geometrical phases in quantal dissipative systems is extended to all time periodic linear systems, including classical linear systems with nonunitary time evolution. Each eigenvector ∣xi(t)〉|{x}_{i}(t)〉∣xi​(t)〉 of the period map modified by a complex scaling exponent \ensuremathθi(t){\ensuremath{\theta}}_{i}(t)\ensuremathθi​(t) has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the ∣xi(t)〉|{x}_{i}(t)〉∣xi​(t)〉 commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.
A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(n l ); the classical property is a shift Delta theta l (I) in the lth angle variable for motion round a phase-space torus with actions I=(I l ); the connection is Delta theta l =- delta gamma / delta n l . Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q 2 , qp and p 2 ; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of a squeezed state. A class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Then their geometric phases are obtained explicitly and found to be $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle. It is shown that the analysis based on the squeezed state approach provides a clear picture of the geometric meaning of the quantal phase.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
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.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.