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arXiv:0807.3121 · Journal of Physics A Mathematical and Theoretical

Geometric phase for non-Hermitian Hamiltonian evolution as anholonomy of a parallel transport along a curve

N A Sinitsyn, Avadh Saxena

Published Sep 2, 2008• 11 citations
arXivHTMLPDF

Abstract

We develop an interpretation of the geometric phase in evolution with a non-Hermitian real-valued Hamiltonian by relating it to the angle developed during the parallel transport along a closed curve by a unit vector triad in the 3D Minkovsky space. We also show that this geometric phase is responsible for the anholonomy effects in stochastic processes considered by Sinitsyn and Nemenman (2007 Europhys. Lett. 77 58001), and use it to derive the stochastic system response to periodic parameter variations.

DOI
10.1088/1751-8113/41/39/392002

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1 A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

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:

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Physical Review A•Dec 1, 1988•3 citations
We develop an interpretation of the geometric phase in evolution with a non-Hermitian real-valued Hamiltonian by relating it to the angle developed during the parallel transport along a closed curve by a unit vector triad in the 3D Minkovsky space. We also show that this geometric phase is responsible for the anholonomy effects in stochastic processes considered by Sinitsyn and Nemenman (2007 Europhys. Lett. 77 58001), and use it to derive the stochastic system response to periodic parameter variations.
A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.
Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.
The complex geometric phases for systems governed by non-Hermitian 3×3 matrix Hamiltonians depending on time through a set of real parameters are constructed. The case that the non-Hermitian three-level system undergoes a cyclic evolution is considered for the class of the Hamiltonians with time-independent off-diagonal matrix elements. The conditions are established whereby the complex geometric phases take their extreme (resonance) values independently of the circuit in the parameter space. The results can be in particular used to study the problem of three-neutrino oscillations with regard to the neutrino refraction and absorption in matter.
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: