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Acta Physica Sinica

CONDITIONS FOR THE EXISTENCE OF NON-ZERO GEOMETRIC PHASE AND ITS FORMULA

Chun-Fang Li, Guo Guang-Can, 中国科学技术大学物理系,合肥230026

Published Jan 1, 1996• 1 citation
Publisher

Abstract

First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.

DOI
10.7498/aps.45.897

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First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.
Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.
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:
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.
To discuss the sufficient condition that the state of a quantum system can be approximated by an eigenstate of the Hamiltonian,the functions f(t)satisfying the condition that〈f(t)|f′(t)〉=0(t∈[0,∞))and∫∞0‖|f′(t)〉‖dteare characterized.At first the general form of a qubit is given,then the exact structures and a necessary and sufficient condition of the components a qubit which satisfying the above adiabatic approximation conditions are proved.Finally some necessary conditions and sufficient conditions of the components of n-dimensional quantum state satisfying the adiabatic approximation are obtained.