Time-dependent gauge transformations and Berry's phase

Theoretical Physics

Jagmeet Singh

J.-Q. Liang, X. X. Ding

Aiping Zhang

A quantum system with time-dependent Hamiltonian which transports slowly will acquire an undynamic phase factor exp{ iγ n(t) } factor during evolution in addition to a dynamic phase factor. By means of time-dependent gauge transformation, we do a thorough study on undynamic phase which emerge in the quantum system. The results show that the system will gain an non-integrable phase when a periodic evolution doesn't accomplished; however, only when the system achieves a periodic evolution, an integrable phase will be acquire, namely Berry phase, and the phase is invariable under periodic time-dependent gauge transformation, thereby it is observable. Accordingly, we do a general interpretation on the topological property of Berry's phase as an example of NMR.

Jeremy I. Musher

JunLi XIN, Jiu‐Qing Liang, JunXia SHEN

Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .