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Physics Letters A

Geometry of quantal adiabatic evolution driven by a non-Hermitian Hamiltonian

Zhaoyan Wu, Ting Yu, Hongwei Zhou

Published Mar 1, 1994• 7 citations
Publisher
DOI
10.1016/0375-9601(94)90922-9

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We study in this paper the time evolution of PT-symmetric non-Hermitian Hamiltonian consisting of periodically driven SU(1,1) generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a PT-symmetric but non-unitary transformation operator in the construction of the non-Hermitian invariant. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time- evolution operator is found analytically. As a consequence of the non-unitarity the ket (|ψ(t)〉) and bra (〈ψ(t)|) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.
Adiabatic chiral state conversion (CSC) is one of the many counterintuitive effects associated with non-Hermitian physics. In quantum systems, numerous works have demonstrated this phenomenon under both non-Hermitian Hamiltonian and Lindblad evolution. However, despite considerable progress, the physical mechanism behind it has been a subject of debate. In this work, we present a unified framework that explains CSC in any non-Hermitian system, encompassing non-Hermitian Hamiltonian, Lindblad, and hybrid settings. Our framework relies on perturbative, non-adiabatic corrections to adiabatic evolution and consistently predicts CSC with only the lowest-order corrections. We demonstrate its efficacy with models of single and coupled dissipative qubits, obtaining analytical solutions for the conversion fidelity. Our analysis further reveals the role of non-perturbative dynamics, which can be present even in apparently slow trajectories. We show that this property can be utilised to considerably enhance state conversion. Finally, we demonstrate that CSC can be observed in a model without the presence of exceptional points.
In this paper the evolution of a quantum system driven by a non-Hermitian Hamiltonian depending on slowly-changing parameters is studied by building an universal high-order adiabatic approximation (HOAA) method with Berry's phase, which is valid for either the Hermitian or the non-Hermitian cases. This method can be regarded as a non-trivial generalization of the HOAA method for closed quantum system presented by this author before. In a general situation, the probabilities of adiabatic decay and non-adiabatic transitions are explicitly obtained for the evolution of the non-Hermitian quantum system. It is also shown that the non-Hermitian analog of the Berry's phase factor for the non-Hermitian case just enjoys the holonomy structure of the dual linear bundle over the parameter manifold. The non-Hermitian evolution of the generalized force harmonic oscillator is discussed as an illustrative example.