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Physical Review A

Evolution of a two-level system with relaxation driven by a time-dependent Hamiltonian: A simple model

Michel Gorlicki

Published Jun 1, 1994• 4 citations
Publisher

Abstract

A time-varying Hamiltonian in a two-level system with different relaxation rates is studied using an effective-Hamiltonian model. Energy sheets are featured. A suited adiabatic basis is provided in which equations of evolution are given for arbitrary paths. Phase factors are discussed for adiabatic loops. The influence of relaxation on the criterion for adiabatic evolution is pointed out.

DOI
10.1103/physreva.49.4339

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A time-varying Hamiltonian in a two-level system with different relaxation rates is studied using an effective-Hamiltonian model. Energy sheets are featured. A suited adiabatic basis is provided in which equations of evolution are given for arbitrary paths. Phase factors are discussed for adiabatic loops. The influence of relaxation on the criterion for adiabatic evolution is pointed out.
We study the time evolution of a quantum system under the adiabatic condition using the path integral formulation. We reobtain Berry's phase for systems with nondegenerate and degenerate energy spectra, recovering the well-known results on the literature, and we verify the behavior of the phases under a change of basis of the energy instantaneous eigenstates. We also present a simple derivation of the Adiabatic Theorem[1] using the current notation. We have a comparative summary between the Schrodinger's and the Heisenberg's pictures when the quantum system is driven by a time-dependent hamiltonian and we show how to aply the Adiabatic Theorem to any operator in the Heisenberg's picture that evolves under adiabactic conditions.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
The general class of models of adiabatic evolution was proposed to speed up the usual adiabatic computation in the case of quantum search problem. It was shown [8] that, by temporarily increasing the ground state energy of a time-dependent Hamiltonian to a suitable quantity, the quantum computation can perform the calculation in time complexity O(1). But it is also known that if the overlap between the initial and final states of the system is zero, then the computation based on the generalized models of adiabatic evolution can break down completely. In this paper, we find another severe limitation for this class of adiabatic evolution-based algorithms, which should be taken into account in applications. That is, it is still possible that this kind of evolution designed to deal with the quantum search problem fails completely if the interpolating paths in the system Hamiltonian are chosen inappropriately, while the usual adiabatic evolutions can do the same job relatively effectively. This implies that it is not always recommendable to use nonlinear paths in adiabatic computation. On the contrary, the usual simple adiabatic evolution may be sufficient for effective use.
Adiabatic passage employs a slowly varying time-dependent Hamiltonian to control the evolution of a quantum system along the Hamiltonian eigenstates. For processes of finite duration, the exact time-evolving state may deviate from the adiabatic eigenstate at intermediate times, but in numerous applications it is observed that this deviation reaches a maximum and then decreases significantly towards the end of the process. We provide a straightforward theoretical explanation for this welcome but often unappreciated fact. Our analysis emphasizes a separate adiabaticity criterion for high-fidelity state-to-state transfer and it points to new effective shortcut strategies for near-adiabatic dynamics.
We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate levels.