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The Journal of Chemical Physics

The interaction representation and nonadiabatic corrections to adiabatic evolution operators. II. Nonlinear quantum systems

Steven D. Schwartz

Published May 22, 1996• 8 citations
Publisher

Abstract

This paper reports further applications of the recently developed interaction representation form of infinite order operator corrections to adiabatic evolution operators. Previous work derived the form of the correction, and applied the methodology to a bilinearly coupled system bath model. In this paper we present results on coupled quantum systems in which the coupling is highly nonlinear. The method is both easy to implement and numerically accurate.

DOI
10.1063/1.471514

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This paper reports further applications of the recently developed interaction representation form of infinite order operator corrections to adiabatic evolution operators. Previous work derived the form of the correction, and applied the methodology to a bilinearly coupled system bath model. In this paper we present results on coupled quantum systems in which the coupling is highly nonlinear. The method is both easy to implement and numerically accurate.
This paper presents a new approach to operator resummation corrections to adiabatic evolution operators. It is shown that an infinite order correction produces an operator that is equivalent to a propagator in the interaction representation. For a problem in which the adiabatic approximation assumes that certain degrees of freedom are held fixed, the interaction representation correction is just the interaction propagator of the coupling for these degrees of freedom. This formulation allows simple physical interpretation and simple mathematical evaluation of the full correction. No power series or cumulant methods are needed. Application to double well splitting when coupled to a bath oscillator shows the approach to be highly accurate.
This paper further investigates quantum activated rate theory from the viewpoint of quantum evolution operators. It is shown that a new adiabatic separation of the quantum system-bath Hamiltonian can, in a single time step, account for quantum turnover behavior at moderate temperatures, and it is also shown how this turnover exponentially vanishes at low temperatures. It is further shown that incorporation of nonadiabatic (interaction representation form) corrections produces quantitatively accurate results at low temperatures, thus extending the applicability of the interaction representation form of nonadiabatic corrections to adiabatic evolution operators.
Adiabatic limit is the presumption of the adiabatic geometric quantum computation and of the adiabatic quantum algorithm. But in reality, the variation speed of the Hamiltonian is finite. Here we develop a general formulation of adiabatic quantum computing, which accurately describes the evolution of the quantum state in a perturbative way, in which the adiabatic limit is the zeroth-order approximation. As an application of this formulation, nonadiabatic correction or error is estimated for several physical implementations of the adiabatic geometric gates. A quantum-computing process consisting of many adiabatic gate operations is considered, for which the total nonadiabatic error is found to be about the sum of those of all the gates. This is a useful constraint on the computational power. The formalism is also briefly applied to the adiabatic quantum algorithm.
This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H+H2 the effective one-dimensional approximation we apply produces essentially exact results.
The scattering properties of any complex scattering potential, , can be obtained from the dynamics of a particular non-unitary two-level quantum system . The application of the adiabatic approximation to yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.
A recently developed quantum correction approach is applied to evaluating the nonadiabatic quantum-mechanical transition rate between Born-Oppenheimer states of a subsystem embedded in a thermal bath of harmonic oscillators. In the first-order perturbation theory, the nonadiabatic rate can be expressed in terms of a quantum-mechanical correlation function, which can be estimated directly from classical data. Application to a popular spin-boson model shows that our results are in excellent agreement with the exact quantum-mechanical results.