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Acta Physica Sinica (Overseas Edition)

Approximation theory of oscillating factor suppressing amplitude in quantum process and its applications

Sun Changpu, Xia-Ji Liu

Published May 1, 1996• 1 citation
Publisher

Abstract

The oscillating factor suppressing amplitude approximation is proposed in this paper as a basic method to study the evolution of a quantum system. By making use of this method, not only the quantum adiabatic approximation is described again, but also both the higher-order corrections for the rotation wave approximation and the influences of other levels on two-level approximation can be analytically investigated with the explicit discussion for the quantitative conditions under which the two approximations hold.

DOI
10.1088/1004-423x/5/5/004

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The oscillating factor suppressing amplitude approximation is proposed in this paper as a basic method to study the evolution of a quantum system. By making use of this method, not only the quantum adiabatic approximation is described again, but also both the higher-order corrections for the rotation wave approximation and the influences of other levels on two-level approximation can be analytically investigated with the explicit discussion for the quantitative conditions under which the two approximations hold.
In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input.
In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input.
The condition for adiabatic approximation are of basic importance for the applications of the adiabatic theorem. The traditional quantitative condition was found to be necessary but not sufficient, but we do not know its physical meaning and the reason why it is necessary from the physical point of view. In this work, we relate the adiabatic theorem to the uncertainty relation, and present a clear physical picture of the traditional quantitative condition. It is shown that the quantitative condition is just the amplitude of the probability of transition between two levels in the time interval which is of the order of the time uncertainty of the system. We also present a new sufficient condition with clear physical picture.
Interactions between a source of light and atoms are ubiquitous in nature. The study of them is interesting on the fundamental level as well as for applications. They are at the core of quantum information processing tasks and quantum thermodynamics protocols. However, even for a two-level atom interacting with a field in rotating wave approximation there exists no exact solution. This touches upon a basic problem in quantum field theory, where we can only calculate the transitions in the time asymptotic limits (i.e., minus and plus infinity), while we are not able to trace the evolution. In this paper we want to get more insight into the time evolution of a total system of a two-level atom and a continuous-mode quantum field. We propose an approximation, which we are able to apply systematically to each order of Dyson expansion, resulting in a greatly simplified formula for the evolution of the combined system at any time. Our tools include a proposed non-Hermitian renormalization method. As a sanity check, by applying our framework, we derive the known optical Bloch equations.
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.