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Physical Review A

Analytic dynamics of coupled two-level systems

Razmik G. Unanyan, Stig Stenholm

Published Sep 20, 2002• 0 citations
Publisher

Abstract

We solve a problem of quantum dynamics of two coupled Schr\"odinger equations. Utilizing a matrix transformation analogous to the Darboux method we find analytic solutions valid in special cases, ``quasiexact solutions.'' The general conditions for matrix Darboux transformations are formulated. The method is applied to a special case of wave-packet dynamics.

DOI
10.1103/physreva.66.032108

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We solve a problem of quantum dynamics of two coupled Schr"odinger equations. Utilizing a matrix transformation analogous to the Darboux method we find analytic solutions valid in special cases, ``quasiexact solutions.'' The general conditions for matrix Darboux transformations are formulated. The method is applied to a special case of wave-packet dynamics.
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Abstract We develop the Darboux procedure for the case of the two‐level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two‐level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two‐level system. Thus, we find three classes of new solutions for the two‐level system and the corresponding new potentials that allow such solutions.
We present a method to calculate exact dynamics of a wave-packet in a quantum two-state problem with Dirac delta coupling. The advantage of our method is that the calculations are done in the time domain. Hence inverting the solutions from other domains (Laplace or Fourier domain) do not intervene us from presenting the solution in the time domain. The initial wave packet is considered to be a Gaussian function. The wave propagation in the quantum state is governed by time-dependent Schr\odinger equation and the coupling is given by non-diagonal element in the matrix representation. We present the exact analytic forms of the wave function for both the states in the time domain.
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville’s theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.