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arXiv:1909.03206

Lindblad dynamics of the damped and forced quantum harmonic oscillator: General solution

H. J. Korsch

Published Sep 7, 2019• 0 citations
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Abstract

The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.

DOI
10.48550/arxiv.1909.03206

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The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.
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We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier transform or chord function as it has been called, it becomes particularly simple to solve the dynamics of the open oscillator in the sense that the dynamics of the system are reduced to the application of an evolution matrix related to the damped motion of the oscillator.
We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.