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arXiv:1403.3686 · Physical Review A

Closed-form solution of Lindblad master equations without gain

Juan Mauricio Torres

Published May 30, 2014• 75 citations
arXivHTMLPDF

Abstract

We present a closed-form solution to the eigenvalue problem of a class of master equations that describe open quantum systems with loss and dephasing but without gain. The method relies on the existence of a conserved number of excitations in the Hamiltonian part and the fact that none of the Lindblad operators describe an excitation of the system. In the absence of dephasing Lindblad operators, the eigensystem of the Liouville operator can be constructed from the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian used in the quantum jump approach. Open versions of spin chains, the Tavis-Cummings model, and coupled Harmonic oscillators without gain can be solved using this technique.

DOI
10.1103/physreva.89.052133

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We present a closed-form solution to the eigenvalue problem of a class of master equations that describe open quantum systems with loss and dephasing but without gain. The method relies on the existence of a conserved number of excitations in the Hamiltonian part and the fact that none of the Lindblad operators describe an excitation of the system. In the absence of dephasing Lindblad operators, the eigensystem of the Liouville operator can be constructed from the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian used in the quantum jump approach. Open versions of spin chains, the Tavis-Cummings model, and coupled Harmonic oscillators without gain can be solved using this technique.
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We study driven and open quantum systems described by a time-periodic Lindblad master equation. In closed systems, the stroboscopic dynamics can always be described by an effective time-independent Floquet Hamiltonian; this idea is the basis of Floquet engineering. However, in the presence of dissipation, the existence of an effective time-independent Floquet Lindbladian is not guaranteed due to the non-unitary nature of the evolution. Using Floquet theory, we construct a well-defined time-independent Floquet Lindbladian in an extended Sambe-Liouville space, transforming the initial time-dependent problem to a static and non-Hermitian eigenvalue problem. For harmonic driving, we introduce a matrix continued fraction method to nonperturbatively resum multiphoton processes and construct an effective Floquet Lindbladian acting only on the physical Liouville space. Compared to other high-frequency expansions, this method has the advantage of providing the whole infinite series expansion at once. Using a resolvent formalism, we show how to obtain a spectral Floquet representation of correlation functions of an open quantum system. As an application, we consider a dissipating two-level system in a linearly polarized field and calculate its resonance fluorescence spectrum. Furthermore, we consider a parametrically driven quantum dot with pump and loss for which we calculate its spectral function and current-voltage characteristics.
It is known that a quantum master equation that describes dissipative relaxation (such as spontaneous decay) can be separated in a part that describes spontaneous jumps and a jump-free part. The jump-free part of the evolution is determined by a non-Hermitian Hamiltonian H, which acts on state vectors, rather than on density operators. We identify a class of systems for which the complete evolution is entirely determined by the jump-free evolution, which is governed by H. Although this class is restricted, it contains a sizable number of systems of physical significance. Among these are two-, three-, and four-state atoms. The method is used to find explicit expressions of time-dependent expectation values in terms of the complex eigenvalues of H. As a first application we evaluate exactly the photon statistics in the driven \ensuremath{\Lambda} system. As a second example of the method, we treat the configuration of a V system driven by two standing waves. This system constitutes an effective beam splitter.
A Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained. This spectral resolution is discussed in depth in the context of the biorthogonal system and the rigged Hilbert space, and the contribution of each eigenprojection to expectation values of physical quantities is revealed. We also construct the ladder operators of the Liouvillian, which clarify the structure of the spectral resolution.
We develop a novel framework to engineer persistent oscillatory modes in Markovian open quantum systems governed by a time-independent Lindblad master equation. We show that oscillatory modes can be created when the Hamiltonian and jump operator can be expressed in the same block-diagonal form. A key feature of the framework is that the dissipator of the Lindblad master equation are generally non-zero. We identify the weak and strong conditions, where the onset of the oscillatory modes is dependent and independent of the parameters of the system, respectively. Our method extends beyond the typical decoherence-free subspace approach, in which the dissipator is zero. We demonstrate the applicability of this framework using various models, showing how carefully tailored system-environment interactions can produce sustained coherent oscillations.