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arXiv:1811.05490 · Physical Review A

Exact solution of time-dependent Lindblad equations with closed algebras

Stefano Scopa, Gabriel Teixeira Landi, Adam Hammoumi, Dragi Karevski

Published Feb 5, 2019• 44 citations
arXivHTMLPDF

Abstract

Time-dependent Lindblad master equations have important applications in areas ranging from quantum thermodynamics to dissipative quantum computing. In this paper we outline a general method for writing down exact solutions of time-dependent Lindblad equations whose superoperators form closed algebras. We focus on the particular case of a single qubit and study the exact solution generated by both coherent and incoherent mechanisms. We also show that if the time dependence is periodic, the problem may be recast in terms of Floquet theory. As an application, we give an exact solution for a two-level quantum heat engine operating in a finite time.

DOI
10.1103/physreva.99.022105

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Time-dependent Lindblad master equations have important applications in areas ranging from quantum thermodynamics to dissipative quantum computing. In this paper we outline a general method for writing down exact solutions of time-dependent Lindblad equations whose superoperators form closed algebras. We focus on the particular case of a single qubit and study the exact solution generated by both coherent and incoherent mechanisms. We also show that if the time dependence is periodic, the problem may be recast in terms of Floquet theory. As an application, we give an exact solution for a two-level quantum heat engine operating in a finite time.
The operation of autonomous finite-time quantum heat engines relies on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what the state of the system is within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses all operating properties of the engine. First, its spectrum dictates the convergence to a limit cycle. Second, the state within the limit cycle is precisely its zero eigenstate, therefore reducing the problem to that of determining the steady state of a time-independent master equation. To illustrate the usefulness of this theory, we apply it to a harmonic oscillator subject to a time-periodic work protocol and time-periodic dissipation, an open-system generalization of the Ermakov-Lewis theory. The use of this theory to implement a finite-time Carnot engine subject to continuous frequency modulations is also discussed.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
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.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
For a closed quantum system described by a time-periodic Hamiltonian, Floquet theorem says that the time evolution operator can be written as $U(t,0)\ensuremath{\equiv}P(t){e}^{\frac{\ensuremath{-}i}{\ensuremath{\hbar}}{H}_{F}t}$ with P(t+T)=P(t)P(t+T)=P(t)P(t+T)=P(t), and HF{H}_{F}HF​ is Hermitian and time independent and called the Floquet Hamiltonian. In this work, we extend the Floquet theorem from closed to open systems described by a time-periodic Lindblad master equation with period $T=\frac{2\ensuremath{\pi}}{\ensuremath{\omega}}$. We derive an expansion for the Lindbladian in powers of $\frac{1}{\ensuremath{\omega}}$. Two examples are presented to illustrate the theorem. We find that the results given by the Lindbladian expansion based on the open system Floquet theorem agree well with the exact dynamics in the high-frequency limit. It may find applications in quantum engineering with systems subject to decoherence.
In this paper, we investigate periodically driven open quantum systems within the framework of Floquet-Lindblad master equations. Specifically, we discuss Lindblad master equations in the presence of a coherent, time-periodic driving and establish their general spectral features. We also clarify the notions of transient and nondecaying solutions from this spectral perspective, and then prove that any physical system described by a Floquet-Lindblad equation must have at least one physical nonequilibrium steady state (NESS), corresponding to an eigenoperator of the Floquet-Lindblad evolution superoperator UF{\mathcal{U}}_{F}UF​ with unit eigenvalue. Since the Floquet-Lindblad formalism encapsulates the entire information regarding the NESS, it in principle enables us to obtain nonlinear effects to all orders at once. The Floquet-Lindblad formalism thus provides a powerful tool for studying driven-dissipative solid-state systems, which we illustrate by deriving the nonlinear optical response of a simple two-band model of an insulating solid and comparing it with prior results established through Keldysh techniques.