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arXiv:1707.05030

Generalized Floquet theory for open quantum systems

C. M. Dai, Hong Li, W. Wang, X. X. Yi

Published Jul 17, 2017• 4 citations
arXivHTMLPDF

Abstract

For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator $Λ(t,0)$ of the system can be factorized as $Λ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}$ with micro-motion operator $\mathcal{D}(t)$ possessing the same period as the external driving, and time-independent operator $\mathcal{L}_{eff}$. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.

DOI
10.48550/arxiv.1707.05030

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For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator Λ(t,0)Λ(t,0)Λ(t,0) of the system can be factorized as Λ(t,0)=D(t)eLefftΛ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}Λ(t,0)=D(t)eLeff​t with micro-motion operator D(t)\mathcal{D}(t)D(t) possessing the same period as the external driving, and time-independent operator Leff\mathcal{L}_{eff}Leff​. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.
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.
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.
Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be nontrivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet generator (describing the stroboscopic evolution) is either again Markovian and of Lindblad type or not. On the other hand, even though in the high-frequency regime a Lindbladian Floquet generator (Floquet Lindbladian) is numerically found to exist, this behavior is, curiously, not correctly reproduced within analytical high-frequency expansions. Here, we demonstrate that a proper Floquet Lindbladian can still be obtained from a high-frequency expansion, when treating the problem in a suitably chosen rotating frame. Within this approach, we can then also describe the transition to a phase at lower driving frequencies, where no Floquet Lindbladian exists, and show that the emerging non-Markovianity of the Floquet generator can entirely be attributed to the micromotion of the open driven system.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
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.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.