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arXiv:2107.05314

Experimental and theoretical characterization of a non-equilibrium steady state of a periodically driven qubit

Yong-Nan Sun, Kimmo Luoma, Zhao‐Di Liu, Jyrki Piilo, Chuan‐Feng Li, Guang‐Can Guo

Published Jul 12, 2021• 0 citations
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Abstract

Periodically driven dynamics of open quantum systems is very interesting because typically non-equilibrium steady state is reached, which is characterized by a non-vanishing current. In this work, we study time discrete and periodically driven dynamics experimentally for a single photon that its coupled to its environment. We develop a comprehensive theory which explains the experimental observations and offers an analytical characterization of the non-equilibrium steady states of the system. We demonstrate that the periodic driving and the properties of the environment can be engineered in such a way that there is asymptotically non-vanishing bidirectional information flow between the open system and the environment.

DOI
10.48550/arxiv.2107.05314

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Periodically driven dynamics of open quantum systems is very interesting because typically non-equilibrium steady state is reached, which is characterized by a non-vanishing current. In this work, we study time discrete and periodically driven dynamics experimentally for a single photon that its coupled to its environment. We develop a comprehensive theory which explains the experimental observations and offers an analytical characterization of the non-equilibrium steady states of the system. We demonstrate that the periodic driving and the properties of the environment can be engineered in such a way that there is asymptotically non-vanishing bidirectional information flow between the open system and the environment.
Periodical driving often leads to nontrivial steady states. In the context of open systems, such states may have nonvanishing currents. In this work, we experimentally study the time-discrete open system dynamics of a two-level system where the system-environment coupling is varied periodically. The two-level system is decoded on the transverse polarization and its environment on the frequency degrees of freedom of a single photon. The dynamics is implemented using linear optics. We find that the periodic driving leads to nonvanishing bidirectional information flow between the open system and the environment. We develop a comprehensive theoretical model which explains the experimental findings.
Periodic driving is used to steer physical systems to unique stationary states or nonequilibrium steady states (NESS), producing enhanced properties inaccessible to nondriven systems. For open quantum systems, characterizing the NESS is challenging and existing results are generally limited to specific types of driving and the Born-Markov approximation. Here, we go beyond these limits by studying a generic periodically driven N -level quantum system interacting with a low-density thermal gas. Exploiting the framework of Floquet scattering theory, we establish general Floquet thermalization conditions constraining the nature of the NESS and the transition rates. Moreover, we examine theoretically the structure of the NESS in the high-temperature limit and find out that the NESS complies, rather surprisingly, with a uniform probability distribution (predicted by the Boltzmann law) for any driving. Numerical calculations illustrate our theoretical elaborations for a simple toy model.
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.
The asymptotic state of a periodically driven many-body quantum system in contact with an environment is investigated. The combined action of the driving and the environment steers the system towards a state being characterized by a time-periodic density operator. To compute this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, find the stroboscopic density operator as its eigen-operator and demonstrate how particle interactions affect properties of the density operator. We illustrate the idea with a periodically rocked open Bose-Hubbard dimer and discuss the relations between the interaction-induced bifurcation transitions in a mean-field dynamics and changes in the characteristics of the quantum many-body state. We argue that Floquet maps can provide insight into the system relaxation towards its asymptotic state and may help to identify the stroboscopic time-independent generator mimicking the action of the original time-dependent one.
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.
Quantum simulation of open quantum systems offers a pathway towards better understanding various non-equilibrium physics that would otherwise be challenging to study. While most open quantum systems studied are modeled as being memory-less (obeying the Markov approximation), real baths generally are influenced by the system-bath interaction, and some systems existing in structured non-Markovian environments can display novel behavior as a result. Here we utilize a trapped ion quantum simulator to simulate a single spin-1/21/21/2 driven-dissipative system with a non-Markovian dissipation channel, and experimentally compare steady-states to those from an analogous Markovian bath. We observe that a non-Markovian dissipative channel can shift the steady-state even for a single qubit, to a regime inaccessible for Markovian dissipation. The techniques used here are compatible with many-body extensions of the model, which can not be simulated efficiently on a classical computer in general. Our work also opens up new possibilities in quantum reservoir engineering beyond the Markovian regime.