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Non-Ergodicity in Open Quantum Systems Through Quantum Feedback

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Non-Ergodicity in Open Quantum Systems Through Quantum Feedback epl draft Non-Ergodicity in open quantum systems through quantum feedback Lewis A. Clark,1;4 Fiona Torzewska,2;4 Ben Maybee3;4 and Almut Beige4 1 Joint Quantum Centre (JQC) Durham-Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle-Upon-Tyne, NE1 7RU, United Kingdom 2 The School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom 3 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, UK, United Kingdom 4 The School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom PACS 42.50.-p { Quantum optics PACS 42.50.Ar { Photon statistics and coherence theory PACS 42.50.Lc { Quantum fluctuations, quantum noise, and quantum jumps Abstract {It is well known that quantum feedback can alter the dynamics of open quantum systems dramatically. In this paper, we show that non-Ergodicity may be induced through quan- tum feedback and resultantly create system dynamics that have lasting dependence on initial conditions. To demonstrate this, we consider an optical cavity inside an instantaneous quantum feedback loop, which can be implemented relatively easily in the laboratory. Non-Ergodic quantum systems are of interest for applications in quantum information processing, quantum metrology and quantum sensing and could potentially aid the design of thermal machines whose efficiency is not limited by the laws of classical thermodynamics. Introduction. { Looking at the mechanical motion applicability of statistical-mechanical methods to quan- of individual particles, it is hard to see why ensembles tum physics [3]. Moreover, it has been shown that open of such particles should ever thermalise. Every particle quantum systems that eventually lose any information moves in a well-defined manner through its available phase about their initial state and whose dynamics result in a space while interacting from time to time, for example unique stationary state are Ergodic [4{6]. For isolated via elastic collisions, with other members of the ensem- many-body quantum systems without an external bath, ble. All equations of motion obey time-reversal symme- the eigenstate thermalisation hypothesis [7{9] has been in- tries and the information about initial phase space loca- troduced to explain when and why they can be described tions is never lost. Although individual particles fail to by equilibrium statistical mechanics. thermalise, macroscopic observables of physical systems Although the dynamics of open quantum systems are arXiv:1611.03716v3 [quant-ph] 30 Jun 2020 usually tend towards constant values, as if the ensemble in general Ergodic, many classical stochastic processes reached a well-defined thermal state. The key to under- are not, particularly in condensed matter systems [10]. standing this seeming contradiction is Ergodicity. Ergodic Hence if classical physics emerges from microscopic quan- systems tend to reach a stationary state, where time aver- tum models, there have to be mechanisms which induce ages are well defined. In such a case, these become equiv- non-Ergodicity. Because of this, Ergodicity breaking is an alent to ensemble averages, which lie at the heart of sta- active area of research in a variety of quantum disciplines tistical physics [1, 2]. [11{14] In this paper, we discuss an example of such a The emergence of Ergodicity and non-Ergodicity in mechanism and show that it can be used to induce non- quantum physics is still the subject of current debate. By Ergodicity in quantum optical systems with spontaneous 1929, von Neumann had proved the Ergodicity of finite- photon emission. sized closed quantum mechanical systems, which evolve For open quantum systems, which evolve with a Lind- according to a Schr¨odinger equation, thereby verifying the bladian master equation, non-Ergodicity seems to require p-1 L. A. Clark et al. ics of an optical cavity with spontaneous photon emission inside an instantaneous quantum feedback loop, as shown in Fig. 1. Ergodicity. { Let us first have a closer look at how to define Ergodicity in open quantum systems. There are a number of ways to define Ergodicity in the literature, with equal validity [25]. In a number of quantum sys- Fig. 1: [Colour online] Schematic view of an optical cavity tems, Ergodicity is considered to be the ability to occupy inside an instantaneous quantum feedback loop. A cavity field the entire phase space. If the dynamics are restricted to is prepared in a coherent state and allowed to freely decay. only a portion of this phase space, this would be a non- Upon detection of a photon, a feedback pulse is applied to Ergodic process. In the following, we instead take a def- the cavity on a very short timescale that can be considered inition which is based on classical statistical mechanics, instantaneous. This pulse has the effect of displacing the cavity motivated by our interest in the emergence of classical field and as such alters its photon emission statistics. physics from quantum processes. Suppose a large number of individual systems generate time-dependent stochastic signals. In the following, we call the dynamics of these the existence of multiple stationary states which typically systems Ergodic when any single, sufficiently long sam- occur only in carefully designed circumstances [4{6] e.g. in ple of the process has the same statistical properties as systems with mutiple symmetries and degeneracies in their the entire process. Let us consider N identically prepared dynamics. Moreover, non-Ergodicity of open quantum systems with stochastic dynamics and denote the ensem- systems can be caused by interaction with non-harmonic ble average of the individual expectation values E (A; T ) thermal environments [11]. In addition, some isolated n of an observable A after a long time T by E(A), which (i.e. closed) many-body quantum systems which evolve ac- implies that cording to the Schr¨odingerequation and can be studied N numerically have been shown to exhibit non-Ergodic dy- 1 X namics after sudden quenches [8, 9]. Very recently, a new E(A) = lim lim En(A; T ) : (1) T !1 N!1 N form of weak Ergodicity breaking with a range of unusual n=1 properties which violate the eigenstate thermalisation hy- Using this notation, the system dynamics are Ergodic pothesis has been shown to occur in constrained quantum when systems and has been interpreted as a manifestation of 1 Z T quantum many-body scars [12, 13]. E(A) = lim dt En(A; t) (2) T !1 T In this paper, we identify another mechanism with the 0 ability to induce non-Ergodic dynamics in open quantum also, for all observables A and for all systems n. In other systems without the need for multiple stationary states. words, system dynamics are Ergodic when ensemble aver- More concretely, we show that open quantum systems can ages and time averages are equivalent for all observables become non-Ergodic in the presence of sufficiently strong and for all possible process realisations [4]. quantum feedback |even when the system possesses only Within this paper, we will have a closer look at the pos- a single well-defined stationary state. This constitutes sible trajectories of open quantum systems with Marko- an interesting observation, since non-Ergodicity and non- vian dynamics. The density matrix ρ of such systems, linear dynamics provide useful tools for processing quan- which represents the quantum state of an ensemble, can tum information [15, 16], quantum metrology and sensing always be described by a Lindbladian master equation [17, 18]. Moreover, it could aid the design of thermal ma- [22, 26]. For a system with only a single decay channel, chines which are not limited by the laws of classical ther- this equation is of the form modynamics [19, 20]. i 1 ρ_ = [H; ρ] + Γ 2LρLy LyL; ρ ; (3) As we shall see below, the individual trajectories of an −¯h 2 − + open quantum system can exhibit different types of dy- where H is the system Hamiltonian, Γ denotes a sponta- namics in the presence of quantum feedback, even when neous decay rate and L is a so-called Lindblad operator. initialised in the same state. For example, some quantum The operator L might simply be an energy annihilation trajectories might reach a fixed point of their dynamics, operator in a system that freely decays. Consequently, while others might evolve further and further away from the time derivative of the expectation value A = Tr(Aρ) it. If the feedback is strong enough, the stationary state of of an observable A equals h i an open quantum system can become repulsive. Instead i 1 of losing any information about the initial state, there can A_ = [A; H] + Γ Ly [A; L] + Ly;A L : (4) be a persistent dependence of ensemble averages in expec- h i −¯h h i 2 tation values on initial conditions. This differs from the Usually the last term in this equation simplifies to the standard use of quantum feedback as a tool to stabilise expectation value of an operator which equals A or is at dynamics [21{24]. To illustrate this, we study the dynam- worst a relatively simple function of A. p-2 Non-Ergodicity in open quantum systems through quantum feedback When adding quantum feedback, the system dynam- As mentioned already above, the dynamics of the den- ics still obey a Lindblad master equation but L is now sity matrix ρI of the cavity in the interaction picture is the product of a unitary operator and the unperturbed given by a master equation. In standard form, this equa- Lindblad operator of the system. When this applies, the tion equals [17] latter terms in the above equation may not simplify, even 1 y y y if they do so in the absence of quantum feedback.
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