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 quantum 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 Schrödinger equation, thereby verifying the bladian master equation, non-Ergodicity seems to require

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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 Schrödingerequation 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 →∞ N→∞ 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 →∞ 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 2LρL† L†L, 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 L† [A, L] + L†,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.

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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 † † † if they do so in the absence of quantum feedback. As a ρ˙I = κ c c, ρI + + ηκ D(β)cρIc D(β) A −2 result, the dynamics of may be governed by an in- † h i + (1 η) κ cρIc , (7) finitely large set of linear differential equations, capable of − effectively generating non-linear dynamics. The accompa- where η is the relevant detector efficiency. Hence the emis- nying non-Ergodicity that we discuss in this paper is thus sion of a photon is registered and triggers a quantum feed- qualitatively different from the non-Ergodicity in other back pulse with probability η. Moreover, 1 η is the proba- open quantum systems [11] and from that in localised bility of an emitted photon leaving the system− undetected. many-body quantum systems with Hamiltonian dynamics The first term in Eq. (7) models the dynamics of the sub- [8,9,12,13]. Moreover, it has been shown already that the ensemble without photon emission, while the second and mechanism which we discuss here can lead to applications the third term model the dynamics of sub-ensembles with in quantum technology, especially in quantum metrology an emission at t. [17, 18]. To predict all possible individual quantum trajectories, An optical cavity with quantum feedback. – To we need to unravel Eq. (7) in a physically meaningful way determine whether a quantum system is Ergodic or not, [27–29]. Doing so, we find that under the condition of no photon emission in (0, t), the initial state ψI(0) of we need to compare the dynamics of its ensemble averages | i with the dynamics of its individual quantum trajectories. the cavity field in the interaction picture evolves with the As an example, we now have a closer look at the experi- non-Hermitian conditional Hamiltonian [17] mental setup which is shown in Fig. 1 and consists of an i † optical cavity inside an instantaneous quantum feedback Hcond = ¯hκ c c . (8) −2 loop. For many of the following calculations, it is advanta- geous to move into an interaction picture with respect to This Hamiltonian reduces the length of the initial state † vector ψI(0) such that the free cavity Hamiltonian H0 = ¯hωcav c c. When doing | i so, we denote state vectors and operators with a subscript 2 i “I.” When using the Schrödingerpicture, state vectors and P0(t) = exp Hcondt ψI(0) (9) −¯h | i operators are subscripted with “S.” Here ωcav is the cavity photon frequency and c denotes the usual cavity photon is the probability for no photon emission in (0, t). More- c, c† annihilation operator with [ ] = 1. over, in the case of an emission at a time t, ψI(t) changes | i In the following, we assume that the detection of a pho- (up to normalisation) into L1 ψI(t) or L2 ψI(t) , respec- | i | i ton triggers a very short-time pulse from a laser resonant tively, with L1 = c and L2 = D(β) c, depending on with the cavity frequency. If this pulse is always taken whether or not a feedback pulse is triggered. Moreover, from the same laser field, it displaces the field inside the the probability density for the emission of a photon at t cavity such that the state vector ψI(t) of the cavity in † | i equals I(t) = κ n(t) where n(t) = c c is the mean cavity the interaction picture changes into D(β) ψI(t) where h i | i photon number. We now have all the tools needed to generate all the possible quantum trajectories of the resonator D(β) = exp β c† β∗ c (5) − in Fig. 1 numerically. Before doing so, let us point out that the quantum state and where β is a complex number that characterises the of the field inside the optical cavity remains coherent at strength of the feedback pulse. For simplicity, we assume all times, if the system was initially prepared in a coher- here that the laser pulse is so strong and short relative ent state α (0) . For example, under the condition of no to the cavity decay rate κ that its effect can be consid- I photon emission| i in a time interval (0, t), the cavity field ered instantaneous. In the Schrödingerpicture, the dis- evolves with the conditional Hamiltonian H and its placement operator is transformed in the standard way by cond (unnormalised state) at time t equals acting the inverse of the time evolution of the free Hamil- tonian H0 upon it. The result of this is that β becomes 0 i ψ (t) = exp Hcondt αI(0) . (10) time dependent and the state vector ψS(t) changes into I | i | i −¯h | i D(βS(t)) ψS(t) with | i To simplify this expression, we denote the state with ex- β (t) = e−iωcavt β . (6) actly n photons inside the resonator by n and take into S | i account that the coherent state αI(0) can be written as In other words, we assume here that every feedback pulse | i ∞ n 1 2 α is generated by a laser pulse with exactly the same (com- − |αI(0)| X I(0) α (0) = e 2 n . (11) I √ plex) Rabi frequency. | i n=0 n! | i

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(a) (c) 100 10 10 ) S ) α S 1 |

α 0 S

0 α | Im ( 1 Im ( 10− 10 − 2 100 10− − 10 0 10 − Re (αS) 100 0 100 1 10 102 1 − Re (αS) t (units of κ− )

(b) (d) 100 10 10 ) S ) α S 1 |

α 0 S

0 α | Im ( 1 Im ( 10− 10 − 2 100 10− − 10 0 10 − Re (αS) 100 0 100 1 10 102 1 − Re (αS) t (units of κ− )

Fig. 2: [Colour online] Evolution of a random sample of ten quantum trajectories subject to the described unravelling in Eqs. (8 - 16) according to a quantum jump simulation of total time t = 10κ−1. We set β(0) = 2, η = 0.5 and have an initial state with α(0) = 2 in (a) and α(0) = −2 in (b). In (a), all but one trajectory diverges from the vacuum state, whereas in (b) only five trajectories diverge. These tracjectories rapidly separate into the whole phase space, as can be seen more clearly in the Schrödinger picture. The non-diverging trajectories asymptotically approach the vacuum state. To see this, we plot |αS(t)| in (c) and (d) for (a) and (b) respectively. Here we see the different times at which various trajectories “escape” from the vacuum, while the remaining ones move increasingly closer to this state.

Applying the time evolution corresponding to the condi- which is the probability to never emit a photon given the tional Hamiltonian Hcond in Eq. (8) to this state is rela- initial state αI(0) . | i tively straightforward. Doing so, we ﬁnd that The reason for the damping of αI(t) in Eq. (13) is that

n not emitting a photon reveals information about the cavity − 1 κt ∞ α 2 state αI(t) which needs to be updated accordingly [27]. 1 2 I(0) e − |αI(0)| X | i αI(t) = e 2 n . (12) If the emission of a photon goes unnoticed, then the state | i √n | i n=0 ! of the cavity remains unchanged, since the coherent states are the eigenstates of L1. However, if a photon triggers a After normalisation, ψ0(t) simpliﬁes to the coherent | I i feedback pulse, the state of the cavity changes from αI(t) state αI(t) with | i | i into 1 − 2 κt αI(t) = e αI(0) . (13) D(β) αI(t) = αI(t) + β (16) | i | i

Moreover, using Eq. (9), we find that the probability P0(t) due to the properties of the displacement operator D in for no photon emission in (0, t) equals Eq. (5). Overall, in the interaction picture, each quantum trajectory is fully described by a series of coherent states −|α (0)|2 1−e−κt P (t) = e I [ ] . (14) αI(t) , which are given by a time dependent complex 0 | i function αI(t). The same applies in the Schrödingerpic- In the limit of t , this equation simplifies to the ture, where each quantum trajectory is fully described by → ∞ probability P0( ), the time dependent complex function α (t) with α (t) = ∞ S S exp( iωcavt) αS(t) and similarly the displacement opera- 2 −|αS(0)| − P0( ) = e , (15) tor is transformed to D(βS(t)). Fig. 2 shows αS(t) for ∞

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100 100

β 0 ϕ 0 ) ) 80 κ 0.2 κ 10 π/5 0.4 2π/5 0.6 60 3π/5 1 0.8 4π/5 1 40 π

0.1 Intensity (units of Intensity (units of 20

0.01 0 0 2 4 6 8 10 0 0.02 0.04 0.06 0.08 0.1 1 1 t (Units of κ− ) t (Units of κ− )

Fig. 3: [Colour online] Time dependence of the photon emission Fig. 4: [Colour online] Time dependence of the photon emission intensity, defined to be the average number of emissions per intensity for different initial states |αI(0)i here with αI(0) = unit time, for different feedback parameters β. As in previous 2eiϕ and varying phase ϕ. All other parameters are as in Fig. 3 figures, we assume αI(0) = 2 and η = 0.5. The figure is the and results are again generated via a quantum jump simulation result of a quantum jump simulation, which averages over 106 averaged over 106 quantum trajectories. This figure illustrates individual quantum trajectories. For relatively small values that there is a strong dependence of the ensemble averages on of β, I(t) tends towards zero. However, as β increases, the the initial state of the resonator. dynamics of the cavity change and the mean number of photons inside the resonator continues to grow in time. where P (α, t) denotes the probability to find a single sub- system of a large ensemble of optical cavities with the random samples of individual quantum trajectories which same initial state at time t prepared in α . Substituting | i have been generated using standard quantum Monte Carlo this form of the density matrix into the master equation simulations [27–29]. Figs. 2(a)–(b) show the evolution in Eq. (7) and using the properties of coherent states, we of ten randomly chosen quantum trajectories, prepared find that in initial states with α(0) = 2 and α(0) = 2 respec- Z − h 2 tively. All other conditions are the same in both setups. ρ˙I(t) = κ dα P (α, t) η α α + β α + β CI | | | ih | In Figs. 2(c)–(d), we plot for αs(t) to show the behaviour | | 1 of the magnitude of the state more clearly. +(1 η) α 2 α α α c† α α − | | | ih | − 2 | ih | Non-Ergodicity. – A closer look at Fig. 2 shows that 1 i α∗ α α c . (18) there are two very distinct types of dynamics. Some quan- −2 | ih | tum trajectories persistently show an overall increase in the mean number of photons inside the cavity and move Looking at the various matrix elements of this operator, further and further away from the vacuum state. In other we find that the cavity possesses a unique stationary state cases, the mean number of photons inside the resonator ρss withρ ˙I(t) = 0. As one would expect, this state is the remains relatively small, decreases further and the sys- vacuum state of the resonator, tem is extremely likely to eventually reach the vacuum ρss = 0 0 . (19) state. This is not surprising, as photon emissions at a | ih | relatively high rate attract more feedback pulses, thereby When reaching this state the time evolution of the res- further increasing the mean number of photons inside the onator comes to a hold. resonator. In the absence of registered emissions and feed- The above-described presence of qualitatively differ- back pulses, the no-photon time evolution continuously ent types of quantum trajectories strongly suggests non- reduces the field amplitude (c.f. Eq. (13)). As we shall Ergodicity. To show that for sufficiently strong quantum see below, the vacuum is the only stationary state of the feedback it is not always possible to deduce the statistical experimental setup in Fig. 1. properties of the cavity from a single run of the experi- Since the cavity field remains always in a coherent state, ment, we now have a closer look at Eq. (15). This equation its density matrix ρI(t) in the interaction picture always shows the probability P0( ) for an optical cavity initially represents a statistical mixture of coherent states. Hence ∞ prepared in the coherent state αI(0) with αI(0) > 0 we can assume without restrictions that | i | | to never emit a photon is always positive (P0( ) > 0) ∞ Z and always smaller than one (P0( ) < 1). The larger ρI(t) = dα P (α, t) α α , (17) 2 ∞ αI(0) , i.e. the larger the initial photon number, the more CI | ih | | |

p-5 L. A. Clark et al. likely it is for the system to experience photon emission. averaged over an ensemble of optical cavities on the com- The fact that P0( ) is always less than one shows that mon initial state αS(0) . In contrast to many other open there is always a non-zero∞ probability that the cavity field quantum systems,| informationi about the initial state may never reaches its vacuum state. The Ergodicity condition persist and may never get lost. This behaviour is illus- of Eqs. (1) and (2) being always the same resultantly does trated in Fig. 4 which shows the time-dependence of the not hold for all possible quantum trajectories. cavity photon emission rate I(t) for different initial states iϕ To show that for sufficiently strong quantum feedback αI(0) with αI(0) = αI(0) e . Although αI(0) and the there is indeed always at least one possible quantum tra- feedback| i parameter β| are| kept the same,| I(t)| does not jectory for which the cavity field never reaches the vacuum converge to a single value. On the contrary, the distance state, we now have a closer look at the dynamics of the between curves that correspond to different values of ϕ mean cavity photon number n(t). Using the master equa- increases in time. This surprising fact has been shown tion in Eq. (7), one can show that the time derivative of already to be useful, for example, in quantum metrology the expectation value of any observable A averaged over protocols where it can be used to overcome the standard an ensemble of systems equals quantum limit [17, 18]. We further note that the generation of this behaviour is 1 D E ˙ † † † dependent on the existance of what effectively constitutes A = κ A, c c + + ηκ c D(β) AD(β)c h i −2 multiple decay channels, i.e. those with feedback and those +(1 η)κ c†Ac . (20) − without. In the case where the channel involving feedback † becomes infintesimally weak (β 0) or is switched off Setting A = c c, this differential equation can be used to (η 0), the system returns to the≈ basic Ergodic case of calculate the mean number of photons n(t) inside the cav- ≈ † free decay towards the stationary state of the vacuum. ity. Using Eq. (16) and the commutator relation [c, c ] = 1 On the other hand, introducing further decay channels by one can show that allowing for multiple feedback protocols could lead to even 2 2 2 more complex behaviour. n˙ (t) = κ 1 η αI(t) + β(t) αI(t) αI(t) , − − | | − | | | | (21) Conclusions. – This paper identifies instantaneous quantum feedback as a mechanism to induce non- if the state of the cavity equals αI(t) . In the absence Ergodicity in open quantum systems. This is achieved | i of any feedback, i.e. for η = 0, this equation simplifies to by altering the Lindblad operators L of the correspond- the simple linear differential equationn ˙ (t) = κ n(t), as ing master equation (c.f. Eq. (4)). As an example, this − it should. However, in the presence of sufficiently strong paper studies an optical cavity inside an instantaneous feedback, the η-term dominates the dynamics of n(t) and quantum feedback loop. This system, which remains al- the mean cavity photon number evolves in a non-linear ways in a coherent state, can be analysed relatively easily fashion, as illustrated in Fig. 3. despite its non-linear dynamics. In the presence of suf- Suppose the cavity is initially in its vacuum state and ficiently strong feedback, the only stationary state of an a small perturbation moves the resonator into a coherent ensemble of optical cavities with quantum feedback can 2 2 state αI with αI β(t) . When this applies, Eq. (21) become repulsive. In general, it is not possible to deduce | i | | | | simplifies to a very good approximation to its dynamics of ensemble averages from individual quantum trajectories. Moreover, it is shown that the dynamics 2 2 n˙ (t) = κ 1 η β(t) αI . (22) − − | | | | of ensemble averages can depend strongly on initial states. Since quantum feedback is relatively easy to implement in with β(t) 2 = β 2. In the presence of sufficiently strong | | | | the laboratory [30], these properties are expected to find feedback, i.e. when useful applications for the processing of quantum infor- 2 mation, for example, when designing quantum versions of η β > 1 , (23) | | Hidden Markov Models [15,16]. The non-linear dynamics this time derivative is always positive. When this condi- of the experimental setup in Fig. 1 has already been used tion applies, the mean photon number n(t) always grows to design novel schemes for quantum-enhanced metrol- in time (cf. Fig. 3) and the vacuum state becomes a repul- ogy [17, 18]. Moreover, quantum feedback-induced non- sive fixed point of the system dynamics. Qualitatively, the Ergodicity might provide an interesting tool for the design behaviour of the resonator field resembles the behaviour of thermal machines whose efficiency is not restricted by of classical systems with feedback. This behaviour can be the laws of classical thermodynamics [19, 20]. seen in Fig. 3, where there is a transition from decay to growth of the photon emission intensity as we transition ∗ ∗ ∗ to higher feedback strengths. Given its non-Ergodicity and the instability of its sta- LAC acknowledges support from the United King- tionary state, it is not surprising to also observe a strong dom Engineering and Physical Sciences Research Coun- dependence of the dynamics of cavity expectation values cil (EPSRC) Award No. 1367108 and also Grant

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No. EP/P034012/1. Moreover, FT and BM acknowl- [18] CLARK, L. A., STOKES, A. and BEIGE, A., Phys. Rev. edge support from a University of Leeds UGRL scholar- A 99 (2019) 022102. ship. In addition, AB was supported by funding from the [19] MEHTA, P. and POLKOVNIKOV, A., Ann. Phys. 332 EPSRC Oxford Quantum Technology Hub NQIT (Grant (2012) 110. No. EP/M013243/1). AB would like to thank Alex Little [20] JAYNES, E. T., Papers on Probability, Statistics and Statistical Physics (Springer Netherlands) 1989. for stimulating discussions. Statement of compliance with [21] CARVALHO, A. R. R. and HOPE, J. J., Phys. Rev. A EPSRC policy framework on research data: This publica- 76 (2007) 010301. tion is theoretical work that does not require supporting [22] WISEMAN, H. M. and MILBURN, G. J., Quantum research data. Measurement and Control (Cambridge University Press) 2010. [23] SCHIRMER, S. G. and WANG, X., Phys. Rev. A 81 REFERENCES (2010) 062306. [24] STEVENSON, R. N., HOPE, J. J. and CARVALHO, A. [1] BIRKHOFF, G. D., Proc. Natl. Acad. Sci. USA 17 R. R., Phys. Rev. A 84 (2011) 022332. (1931) 656. [25] ZELDITCH, S., in Encyclopedia of Mathematical [2] VON NEUMANN, J., Proc. Natl. Acad. Sci. USA 18 Physics, editted by FRANC¸OISE, J.-P., NABER, G. L. (1932) 70. and TSUN, T. S. (Academic Press, Oxford) 2006, pp. [3] VON NEUMANN, J., Z. Phys. 57 (1929) 30. 183-196. [4] VAN KAMPEN, N. G., Stochastic processes in Physics [26] LINDBLAD, G., Comm. Math. Phys. 48 (1976) 119. and Chemistry (Elsevier Science B.V., Amsterdam) [27] HEGERFELDT, G. C., Phys. Rev. A 47 (1993) 449. 1992. [28] DALIBARD, J., CASTIN, Y. and MØLMER, K., Phys. [5] MANDEL, L. and WOLF, E., Optical coherence and Rev. Lett. 68 (1992) 580. quantum optics (Cambridge University Press, Cam- [29] CARMICHAEL, H., An Open Systems Approach to bridge) 1995. Quantum Optics, Lecture Notes in Physics, Vol. 18 [6] CRESSER, J. D., Ergodicity of Quantum Trajectory De- (Springer-Verlag Berlin) 1993. tection Records in Directions in Quantum Optics by H. [30] REINDER, J. E., SMITH, W. P., OROZCO, L. A., J. Carmichael, R. J. Glauber and M. O. Scully (Editors), WISEMAN, H. M. and GAMBETTA, J., Phys. Rev. A Lecture Notes in Physics (Springer-Verlag, Berlin) 2001, 70 (2004) 023819. p. 358. [7] SREDNICKI, M., Phys. Rev. E 50 (1994) 888. [8] POLKOVNIKOV, A., SENGUPTA, K., SILVA, A. and VENGALATTORE, M., Rev. Mod. Phys. 83 (2011) 863. [9] D’ALESSIO, L., KAFRI, Y., POLKOVNIKOV, A. and RIGOL, M., Adv. Phys. 65 (2016) 239. [10] PALMER, R. G., Advances in Physics 31 (1982) 669. [11] PARRA-MURILLO, C. A., BRAMBERGER, M., HU- BIG, C. and DE VEGA, I., Open quantum systems in thermal non-ergodic environments, submitted (2020); arXiv:1910.10496. [12] TURNER, C. J., MICHAILIDIS, A. A., ABANIN, D. A., SERBYN, M. and PAPIC, Z., Nat. Phys. 14 (2018) 745. [13] BULL, K., MARTIN, I. and PAPIC, Z., Phys. Rev. Lett. 123 (2019) 030601. [14] MATIN, S., PUN, C.-K., GOULD, H., and KLEIN, W., Phys. Rev. E 101 (2020) 022103. [15] CLARK, L. A., HUANG, W., BARLOW, T. M. and BEIGE, A., Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback, in ISCS 2014: Interdisciplinary Symposium on Com- plex Systems, Emergence, Complexity and Computa- tion 14, (Springer Verlag, Heidelberg) 2015, p. 143; arXiv:1406.5847. [16] ADHIKARY, S., SRINIVASAN, S., GORDON, G. and BOOTS, B., Expressiveness and Learning of Hidden Quantum Markov Models, Proceedings of the 23rdInter- national Conference on Artiﬁcial Intelligence and Statis- tics (AISTATS) 2020 (Palermo, Italy) PMLR: Vol. 108, 2020; arXiv:1912.02098. [17] CLARK, L. A., STOKES, A. and BEIGE, A., Phys. Rev. A 94 (2016) 023840.

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