Les Houches Quantum Optomechanics School 2015 Lecture Notes

Transcription

1 Les Houches Quantum Optomechanics School 2015 Lecture Notes Pierre Meystre Department of Physics and College of Optical Sciences, University of Arizona 1

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3 Preface This is a set of rough notes for my lectures at the Les Houches 2015 Summer School on Optomechanics. They are incomplete, suffer from notational and other inconsistencies, and are in need of substantial improving and proofreading. Also, the current list of references is incomplete and inadequate at this point. This document is not meant for distribution past the small group of Summer School attendees in its present preliminary form.

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5 Contents 1 Quantum optomechanics, thermodynamics, and heat engines Introduction Quantum work Quantum trajectories and continuous measurements Quantum heat engines Photon-phonon polaritons A quantum optomechanical heat engine Quantum fluctuations Heat pump based on polariton fluid Outlook 38 References 41

6 1 Quantum optomechanics, thermodynamics, and heat engines 1.1 Introduction Cavity optomechanics (Aspelmeyer et al. 2012, Meystre 2013, Aspelmeyer et al. 2014) provides a universal tool to achieve quantum optical control over the mechanical motion - or conversely mechanical control over the optical or microwave field - in devices spanning this vast parameter space. It covers a wide spectrum of systems from nanometer-sized devices, comprising as little as 10 7 atoms, to nano- and micromechanical structures and to the centimeter-sized mirrors used in gravitational wave detectors, weighing up to several kilogram. Because their mechanical elements can be functionalized by coupling their motion to other physical systems, including for example photons, spin(s), charges, atomic systems, or qubits, they offer a rich potential both for applications and in basic science. From the point of view of applications, optomechanical techniques in both the optical and microwave regimes readily provide motion and force detection near the standard quantum limit. The force sensitivity of these systems now exceeds N/ Hz, an astoundingly weak force that corresponds to the gravitational attraction between a person in Los Angeles and another in New York. Micromechanical oscillators also offer a route to the development of new tests of of the foundation of physics, including tests of quantum theory at unprecedented size and mass scales. For instance, spatial quantum superpositions of massive objects could be used to probe various theories of decoherence, i.e. the transition from quantum behavior to classical behavior. It is widely accepted that quantum mechanics is the fundamental theory of nature, yet in everyday life we don t observe the remarkable quantum effects that we can achieve with small ensembles of atoms or with photons under exquisitely controlled conditions. We cannot make a car be in two places at the same time, or, in the famous example of Schrödinger, we cannot have a cat that is both alive and dead at the same time. Even large quantum condensates such as a cup of superfluid helium do not display macroscopic superposition states and entanglements. The everyday world seems to be most definitely governed by the laws of classical physics, not by quantum mechanics. This is extremely puzzling, because if the quantum mechanical description of nature is more fundamental than its classical description, then quantum mechanics should govern not just the microscopic world, but the macroscopic world as well.

7 2 Quantum optomechanics, thermodynamics, and heat engines Romero-Isart et al. (2011) have recently considered a method to prepare and verify spatial quantum superpositions of a nanometer-sized object separated by distances comparable to its size. It is hoped that such experiments will eventually be able to operate in a parameter regime where it will be possible to test various proposed mechanisms beyond quantum mechanics that have been advanced to explain the washing out of quantum properties in macroscopic objects. It will be exciting indeed to see these proposed experiments being realized and start answering questions that have surrounded quantum mechanics and its interpretation since its early days, nearly 100 years ago. These lectures address another area which quantum optomechanics is also in a position to contribute significantly, quantum thermodynamics. It can loosely be defined as the study of thermodynamics in regimes where quantum mechanical noise coexists with thermal noise. It some ways it is an extension of stochastic thermodynamics, which deals with the consistency of macroscopic thermodynamic quantities such as work, heat, and entropy with the random, erratic motion of small systems. Small classical systems are not within he scope of macroscopic thermodynamics, because they are dominated by fluctuations one example being a colloidal particle immersed in a viscous fluid or a laser trap, other examples include biomolecules such as RNA or DNA manipulated by optical tweezers. In these cases, much progress has recently be made toward understanding how thermodynamic quantities such as work, heat and entropy can be defined along individual stochastic trajectories. There is now a first-law like energy balance equation, introduced by Sekimoto (2007), and also a definition of entropy along single trajectories. Quantum thermodynamics presents however a number of additional challenges and open questions that remain largely unresolved at this time. They include proper definitions of work and heat, especially for open quantum systems, and the backaction of quantum measurements on thermodynamic quantities, one example being the efficiency of heat engines. Further fascinating questions include whether quantum fluctuations can be exploited to provide advantages over classical fluctuations, e.g. in engine efficiency, using for example squeezed vacua, the roles of quantum coherence and entanglement, as well as fundamental issues related to thermalization in closed systems. The definition of thermodynamic quantities in the quantum context presents conceptual challenges (Allahverdyan and Nieuwenhuizen 2005, Esposito and Mukamel 2006, Boukobza and Tannor 2006, Talkner et al. 2007). Much attention has been devoted to the proper definition and the quantum statistical properties of quantities such as heat, work and entropy (Fusco et al. 2014, Dorner et al. 2012, Mascarenhas et al. 2014, Appolaro et al. 2014, Campisi et al. 2013, Joshi and Campisi 2013, Smacchia and Silva 2013, Saira et al. 2012, Batalhão et al. 2014, Jarzynsky 2015, Hänggi and Talkner 2015, An et al. 2015, Suomela et al. 2015, Salmilehto et al. 2014, Frenzel et al ) In closed quantum systems work may be defined in terms of a two-time measurement scheme (Esposito et al. 2009, Campisi et al. 2011, Talkner et al. 2008, Mukamel 2003) or, in a recently proposed alternative approach, of a single projective measurement (Roncaglia et al. 2014). However the situation is less clear for open quantum systems, where there are still unsettled questions regarding the definition of and experimental

8 Quantum work 3 implementation of measurements of work and heat (Esposito et al. 2009, Campisi et al , Ritort 2009 ) due to the lack of energy conservation in the reservoir(s). In this context quantum stochastic thermodynamics (Horowitz 2012, Esposito et al. 2007, Esposito et al. 2010), like its classical counterpart (Seifert 2005, Schmiedl et al. 2007, Schmiedl and Seifert 2007, Seifert 2008), offers an interesting framework to discuss thermodynamic properties and simulate numerically the system behavior. In these lecture notes I consider the specific example an an optomechanical quantum heat engine (QHE) to address some of these issues, using the specific example of a quantum optomechanical heat engine as an example. Section 2 sets the stage by briefly reviewing some of the outstanding questions underlying a proper definition and issues associated with the measurement of quantum work. As we shall see, and not surprisingly, quantum measurements are a key element of that discussion. Section 3 then outlines the key steps leading to the formulation of continuous measurements in terms of stochastic Schrödinger equations. Turning specifically to QHE, Section 4 reviews their main characteristics, comparing and contrasting thermodynamic processes and engine cycles in the classical and the quantum regimes. The working substance of the heat engine that we consider is a photon-phonon polariton fluid. We review the main properties of these quasiparticles in Section 5 before introducing the specific optomechanical QHE system in Section 6. Section 7 then turns to the quantum properties of the engine, including quantum fluctuations in the work that it produces, and discusses two explicit continuous measurement schemes. Section 8 briefly expands the discussion to a related system, a polariton heat pump, illustrating the general advantages of polaritonic working substances in quantum thermodynamics. Finally, section 9 is a conclusion and outlook. 1.2 Quantum work An important question in quantum thermodynamics is the definition and measurement of work and its quantum characteristics, including its fluctuations. Work is not an operator, its value depends on the protocol followed to go from one point to the other in state space. As such, it is characterized by a process, not only by the instantaneous state of the system. An additional open question is to characterize the impact of measurement backaction on the thermodynamic characteristics of the system. For isolated quantum systems work can be determined unambiguously via a two-time measurement process (Esposito et al. 2009, Campisi et al. 2011), but this approach is problematic for open quantum systems since it would require additional measurements on the reservoirs. In order to circumvent this issues to the extent possible, we will adopt in these notes an experimentally realizable operational approach based on stochastic quantum thermodynamics (Horowitz 2012). However, to set the stage for this discussion it is useful to first review some important concepts and some of the key ideas being currently pursued to tackle this problem. In classical thermodynamics, the expression of the first law is du = dq + dw, (1.1)

9 4 Quantum optomechanics, thermodynamics, and heat engines where U, Q, and W, are energy, heat, and work, respectively. This law states that the energy exchanged by a system in a transformation is divided between work W and heat Q. To obtain a quantum version of this expression, we express the average energy in terms of the eigenstates of the Hamiltonian H as U = H = i p i E i, (1.2) where E i is the energy of the eigenstate i, with corresponding occupation probability p i. An infinitesimal change in energy is then given by du = i dp i E i + i p i de i. (1.3) One can identify the first term on the right-hand side of this equation as the infinitesimal heat transferred, and the second as the infinitesimal work performed (?), dq = i dw = i dp i E i, (1.4) p i de i. (1.5) The heat transferred to or from the quantum system corresponds to a change in the populations p i without change of the energy eigenvalues, while the work done on or by that system corresponds to a redistribution of the energy eigenvalues. These quantum expressions of the infinitesimal average heat and work variations are consistent with their definitions in classical thermodynamics and statistical physics. That is, the heat exchange results in a change in the statistical distribution of the microstates of different energies while the work is a change in the energy structure of the system. But how do we measure the work? A key issue is that work is that it is not an observable, and its determination requires need two-time measurements. [Note however the recent paper by Roncaglia et al. (2014), who considered work measurement as a generalized quantum measurement. To address this question Solinas and coworkers (Solinas et al. 2013) consider work in analogy to the classical case as an integral over the injected power during the evolution of the system. Work is then defined as the energy difference between the initial and final states of the system plus the heat released to the heat bath. To see how this works we consider a system described by a Hamiltonian Ĥ(λ) that can be varied by a control parameter λ(t) and is coupled to reservoir by some Liouvillian L( ˆρ), ˆρ being its reduced density operator. The injected power operator and average power are ˆP = dĥ dt λ = dĥ (1.6) dt and ˆP = dĥ dt (1.7)

10 so that W = But Ĥ = Tr(ˆρĤ), so that d Ĥ dt = Tr For a dissipative system we have so that It follows that or tf 0 ˆP (t ) dt = tf 0 Quantum work 5 dt d Ĥ dt. (1.8) ( ) ( ) dˆρ ˆρ dt Ĥ + ˆρdĤ = Tr dt t Ĥ + ˆP. (1.9) ˆρ t = ī [Ĥ, ˆρ] + L(ˆρ) (1.10) h ( ) ( ) dˆρ i ( ) Tr dt Ĥ = Tr [Ĥ, ˆρ]Ĥ + L(ˆρ)Ĥ = Tr L(ˆρ)Ĥ. (1.11) h W = tf 0 dt [ W = Ĥ(t f ) Ĥ(0) tf ] dĥ dt + Tr(L)(ˆρ)Ĥ) so that one can identify the average work W and heat Q as 0 0 (1.12) ( ) Tr L(ˆρ)Ĥ, (1.13) W Ĥ(t f ) Ĥ(0), (1.14) tf ( ) Q Tr L(ˆρ)Ĥ. (1.15) This shows that the average heat is associated with energy exchange with the environment during the evolution process, and depends on the particular realization of the evolution trajectory. Using quantum trajectory methods, in analogy with the approach used in stochastic thermodynamics, it is possible to use such two-time measurement schemes to evaluate the fluctuations of these quantities as well. To illustrate the idea behind this approach more concretely the next subsection reviews the main ingredient of the Monte Carlo wave function method. Its application to two specific examples considered by Horowitz (2012) and by Hekking and Pekola (2013), the first one of a driven harmonic oscillator coupled to a reservoir of two state systems, and the second example a two-state system coupled to a reservoir of harmonic oscillators, is then outlined in section Monte Carlo Wave Function Method The Monte Carlo wave functions method provides a tool to determine the evolution of an open system to a thermal reservoir. It proceeds by generating a large series of wave functions representative of the dynamics of single realizations of the system, and carries out a statistical average in the end to obtain its density operator evolution. But

11 6 Quantum optomechanics, thermodynamics, and heat engines in contrast to the situation for closed systems, these wave functions are intrinsically stochastic, resulting from the combination of a nonhermitian Schrödinger-like evolution and random quantum jumps. Because that method may not be completely familiar, and also informs the discussion of weak measurements of the following sections, we briefly review its main ingredients before returning to the specific problem at hand. For more details see e.g. Meystre and Sargent (2007). The general form of the master equation describing the evolution of the reduced density operator for a system coupled to a general markovian reservoir is ˆρ = ī h [Ĥs, ˆρ] + L[ˆρ] (1.16) where Ĥs is the system Hamiltonian, and the Liouvillian L[ˆρ] that accounts for dissipation is of the general Lindblad form L[ˆρ] = 1 2 (Ĉ i Ĉi ˆρ + ˆρĈ i Ĉi) + Ĉ i ˆρĈ i, (1.17) i i where the Ĉi s are system operators. The explicit form of these operators depends on the coupling of the system to its thermal reservoir(s). For a two-state system coupled linearly to a bath of harmonic oscillators, for example, they would be ˆσ + and ˆσ. The starting point of the Monte Carlo wave function method is to rewrite the master equation (1.17) as dˆρ dt = ī ) (Ĥeff ˆρ h ˆρĤ eff + L jump ˆρ, (1.18) where we have introduced the effective non-hermitian Hamiltonian Ĥ eff Ĥs i h Ĉ i 2 Ĉi (1.19) and the jump Liouvillian L jump [ˆρ] i i Ĉ i ˆρĈ i. (1.20) One can justify identifying L jump [ˆρ] with quantum jumps by returning to a state vectors description of the problem instead of density operators, expressing the density operator was introduced as a statistical mixture of state vectors, ˆρ = ψ P ψ ψ ψ, (1.21) the summation over ψ resulting from a classical average over the various states that the system can occupy with probability P ψ. Introducing that form of ˆρ into the master equation (1.18) gives [ P ψ ψ ψ + ψ ψ = ī ) (Ĥeff ψ ψ h ψ ψ Ĥ eff + ] Ĉ i ψ ψ Ĉ i. ψ i (1.22) If we restrict ourselves to a single representative state vector ψ in the mixture, we recognize that the first term on the right-hand side of this equation can be simply

12 Quantum work 7 interpreted as resulting from the Schrödinger-like evolution of ψ under the influence of the nonhermitian Hamiltonian H eff. Things are more tricky for the second term, however. This is clearly not a Schrödinger-like term, Rather, it seems to result from a discontinuous evolution whereby the state ψ is projected, or jumps, onto one of the possible states ψ ψ i = Ĉi ψ. This is the observation that motivates calling L jump [ˆρ] a quantum jump Liouvillian. The decomposition of the evolution of the representative state vector ψ into a Schrödinger-like part and a quantum jumps part suggests an elegant way to solve master equations by carrying out an ensemble average over the evolution of a large number of such state vectors. It proceeds numerically by first selecting an arbitrary state vector ψ out of the initial ensemble, and evolving it for a short time δt under the influence of H eff only. For sufficiently small time intervals, this gives ψ(t + δt) = ( 1 iĥeffδt h ) ψ(t). (1.23) Because of the nonhermitian nature of Ĥeff, though, ψ(t + δt) is not normalized. The square of its norm is ( ) ( ) ψ(t + δt) ψ(t + δt) = ψ(t) 1 + iĥ eff δt 1 iĥeffδt ψ(t) = 1 δp, (1.24) h h where to lowest order in δt δp = ī h δt ψ(t) Ĥeff Ĥ eff ψ(t) = δt i ψ(t) Ĉ i Ĉi ψ(t) i δp i. (1.25) Of course the full master equation evolution does preserve the norm. The lack of norm preservation that we now encounter results from the fact that we have so far ignored the effects of L jump [ˆρ]. The missing norm δ p is contained in the states ψ i resulting from the jumps part of the evolution. We interpret this as a result of the fact that L jump [ˆρ] projects the system into the state ψ i = Ĉi ψ with probability δp i such that i δp i = δp. Hence the next step of a Monte Carlo simulation consists in deciding whether a jump occurred or not. Numerically, this is achieved by choosing a uniform random variate 0 r 1. If its value is larger than δp, no jump is said to have occurred, and the next integration step proceeds from the normalized state vector ψ(t + δt) = ψ(t + δt) ψ(t + δt). (1.26) If on the other hand r δp, a jump is said to have occurred. The state vector ψ(t+δt) is then projected to the normalized new state ψ(t + δt) = Ĉi ψ(t) Ĉi ψ(t) = δt Ĉ i ψ(t) (1.27) δp i with probability δp i /δp, and this state is taken as the initial condition for the next integration step. The procedure is then repeated for as many iterations as desired,

13 8 Quantum optomechanics, thermodynamics, and heat engines and yields a possible time evolution of the initial state vector ψ, sometimes called a quantum trajectory. Clearly, the random nature of the jumps implies that a different trajectory will be obtained in another simulation from the same initial state. In some cases it is possible to interpret the reservoir to which the small system is coupled as a measurement apparatus, and the single quantum trajectories may be interpreted as typical of a single sequence of measurements on the system. This is the idea underlying the approaches of Horowitz (2012) and Hekking and Pekola (2013) to the quantum work measurement problem Examples 1. Damped forced harmonic oscillator. In this example, considered by Horowitz (2012), a simple harmonic oscillator parametrized by an externally controlled force f = 2/(mω)νt and described by the Hamiltonian Ĥ(f) = ˆp2 2m mω2ˆx 2 mω 2 f ˆx (1.28) with mass m, position ˆx and momentum ˆp, interacts with a thermal reservoir of twolevel atoms. The work on the oscillator is measured at the beginning and the end of the protocol by making projective measurements on the state of the atoms.the Hamiltonian of the single particles is Ĥ A = hωˆσ +ˆσ (1.29) where ˆσ = g e, and ˆσ + = e g, with g the ground state and e the excited state. They are individually coupled to the oscillator through the interaction Hamiltonian V (f t ) = hλ(ā t ˆσ + ā tˆσ + ) (1.30) where ā t = mω 2 ( ˆx f + iˆp ) + iν/ω. (1.31) mω and the index t labels the raising operator at time t when the force has the value f t. Before interacting with the oscillator, each atom is prepared either in the ground state g or the excited state e, with probabilities r g = r e = 1, 1 + e β hω (1.32) e β hω, 1 + e β hω (1.33) with β = k B T. The interaction between the atoms and the oscillator is assumed to be weak and short enough that it can be described by first-oder perturbation theory. The

14 Quantum work 9 probability to observe a jump of one of the atoms from g to e after they interact with the oscillator for a short time dt about t is then given by where p ge = h 2 λ 2 r g ā tā t t dt gr g ā t(t)ā t t dt (1.34) g = h 2 λ 2 δt (1.35) is the jump rate, δt < dt being the interaction time of individual atom with the oscillator, and the subscipt t indicating that the expectation value is taken at time t. Similarly the probability to oberve a jump of an atom from state e to g is p 10 = gr 1 â(t)â (t) dt, (1.36) and the probability of observing no transition is 1 p ge p eg. Since the probability to observe a jump scales at dt the process is Poissonian, and we can introduce two Poisson increments dn + and dn corresponding to an atom jumping up or down, and the oscillator jumping correspondingly down or up. The increments are a sequence of random numbers that are either 1 or 0, so that with ensemble expectation values (dn + ) 2 = dn + ; (dn ) 2 = dn (1.37) E[dN + ] = gr g â â t, (1.38) E[dN ] = gr e ââ t. (1.39) This permits to describe the evolution of the system in terms of the stochastic Monto Carlo Schrödinger equation, as oultined in section Heat is the energy exchanged with the thermal reservoir of two-level atoms, hence the heat absorbed by the oscillator is identified as the energy released by the atoms. In particular each time an atom jumps for its ground state g to its excited state e for a single trajectory ψ the atom has absorbed hω of energy, and in the reverse case it has released that same amount of energy. Therefore for a given trajectory the heat increment absorbed by the oscillator between t and t + dt is and dq t [ψ] = t 0 Q t [ψ] = hω = hω[dn t dn + t ], (1.40) t 0 [dn t dn + t ]. (1.41) Similarly, the internal energy U t [ψ] = ψ t H(f t ) ψ t and is found to be, (Horowitz (2012)), du t [ψ] = dtf t ψ t f H(f t ) ψ t [ ] 1 dtgr g ψ t 2 {H(f t)ā tā t } H(f t ) ā tā t t ψ t

15 10 Quantum optomechanics, thermodynamics, and heat engines [ ] 1 dtgr e ψ t 2 {H(f t)ā t ā t} H(f t ) ā t ā t t ψ t (1.42) [ ] [ ] + dnt ā t H(f t )ā t t ā t ā H(f t ) t + dn t + ā th(f t )ā t t t t ā H(f t ) t, tā t t where {Â, ˆB} =  ˆB + ˆB is for the anticommutator of  and ˆB. Finally, the work performed during the interval for t to t + dt is dw t [ψ] = du t [ψ] dq t [ψ], (1.43) that is, is is the change in energy not accounted by the heat. 2. Driven damped two-state system. In this second example, due to Hekking and Pekola (2013), a driven two-state system described by the Hamiltonian Ĥ s = hω 0ˆσ z /2 + λ(t)(ˆσ + + ˆσ ), (1.44) where ˆσ z = e e g g and λ(t) is a control parameter, is linearly to a bath of harmonic oscillators by the interaction Hamiltonian Ĥ c = µ (c µˆσ +ˆbµ + h.c.). (1.45) The annihilation and creation operators ˆb µ and ˆb µ of the mode of frequency ω µ satisfy the familiar bosonic commutation relations [ˆb µ, ˆb µ] = δ µ,µ, and c µ are coupling constants. Hekking and Pekola propose to evaluate the work by making two measurements, one before and one after the driving period. They then use as Horowitz (2012) a Monte Carlo wave function approach to simulate a number of realizations of the protocol, an approach that allows them to obtain the quantum statistics of the work distribution in the system. Assuming that at time t the two-state system is in the arbitrary superposition state ψ(t) = [a(t) g + b(t) e ] 0, (1.46) the probabilities Γ and Γ that it either absorbs or emits a photon into the bath in a time t + t short enough that at most one photon is exchanged with the bath are Γ = 2π h Γ = 2π h (n µ + 1) c µ 2 δ(ω 0 ω µ ), µ n µ c µ 2 δ(ω 0 ω µ ), (1.47) µ where n µ = [exp hω µ /k B T ) 1] 1. In case the two-state system does not perform a jump, the evolution is then governed by the effective non-hermitian Hamiltonian

16 Quantum trajectories and continuous measurements 11 Ĥ = Ĥs i hγ e e i hγ g g. (1.48) If it makes a quantum jump, then it winds up in state e if it absorbs a photon, and in state g if it emits a photon. The probability that that system undergoes any jump is p = t[ a(t) 2 Γ + B(t) 2 Γ. (1.49) The dynamics of the system can then be simulated by a standard Monte Carlo wave function approach, starting form some initial condition and picking a random number ɛ between 0 and 1. If that number is less that one during the interval t then the system will have either emitted a photon with probability b(t) 2 Γ /[ a(t) 2 γ + B(t) 2 γ ] or absorbed it with probability a(t) 2 Γ /[ a(t) 2 γ + B(t) 2 γ ]. By repeating such Monte Carlo simulations many times, and with a proper choice of initial conditions that reproduce when averaged over the initial density operator of the system one reproduces the evolution for the ensity operator of the system coupled to the bath, with the benefit of generating a set og quantum trajectories typical of what would be observed in single runs of an experiment. Hekking and Pekola evaluate the work for a specific realization by making two measurements, one before and one after the driving period. These measurements are realized by the detection (projective measurement) of the last photon emitted to the environment, or absorbed by the system, in a projective measurement. If both the first and the second measurement show that the state of the system was, say, the ground state g then the internal energy of the system was not changed for that particular realization. The other two possible outcomes are U = ± hω 0. During that drive heat is taken from or given to the environment during the quantum jump events. The quantum statistics of the work are directly obtained form from a statistical analysis of the results of a large number of stochastic Monte Carlo wave function simulations. 1.3 Quantum trajectories and continuous measurements The two examples discussed in the previous section exploit projective measurements on the bath to extract information on the work acting on the system. In practice this is not easy to do, though. Thermal baths are normally assumed to be large systems that remain essentially unchanged due to their interaction with the system, so measuring a small change, say, in their occupancy, is no small challenge. Alternatively one could think of the two-level atoms used in the first example as a probe beam on which projective measurements are performed to learn about the system. Even then, though, this is not completely without difficulties: quantum measurements on a probe usually have a backaction on the system, and that backaction impacts its subsequent evolution and the result and accuracy of successive measurements. So an assumption implicitly made in such projective measurement schemes is that subsequent series of measurements will be done on newly prepared systems. There can however be additional benefits and useful information to be gained in monitoring a single system repeatedly, without having to reset it. One way to do that is to observe the system gently, so as not to perturb its subsequent evolution too significantly. Quantum non-demolition measurements can achieve this goal, but they are not necessarily easy to realize. An alternative approach is to make a series

17 12 Quantum optomechanics, thermodynamics, and heat engines of weak observations in a non-projective fashion. This section discusses how to do that, and outlines the main steps of derivation of a stochastic formalism that leads to the description of continuous quantum measurements in terms of stochastic master equations of stochastic Schrödinger equations. This is taken directly from the excellent tutorial presentation of Jacobs and Steck (2006), which the reader is referred to for details. Projective measurements use a set of projection operators { ˆP n = n n } that describe what happens in one of the possible outcomes of the measurement. If the state is ˆρ = ψ ψ with ψ = n c n n then the n-th possible final state is ˆρ f = n n = ˆP n ˆρ ˆP n Tr(ˆP n ˆρˆP n ) (1.50) with probability P (n) = Tr( ˆP n ˆρ ˆP n ) = c n 2. As discussed by Jacobs and Steck (2006) every measurement may be described in a similar fashion by generalizing the set of operators P n. If we pick any set of m max operators ˆΩ m with the restriction M max m 1 ˆΩ m ˆΩ m = I, (1.51) with I the identity operator, then it is possible to design a measurement with N possible outcomes, with probabilities ˆρ f = ˆΩ m ˆρˆΩ m Tr[ˆΩ m ˆρˆΩ m], (1.52) P (m) = Tr[ˆΩ m ˆρˆΩ m], (1.53) the total probability of obtaining a result in the range [a, b] being P (m [a, b]) = [ b b ] Tr[ˆΩ m ˆρˆΩ m] = Tr ˆΩ ˆΩ m m ˆρ. (1.54) m=a These generalized measurements are referred to as Positive Operator-Valued Measures, or POVM. They can be implemented by performing a unitary interaction between the system to be characterized and an auxiliary system and then performing a von Neumann measurement on that system. In continuous measurement information is continuously extracted from the system. To construct such a measurement scheme we divide time into a sequence of small intervals t and perform a weak measurement during each of them. We denote the observable to be monitored by the hermitian operator ˆX and although this is not necessary we assume for simplicity that ˆX has a continuous spectrum of eigenvalues {x}, with eigenstates as x, so that x x = δ(x x ). Again, we are not interested in making projective measurements that would leave ˆX in one of its eigenstates. Rather, we m=a

18 Quantum trajectories and continuous measurements 13 consider weaker measurements characterized by a positive operator valued measure (POVM) Â(α), with ( ) 1/4 4k t + Â(α) = e 2k t(x α)2 x x dx, (1.55) π which is a Gaussian-weighted sum of projectors onto the eigenstates of ˆX that provides only partial information about the observable. Here α is a continuous index, such that the spectrum of measurement result is a continuum labeled by α. As we shall see, the parameter k can be understood as a measure of the measurement strength. Continuous measurements result from taking the limit t 0. In practice the measurements are realized by coupling the system to a measuring apparatus through an interaction proportional to ˆX and some observable of the measuring device, which is then determined by a projective measurement. In the quantum heat engine that we consider later in these notes, the measuring apparatus will be a low density beam of two-level atoms that are either resonant (absorptive case) or off-resonant (dispersive case) with the intracavity field. For the initial state ψ = ψ(x) x dx we have P (α) = Tr[Â(α) Â(α) ψ ψ ] and α = αp (α)dα = αtr[â(α) Â(α) ψ ψ ] = x ψ(x) 2 dx = ˆX. (1.56) The probability density for obtaining the measurement outcome α is then found to be (Jacobs and Steck (2006)) P (α) = T r[â(α) x x  (α)] = 4k t π + ψ(x) 2 e 4k t(α x)2 dx. (1.57) For t sufficiently small the Gaussian is much broader than ψ(x), so that one can approximate ψ(x) 2 by a delta function centered at the expectation value ˆX. We then have that 4k t 2 ˆX ) P (α) e 4k t(α. (1.58) π It follows that we can write α as the stochastic quantity α = ˆX + w 8k t (1.59) where w is a zero-mean, Gaussian random variable of variance t. This implies that its root mean square scales as ( t) 1/2 and its variance scales as t. It is the stochastic nature of α that accounts for the random nature of the quantum successive measurements. The larger k, the smaller the fluctuations in the measurement outcomes. In the infinetisimal limit we write t dt and δw dw. The stochastic variable w(t) is then referred to as a Wiener process, a random walk with arbitrary small, independent steps taken arbitrarily often. Importantly, the Wiener differential dw

19 14 Quantum optomechanics, thermodynamics, and heat engines satisfies the Itô rule dw 2 = dt. This might appear surprising since dw is a stochastic quantity but dt is not, and neither is dw 2. This is discussed in detail by Jacobs and Steck (2006). These results permit to numerically determine the evolution of the wave function subject to measurements characterized by the POVM Â(α) at each time step, with the stochastic infinitesimal change of the quantum state following a single measurement given by ψ(t + t) Â(α) ψ(t) e 2k t(α ˆX) 2 ψ(t). (1.60) Expanding the exponential to first order in t dt and to second order in the Wiener process dw (due to the Itô rule dw 2 = dt) and normalizing ψ(t + dt) finally gives the stochastic Schrödinger equation d ψ = [ k( ˆX ˆX ) 2 dt + 2k( ˆX ˆX )dw] ψ(t). (1.61) For example, for measurements on a single-mode field amplitude one find, taking ˆX = (â + â ) one finds (Imoto et al. 1990, Jacobs and Steck 2006) (â â â + â â + â + â 2 d ψ j (t) = {[ ī hĥab 1 2 λ a )] dt 4 + ( λ a â â + ) } â dw ψ j (t), (1.62) 2 where λ a measure the strength of the measurement, while for ˆX = â â with measurement strength λ d the stochastic Schrödinger equation reads (Ueda et al. 1992) d ψ = {[ īhĥab 12 ] λ d (ˆn a ˆn a ) 2 dt + } λ d (ˆn a ˆn a )dw ψ(t). (1.63) These equations will reappear in Section 1.7 to characterize the work of a quantum optomechanical heat engine and its fluctuations. 1.4 Quantum heat engines We have now prepared the ground in sufficient detail to begin a quantitative discussion of quantum heat engines (QHE). These are engines that use quantum matter rather than classical matter as their working substance. An excellent introduction to these systems in provided by Quan et al. (2007). This section briefly summarizes some of its most salient points Quantum thermodynamic processes We have already mentioned the first law of thermodynamics, and how it allows one to separate average work differential and average heat differential in terms of the eigenenergies of the working substance and their probabilities of occupation. This simple identification allows one to identify the basic quantum thermodynamic processes: quantum isothermal process, quantum isochoric process, and quantum adiabatic process. We then discuss how they enter into two of the most familiar cycles used in QHE, Carnot cycles and quantum Otto cycles.

20 Quantum heat engines 15 Quantum isothermal processes are similar to classical isothermal processes. The working substance, be it classical or quantum, is in contact with a heat bath at constant temperature and can perform work to the outside by absorbing heat form that bath. In that process both the energies E i and their occupation probabilities p i in expression (1.2) U = Ĥ = i p i E i, (1.64) need to change simultaneously to account for the work performed and heat exchange with the thermal bath. Quantum isochoric processes share with their classical counterparts the property that no work is performed in them, while heat is exchanged with the thermal bath. Classically, this corresponds to keeping to volume of the working substance constant. In quantum isochoric processes the occupation probabilities p i change, but the eigenenergies E i remain constant. Quantum adiabatic processes are somewhat more subtle, because of the difference between classical and quantum adiabaticity. In the classical adiabaticity there is no heat exchange with reservoirs, dq = 0, but work can be performed. Quantum adiabaticity requires something additional, namely that the occupation probabilities p i remain constant. As a result, quantum adiabatic processes are also adiabatic in the familiar, classical sense, but classical adiabatic processes, need not satisfy quantum adiabaticity. Because of the differences in which thermodynamic processes are characterized it is not particularly useful to try to represent quantum heat engines in familiar P-V diagrams. The commonality between them is best visualized in S-T diagrams instead, to which we turn to discuss Carnot and Otto engine cycles Carnot and Otto engine cycles Like their classical counterparts, quantum Carnot engines consist of two isothermal processes and two adiabatic processes. During the isothermal expansion processes the quantum working substance is kept in contact with a heat bath, with its energy levels changing slowly enough that it is keeps in constant thermal equilibrium with the bath. The quantum adiabatic processes keep the populations of the energy levels constant, but change the values of these energies. As discussed by Quan et al. (2007), ideally all energy gaps need to change by the same ratio in these processes, and this ratio must be equal to the ratio of the temperatures of the heat baths. If that is not the case then a thermalization of the working substance is inevitable before the subsequent isothermal process. In this step the combined system plus bath entropy increases, indicating that it is not reversible. Quantum Otto cycles consist of two isochoric and two quantum adiabatic processes. In the isochoric strokes no work is done, but heat is absorbed. Importantly the heat bath and the working substance are not always in thermal equilibrium, so that the process is not thermodynamically reversible. Contrary to quantum Carnot engines, there is no constraint on maintaining the ratio of the energy gaps during the adiabatic processes.

21 16 Quantum optomechanics, thermodynamics, and heat engines Fig. 1.1 Comparison of the classical and quantum properties of basic thermodynamic processes. Quantum Heat Engines are actually not a new topic. As early as 1959 Scovil and Schultz-Dubois (1959), see also Geusic et al. (1967), discussed the three-level maser as an example of a QHE and introduced the concept of negative temperatures in these systems. Recently a number of other schemes of QHE have been proposed, motivated in part by advances in ultracold science, single atom and ion manipulation and nanofabrication. For example, Abah et al. (2012) proposed a scheme to realize a nanoscale heat engine with a single ion, the idea being to confine that ion in a linear Paul trap with tapered geometry and coupling it to engineered laser reservoirs to realize an Otto cycle. Fialko and Hallwood (2012) considered theoretically an isolated system using cold bosonic atoms confined to a double well potential that is created by splitting a harmonic trap with a focused laser. The system shows thermalization and is predicted to be able to operate as a heat engine with a finite quantum heat bath, thereby addressing fundamental questions related to the issue of the thermalization of isolated quantum and the so-called eigenstate thermalization hypothesis. More recently, Bergenfeldt et al. (2014) analyzed the use of hybrid microwave cavities as QHE, a possible realization consisting of two macroscopically separated quantumdot conductors coupled capacitively to the fundamental mode of a microwave cavity. A QHE consisting of a photon gas inside an optical cavity as the working fluid and

22 Photon-phonon polaritons 17 multi-atom coherent atomic systems as the fuel was also recently analyzed by Hardal and Müstecaploglu (2015). 1.5 Photon-phonon polaritons In the next sections we will discuss an optomechanical QHE and an optomechanical heat pump whose working fluid consists of photon-photon polaritons, the normal modes of a coupled system of photon and phonon fields. Since they are central to that discussion we first review their main properties. We consider a generic optomechanical system consisting of a Fabry-Pérot resonator with a compliant end mirror of effective mass m and frequency ω m driven by the radiation pressure from a single-mode intracavity field of frequency ω c. We assume that the system is driven by a classical pump filed of amplitude α in and frequency ω p and has reached a mean-field steady state with intracavity field α and normalized mirror displacement x/x zpt = β, where x zpt = ( h/2mω m ) 1/2 is the zero-point mirror displacement. For small optical damping rates κ p we have α α in / p and β g 0 α 2 /ω m. Linearizing the optomechanical interaction about α the effective optomechanical Hamiltonian simplifies to (see e.g. Aspelmeyer et al. 2014) Ĥ 0 H ab = h â â + hω mˆb ˆb + hg(ˆb + ˆb )(â + â ). (1.65) Here G = g 0 α is taken to be real and positive, g 0 is the single-photon optomechanical, â is the photon annihilation operator for the quantum fluctuations of the optical mode ˆb describes the quantum fluctuations of the mechanics, and = ω p ω c 2g 0 β, (1.66) is the effective detuning between the optical pump and the cavity mode, and we use the notation Ĥab in the following when working in the bare modes representation Polariton spectrum The Hamiltonian (1.65) can be diagonalized in terms of two bosonic normal modes, or polaritons, with annihilation operators  and ˆB as Ĥ 0 ĤAB = hω A   + hω B ˆB ˆB + const., (1.67) where ĤAB will refer to the polariton picture expression in the following, with corresponding eigenfrequencies (Zhang et al. 2014b) ω A = ω 2 m + ( 2 ω 2 m) 2 16G 2 ω m, (1.68) ω B = ω 2 m ( 2 ω 2 m) 2 16G 2 ω m (1.69) At the avoided crossing = ω m they simplify to ω A,B = ω m 1 ± 2G ω m, (1.70) indicating that the minimum frequency difference between the branches A and B s proportional to G/ω m.

23 18 Quantum optomechanics, thermodynamics, and heat engines E ħω m ω A 1.0 Phonon 0.8 ω B Photon p Ω m Fig. 1.2 Frequencies of the two polaritons (normal modes) of the optomechanical system for G/ω m = 0.05 in the red-detuned case p < 0. The dashed curves correspond to the frequencies of the bare photon and phonon modes. Importantly, for /ω m we have ω A and ω B ω m, so that the branch A describes a photon-like excitation and the branch B a phonon-like excitation. In contrast, for ω m < < 0 we have 2G ω A ω m (1 2 ) +, ( 2 ωm)ω 2 m ( ω B 1 2G2 ω m ( 2 ωm) 2 ). (1.71) That is, the polariton A is then phonon-like while B is photon-like as 0 ( ). For small dimensionless optomechanical couplings g 1 and detunings δ 1 the normal mode operators are given in terms of the bare mode operators by and ˆN A =   =  = [1 + 2δg2 (δ 1) 2 ˆB = g 1 + δ â + ] â [1 + 2δg2 (δ 1) 2 g δˆb ] ˆb + g 2 δ(1 δ 2 )â + g 1 δˆb, (1.72) g 1 δ â + g2 δ δ 2 1ˆb, (1.73) ] [1 + 4δg2 (δ 1) 2 â â + 2(1 + δ2 )g 2 ( ) 2 g ˆb ˆb + (δ 1) 2 (1.74) 1 δ g 1 + δ (â ˆb + ˆb â) + ˆN B = ˆB ˆB = [ 1 + 4δg2 (δ 1) 2 g 2 δ(1 δ 2 ) (â2 + â 2 ) + g 1 δ (âˆb + ˆb â ) g2 1 δ 2 (ˆb 2 + ˆb 2 ), ] 2(1 + δ ˆb ˆb 2 )g 2 ( ) 2 g + (δ 1) 2 â â + (1.75) 1 δ

24 Photon-phonon polaritons 19 + g 1 + δ (â ˆb + ˆb â) + g2 1 δ 2 (â2 + â 2 ) + g 1 δ (âˆb + ˆb â ) + g2 δ δ 2 1 (ˆb 2 + ˆb 2 ), To lowest order one can neglect the intermode correlations and squeezing terms appearing in these expressions. This approximates the mean polariton numbers by the mean thermal occupations of the optical reservoir n a ( n a 0 at optical frequencies) and of the mechanical reservoir n b, respectively. When the optomechanical coupling is small but finite, though, all terms in Eqs. (1.75) and (1.76) contribute, and the steady-state polariton populations deviate from thermal equilibrium. For 1 < δ < 0, the expressions for  and ˆB are simply interchanged Dissipation In addition to the unitary evolution governed by the Hamiltonian (1.65), the optical mode and the mechanical mode also suffer from dissipation at rates κ and γ, respectively. Since for high-q mechanical oscillators the effect of Brownian thermal motion can be described by a familiar Lindblad form (?), the master equation describing these two dissipation channels is dˆρ dt = ī h [Ĥ0, ˆρ] + κ( n a + 1)L[â]ˆρ + κ n a L[â ]ˆρ + γ( n b + 1)L[ˆb]ˆρ + γ n b L[ˆb ]ˆρ, (1.76) where L[ˆx]ˆρ = ˆxˆρˆx 1 2 ˆx ˆxˆρ 1 2 ˆρˆx ˆx. (1.77) The two polariton modes are coupled via the Lindblad superoperators L[â] and L[ˆb], so that it is not possible to define two uncoupled master equations for the normal modes A and B. To illustrate how this impacts their dissipation we consider the simple case the normal mode A vanishes throughout the Otto cycle, ˆρ ˆρ B 0 0 A. In this simple case ĤAB ĤB = hω B ˆB ˆB and one finds that the dissipation of mode B is governed by the master equation (Zhang et al. 2014b) dˆρ B dt = ī h [ĤB, ˆρ B ] + Γ B ( N B + 1)L[ ˆB]ˆρ B + Γ B NB L[ ˆB ]ˆρ B + Γ B MB J [ ˆB]ˆρ B + Γ B M B J [ ˆB ]ˆρ B. (1.78) Here N B = κ( n a + 1) V κ n a U γ( n b + 1) V γ n b U 22 2 κ( U 12 2 V 12 2 ) + γ( U 22 2 V 22 2, (1.79) ) M B = κ(2 n a + 1)V 12 U 12 + γ(2 n b + 1)V 22 U 22 κ( U 12 2 V 12 2 ) + γ( U 22 2 V 22 2 ), (1.80) Γ B = κ( U 12 2 V 12 2 ) + γ( U 22 2 V 22 2 ), and U ij and V ij are the elements of the submatrices U and V of the Bogoliubov transformation.

25 20 Quantum optomechanics, thermodynamics, and heat engines The key point is that the master equation (1.78) differs form the bare-mode master equation through the appearance of the additional Lindblad operator J [ˆx]ˆρ = ˆxˆρˆx 1 2 ˆxˆxˆρ 1 ˆρˆxˆx, (1.81) 2 which indicates that the polariton mode is effectively coupled to a squeezed thermal reservoir (?). The presence of a squeezed reservoir implies that the steady state of mode B is not a thermal state. Rather, it is a state that is in some sense hotter than the corresponding thermal reservoir. Its steady state population is with N B = ˆB ˆB s = ˆX 2 s + Ŷ 2 s 1, (1.82) 2 ˆX 2 s = N B M B + 1 2, (1.83) Ŷ 2 s = N B + M B + 1 2, (1.84) ˆX and Ŷ being its quadrature operators. For the case δ i < 1 and n a = 0 we find to second order in g that [ N B = 1 + 4δ ig 2 ] κ γ(δi 2 n b + κ ( ) 2 g, (1.85) 1)2 γ 1 δ i Γ B = γ + (γ κ) (δ 2 i 4g 2 δ i, (1.86) 1)2 Both the steady population and the effective decay rate of the polariton B are therefore close to the phonon case for δ i. On the other side of the avoided crossing, 1 < δ f < 0, one finds likewise that they approaches the values of the bare phonon mode for δ f 0 ( ). 1.6 A quantum optomechanical heat engine Armed with the information of the previous sections we can now turn to the specific example of an optomechanical QHE.(Zhang et al. 2014a) In that engine the working substance is a polariiton fluid, By adiabatically switching its nature from phonon-like and the photon-like, and enabling thermalization with the corresponding hot and cold reservoirs, it is possible to operate it on a quantum Otto cycle. This can be achieved experimentally by varying the detuning p while keeping the intracavity optical field α constant. Provided that nonadiabatic transitions between the two polariton branches A and B can be avoided, each can be individually associated with a different Otto cycle The four strokes We assume that the optomechanical system is initially in thermal equilibrium at large red detuning i, in dimensionless units δ i i /ω m, so that the phononlike lower polariton branch B is in thermal equilibrium with a reservoir at effective

26 A quantum optomechanical heat engine 21 temperature T Bi for all practical purposes the temperature of the phonon heat bath. Similarly, the photon-like upper polariton branch A is in thermal equilibrium with a reservoir at temperature T Ai 0K, so that the initial polariton population are related by ˆN B i ˆN A i. The first stroke of the cycle is an adiabatic change of from its initial value to small negative final value, δ f f /ω m 0 ( ). This step should occur fast enough that the interaction of the system with the thermal reservoirs can be largely neglected, yet slowly enough that nonadiabatic transitions between the two polariton branches remain negligible. Ideally, at the end of the stroke the lower-branch polariton becomes photon-like. In the second stroke it is then allowed to reach thermal equilibrium with a reservoir at the temperature T Bf 0K of the photon reservoir. The third stroke involves sweeping the detuning back to its initial large negative value. Again, this step has to be fast enough to avoid thermalization, but slow enough to avoid nonadiabatic transitions. The fourth and final stroke is the rethermalization at fixed detuning δ i to the temperature T Bi for the lower polariton branch and to T Ai for the upper branch. During the first stroke the phonon-like thermal excitations, which are initially large due to the contact with a thermal reservoir that is essentially at the temperature of the mechanics, become photon-like. During this step the amplitude of vibrations of the mechanics decreases, with excess energy transferred to the intracavity field. As a result of the increased radiation pressure the resonator length increases slightly. It is at this point that the mechanical work on the oscillator is produced, but this work is very small due to the disproportion between the steady amplitudes and the quantum fluctuations of the photon and phonon fields. During the thermalization step of stroke 2 the population of photon-like excitations decays at rate κ, with the cavity length unchanged. In stroke 3 the remaining photon-like polariton excitations are then turned back to phonon-like. This results in a small contraction of the cavity length and costs small amount of work. In the final stroke 4 the polariton population, now back to phonon-like, grows back to its initial value via thermal contact with the hot mechanical reservoir Average work and efficiency Classically the heat exchanged and the work performed are given by W 1,α = E 2,α E 1,α, W 3,α = E 4,α E 3,α, Q 2,α = E 3,α E 2,α and Q 4,α = E 1,α E 4,α with W 1,α + W 3,α + Q 2,α + Q 4,α = 0, where E i,α, i = 1,..., 4 and α = {A, B} are the energies of the system at the nodes of the four stokes. As defined in Eq. (1.2), we define the work as positive if the change in internal energy of the system is positive. In the absence of heat exchange this corresponds to work performed on the system. The work performed by the system is its opposite, W out W, (1.87) The efficiency of cycle B is defined by the ratio between the total work and the input heat (?) η B = W B,tot Q 4,B = 1 ω Bf ω Bi. (1.88) Figure 1.3 shows η B and W B,tot as a function of δ f and the dimensionless interaction

27 22 Quantum optomechanics, thermodynamics, and heat engines Fig. 1.3 (Color online) Contour maps of the thermal efficiency and the total work (absolute value) of the B-branch Otto cycle for δ i = 3 with value legends aside. The thermal mean population of the normal mode B is calculated through a numerical Bogoliubov transformation with the thermal mean photon and phonon number, n a = 0 and n b = 10, respectively. The white region is mechanically unstable. strength g = G/ω m. Its expression is evaluated in detail by Zhang et al. (2014b). The calculations are straighforward and we don t reproduce them here. To second order in g 2, and with δ i large and negative and δ f negative close to zero one finds η 1 ( δ f 2g 2 ), which is maximum for g 2 = δ f /2. The total average work is also readily found to be W B,tot hω m ( δ f 2g 2 1)[(1 2g 2 ) n b g 2 ]. It is minimum for g 2 = δ f /4 hω m /(8k B T b ), assuming that the phonon temperature T b is high enough that n b (k B T b /( hω m ) 1/2. (Remember that W B,tot < 0.)This yields the efficiency at maximum power ( pf η P = 1 + hω ) m, (1.89) 2ω m 4k B T b which, with the help of a simple inequality, gives h( pf ) η P < 1. (1.90) 2k B T b This is the quantum version of the classical Curzon-Ahlborn efficiency limit, 1 Tlow /T high.(curzon and Ahlborn 1975) Polaritons vs. bare modes So far, our discussion has been based on the polariton picture. This energy representation is naturally required for the study of its thermodynamical properties. As we have seen in Sections 1.2 and 1.4 thermodynamics can only be meaningfully intoduced when considering the energy eigenstates of the system, the polaritons, and hence the application of thee concepts to the bare modes is only qualitative. However, the understanding of the various strokes in the bare photon and phonon modes is also of interest, For example, the adiabatic evolution means that the system as a whole has

28 A quantum optomechanical heat engine 23 no heat exchange with the environment, but heat can of course be exchanged between its subsystems. Also, in practice measurements on the system are normally performed on its bare modes, in particular the photon field, since there is no such thing as polariton counter. So, it is important to also understand what happens at the level of the subsystems. Polariton B Phonon b V Photon a Q b Q a W B Control field (classical) Fig. 1.4 Hierarchical structure of the quantum engine where the coupled photon and phonon modes constitute a polariton mode which exchanges heat and work with the external control field. To show how this works, we analyze the hierarchical structure of the QHE during the first stroke of the engine, see Fig (Zhang et al. 2014b) The first level is the bare mode picture, with the photon and phonon modes coupled by the linearized optomechanical interaction ˆV ab = G(ˆb + ˆb )(â + â ). (1.91) The second level is the dressed picture, where the system is described in terms of the noninteracting polaritons A and B. The third level, finally, includes the external controls. They are the driving optical field, the steady cavity field α, and the normalized displacement β. The temperatures of the photon and the phonon reservoirs should also be present but they can be ignored during during the isentropic strokes. In the following we ignore for simplicity the polariton A, whose population remains negligible throughout the cycle, so that ĤAB ĤB = hω B ˆB ˆB. In the polariton picture we then have d ĤB = Tr[dˆρ B Ĥ B ] + Tr[ˆρ B dĥb], (1.92) where ˆρ B is the density matrix of the normal mode B. Since the transformation is adiabatic, dˆρ B = 0, so that dq B = 0 and

29 24 Quantum optomechanics, thermodynamics, and heat engines In contrast, in the bare picture du = Tr[ˆρ B dĥb] = dw B. (1.93) d Ĥab = d Ĥa + d Ĥb + d ˆV (1.94) = Tr[dˆρ a Ĥ a ] + Tr[ˆρ a dĥa] + Tr[dˆρ b Ĥ b ] + Tr[ˆρ b dĥb] + Tr[dˆρ ab ˆV ] + Tr[ˆρab d ˆV ], where ˆρ ab is the density matrix of the two-mode system and ˆρ a and ˆρ b are the reduced density matrices of the photon and phonon mode, respectively. Since H b and V are constant, we have from the quantum definitions of the average work and heat differentials d Ĥab = dq a + dw a + dq b + Tr[dˆρ ab V ] (1.95) and, considering the change of the populations of the photon and phonon modes in the first stroke, dq a = Tr[dˆρ a Ĥ a ] > 0, (1.96) dq b = Tr[dˆρ b Ĥ b ] < 0. (1.97) The evolutions of the photon and phonon subsystems are therefore neither adiabatic nor isentropic. Furthermore, since the initial population of the photon mode is zero, we have dw a = Tr[ˆρ a dh a ] = 0 and dw B = dq a + dq b + Tr[dˆρ ab ˆVab ], (1.98) where the last term is the change in quantum correlations between the photon and phonon fields: This is initially zero for a product of thermal states, but becomes finite as a result of the optomechanical coupling. This term, which does not have a corresponding classical thermodynamical quantity, is much smaller than dq a and dq b for the weak optomechanical couplings considered here. Summarizing, then, the phonon mode releases heat, part of which is absorbed by the photon field, a small amount contributing to the generation of quantum correlations, and the rest being absorbed by the external control field Numerical simulations To simulate all aspects of the proposed Otto cycle, including nonadiabatic transitions between the lower and upper polariton branches and the effects of dissipation, we have solved numerically the full master equation (1.76). Here we summarize some of the main results, more details can be found in Zhang et al. (2014). Figure 1.5 shows the dynamics of the population of the bare modes a and B and of the polariton modes A and B for a full loop of the Otto cycle. We assume a small thermal mean phonon number n b = 4 and truncate the number state representation at N = 30. For a mechanical resonator of frequency ω m = 2π 200 MHz, it corresponds to a phonon reservoir temperature T b = 45 mk, which is in the parameter range of present optomechanical experiments. The mean photon number is zero. The initial detuning is δ i = 3 and we choose the final detuning δ f = 0.4 to avoid the unstable region near the cavity resonance. All other parameters are given in the figure caption.

30 Quantum fluctuations 25 Fig. 1.5 (Color online) Time evolution of the number of photons (blue dashed line), phonons (red solid line), polariton A (black dotted line) and B (pink dot-dashed line) in a loop of the Otto cycle. Here G = 0.2ω m, κ = 0.03ω m, and γ = 10 3 ω m, all normalized to ω m = 2π 200 MHz.. The stroke times are τ 1 = τ 3 = 25ωm 1, τ 2 = 50ωm 1, and τ 4 = 10 4 ωm 1. For the small optomechanical coupling considered here it is appropriate to start from the factorized thermal state ˆρ sys (0) = ˆρ a th ˆρb th, where ˆρa th and ˆρa th are the thermal states of the photon and the phonon mode. In the first stroke the population of the polaritons A and B initially coincides with the mean photon and phonon number, respectively. During the change in detuning, in the ideal case, the photon and the phonon mode would exchange their population while keeping the polariton numbers constant. In practice, the variation of the detuning is not quite slow enough to avoid transitions between the two polariton branches. Meanwhile, the cavity and mechanical damping rates result in a small decay and increment of the polariton B and A, respectively. In the second stroke, the photon-like polariton B decays fast due to the cavity decay, while the thermalization of the phonon-like polariton A, at rate γ, is negligible. This step is also characterized by Rabi oscillations between photon and phonon populations due to the optomechanical coupling. The polariton B then recovers its phononlike properties in the third stroke and finally thermalizes back to its initial population at the end of the last stroke (not shown, at t > 10 4 ωm 1 ). The total cycle takes a time of the order 10 4 s. Note that as the population of polariton A remains small throughout the whole cycle, its effect on the engine can be safely neglected as we did in the previous section. 1.7 Quantum fluctuations We already mentioned that it is not possible to directly monitor the occupation of the polariton mode B since it consists of quasiparticles that are coherent superpositions of photon and phonon states. What is experimentally accessible instead is the intracavity optical field. In this section we show that the average work (1.102) can be expressed in terms of the mean intracavity photon number. Building on this observation, we then consider continuous quantum measurements that permit to monitor that field and its fluctuations without having to reset the system after each measurement.

31 26 Quantum optomechanics, thermodynamics, and heat engines Classical work measurement Neglecting for simplicity adiabatic transitions between the two polariton modes and dissipation, and since N B = 0 after thermalization with the optical reservoir at T = 0 we restrict for now the determination of the work to the first adiabatic stroke, The average work is then simply W = dw = Tr[ˆρ AB (dĥab)] = N B hdω B. (1.99) f i dw = h[ω B ( f ) ω B ( i )] N B, (1.100) where i and f are the initial and final detuning.(dong et al. 2015a) Alternatively one can also work in the bare modes representation, where the photon and phonon distributions are time dependent as we have seen and dw = Tr[ˆρ ab (dĥab)] = h n a d. (1.101) Here we have used the fact that the only term in the Hamiltonian (1.65) with a dependence on the detuning is the free field part h â â, and n a = â â is the average number of excitations in the photon mode â. In this picture the average work is given by f W = h n a ( )d. (1.102) i If the stroke is perfectly adiabatic the values of the average work obtained from expressions (1.100) and (1.102) are equal. However if either the optical or the mechanical damping is significant on the time scale of the adiabatic stroke, or if the variation of the optical detuning induces non-adiabatic transitions, then Eq. (1.100) is no longer exact. The expression (1.102) remains however valid, as confirmed by considering an operational way to measure that work to which we now turn. â ˆb y Fig. 1.6 (Color online) Schematic setup for a classical measurement of the output work, whereby the work performed by radiation pressure acting on the mirror of large mass M can be stored in the control system. See text for details.

32 Quantum fluctuations 27 Experimentally the detuning (t) can be varied in principle through a change in cavity length, (t) (y) = 0 g M y (1.103) where y is a classically controlled length change, assumed small compared to the total cavity length. In this expression g M g/y M is the optomechanical coupling normalized to the mirror zero-point motion y M, and 0 is a nominal detuning. We can then express W in terms of the spatial integral of the position-dependent radiation pressure force F rp (y) as W = yf y i F rp (y)dy (1.104) where F rp (y) = hg M n a (y). This suggests the scheme to measure the classical work output of the engine illustrated in Fig Here the optomechanical resonator comprises in addition to the oscillating end mirror driven by radiation pressure a classical input mirror of very large mass M whose position y is controlled externally by the potential V (y) provided by a piezoelectric element, thereby controlling the detuning (y) in the presence of the radiation force F rp. The total system Hamiltonian is then Ĥ m = Ĥab + H M, (1.105) where Ĥab is the optomechanical Hamiltonian expressed in the bare modes picture with = (y), and H M = p2 + V (y) (1.106) 2M is the classical Hamiltonian for the massive control mirror. The classical equations of motion for that mirror are dy dt = H m p = p M, (1.107) dp dt = H m y = V (y) F rp, y (1.108) where H m is the classical limit of Ĥm. If M is large enough that it can be considered as essentially infinite compared to all other optomechanical elements we have dy/dt dp/dt 0. That is, the force exerted by the control system balances the expectation value of the radiation pressure force, V (y) = F rp. (1.109) y This shows that provided the kinetic energy of the large mirror remains essentially zero, all work performed by the photons is converted to the control potential energy and can be measured in that way. This confirms the intuitive result that the work can be measured by sensing the radiation pressure force, which is of course proportional to the mean number of intracavity photons.

33 28 Quantum optomechanics, thermodynamics, and heat engines Continuous quantum measurements Building on that intuition we now consider now two types of continuous quantum measurements that permit to monitor both the mean intracavity photon number and its fluctuations.(dong 2015a) Absorptive measurements. Consider first the resonant situation where a low density probe beam of two-state systems of transition frequency ω 0 = ω c, and prepared in their ground state g is injected inside the optical cavity, see Fig. 3(a). The atom-field coupling is given in the rotating wave approximation by ˆV a = hg a (â ˆσ + âˆσ + ) (1.110) where g a is the single-photon Rabi frequency of the transition and ˆσ ij = i j. As we have seen in Section 1.3, for each stochastic quantum trajectory j, the effect of the continuous measurements is described by the stochastic Schrödinger equation (1.62) d ψ j (t) = {[ ī hĥab 1 2 λ a (â â â + â â + â + â 2 )] dt 4 + ( λ a â â + ) } â dw ψ j (t), (1.111) 2 where dw is a Wiener process, λ a = g 2 aτ is a measure of the strength of the measurement, and τ is the transit time of an individual atom through the resonator (Imoto et al. 1990, Jacobs and Steck 2006). The term proportional to λ a on the right-hand side of Eq. (1.62) accounts for the additional dissipation channel of the intracavity field resulting from the absorption of photons by the successive atoms, and the term proportional to λ a describes the stochastic changes of the intracavity field about its expected value â + â resulting from the stochastic measurement outcomes. Dispersive measurements. When the interaction between the two-level atoms and the intracavity field mode is off-resonant we can adiabatically eliminate of the upper electronic state to obtain the effective Hamiltonian ˆV a = hg d â â(ˆσ ee ˆσ gg ) = hg d â â(ˆσ + + ˆσ + ), (1.112) where ± = ( e ± g )/ 2, and g d = ga/2δ 2 is the off-resonant effective Rabi frequency coupling between the atoms and the intracavity intensity, with δ = ω c ω eg. The atoms are now prepared in the superposition + of the ground and excited states, and information on the intracavity field is inferred from a change in phase of the atomic state. In that situation the effect of the measurements on the optical field is described by the stochastic Schrödinger equation (1.63) d ψ = {[ īhĥab 12 ] λ d (ˆn a ˆn a ) 2 dt + } λ d (ˆn a ˆn a )dw ψ(t), (1.113) (Ueda et al. 1992), where λ d = g 2 d τ.

34 Quantum fluctuations 29 (a) (b) Fig. 1.7 (Color online) Schematic setup for a continuous quantum measurement of the output work with a beam of two-level atoms. a) Absorptive measurement: the cavity field is resonant with the atomic transition and the coupling induces real oscillations in the atomic population, which result in the loss of intracavity photons. b) Dispersive measurement: the cavity mode frequency is far off-resonant from the two-level atom transition frequency, resulting in a dispersive interaction that only modifies the phase of the atomic ground state wave function. See text for details. This equation also comprises two contributions, the second one accounting for the stochastic changes of the mean intracavity intensity ˆn about its expected value resulting from successive measurements. But because of the quantum non-demolition nature of the non-resonant atom-field interaction for the mean photon number â â, the dissipative channel of Eq. (1.63) is now replaced by a number conserving term that results in additional damping of the phase of the optical field. Importantly, for the specific QHE considered here the effective interaction (1.112) couples the polariton branches A and B and transfers excitations between them. This is in contrast to absorptive measurements, in which case the interaction (1.110) does not significantly couple the two normal modes. Still, both measurement schemes result in the appearance of additional photon loss channels that limit the amount of extractable work. In both cases these measurement back-action mechanisms may be viewed as heat exchange between the engine and the environment. Statistics of work from quantum trajectories. Solving the stochastic Schrödinger equations (1.62) and (1.63) repeatedly generates a set of trajectories ψ j (t) that can be used to evaluate the statistics of any field observable. Just like in the examples of Section 1.2.2, for each trajectory we can use this information to operationally define a stochastic variable associated with the work along that specific quantum trajectory for the time interval t i to t f as (Horowitz 2012, Solinas et al. 2013)