Autonomous Rotor Heat Engine
Alexandre Roulet, Stefan Nimmrichter, Juan Miguel Arrazola, Stella Seah, Valerio Scarani
AAutonomous Rotor Heat Engine
Alexandre Roulet, Stefan Nimmrichter, Juan Miguel Arrazola, Stella Seah, and Valerio Scarani
1, 2 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
The triumph of heat engines is their ability to convert the disordered energy of thermal sourcesinto useful mechanical motion. In recent years, much effort has been devoted to generalizing ther-modynamic notions to the quantum regime, partly motivated by the promise of surpassing classicalheat engines. Here, we instead adopt a bottom-up approach: we propose a realistic autonomousheat engine that can serve as a testbed for quantum effects in the context of thermodynamics. Ourmodel draws inspiration from actual piston engines and is built from closed-system Hamiltoniansand weak bath coupling terms. We analytically derive the performance of the engine in the classi-cal regime via a set of nonlinear Langevin equations. In the quantum case, we perform numericalsimulations of the master equation. Finally, we perform a dynamic and thermodynamic analysis ofthe engine’s behaviour for several parameter regimes in both the classical and quantum case, andfind that the latter exhibits a consistently lower efficiency due to additional noise.
I. INTRODUCTION
Historically, the initial goal of thermodynamics was tounderstand how to convert heat into useful mechanicalmotion, and it was only once this goal was achieved withthe rise of steam engines that there was an interest in ex-ploring the theoretical limitations to their efficiency [1].As our ability to control quantum systems progresses, ithas now become interesting to study thermal machineswhere quantum effects are relevant. However, in thiscase, the historical order has been reversed, with earlierwork focusing on the thermodynamic limitations of quan-tum machines [2–4]. More recently, new proposals forthermal machines that do not require external sources ofwork have been made. These include absorption refriger-ators [5–9], heat engines [10–15], as well as thermoelectricdevices converting between temperature bias and electri-cal current or voltage bias [16–20]. Additionally, therehas been significant interest in determining the extentto which quantum effects such as squeezing or coherencemay help surpass classical limits such as the Carnot ef-ficiency [21–26], or complicate classical notions such aswork [27, 28].In this work, we consider the problem of designing aquantum heat engine that achieves its goal of convertingheat into the useful mechanical motion of a system. Inparticular, our goal is to devise a self-contained enginethat autonomously converts heat from a thermal bathinto motion in a single rotational degree of freedom. Wechoose to study a rotor to benefit from the useful fea-tures of rotational motion: it is inherently periodic, itcan in principle be coupled through other systems suchas gears and pistons into other types of motion, and it canbe used to drive electric generators. Additionally, unlikethe motion of oscillators, rotational motion has a mean-ingful sense of directionality [29] through the distinctionbetween clockwise and counter-clockwise rotation. Therotation frequency is not upper bounded, as would be thecase in finite dimensional systems, and it unambiguouslydisplays the amount of useful energy stored in the rotor
FIG. 1. Autonomous rotor heat engine. A harmonic mode ispushing down a piston attached to a rotor through radiationpressure. Concurrently, the angular position ϕ of the rotor(defined relative to the upper turning point) modulates thecoupling of the mode to baths at respective occupations ¯ n H and ¯ n C . This leads to a preferred clockwise motion of therotor. Note that the specific implementation of the valvesdepicted here realizes exactly the modulation functions (5).The terms κ , g , and I denote the bath thermalization rate,the torque per excitation in the mode, and the moment ofinertia, respectively. degree of freedom.We design a simple opto-mechanical engine where therotor is coupled to a single harmonic mode through ra-diation pressure, which is in turn coupled linearly to twobaths at different temperatures. In analogy to the hotgas pushing down the piston of an actual car engine, themode serves as the working medium transferring heatfrom the hot to the cold bath in sync with the angularmotion of the rotor. We characterize analytically the en- a r X i v : . [ qu a n t - ph ] J un gine’s operation in the classical regime and we find thatit functions as desired. We then describe how these equa-tions of motion can be solved, analytically in the classicalcase and numerically in the quantum regime, and studythe dynamics for different values of the relevant param-eters, focusing on the transient behaviour of the engineas it accelerates from rest. We then perform a thermo-dynamic analysis of the engine by computing the workoutput, heat input and efficiency of the engine in boththe classical and quantum regimes. We conclude by dis-cussing the significance of the various engine parametersand the role of quantum effects. II. ROTOR HEAT ENGINE
Design.–
The guiding principle in our design of the ro-tor heat engine is that it should unambiguously achieveits intended goal of converting heat into useful mechani-cal motion of the rotor. To be precise, we demand that(i) the engine is autonomous, (ii) the rotor draws energyexclusively from a thermal source, and (iii) the rotor un-dergoes useful directional motion, i.e. it has a well-definedangular momentum increasing with time. We allow foran initialization of the rotor at a well localized angle,but do not permit the use of external control fields andtime-dependent Hamiltonians for the dynamics, contraryto previous works in the literature, e.g. Refs. [3, 13, 21–23, 30].Our engine model is sketched in Fig. 1. In its initialconfiguration, the harmonic working mode is in contactwith a hot reservoir, which causes it to thermalize to theaverage excitation number ¯ n H at a rate κ . Radiationpressure then pushes on the piston, which exerts torqueon the attached rotor and causes it to spin clockwise.Once the piston passes its bottom turning point, the ra-diation pressure starts to push against the spin, whichwould drive the system into pendular motion. To pre-vent this, the working mode is now brought into contactwith a cold reservoir that decreases the average excita-tion number to ¯ n C in order to suppress radiation pressureuntil the upper turning point is reached and the thermalcontact switches from the cold to the hot bath again.This modulated, angle-dependent thermal coupling be-tween the working mode and the two reservoirs is whatkeeps the wheel spinning clockwise, gaining momentumwith every round-trip. Note that this modulation doesnot conflict with our aim of building an autonomous en-gine. Indeed, our engine is analogous to a car enginewhere a crankshaft spins due to contact with an oscil-lating piston, and a synchronized camshaft controls theopening of the inlet and exhaust valves. In our heat en-gine, the rotor plays the role of the crankshaft while themodulation of the coupling to the bath plays the role ofthe camshaft. Model.–
Classically, the rotor degree of freedom is de-scribed by an angle variable ϕ ∈ [0 , π ) and an angularmomentum component L z perpendicular to the plane of rotation. However, in the quantum version of the rotor,the bound spectrum of the angle operator ˆ ϕ implies adiscrete spectrum of ˆ L z , and Hermitianity must be en-forced by imposing periodic boundary conditions [31],i.e. we must work with strictly periodic functions of theangle. In particular, we use the commutation relation[ e i ˆ ϕ , ˆ L z ] = − (cid:126) e i ˆ ϕ .The Hamiltonian of the engine in a frame rotating atthe mode frequency is given byˆ H S = ˆ L z I + (cid:126) g ˆ a † ˆ a cos( ˆ ϕ ) , (1)where I is the moment of inertia of the rotor, g is theopto-mechanical coupling strength and ˆ a is the annihila-tion operator of the mode. Importantly, the mode fre-quency does not enter the description as the radiationpressure term is only proportional to the mode occupa-tion ˆ a † ˆ a . Hence, we are free to match the mode frequencyto the requirements of weak linear bath coupling and ofthe preferred physical implementation.The Hamiltonian ˆ H S may for instance describe an opti-cal Fabry-Perot cavity where one of the mirrors is rigidlyattached to the rotor and allowed to move along the x direction [32]. The radiation pressure acting on the mir-ror will then always try to push the rotor away from thecavity, in proportion to the number of photons in thecavity. Although in this work we are concerned with atheoretical study of the engine, a direct opto-mechanicalrealization of the engine model can be envisaged given therecent experimental advances in the rotational control ofnanorods trapped in a cavity field [33–35]. Other phys-ical realizations are also conceivable, where the workingmode is not restricted to an optical field mode and wherethe angular variable might be associated to a phase de-gree of freedom instead of the mechanical gear depictedin Fig. 1.We can now describe the interaction of the mode withits environment, consisting of the hot (H) and cold (C)baths. Specifically, we describe each bath as an ensembleof harmonic oscillatorsˆ H T = (cid:90) ∞−∞ d ω (cid:126) ω ˆ b † T ( ω )ˆ b T ( ω ) , (2)with correlation functions (cid:68) ˆ b † T ( ω )ˆ b T ( ω (cid:48) ) (cid:69) = ¯ n T δ ( ω − ω (cid:48) ) , (3) (cid:68) ˆ b T ( ω )ˆ b † T ( ω (cid:48) ) (cid:69) = (¯ n T + 1) δ ( ω − ω (cid:48) ) , where T = H, C and ¯ n T is the associated thermal occu-pation at the mode frequency. In the Schr¨odinger frame,the interaction between the mode and the two baths isthen described byˆ H B − S = i (cid:126) (cid:88) T = H,C (cid:90) ∞−∞ d ω γf T ( ˆ ϕ ) (cid:104) ˆ b † T ( ω )ˆ a − ˆ a † ˆ b T ( ω ) (cid:105) , (4)where we neglect the variation of the coupling constant γ and of the thermal occupation ¯ n T around the modefrequency. At this point, the only non-standard featureof our model for the baths is the modulation of the cou-pling via the functions f T ( ˆ ϕ ). This synchronicity is whatbreaks the symmetry, allowing the engine to be pushedharder than it is slowed down. Naturally, this internalclock cannot be used to construct a perpetual machineof the second kind, namely a heat engine that would runon a single heat bath at a fixed temperature. Indeed,the presence of the cold bath is of utmost importance inorder to extract heat from the mode and hence lower theradiation pressure. We note that the crucial role of thisinternal clock for building autonomous machines has alsobeen pointed out recently in the context of solar cells [36].When engineering the modulating function f H ( ϕ ), itis sufficient to ensure that the mode is coupled stronglywith the hot bath in the interval 0 < ϕ < π but onlyweakly coupled in the interval π < ϕ < π , and viceversa for f C ( ϕ ). For simplicity, we consider the followingmodulating functions f H ( ˆ ϕ ) = 1 + sin( ˆ ϕ )2 , f C ( ˆ ϕ ) = 1 − sin( ˆ ϕ )2 . (5)This specific choice is motivated by the requirement ofworking with periodic functions of the angle in the quan-tum regime. Nevertheless, the results obtained will notbe qualitatively affected by a different choice of func-tions, as long as they alternate with negligible overlapas described above. A somewhat closer resemblance tothe engine of a car can be achieved by working with cou-pling functions of narrow support, say, on small windowsaround the angles ϕ = 0 (H) and π (C). Efficient oper-ation would then require sufficiently fast thermalizationwithin the respective time windows.To conclude the presentation of our model, we em-phasize that while the choice of modulating the couplingrate via the rotor’s angular position is genuinely novel,the Langevin equations and the master equation can stillbe obtained following the original derivations presentedin [37]. This will allow us to trace out the bath degreesof freedom and describe their influence in terms of aneffective thermalization rate κ = 2 πγ [38]. For thistreatment to be valid, the latter is assumed to be smallcompared to the (freely adjustable) mode frequency andthe spectral variation in the bath coupling. III. CLASSICAL REGIME
Nonlinear stochastic dynamics.–
We start by studyingthe dynamics of the rotor heat engine in the classicalregime. To this end, we consider the classical limit of thequantum Langevin equations for the rotor coordinates and the complex mode amplitude [37],d ϕ = L z /I d t, (6)d L z = (cid:126) g | a | sin( ϕ ) d t − (cid:88) T = H,C (cid:126) √ κ ¯ n T f (cid:48) T ( ϕ ) Im ( a ∗ d w T ) , d a = − [ ig cos( ϕ ) + κ ( ϕ ) / a d t − (cid:88) T = H,C √ κ ¯ n T f T ( ϕ ) d w T . The second and third line are stochastic differential equa-tions in Itˆo form [39]. For a fixed angle ϕ , the thirdline describes the thermalization of the mode with thetwo baths. Here w H and w C are complex Wiener pro-cesses, i.e. continuous stochastic processes with inde-pendent time increments d w T that take complex valuesfollowing a normal distribution with | d w T | = d t . Theycorrespond to the noise incoming from the baths, with κ ( ϕ ) = κ (cid:2) f H ( ϕ ) + f C ( ϕ ) (cid:3) (7)the overall decay rate of the mode intensity. For ourchoice of functions, we have κ/ ≤ κ ( ϕ ) ≤ κ , such thatthe mode is always in contact with a thermal bath forany position of the rotor. Note that the noise input alsoaffects the angular momentum of the rotor directly. Infact, the stochastic term in the second line of (6) can beunderstood as the classical counterpart of the quantummeasurement backaction due to the angle-dependent cou-pling to the baths. We omit it in the following classicalassessment of the engine performance, but we will seelater that it accounts for a part of the additional noise inthe quantum version of the engine.In describing the dynamics of the engine, we are in-terested in the evolution of statistical quantities like theaverages and variances of the random variables describedin the equations of motion (6). However, solving them isnot a straightforward matter, especially given the non-linear form of the radiation pressure term driving theangular momentum L z . In order to tackle this problem,we proceed in three steps: (i) reduce the two Wienerprocesses to a single one (ii) derive the equation of mo-tion for the mode intensity, from which we can simulateefficiently the dynamics (iii) perform an adiabatic elimi-nation of the mode, which will allow us to obtain compactanalytical results.As a first step, we thus make use of the fact that thesum of two independent Wiener processes can be ex-pressed as a single effective Wiener process w eff , namely (cid:88) T = H,C √ κ ¯ n T f T ( ϕ ) d w T = (cid:112) κ ( ϕ )¯ n ( ϕ ) d w eff . (8)In the classical regime, the two baths can thus be re-duced to a single bath with an effective thermal occupa-tion modulated by the rotor’s position¯ n ( ϕ ) = f H ( ϕ )¯ n H + f C ( ϕ )¯ n C f H ( ϕ ) + f C ( ϕ ) . (9)As we shall see later, this intuitive simplification doesnot generalize to the quantum regime where coherencebetween different angles may occur. (a) = 100 g = g Time t p hg=I A n g u l a r f r e q u e n c y L z = I p h g = I (b) = 100 g = g Time t p hg=I A n g u l a r p o s i t i o n ' = : (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:1) (cid:2)(cid:2) (cid:4)(cid:5)(cid:6)(cid:7) (cid:3)(cid:4) (cid:8) (cid:4) (cid:5) (cid:6) (cid:7) (cid:3) (cid:4) (cid:9) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) FIG. 2. Classical simulation of the engine dynamics. Panels (a) and (b) depict, respectively, the angular momentum and theunwrapped angle of the rotor for two exemplary cases of fast ( κ = 100 g ) and slow ( κ = g ) thermalization. The lines representthe mean values for a numerical sample over 10 trajectories, the shaded areas cover two standard deviations. Red and blueareas on the top indicate the sign of sin (cid:104) ϕ ( t ) (cid:105) for κ = 100 g , i.e. where the working mode is mostly in contact with the hot andthe cold bath. Panel (c) shows the angular two-time correlation function S ϕϕ ( t , t ) for κ = 100 g . S ϕϕ ( t , t ) = 1 implies thatthe angular position at time t can be predicted from its value at time t (and vice versa) while S ϕϕ ( t , t ) = 0 correspondsto uncorrelated angular random variables. All simulations start with the rotor at rest ( L z (0) = 0, ϕ (0) = π/ I = (cid:126) /g ), andreservoirs at (¯ n H , ¯ n C ) = (1 , Given that the phase of the mode does not impact thedynamics of the rotor heat engine, the model can be re-duced further by solely considering the mode intensity | a | , whose Itˆo stochastic differential equation can be de-rived from the Fokker-Planck equation [39, 40],d ϕ = ( L z /I )d t, d L z = (cid:126) g | a | sin( ϕ ) d t (10)d | a | = − κ ( ϕ ) (cid:2) | a | − ¯ n ( ϕ ) (cid:3) d t + (cid:112) | a | κ ( ϕ )¯ n ( ϕ ) d W, where the characteristic frequency of the engine motionis (cid:112) (cid:126) g/I . Here W is a single real-valued Wiener process(d W = d t ), the only remaining source of randomnessthat enters the classical backaction-free model of the en-gine. Note that the Itˆo calculus used here implies thatthe dynamical variables | a | ( t ), L z ( t ) and ϕ ( t ) are non-anticipating functions of the noise [39], i.e. they are in-dependent of the behaviour of the Wiener process W inthe future of t . Numerical simulation.–
Typical results from a numeri-cal Monte Carlo integration of the classical engine model(10) are shown in Fig. 2 for exemplary cases of fast andslow thermalization (see [41] for animated trajectories).Starting from the rotor at rest ( L z (0) = 0, ϕ (0) = π/ (cid:104) L z (cid:105) [green line in panel (a)] shows that theengine is accelerating clockwise. Moreover, the noise(shaded area) is relatively small and does not impact sig-nificantly the performance of the engine, which is in con-trast to the case of slow thermalization (grey). Lookingat the angle variable in panel (b) yields the same conclu-sion, with an additional subtlety. Indeed, while it is truethat the one-dimensional unbounded coordinate ϕ growswith negligible relative noise, the actual angle coordinateof the rotor is defined up to a multiple of 2 π . As a conse-quence, as soon as the standard deviation ∆ ϕ is of order π , the distribution of angles will essentially appear flat.Physically speaking, this means that one will not be ableto infer the exact angle of the rotor from a known value that lies several cycles in the past. This can be seen inFig. 2(c) where we plot the two-time correlation functionof the periodic angle variable [42] S ϕϕ ( t , t ) = R (cid:2) ϕ ( t ) − ϕ ( t ) (cid:3) − R (cid:2) ϕ ( t ) + ϕ ( t ) (cid:3)(cid:114)(cid:16) − R (cid:2) ϕ ( t ) (cid:3)(cid:17)(cid:16) − R (cid:2) ϕ ( t ) (cid:3)(cid:17) (11)where R [ φ ] = (cid:104) cos φ (cid:105) + (cid:104) sin φ (cid:105) . The width of S ϕϕ withrespect to | t − t | determines the time window of phasestable motion, before the angular position is completelydiffused. However, the fact that the phase stability of themotion is bounded by thermal fluctuations only poses apractical limitation if they are able to stop or reverse theaverage spinning direction. In other words, the relevantconditions for a steady operation of the engine are thatthe relative spread ∆ L z / (cid:104) L z (cid:105) in the angular momentummust be small and that the two-time correlations S ϕϕ extend over more than one cycle of rotation. Adiabatic elimination.–
The equations for the classi-cal description are exact so far. In order to charac-terize analytically the engine’s operation, we now adi-abatically eliminate the mode variable | a | ( t ). Specifi-cally, we assume that thermalization occurs on a muchshorter timescale than the motion of the rotor such thatthe mode intensity will assume its mean value (9) al-most instantaneously for each angle ϕ ( t ). To be ex-plicit, let us separate the small noise deviations from themean in the mode intensity, | a | ( t ) = ¯ n [ ϕ ( t )] + ε a ( t ),and insert it into the equation of motion (10). We ob-tain d ε a = − κ ( ϕ ) ε a d t + (cid:112) κ ( ϕ )¯ n ( ϕ )(¯ n ( ϕ ) + ε a )d W , arandom variable that contains information of the rotortrajectory integrated over a time scale of 1 /κ . At lowrotation speed L z /I (cid:28) κ , we can neglect this short-timememory effect and assume that any function of the angle ϕ ( t ) is non-anticipating for ε a ( t ).This approximation, which we use to derive the ana-lytical results below, corresponds to the desired regimeof operation for the engine (if an external load were at-tached to the piston). Indeed, it is precisely when themode is given sufficient time to thermalize with the bathsthat heat can be extracted to create the required bias inradiation pressure. Optimal performance is thus reachedin the limit κ → ∞ , whereas κ -dependent corrections areexpected to appear for κ ∼ (cid:112) (cid:126) g/I or κ ∼ (cid:104) L z (cid:105) /I .As the rotor accelerates, it will eventually enter thislatter regime where the approximation breaks down andthe angular frequency saturates. This situation wouldbe analogous to a car engine that could not follow theopening and closing of its valves given the fast rotation ofthe camshaft. However, a car engine would rarely operatein such a regime as the targeted velocities under load arekept well below the intrinsic thermalization rates. Thesaturation can already be seen in Fig. 2 (a) for the case κ = g . Additionally, other effects would start to play arole at high speeds, such as friction of the rotor which isnot included in the present model since it is difficult toinclude rigorously, although it could be done for examplefollowing the results of Ref. [43].At this point, we have all the necessary information toderive the rates at which the average angular momentum (cid:104) L z (cid:105) and its variance ∆ L z increase as a function of time.They read˙ (cid:104) L z (cid:105) = (cid:126) g (cid:104) sin( ϕ )¯ n ( ϕ ) (cid:105) F.R. → (cid:126) g (1 − √ n H − ¯ n C ) , (12)˙∆ L z = 2 (cid:126) g (cid:28) sin( ϕ )¯ n ( ϕ ) (cid:16) L z − (cid:104) L z (cid:105) + (cid:126) g sin( ϕ )¯ n ( ϕ ) κ ( ϕ ) (cid:17)(cid:29) F.R. → (cid:126) g κ (cid:20) (1 − √ n H + ¯ n C ) + 38 √ n H − ¯ n C ) (cid:21) , where F.R. → stands for the limit when the gain in angularmomentum per cycle is small enough so that the quanti-ties can be averaged over one round-trip of free rotation, (cid:104) g ( ϕ ) (cid:105) F.R. → / π (cid:82) π d ϕ g ( ϕ ). This yields compact ana-lytical expressions in spite of the nonlinear dynamics (seeFig. 3 for a comparison with the exact dynamics).Following our intended goal, the average angular mo-mentum (cid:104) L z (cid:105) is driven in proportion to the difference inthermal occupation of the baths, which in turn drivesthe average angle coordinate (cid:104) ϕ (cid:105) . Inevitably, heat alsoenters the system in the form of noise, limiting the phasestability and accumulating uncertainty in the angle as afunction of time. Classically, this happens at finite tem-peratures even in the absence of driving, ¯ n H − ¯ n C = 0.Indeed, the angular momentum variance ∆ L z acquirescontributions from the difference as well as the sum inthermal occupation of the baths. However, it grows lin-early in time as does (cid:104) L z (cid:105) , which implies that the relativenoise ∆ L z / (cid:104) L z (cid:105) decreases over time. In other words, therotor behaves as a clock with a steadily improving signal-to-noise ratio. Work output.–
By looking at the dynamics, we haveshown that the classical version of the rotor heat engine (a)
Time t p hg=I h _ L z i = h g (b) Time t p hg=I -0.2-0.100.10.2 _ " L z = h g p h g I FIG. 3. Rate of increase of (a) the average angular momentum (cid:104) L z (cid:105) and (b) its variance ∆ L z for the parameters of Fig. 2.The green solid line corresponds to the direct computation ofthe derivative from the simulated dynamics. The dotted anddashed line are obtained from the analytical rates given inEq. (12), respectively before and after taking the limit of freerotation. As expected, the latter description is only valid oncethe engine has started and the gain in angular momentumwithin each cycle is sufficiently small. is achieving its goal of extracting useful directional mo-tion from the heat transfer between the baths. How wellit performs its task can be further quantified in termsof energy flows. In each engine cycle, the working modeis extracting a positive net amount of useful rotationalenergy by pushing via the radiation pressure force ontothe piston. The latter plays the role of a flywheel [44]that stores this energy, e.g. for later extraction by exter-nal loads. The corresponding mechanical work generatedalong the cycle is given by δ W = F d x = (cid:126) g | a | sin( ϕ )d ϕ. (13)where x = − x cos( ϕ ) is the vertical piston positionwhich determines the volume of the working mode. Thepressure is given by the force that pushes downwards,i.e. in the positive x -direction. It is proportional tothe mode intensity [32, 45], F = (cid:126) g | a | /x , as fol-lows from the optomechanical potential in (1). In theideal case of fast thermalisation where | a | → ¯ n ( ϕ ), wecan integrate over one cycle to obtain the upper bound W cyc = (cid:126) gπ (2 − √ n H − ¯ n C ) on the mean work out-put per cycle. Figure 4 illustrates the engine cycle ina pressure-volume diagram, comparing the ideal cycle(solid line, clockwise rotation) to a snapshot of 10 sim-ulated trajectories based on the same parameters as inFig. 2. The area enclosed by the data points yields theaverage work output, upper bounded by the ideal case(solid line). The data for κ = 100 g (green outer cycle)follows the ideal curve closely and outputs almost max-imum work (98%), whereas the data for κ = g (grey Radiation pressure Fx = hg P i s t o np o s i t i o n x = x -101 FIG. 4. Engine cycle in a pressure-volume diagram, wherethe mode volume grows linearly with the piston position x .The solid line represents the ideal cycle running clockwiseand producing W cyc , where the cavity has time to thermalize | a | → ¯ n ( ϕ ). The color indicates the sign of sin φ , i.e. thedominant bath coupling. The markers show the mean radi-ation pressure in 100 bins of ϕ mod 2 π , sampled from 10 trajectories evolved to the time gt = 30, with the parametersof the fast (outer cycle) and slow (inner cycle) cases chosenas in Fig. 2. inner cycle) performs significantly worse (27%) since theworking mode already lacks time to thermalize with eachbath. This confirms our previous observation in Fig. 2(a)that the mechanical output of the engine deteriorates inthe saturated regime of fast rotation.A direct consequence of the autonomous engine designis that the cycle duration is fluctuating and decreasingover time, as opposed to externally driven Otto cycleswhere the clock reference is provided by the machine op-erator [23, 46]. It will prove expedient to measure theengine’s work performance in terms of the mean outputpower P W = (cid:28) δ W d t (cid:29) = (cid:126) gI (cid:10) | a | sin( ϕ ) L z (cid:11) (14) ≈ (cid:126) gI (cid:104) ¯ n ( ϕ ) sin( ϕ ) L z (cid:105) F.R. → W cyc π (cid:104) L z (cid:105) I , i.e. the rate at which work is performed on the piston[47]. It is a function of the angular frequency in the sameway the horsepower of a car engine is a function of thenumber of revolutions per minute (RPM). In the limit offree rotation and fast thermalization, the mean outputpower is simply given by the work per cycle divided bythe average period of rotation.Notice that the output power gives the average rate atwhich the kinetic energy L z / I of the rotor increases dueto radiation pressure, as described by the backaction-freedynamics (10). This means that a finite output powerdoes not guarantee that the engine is spinning in a fixeddirection, a prerequisite for the generated motion to beuseful. Using (14) as a figure of merit for the engineperformance is thus only meaningful in combination with (cid:104) L z (cid:105) (cid:38) (cid:10) L z (cid:11) . Heat input.–
Having identified the work output δ W inEq.(13) as the change of the working mode energy caused by the moving piston, we can identify heat as the energychange resulting from a change in mode occupation | a | .In accordance with the first law of thermodynamics, theenergy change d E = d( (cid:126) ω ( ϕ ) | a | ) due to heat transfer isthus δ Q = d E + δ W = (cid:126) ω ( ϕ ) d | a | , (15)where ω ( ϕ ) = ω + g cos( ϕ ) is the mode’s resonance fre-quency modulated as a function of the piston position.Note that we consider the standard regime of cavity op-tomechanics [32, 45] where this modulation of the mode’sboundary condition, which gives rise to the optomechan-ical potential in (1), is accounted for to first order in∆ ω = 2 g (cid:28) ω .To compute the efficiency of the engine, we must distin-guish between the heat input δ Q H from the hot reservoirand the output into the cold. The change in the mode oc-cupation d | a | corresponding to its thermalization withthe hot bath is given by the respective dissipator plusnoise,d H | a | = − κf H ( ϕ ) (cid:2) | a | − ¯ n H (cid:3) d t (+ noise) , (16)where the noise term will average out in the following. Inan idealized scenario of clearly separated work and heatstrokes [22, 23], where the mode is allowed to thermal-ize with each reservoir separately and is entirely isolatedfrom them when performing work, the mean input of thehot bath would add up to (cid:126) ω H (¯ n H − ¯ n C ) per cycle, corre-sponding to the hot thermalization stroke at a frequency ω H [23]. In contrast, our choice of overlapping couplingfunctions f C ( ϕ ) and f H ( ϕ ) leads to a greater averageheat consumption since the simultaneous interaction withboth reservoirs implies a balance of heat flows, where thehot bath input (16) must counteract losses into the coldbath in order to maintain the average occupation ¯ n ( ϕ ),similar to the Stirling engine of Ref. [15]. Hence themode never reaches the highest mean value ¯ n H except at ϕ = π/
2. We quantify the average heat consumption asa function of time in terms of the input power P H = (cid:28) δ Q H d t (cid:29) = (cid:126) κ (cid:10) ω ( ϕ ) f H ( ϕ )(¯ n H − | a | ) (cid:11) (17) ≈ (cid:126) κ (cid:10) ω ( ϕ ) f H ( ϕ )(¯ n H − ¯ n ( ϕ )) (cid:11) F.R. → (cid:126) ω κ √ − /
44 (¯ n H − ¯ n C ) . Here the second line is a good approximation in theregime of fast thermalization, contrary to a scenario ofseparated strokes of hot and cold thermalization.
Efficiency.–
The thermal efficiency of the engine in thenon-stationary case without external load is η = P W P H F.R. → gω (cid:104) L z (cid:105) Iκ − √ √ − / . (18)Our autonomous design implies that the efficiency is afunction of the parameter (cid:104) L z (cid:105) /Iκ . In particular, thefaster the engine spins, the better its efficiency, as shownin Fig. 5. This behavior is valid up to the regime where (cid:104) L z (cid:105) /I ∼ κ , in which case the adiabatic elimination doesnot hold anymore, as discussed previously. In fact, thisdependence is a feature shared with actual car engines,for which the efficiency typically follows a bell-shapedcurve as a function of the RPM [48]. In our case, higherefficiencies could in principle be obtained by adapting theprofiles f H ( ϕ ), f C ( ϕ ) and/or the thermalization rate κ as a function of the angular frequency L z /I .The efficiency (18) is inherently small, being propor-tional to both 2 g/ω (cid:28) (cid:104) L z (cid:105) /Iκ (cid:28)
1. How-ever, given the absence of explicit temperatures, a natu-ral question that arises is whether our engine is boundedby the Carnot efficiency η Carnot = 1 − T C /T H with thetemperatures T H > T C associated to the two baths. Toanswer this question, we first note that given the mod-ulation of the mode frequency ω ( ϕ ), the correspondingthermal occupation¯ n T ( ω ) = ( e (cid:126) ω/k B T − − (19)is also slightly fluctuating along the cycle, which we haveneglected in our model. In order for the engine to pro-duce work, the lowest possible hot occupation number¯ n H ( ω + g ) must be greater than the highest possiblecold occupation ¯ n C ( ω − g ) number. Inserting this intoEq. (19) leads to the condition ω − gT C > ω + gT H (20)which in turn implies η Carnot > − ω − gω + g = 2 gω + O (cid:34)(cid:18) gω (cid:19) (cid:35) . (21)As shown in Fig. 5, the efficiency of the engine re-mains below this bound in both the regime of operation (cid:104) L z (cid:105) /Iκ (cid:28)
1, described by Eq. (18), as well as in thesaturated regime where the efficiency starts to drop asexpected.
IV. QUANTUM REGIME
Master equation.–
The master equation governing thedynamics of the quantum heat engine is given by [37]˙ˆ ρ = − i (cid:126) (cid:104) ˆ H S , ˆ ρ (cid:105) (22)+ (cid:88) T = H,C κ (¯ n T + 1) D [ f T ( ˆ ϕ )ˆ a ] ˆ ρ + κ ¯ n T D (cid:2) f T ( ˆ ϕ )ˆ a † (cid:3) ˆ ρ, where D [ ˆ O ]ˆ ρ = ˆ O ˆ ρ ˆ O † − (cid:110) ˆ O † ˆ O , ˆ ρ (cid:111) is the Lindblad su-peroperator [49].As stated before, the mode frequency does not enterthe description as the radiation pressure term is only pro-portional to the mode occupation. This means that we Carnot Bound
10 50 100 500
Time t p hg=I E / c i e n c y ! = g FIG. 5. Efficiency of the engine in units of 2 g/ω for the pa-rameters of Fig. 2. The green solid line corresponds to thedirect computation of the efficiency from the simulated dy-namics, using the expressions (14) and (17) before the adia-batic elimination and the free-rotation limit. The black lineis obtained from the analytical result given in Eq. (18) andis plotted in the regime of operation (solid) and in the satu-rated regime (cid:104) L z (cid:105) /Iκ ≥ − (dotted). At short times, theefficiency oscillates as the free-rotation limit is not valid, whilein the long-time limit the efficiency deviates from (18) due tothe mode not thermalizing fast enough. can match the frequency to the weak-coupling conditionrequired for the validity of the master equation. In Ap-pendix A, we derive this master equation using a weak-coupling approach, and we check that the Lindblad termsare consistent with a thermodynamic interpretation ofthe engine dynamics in terms of entropy flows [50–52].In contrast to the classical regime, the dissipative cou-pling of the mode to the reservoirs cannot be reduced toan effective single-bath term. This is a manifestation ofquantum coherence in the angle coordinate ϕ , since off-diagonal matrix elements (cid:104) n, ϕ | ˆ ρ | n, ϕ (cid:48) (cid:105) are influenced byboth reservoirs simultaneously. In fact, the Lindblad dis-sipators will lead to decoherence in the angle coordinate,as the exchange of photons with the reservoirs revealsinformation about the piston position, i.e. constitutes acoarse angle measurement by the environment. Initialization.–
For the rotor to start spinning in theright direction, we initialize its angular position in the re-gion ϕ ∈ [0 , π ] where the mode is predominantly coupledto the hot bath. This implies that any convex combina-tion of energy eigenstates – such as a thermal distribution– should be avoided, since in this case the rotor is com-pletely delocalized. Alternatively, one could start with aninitial displacement in momentum. However, this wouldrequire an external energy source, just like the battery ofa starting car engine, which we avoid in our autonomousmodel.Here we select an initial pure state given by the peri-odic von Mises wavefunction ψ i ( ϕ ) = e k cos( ϕ − µ ) (cid:112) πI (2 k ) , (23)where I (2 k ) is a modified Bessel function. With thischoice, the angular position of the rotor is localized (a) Time t p hg=I A n g u l a r f r e q u e n c y L z = I p h g = I (b) : /2 : : /2 2 : Angular position ' (modulo 2 : ) T i m e t p h g = I (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . (d) Time t p hg=I A n g u l a r f r e q u e n c y L z = I p h g = I (e) : /2 : : /2 2 : Angular position ' (modulo 2 : ) T i m e t p h g = I (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) . . . . FIG. 6. Simulation of the engine dynamics for
Ig/ (cid:126) = 100 k (first row) and Ig/ (cid:126) = k (second row). All the results obtainedfrom the quantum model (in yellow) are compared to the classical backaction-free case (in green). Panels (a) and (d) depictthe angular momentum while (b) and (c) show the evolution of the angular distribution. Panels (c) and (f) show the angulartwo-time correlation function S ϕϕ ( t , t ), which we symmetrize in the quantum regime to obtain a real quantity (note thatthe classical version is always real and symmetric). The grey area in (d) shows the uncertainty predicted by a classical modelincluding backaction noise. The engine parameters are ( k = 10, µ = π/ κ = 10 / (cid:112) (cid:126) g/I ). around a mean value µ with a spread determined by theparameter k . For a sufficiently localized angle, i.e. for k (cid:29)
1, the angle distribution is approximately Gaussian,with a standard deviation 1 / √ k (cid:28) π/
2. The corre-sponding angular momentum distribution covers a spec-trum of order √ k quanta around its zero average. Nofurther initialization is required, and any net rotation ofthe rotor will come entirely from the engine dynamics.Ideally, the rotor will evolve in such a way that its aver-age angular momentum (cid:68) ˆ L z (cid:69) increases with time, whilethe relative spread ∆ L z / (cid:68) ˆ L z (cid:69) remains small. In con-trast to the classical model, however, quantum mechanicswill impose additional constraints for the performance ofthe engine. In particular, for the rotor to start spinning,the initial acceleration of the engine must exceed the freedispersion of the rotor. This is roughly the case if the ini-tial spread in kinetic energy, ∆ E (0) ≈ (cid:126) k/ I , is smallcompared to the gain (cid:126) g ¯ n H in potential energy during thefirst half-cycle, Ig ¯ n H / (cid:126) (cid:29) k . The larger Ig ¯ n H / (cid:126) is, themore the quantum rotor approaches classical behaviourand the less it is prone to free dispersion. Numerical simulations.–
We employ two numericalmethods to simulate the dynamics of the quantum en-gine. One is direct integration using the QuTip packagein Python over a truncated Hilbert space [53]. Specifi-cally, we restrict the rotor Hilbert space to angular mo-mentum quantum numbers m min ≤ m ≤ m max , withsuitably chosen bounds to cover the occupied spectrum at all times. The thermal mode is allowed to have atmost 8 excitations, which is sufficient given that we oper-ate with (¯ n H , ¯ n C ) = (1 ,
0) throughout. Higher reservoirtemperatures are computationally expensive but do notprovide further insight into the engine’s operation. Wechoose ¯ n C = 0 for simplicity, but of course this shouldbe interpreted as a negligibly small mean photon num-ber at appropriate mode frequencies and temperatures, (cid:126) ω (cid:29) k B T C . We consider 0 < ¯ n C (cid:28) Quantum vs. Classical.–
The results of the numericalsimulations for both the classical backaction-free modeland the quantum model are summarized in Fig. 6 for twodifferent choices of parameters. We have selected valuesof the moment of inertia I and the coupling strength g such that Ig/ (cid:126) (cid:29) k in one case, while Ig/ (cid:126) = k in theother case. Additionally, these values are set such thatthe classical dynamics are essentially unchanged in bothcases, allowing us to showcase how properties that areirrelevant classically become meaningful in a quantumsetting.As discussed previously in reference to Eq. (23), dueto the uncertainty principle, it is impossible to perfectlylocalize both the angular position and angular momen-tum of a quantum rotor. In order to make a fair com-parison and mimic this in the classical case, we initializethe rotor’s angular position and angular momentum ina Gaussian probability distribution, with mean π/ / √ k for the angular position, andmean 0 and standard deviation (cid:112) k/ k (cid:29)
1. This initialization of the classical rotor results ina free dispersion of the angle coordinate over time similarto the quantum case, and so it allows us to distinguishquantum effects that arise due to initialization from ef-fects that originate from the rotor’s interaction with theworking mode.As seen in Fig. 6(a), the quantum engine shows al-most identical behaviour for the angular frequency as inthe classical case when
Ig/ (cid:126) (cid:29) k , as expected. However,for Ig/ (cid:126) = k in panel (d), the quantum model yields amuch larger variance around the mean value, even thoughthe latter still increases steadily in time. We can alsoexamine the distribution of the angular position of therotor as a function of time. Here we see that there is sig-nificantly less broadening when Ig/ (cid:126) (cid:29) k in panel (b),whereas the angular distribution is almost flat before itcompletes one revolution for Ig/ (cid:126) = k in panel (e). Clas-sically, this is explained by different spreads in angularfrequency for a given spread in angular momentum. Inthe quantum case, however, the additional noise contri-butions not only broaden the angle distribution further,but also impact the phase stability of the rotor engine.This is shown in panels (c) and (f), where the symmet-ric two-time correlation function S ϕϕ ( t , t ) is plotted forthe quantum (upper triangle) and classical (lower trian-gle) cases. While the phase stability of the classical en-gine survives in the regime Ig/ (cid:126) = k , correlations dropalmost instantaneously in the quantum case. Classical backaction.–
In fact, in the two regimes wehave explored, the amount of noise in the quantum caseis strictly larger than in the classical one. This addi-tional uncertainty arises due to the combined effect ofmeasurement backaction noise and of vacuum fluctua-tions contributing to the noise input of the hot and thecold bath. While the latter is a genuine quantum feature,backaction noise can be accounted for in a classical enginemodel. Specifically, the difference to the backaction-freemodel (10) is an additional noise term in the equationfor the angular momentum variable,d L z = (cid:126) g | a | sin( ϕ ) d t (24) − (cid:126) (cid:114) κ | a | (cid:110) ¯ n C [ f (cid:48) C ( ϕ )] + ¯ n H [ f (cid:48) H ( ϕ )] (cid:111) d U. Here, d U stands for the increment of a second, indepen-dent Wiener process, (cid:104) d U d W (cid:105) = 0 (See Appendix B fora derivation based on the classical Langevin equations(6)). For an exemplary comparison, the uncertainty pre-dicted by this model is evaluated and depicted as the (a)
11 12 13 14 15
Time t p hg=I E / c i e n c y ! = g P H = h! P W = hg (b)
11 12 13 14 15
Time t p hg=I P o w e r P = p h g = I FIG. 7. Performance of the engine in the regime
Ig/ (cid:126) = k of Fig. 6 where quantum effects play a role. The quantummodel (in yellow) is compared to the classical results averagedover 5 × trajectories with (black dashed) and withoutbackaction noise (green). (a) Efficiency in units of 2 g/ω . (b)Heat input power (top three lines) and work output power(bottom three lines). grey-shaded area in Fig. 6(d). It accounts for about halfof the excess noise of the quantum model for the specificlow-temperature parameters considered here. At hightemperatures, ¯ n C , ¯ n H (cid:29)
1, we expect a good match withthe quantum prediction, even in the regime of low iner-tia where the additional uncertainty would mainly comefrom the backaction noise.
Quantum efficiency.–
In Figure 7, we compare thequantum and the classical version of the engine in termsof the thermal efficiency (18). It is defined in terms ofthe mean work output power (14) and heat input power(17), which carry over to the quantum model if we replacethe rotor and mode coordinates by the respective opera-tors and symmetrize the product of ˆ L z and ˆ ϕ -dependentterms for hermiticity. The input power follows from themaster equation (22) if we consider only the dissipator tothe hot bath for the time derivative of the cavity energy.We define the power P C of the heat flow dumped intothe cold bath in the same way, P H,C = (cid:126) κ (cid:10) ω ( ˆ ϕ ) f H,C ( ˆ ϕ )(¯ n H,C − ˆ a † ˆ a ) (cid:11) . (25)The mechanical output power reads as P W = (cid:126) g I (cid:68) ˆ a † ˆ a (cid:110) sin( ˆ ϕ ) , ˆ L z (cid:111)(cid:69) , (26)which also corresponds to the change of the rotor’s kineticenergy due to radiation pressure (excluding the contribu-tion of backaction noise). In total, a dynamical version ofthe first law [55] can be formulated for the time derivativeof the work mode energy in the quantum case, ∂ t (cid:10) (cid:126) ω ( ˆ ϕ )ˆ a † ˆ a (cid:11) = P H + P C − P W . (27)0Note that, in order to make sense of the work outputstored in the rotational motion, we define the energy bal-ance with respect to the work mode subsystem excludingthe rotor degree of freedom. Hence we do not accountfor the increase of its kinetic energy due to backaction,as given by the power P B = dd t (cid:42) ˆ L z I (cid:43) − P W (28)= (cid:126) κ I (cid:28)(cid:20) ¯ n H + ¯ n C a † ˆ a + 1) + ˆ a † ˆ a (cid:21) cos ( ˆ ϕ ) (cid:29) ≥ . While we have seen that it can be responsible for a sig-nificant noise contribution to the motion of low-inertiarotors, P B will be much smaller than the direct heat flows(25) in and out of the work mode for ω (cid:29) (cid:126) /I . Hence wecan safely operate with P H as the input power to assessthe quantum efficiency.We also remark that the intrinsic definition of P W ,which reflects the growth in kinetic energy of the ro-tor, will in general overestimate the amount of work thatcould be extracted by an external load in the low-inertiaquantum regime. Indeed, the large uncertainty aroundthe average angular momentum in Fig. 6(d) indicates asignificant difference between the rotor’s average kineticenergy (cid:68) ˆ L z (cid:69) / I and the energy (cid:68) ˆ L z (cid:69) / I associated toits average directional motion. Formally, the latter couldbe extracted by applying a unitary momentum displace-ment operator, leaving behind passive energy in the formof momentum noise [56, 57]. The distinction turns irrel-evant as the momentum signal-to-noise improves in thehigh-inertia regime.We use the low-inertia parameter regime of before( Ig/ (cid:126) = k ) for a pronounced difference between the quan-tum and the classical models. The yellow line in Fig. 7(a)represents the quantum result, which is systematicallybelow the classical results with (black dashed) and with-out (green) backaction. In fact, as shown in Fig. 7(b),the classical version of the engine is consistently extract-ing more work from less input heat. The reason lies inthe additional vacuum fluctuations acting on the workingmode and deteriorating correlations between the numberoperator ˆ n and the rotor observable sin( ˆ ϕ ). The signof the latter encodes whether the rotor is predominantlycoupled to the hot or the cold reservoir, and it thus getscorrelated with ˆ n as the engine operates. These correla-tions enter the mean input (17) and output (14) powerand are affected by both the backaction noise and thevacuum fluctuations, as is reflected by the lower efficien-cies in Fig. 7 compared to the classical backaction-freeresult. The classical and quantum efficiencies convergein the regime of high inertia ( Ig/ (cid:126) (cid:29) k ) and high tem-perature (¯ n C (cid:29)
1) where neither backaction nor vacuumnoise play a role.Overall, the quantum heat engine operates best when itapproaches the classical limit of a large moment of inertiatogether with a large coupling strength. Free dispersion, vacuum fluctuations, and backaction noise do not directlyaffect the average mechanical output, but they can ruinthe phase stability of the rotor and are thus problem-atic for the proper functioning of the engine. Our resultsshowcase the importance of studying the actual dynam-ics of heat engines and of addressing countermeasures toquantum sources of noise.
V. CONCLUSION
Inspired by actual piston engines, we have proposedan autonomous rotor heat engine described by standardHamiltonians. By solving the underlying equations ofmotion in the transient regime where the rotor acceler-ates from rest, we have shown analytically that the enginefunctions as desired in the classical regime, while identify-ing the best regimes of operation. We have also exploredthe role of quantum effects, and our results show that, inthe case of our engine, they mainly give rise to additionalnoise in the motion of the rotor. It is a relevant questionwhether other quantum effects such as entanglement andcoherence can lead to a better performance compared toa fully classical version of the engine. To this end, our ro-tor heat engine provides a suitable testbed for the variousnotions and concepts that have been put forward in thecontext of quantum thermodynamics. For example, thestudy of Otto-cycle-type of modulating functions, as wellas quantum-engineered initial states of the rotor, couldopen new room for improvement of the engine.With our results, our aim is to generate new theoret-ical insights into how thermal energy can be convertedautonomously into useful mechanical motion, and to un-derstand the role that quantum mechanics plays in ther-mal machines. With respect to an implementation of theengine, the technological challenge is to simultaneouslybuild a system where the thermalization rate is fast com-pared to the angular frequency of the rotor, where thedissipation of the rotor is negligible on the timescale ofoperation of the engine, and where the frequency of themode is significantly larger than the coupling to the en-vironment. Typical high-Q cavities are well suited toimplement the working mode [32]. As for the rotor de-gree of freedom, it is possible to consider non-mechanicalsystems where a phase variable plays the role of the an-gular position, such as the flux in electric circuits basedon Josephson junctions.Finally, we note that an analogy can be drawn betweenour engine and the one-dimensional motion of a parti-cle in a periodic potential generated by, say, a standing-wave cavity field [58]. In this context, the accelerationof the rotor can also be understood as the reverse of aSisyphus cooling scheme, where the rotor runs effectivelymore down- than uphill on the potential energy curve[59, 60].1
ACKNOWLEDGMENTS
We thank Paul Skrzypczyk, Dario Poletti, Colin Teo,and Marc-Antoine Lemonde for fruitful discussions. Thisresearch is supported by the Singapore Ministry ofEducation through the Academic Research Fund Tier3 (Grant No. MOE2012-T3-1-009); by the NationalResearch Foundation, Prime Ministers Office, Singa-pore, through the Competitive Research Programme(Award No. NRF-CRP12-2013-03); and by both above-mentioned source, under the Research Centres of Excel-lence programme.
Appendix A: Weak-coupling derivation of themaster equation
Here we rederive the master equation (22), which wasstated following the Gardiner-Collett derivation [37] ap-plied to each bath. Following the standard Born-Markovsecular approach [54, 61], we switch to the interactionpicture with respect to the free system and bath Hamilto-nians (1) and (2), and we rewrite the interaction Hamil-tonian (4) as ˆ H I ( t ) = i (cid:126) (cid:80) T = H,C (cid:104) ˆ A T ( t ) ˆ B † T ( t ) − H . c . (cid:105) with ˆ A T ( t ) = e i ˆ H S t/ (cid:126) ˆ af T ( ˆ ϕ ) e − i ˆ H S t/ (cid:126) , (A1)ˆ B T ( t ) = (cid:90) ∞ d ω Γ( ω )ˆ b T ( ω ) e − i ( ω − ω ) t . Contrary to the simplified Gardiner-Collett treatment,we still keep the restriction to positive physical frequen-cies and account for the frequency dependence of thecoupling rate density Γ( ω ) at this point. Notice thatthe secular approximation is already implied as only res-onant interaction terms are considered; for this we mustassume that the modulation of the bare mode frequencyby radiation pressure is negligible, g (cid:28) ω .In the Born approximation, we describe the two inde-pendent baths by stationary Gibbs states ˆ σ T in the for-mal solution for the time evolution of the reduced systemstate ˆ ρ I ( t ),˙ˆ ρ I ( t ) = − (cid:90) t d t (cid:48) (cid:126) tr H,C (cid:110)(cid:104) ˆ H I ( t ) , (cid:104) ˆ H I ( t (cid:48) ) , ˆ ρ I ( t (cid:48) )ˆ σ H ˆ σ C (cid:105)(cid:105)(cid:111) . (A2)Using the fact that the bath modes at different frequen-cies are uncorrelated, i.e. the noise correlation functions(3) with a frequency-dependent occupation ¯ n T ( ω ), thenon-vanishing bath correlation functions are given by (cid:68) ˆ B † T ( t ) ˆ B T ( t (cid:48) ) (cid:69) = (cid:90) ∞ d ω Γ ( ω )¯ n T ( ω ) e i ( ω − ω )( t − t (cid:48) ) ≈ κ n T ( ω ) δ ( t − t (cid:48) ) , (A3) (cid:68) ˆ B T ( t ) ˆ B † T ( t (cid:48) ) (cid:69) ≈ κ n T ( ω ) + 1] δ ( t − t (cid:48) ) . In the Markov approximation step, we omit the Lambshift, and we restrict to the coarse-grained time scales ofthe relevant system dynamics. We are allowed to do soif the latter is much slower than the inverse correlationtimes 1 /t H,C of the two reservoirs, which in turn mustbe much smaller than ω . Specifically, we consider theengine to operate in a regime where the effective thermal-ization rate κ marks the fastest time scale in the evolutionof the system state. Hence the Markov approximation re-quires that κ (cid:28) /t H,C (cid:28) ω . For free-space radiation,the typical correlation time depends on the temperature, t H,C ∼ (cid:126) /k B T H,C , which is typically much shorter thanachievable decay times of cavity modes at realistic tem-peratures T H,C .The Markov assumption amounts to using the noisecorrelation functions (3) with a constant γ and ¯ n T =¯ n T ( ω ) in the Gardiner-Collett approach, and it allows usto approximate (A2) by setting the upper integral boundto infinity and replacing ˆ ρ I ( t (cid:48) ) → ˆ ρ I ( t ) in the integrand.The resulting time-local master equation reduces to˙ˆ ρ I ( t ) ≈ κ (cid:88) T = H,C (cid:110) (¯ n T + 1) D (cid:104) ˆ A T ( t ) (cid:105) ˆ ρ I ( t )+¯ n T D (cid:104) ˆ A † T ( t ) (cid:105) ˆ ρ I ( t ) (cid:111) , (A4)which is equivalent to (22) in the Schr¨odinger frame (ro-tating at the bare mode frequency ω ).In general, a heuristic description of system-bath cou-plings to different temperatures by simply adding the as-sociated Lindblad dissipators to the master equation maylead to a violation of the second law of thermodynamics[50]. This does not happen here, as we have checked bycomputing the von Neumann entropy S ( t ) = − tr( ρ ln ρ )of the engine over time. The intrinsic entropy productionrate is obtained by taking the time derivative and sub-tracting the contributions associated to the heat flows inand out of the two reservoirs,˙ S int = d S d t − P H k B T H − P C k B T C . (A5)Assuming that the bare mode frequency ω exceeds therotation frequency quantum (cid:126) /I by several orders of mag-nitude, we omit the additional backaction heating term(28) that would act directly on the rotor. Negativity ofthe entropy production rate would indicate a violation ofthe second law [55]. Notice however that, in the absenceof external loads or rotor dissipation, the combined sys-tem of work mode and rotor will not evolve towards astationary (Gibbs) state. In fact, we rather observe thatthe entropy of the engine grows steadily over time as theengine spins up and more and more energy is stored inthe rotor.To evaluate (A5), finite temperatures are now explic-itly required, in particular ¯ n C >
0. We started fromthe parameter regime explored in Sec. IV and varied¯ n C = 10 − , − , − . In all cases, the intrinsic en-tropy production rate was consistently positive. Fig. 8illustrates the case of ¯ n C = 10 − , whose results for the2 FIG. 8. (a) Entropies of the quantum engine in the regime
Ig/ (cid:126) = k of Fig. 6, with ¯ n C = 10 − . The von Neumannentropy of the engine state in units of k B (black) is comparedto the entropies of the reduced work mode and rotor states(yellow and green), and to the sum of both (black dotted). (b)Rate of entropy change (black dotted) versus entropy flowsin and out of the engine (red and blue) in units of k B κ atshort times. The black solid line shows the intrinsic entropyproduction rate (A5). It is always positive, with a minimumvalue of 7 . × − k B κ at t = 0 . (cid:112) I/ (cid:126) g . relevant engine observables discussed in the main text arepractically indistinguishable from the zero-temperatureidealization. Panel (a) depicts the von Neumann entropy S ( t ) of the engine state (black) and of the reduced statesof work mode and rotor (yellow and green). The sum ofthe latter (dotted) evolves slightly above former, indicat-ing some correlation between rotor and work mode. In(b) we plot the intrinsic entropy production rate (blacksolid) and compare it to the individual terms in (A5),viz. the overall change d S / d t (dotted), and the entropyflows P H /k B T H and P C /k B T C from the two baths (redand blue). Appendix B: Classical engine model with backaction
We can simplify the classical Langevin equations (6)by expressing the working mode in terms of its action-angle variables (number and phase), i.e. a = a r + ia i = √ n exp( iθ ) with n = | a | and θ = arctan( a i /a r ). Noting that each complex Wiener process is a linear combinationof two independent real-valued Wiener processes, d w T =(d X T + i d Y T ) / √ X T = d Y T = d t , we obtaind ϕ = ( L z /I )d t (B1)d L z = (cid:126) gn sin( ϕ )d t − (cid:126) (cid:88) T √ κn ¯ n T f (cid:48) T ( ϕ ) [cos( θ )d Y T − sin( θ )d X T ] , d n = − κ ( ϕ ) [ n − ¯ n ( ϕ )] d t − (cid:88) T √ κn ¯ n T f T ( ϕ ) [cos( θ )d X T + sin( θ )d Y T ] . The last equation is obtained using the Itˆo rule, d n = a d a ∗ + a ∗ d a +d a d a ∗ . We omit the equation for the phase θ as it turns out to be irrelevant. In fact we notice thatthe square-bracketed terms in (B1) define a rotation ofd X T and d Y T to two new mutually independent Wienerprocesses given the non-anticipating phase angle θ , (cid:18) d W T d V T (cid:19) = (cid:18) cos( θ ) sin( θ ) − sin( θ ) cos( θ ) (cid:19) (cid:18) d X T d Y T (cid:19) , (B2)where (cid:104) d W T d V T (cid:105) = 0 and d W T = d V T = d t . Hence therotor dynamics does not depend on θ .Next we simplify further by combining the noise in-puts of the hot and the cold bath in the same way as in(8). This leaves us with only two independent Wienerprocesses U, W , andd L z = (cid:126) gn sin( ϕ )d t (B3) − (cid:126) (cid:114) κn (cid:110) ¯ n H [ f (cid:48) H ( ϕ )] + ¯ n C [ f (cid:48) C ( ϕ )] (cid:111) d U, d n = − κ ( ϕ ) [ n − ¯ n ( ϕ )] d t − (cid:113) κn [¯ n H f H ( ϕ ) + ¯ n C f C ( ϕ )]d W. The second equation for the mode intensity correspondsto the one in (10). The angular momentum equation,which contains the backaction noise, equals (24). 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