Quantum field heat engine powered by phonon-photon interactions

We present a quantum heat engine based on a cavity with two oscillating mirrors that confine a quantum field as the working substance. The engine performs an Otto cycle during which the walls and a field mode interact via a nonlinear Hamiltonian. Resonances between the frequencies of the cavity mode and the walls allow to transfer heat from the hot and the cold bath by exploiting the conversion between phononic and photonic excitations. We study the time evolution of the system and show that net work can be extracted after a full cycle. We evaluate the efficiency of the process.

Introduction.-Quantumthermodynamics studies physical processes at the quantum scale through the lens of thermodynamics [1][2][3].The overall aim of this field of research is to extend concepts initially developed in the classical theory, such as heat, work and thermodynamic efficiency, into the quantum domain where purely quantum features can be exploited [4,5].These include quantum correlations [6], quantum coherence [7] and vacuum fluctuations [8,9].One task is to propose and characterize novel thermodynamic cycles by taking advantage of the nonclassical nature of the working substance to extract work for different tasks [10,11].The interest in studying thermodynamical cycles at the quantum level goes beyond the mere possibility of reaching higher degrees of efficiency but aims towards miniaturization of future thermal machines based on quantum systems.
The quantum Otto cycle is a thermodynamic cycle that enjoys relative ease of theoretical implementation in a quantum framework compared to other cycles, especially if implemented in cavity-optomechanics [12][13][14][15][16][17].It consists of a combination of two thermodynamic "strokes": i) the isochoric transformation, performed by maintaining constant the spacing of the energy levels of the system during thermalization with the bath; ii) the adiabatic transformation, where the thermally isolated system evolves with the number of excitations kept constant.This simple cycle has been implemented and studied in the context of finite-time Otto cycles [18][19][20], Ottoengine power generation [21][22][23], heat engines with interacting systems [24], and quantum heat engines based on phononic fields in Bose-Einstein condensates [25].
In this work we study the quantum Otto cycle in the context of quantum optomechanics [26].The system consists of a cavity-optomechanical setup where two movable mirrors confine a quantum field.The mirror and field modes strongly interact via phonon-photon vacuum fluctuations, and the mirrors are also coupled individually to a thermal bath.We employ our system to show that the proposed quantum field heat engine can generate power in finite time after each cycle, and we estimate the efficiency of such process.We note that the platforms considered here have already allowed for the experimental observation of thermal-phonon hopping, i.e, the exchange of thermal energy between individual phonon modes [27].
In contrast to the standard quantum Otto cycle, the cavity mode here plays the role of working substance and does not interact with the baths directly.Instead, the interaction is mediated by the quantized position degree of freedom of the respective wall.More precisely, each wall is connected to a thermal bath: the first wall (called W1) interacts with a cold bath at temperature T c , while the second wall (called W2) interacts with a hot bath at temperature T h .The transfer of thermal excitations from the bath to the cavity mode occurs whenever the frequency of the cavity is resonant with the frequency of the corresponding wall.The field modes are driven by an external drive, which controls the length of the cavity.
We stress that the photon-phonon interaction occurs beyond linearization, thereby retaining the dynamical-Casimir-like three-body terms in the Hamiltonian [28][29][30][31][32][33], with the ambition of characterizing the thermodynamic performance of the system in its full nonlinear regime.This includes also the multimode character of the interaction [34].Moreover, in contrast to standard studies of four-stroke thermal machines that consider the single strokes separately, we employed the master equation formalism in order to run two consecutive cycles as a function of time.The need for the master equation approach throughout the whole dynamics is explained by the fact that we effectively change the length of the cavity in a time-dependent way through the external drive, thereby controlling the heat transfer between the cavity and the baths through the walls.
The quantum model.-Thesystem is composed of two movable walls that confine an uncharged massless scalar quantum field and interact with a local bath.Our choice of field is a good approximation for a single-polarization version of a confined electromagnetic spin-1 field [35].We further simplify the setup by considering a one dimensional cavity, which allows us to obtain the system Hamiltonian Ĥs following the standard procedure of solving the classical field equations and then quantizing [31].The system is schematically represented in Fig. 1.The Hamiltonian can be split as usual as Ĥs = Ĥ0 + ĤI , where Ĥ0 = ω 1 b † 1 b1 + ω 2 b † 2 b2 + ω c â † â is the bare Hamiltonian and H I is the interaction Hamiltonian.The latter has been previously obtained [26], and reads Here, â, â † are the photonic operators for the cavity-field mode, while bj , b † j (j = 1, 2) are the phononic operators for the walls-field mode.The operators satisfy the canonical commutator relations [â, â † ] = 1 and [ bj , b † j ] = δ jj .Furthermore, ω j are the frequencies of the two movable walls (ω 1 < ω 2 for convenience), whereas ω c is the frequency of the cavity field mode.In addition, the coupling constants g j quantify the strength of the coupling between the field mode and the jth-wall (see Appendix).Finally, eigenvalues and states will be labelled by l, m, n ∈ N, which stand for l excitations of the field mode, m excitations of W1 and excitations n of W2.Throughout this work we assume that = c = k B = 1.
The interaction Hamiltonian Eq. (1) contains three types of terms: (i) The radiation pressure â † â( bj + b † j ), paramount in standard optomechanics [36,37], which shifts the cavity frequency ω c in case of coherent motion of the wall.(ii) The excitation transfer terms â2b † j + (â † ) 2b j , which convert single-phonon excitations into photon pairs (and vice versa).In other words, they convert mechanical and electromagnetic energy into each other.In order for the photons to appear and contribute during the dynamics, resonance conditions kω j = 2ω c with k ∈ N involving high-frequency movable walls must be fulfilled [30].Such resonances can be achieved in current optomechanical setups using real movable mirrors [38].However, they play the most significant role in experimental platforms based on superconducting circuits [39][40][41][42].(iii) The counter-rotating terms â2b j + (â † ) 2b † j , which generally contribute to modifying the energy-density of the cavity field (because of the quantum wall fluctuations [29]) and are also responsible for the non-conservation of the particle number.Counter-rotating terms are involved in virtual processes that would allow to observe higher-order coherent processes in cavity-optomechanics [43,44].
The interaction between the bosonic modes not only inevitably alters the structure of the energy levels with respect to the bare ones, but it also lifts the degeneracy in the presence of resonances.To this end we diagonalize the Hamiltonian H s numerically and plot the energy levels in Fig. 2  We use open quantum system dynamics to compute all quantities of interest [45].This requires us to solve the master equation for the density operator ρ representing the state.Since we are considering a strongly in-teracting system, we employ the tools developed in the literature [31,46,47].In particular, we employ the Lindblad equation ρ = −i[ Ĥ, ρ] + LD ρ, where LD indicates the Lindblad superoperator expressed in the dressed base (see Appendix).Here we assume that the three subsystems are coupled to three different baths: W1 is coupled to a cold bath with damping rate γ 1 and temperature T c ; W2 is coupled to a hot bath with damping rate γ 2 and temperature T h ; the cavity mode interacts with its own bath with damping rate κ 0 temperature T 0. This is a reasonable assumption for realistic implementations.
Quantum Otto cycle.-Themain idea of this work is to perform the quantum Otto cycle by using the two walls as bosonic channels to perform the heat transfer between the hot and cold baths through the cavity mode by exploiting the mechanical-electromagnetic energy conversion.We present here the framework employed to achieve our goal.
Any heat engine can be characterized by evaluating the output power P and the efficiency η of a single thermodynamic cycle.Thus, we consider the total Hamiltonian Ĥtot (t) = Ĥs + Ĥdr (t) that includes the timedependent drive term Ĥdr (t).This contribution is expressed in the dressed picture and periodically drives the cavity frequency from ωc,1 to ωc,2 and vice versa, simulating the physical process of compression and expansion of the cavity.This can be understood from the fact that the frequency of a trapped field mode decreases or increases by respectively enhancing or reducing the length of the cavity.Thus, changes in frequency simulate changes in length.Concretely, we have Ĥdr (t) = f (t)∆ω Â † Â, where ∆ω = ωc,2 − ωc,1 , Â, Â † are the cavity dressed operators obtained by diagonalizing the Hamiltonian, and f (t) is a periodic smooth step function (see Appendix).Such step functions are commonly employed in circuit quantum electrodynamics (QED) [48][49][50][51][52][53].Furthermore, time dependent drives have also been considered in studies of nonlinear optomechanics [54][55][56].
In order to investigate the thermodynamic features of the system, we define the change of internal energy ∆U(t) = Tr[ Ĥtot (t)ρ(t)] − Tr[ Ĥtot (0)ρ(0)], the change in heat ∆Q(t) = t 0 dτ Tr Ĥtot (τ ) ρ(τ ) , and the change in work ∆W(t) = t 0 dτ Tr Ḣtot (τ )ρ(τ ) .These three quantities satisfy the first law of thermodynamics in its quantum formulation: ∆U(t) = ∆Q(t) + ∆W(t). ( We then provide a formal definition for the output power P and the efficiency η of a thermodynamical process as where W out and Q in are, respectively, the work provided and the heat absorbed by the system.These are the main expressions evaluated in our work. We now present our quantum Otto cycle, which is composed of a preliminary phase and four steps: (0) Initialization: the whole system is prepared in its vacuum state ρ(0) = |0 0| and we assume that the cavity is initially coupled to W1 by means of the resonance condition ω c,1 = ω 1 /2 (see Fig. 1); (a) Cold isochoric: Thermal cold phonons from W1 are converted into photons, until the subsystem W1+cavity is thermalized; (b) Adiabatic compression: The external drive shifts the field frequency from ωc,1 to ωc,2 ; (c) Hot isochoric: The newly activated resonance ω c2 = ω 2 /2 facilitates the excitation transfer between the cavity mode and W2, during which hot phonons are converted into photons; (d) Adiabatic expansion: The drive changes the cavity mode frequency back to the resonance regime ω c,1 = ω 1 /2.After this step, the system is ready to restart from the cold isochoric stroke of step (a).
We now offer a few remarks on the properties of our cycle and the differences with the standard quantum Otto cycle already known in the literature [10].Recall that a quantum isochoric transformation occurs whenever the population of the system changes while thermalizing with a bath and maintaining the energy level spacing constant, while a quantum adiabatic transformation occurs whenever both the energy level spacing and the temperature are tuned such that the population does not vary throughout the process [10,57].The requirements for a transformation to be called "quantum adiabatic" dramatically differ with respect to the classical case.Indeed, while a classical adiabatic transformation simply forces the system to be thermally isolated throughout the process, namely not coupled to any bath, the condition of quantum adiabaticity additionally requires the stability of the population [10].In this sense, the latter is conceptually more limiting and challenging to realize in an experiment.Furthermore, any quantum adiabatic transformation has to be performed very slowly, namely without generating any output power [21,22,58].
Importantly, since the frequency change occurs much faster than the effective coupling, the system does not have time to modify its eigenstates, which implies that we do not need to diagonalize the Hamiltonian at all times.Therefore, we can maintain the same eigenbasis, hence the same dressed operators, throughout the whole dynamics.Moreover, the interaction between the cavity and each wall occurs not only at the minimum point of the level avoidance but also weakly in its proximity.It follows that this rapid jump between ωc,1 and ωc,2 is necessary to ensure the classical adiabaticity of the process [10], namely simulating the deactivation of the interaction between the cavity and the bath mediated by the wall.In this quantum description, the two walls are thermalized via different baths.In particular, the interaction between a specific bath and the cavity occurs once the resonance condition with the relative wall is imposed, and it is halted once the system is driven off-resonance.
We have W2 at frequency ω 2 that thermalizes via a hot bath at T h , while W1 has a lower frequency ω 1 < ω 2 and it is thermalized via a cold bath at T c < T h .The cavity interacts with its own bath at T 0.
Numerical analysis.-Wehave solved numerically the master equation and therefore calculated the time evolution of the average internal energy ∆U(t) directly, while we have computed the average work ∆W(t) by integrating the expression of the power P(t) ≡ Tr[ρ(t) Ḣtot (t)] = Tr[ρ(t) Ḣdr (t)] = ḟ (t)∆ωTr[ρ(t) Â † Â] defined in Eq. ( 3).The average heat change ∆Q(t) can be easily derived employing the first law of thermodinamics in Eq. ( 2).We have used the parameters indicated in Fig. 3, and our choice of frequencies has followed two important criteria: we need to clearly distinguish the two avoided levels in order to perform the jump, but at the same time we must avoid any degeneracy with other possible resonances in order to prevent unwanted heat flows that could reduce the efficiency.Our results can be found in Fig. 3.
We now discuss our findings.Once the dynamics start, a transient phase occurs in which both walls absorb thermal excitations from their own baths.However, while the interaction between W1 and the cavity converts part of the thermal phonons into photons, W2, which is momentarily not interacting with the cavity, thermalizes.In Fig. 3a, ∆U = 0 corresponds to the energy of the system at the end of this transient phase.Recall that thermalization of the cavity while interacting with W1 is defined as stroke (a).During stroke (b), i.e., during compression, the cavity absorbs work from the drive and increases ωc,1 to ωc,2 , eventually starting to absorb thermal excitations from the hot wall and converting them into photons [which defines stroke (c)].This causes a drastic enhancement of the photon population at the expense of part of the phonon population, as seen in Fig. 3b.At the same time, the population of the cold wall, now off-resonance, increases and W1 thermalizes completely.Once the internal energy becomes constant and the populations stop fluctuating, we perform the rapid expansion of the cavity as described in stroke (d), which causes the system to release an amount of energy which is higher than the one initially absorbed, a key feature of a properly functioning heat engine.This net gain becomes evident by looking at ∆W in Fig. 3a.After this last stroke, the cavity thermalizes with W1 again and the system reaches its initial configuration: a new cycle can now take place.We conclude that, after each cycle, we can extract a net amount of work using our system.
At this point, we wish to provide a measure of quality for the proposed cycle.This can be done by means of the efficiency defined in Eq. ( 3), which we compute using the parameters employed in the figures.We find an efficiency of η=0.376, which can be compared to the Carnot limit η C = 1 − T c /T h =0.625 as a benchmark for the performance of the machine.Further benchmarking could be carried out by evaluating the efficiency at max- imum power [58].Such analysis would require a detailed numerical optimization of this efficiency as a function of the frequencies of the two walls, and it could include the activation of further resonances during the adiabatic transformations.This is beyond the scope of the present work, and it is left for future investigations.
We now provide a few remarks on our proposal.As mentioned before we note that, once the resonance with the wall j is implemented, the interaction between wall j and the cavity mode is given by terms (â † ) 2b j + â2b † j in ĤI , which mediate the phonon-photon conversions.From a physical point of view this means that the degree of freedom that thermalizes during any isochoric stroke is not the cavity mode alone, but is instead the combination of the cavity mode and the activated wall.In other words, during the isochoric transformation the change in photon population is not a direct consequence of the thermalization of the cavity mode with the relative bath, instead it is a consequence of the partial conversion of thermal phonons into photon pairs.This process continues until the system wall-cavity reaches the steady state, as can be observed in Fig. 3b.Thus, the number of photons at the end of any isochoric process will be less than N = (e ωc/T0 − 1) −1 as predicted by Bose-Einstein statistics and expected in the Otto cycle with a single-mode as working substance [59,60].In particular, the photon population in Fig. 3b can be compared with the average number of photons for a single cavity mode with frequency ωc,1 = 1.01 and temperature T c = 0.15, which is N = 1.2 • 10 −3 , and the population of a singlecavity mode with frequency ωc,2 = 1.31 and temperature T h = 0.4, which is N = 3.96 • 10 −2 .It can be seen that the photon population in our system is about 94% less than what expected by a direct thermalization with the bath at the end of the cold isochoric, and about 95% less at the end of the hot isochoric.This suggests that we can run our machine with fewer resources.
Furthermore, we notice that, during the isochoric strokes, quantum excitations are extracted from the thermal fluctuations of the walls and converted into photons without requiring the walls to perform any classical motion.This is in contrast to the semiclassical description of our setup, wherein walls are treated as classical degrees of freedom [33].The phonon-photon conversion occurs between two quantum channels by simply activating the resonance ω i = 2ω c,i (i = 1, 2).The possibility of extracting thermal excitations from the walls in case of no coherent motion is a consequence of the fact that the two mirrors operate in the quantum regime [31,61].
Finally, we stress that, in contrast to other works studying optomechanical systems in a quantum thermodynamic framework [12,13], we employ a nonlinear Hamiltonian that includes the nonlinear interaction between the movable walls and the electromagnetic mode.In particular, the entire dynamics of our system is based on the conversion of phonons into photons and vice versa.To our knowledge, this effect has never been considered in a quantum thermodynamic context.This highlights the importance of moving beyond the linearized regime.
Conclusions.-In this work we proposed a quantum heat engine based on a cavity system composed of a scalar field trapped by two fluctuating walls that performs an Otto cycle.In our setup, we exploit the phononphoton conversion mechanism to let the working substance, namely the cavity mode, exchange heat with the thermal baths during the cycle.We demonstrated that it is possible to extract net work using carefully modulated resonances, and we have evaluated the overall efficiency of the cycle.We believe that this work opens the way to the systematic study of quantum field thermodynamic engines, to be used for fundamental science, as well as the development of novel quantum technologies.
In the end, this procedure yields the Hamiltonian Ĥ(t) = Ĥ0 + ĤI , where each term reads In order to simplify the notation, we introduced the wave vector k(n w ) = πn w /L w , with w = x, y, z, the quadrature position operators Xb,j = 1 2 ( b † j + bj ), with j = 1, 2, and Xn = 1 2 (â † n + ân ).We note that the ambiguity on the negative sign in ĤI is solved by including all terms of the Taylor expansion with respect to δL [28].
For our purposes, such Hamiltonian can be drastically simplified by assuming: 1.The constraint L x L y , L z on the magnitude of the length of the edges of the piston.This is motivated by the fact that there are no excitations initially present in the y and z degrees of freedom.Given the higher energy required to excite these degrees of freedom (since the energy gaps are inversely proportional to the corresponding length) it is reasonable to assume that they will remain unexcited.Thus, we can drop the first term in the square brackets and obtain Hamiltonian (1) in the main text.
2. The presence of only one cavity mode, say the fundamental mode ω 1,1,1 = ω c .This is a reasonable assumption, since other cavity modes with n x > 1 are initialized in the vacuum state.
Under such assumptions, the system Hamiltonian is reduced to Eq. ( 1) of the main text.
The master equation in the dressed picture and the dressed operators In this section we want to review the necessary tools to describe the time evolution of the quantities of interest.Working in the Schrödinger picture, we first need to solve the master equation for the density operator; however, since we are dealing with a strongly interacting system, the best way to proceed is to employ the master equation in the dressed picture [46].To do this, first let us introduce the transition amplitudes for the canonical position operators calculated on the dressed basis: c ij = i|â + â † |j , u ij = i| b1 + b † 1 |j and v ij = i| b2 + b † 2 |j , where the state |i is the i-th eigenstate of the Hamiltonian with eigenenergy E i .
We assume that the three subsystems are coupled to three different baths: in particular, the wall 1 is coupled to a cold bath with damping rate γ 1 , the wall 2 is coupled to a hot bath with damping rate γ 2 .Finally, we also assume that the cavity always weakly interacts with its own bath at T 0, with damping rate κ 0. Although it does not play an active role in our dynamics, the interaction with the third bath whose temperature is not exactly zero is expected in experimental scenarios and prevents the violation of the third law of thermodynamics.
The master equation in the dressed picture has been obtained before [46,47], and it reads where we have defined Lx ρ = y j,i>j that needs to be supplemented by the quantities x = c, u, v, the rates of losses y = γ 1 , γ 2 , κ, the thermal excitation numbers n ij (T ) = (e (Ei−Ej )/T − 1) −1 , the superoperators and the transition operators Pij = |i j|.

FIG. 1 .
FIG. 1. Pictorial representation of the cavity performing the quantum Otto cycle: two movable walls, coupled to two baths at different temperatures, confine a cavity mode, which is weakly coupled to a bath at T 0. The four panels describe the four strokes of the cycle: (a) cold isochoric, (b) adiabatic compression, (c) hot isochoric, (d) adiabatic expansion.