Nonequilibrium thermodynamics in cavity optomechanics

Abstract

Classical thermodynamics has been a great achievement in dealing with systems that are in equilibrium or near equilibrium. As an emerging field, nonequilibrium thermodynamics provides a general framework for understanding the nonequilibrium processes, particularly in small systems that are typically far-from-equilibrium and are dominated by thermal or quantum fluctuations. Cavity optomechanical systems hold great promise among the various experimental platforms for studying nonequilibrium thermodynamics owing to their high controllability, excellent mechanical performance, and ability to operate deep in the quantum regime. Here, we present an overview of the recent advances in nonequilibrium thermodynamics with cavity optomechanical systems. The experimental results in entropy production assessment, fluctuation theorems, heat transfer, and heat engines are highlighted.

1. Introduction

Classical thermodynamics is an essential branch of physics, which studies heat, work, and temperature, as well as their relationships, typically in large systems that are confined to equilibrium or near equilibrium states. Most natural systems, on the other hand, are not in thermodynamic equilibrium, i.e. are in nonequilibrium states. Nonequilibrium thermodynamics was developed over a century ago to understand the nonequilibrium situations, such as the measure of irreversibility, but it is still a task in progress rather than a well-established discipline [1].

The focus of this article is on the recent developments of nonequilibrium thermodynamics in small systems. As the size of a system decreases, the impact of its surroundings increases and the fluctuations become fundamentally more significant, often quite far from equilibrium, prompting the fields of stochastic thermodynamics [2], [3], [4] and quantum thermodynamics [5,6], depending on whether the fluctuations are thermal or quantum. Fluctuation theorems [7,8], the relation to quantum information [9,10], and stochastic/quantum thermodynamic machines [11], [12], [13] are just a few of the fascinating topics related to stochastic and quantum thermodynamics. Various platforms for studying stochastic and quantum thermodynamics have been developed as a result of advances in material science and laser technologies, including cold atoms [14,15], trapped ions [16], biological molecules [17], superconducting circuits [18,19], and cavity optomechanical systems [20,21].

Cavity optomechanics, which couples electromagnetic degrees of freedom with mechanical motions via radiation pressure, is promising for both fundamental and applied research [22]. A prototype cavity optomechanical system with one fixed mirror and one vibrating end mirror is illustrated in Fig. 1. The optical cavity enhances the radiation pressure force and provides a feedback mechanism, which allows one to control the motion of the mechanical resonator at macroscopic or mesoscopic scales. Much remarkable progress has been made in the field of cavity optomechanics, such as ground-state cooling of mechanical resonators [23], [24], [25], ultrasensitive motion detections [26], [27], [28], and manipulation of photons and phonons at quantum levels [29], [30], [31].

Fig. 1.

Schematic of a typical cavity optomechanical system with a fixed mirror and a movable end mirror. The optical cavity is driven by a laser field. The radiation pressure force F is used to manipulate the motion of movable mirror.

Cavity optomechanics has recently been recognized as an excellent platform for studying nonequilibrium thermodynamics. On one hand, cavity optomechanical systems combine the high quality mechanical resonators with highly sensitive optical detection in an optical cavity, allowing the systems to operate in the underdamped regime and perform real-time single trajectory measurements. On the other hand, cavity optomechanical systems, by taking advantage of the optomechanical interaction, have high controllability and scalability, as well as the ability to achieve motional quantum ground state, allowing researchers to investigate the underlying physical mechanisms under a variety of parameter conditions.

In this article, we provide an overview of the recent advances of nonequilibrium thermodynamics in cavity optomechanical systems, with a focus on the experimental results. Four aspects of nonequilibrium thermodynamics are covered, i.e., thermodynamic quantities [4,20,[32], [33], [34]], fundamental principles [35,36], physical phenomena [21,[37], [38], [39]], and practical applications [40], [41], [42], [43], [44], as shown in Fig. 2. The remainder of this paper is organized as follows. The assessment of entropy production (an essential thermodynamic quantity) in a cavity optomechanical system is presented in Section 2. We introduce the experimental progress of fluctuation theorems (fundamental principles) with optomechanics in Section 3. In Section 4, we present the recent demonstrations of heat transfer (physical phenomenon) in optomechanical systems. The implementation of an optomechanical heat engine (practical application) is discussed in Section 5. The conclusion and outlook can be found in Section 6.

Fig. 2.

Recent experimental works on various platforms besides cavity optomechanics are selected as examples for four aspects of nonequilibrium thermodynamics, i.e. thermodynamic quantities [4,20,[32], [33], [34]], fundamental principles [35,36], physical phenomena [21,[37], [38], [39]], and practical applications[40], [41], [42], [43], [44].

2. Entropy production

In thermodynamics, entropy production is a crucial physical quantity that is intimately linked to the fundamental laws of thermodynamics. Entropy production is a measure of irreversibility that can be generated by any finite-time thermodynamic process [45]. It is closely related to fluctuation theorems, information thermodynamics, heat transport, and operation of heat engines. As a result, assessing irreversible entropy production is one of the most important tasks in nonequilibrium thermodynamics.

2.1. Assessment of entropy production in an optomechanical cooling process

The change of system entropy dS in a thermodynamics process can be attributed to the flow of entropy ϕ between the system and its surroundings as well as a contribution from the irreversible entropy production π, i.e. dS = π ‒ ϕ. It is often convenient to express the result in terms of the entropy production rate, thus the second law of thermodynamics can be written as [32]

$$\text{dS}/\text{dt}=\Pi \left(t\right)-\Phi \left(\mathrm{t}\right)$$ (1)

where Π(t) represents the irreversible entropy production rate and Φ(t) depicts the entropy flux from the system to the environment. Π_s and Φ_s are the values of these quantities as the system approaches a nonequilibrium steady state (NESS), with Π_s = Φ_s > 0. Entropy is produced and exchanged with the local baths at the same rate under such circumstances. Both quantities vanish in thermal equilibrium, i.e. Π_s = Φ_s = 0. Therefore, the entropy production rate directly connects to the irreversibility of a process and reveals the nonequilibrium characteristics of a system.

The experimental assessment of entropy production in NESS was realized by Brunelli et al in a cavity optomechanical setup, with a micromechanical oscillator coupled to the field mode of an optical Fabry-Perot cavity, as shown in Fig. 3a [32]. One of the cavity mirrors is a mechanical cantilever. The intracavity photon number is coupled to the position of the cantilever via radiation pressure. The optical cavity is driven by a red detuned laser field, and the sideband cooling is observed. The light reflected off the cavity via homodyne detection is used to analyze the entropy production rate of the system in the cooling process.

Fig. 3.

(a) The optomechanical setup of a mechanical oscillator ($\delta {\widehat{q}}_b$) coupled to an optical cavity field ($\delta {\widehat{q}}_a$). (b) The system can be treated as two quantum harmonic oscillators with frequencies ω_a and ω_b that are linearly coupled with a coupling strength g_ab. Each harmonic oscillator is coupled to their independent local thermal baths at temperature T_a and T_b, with the corresponding coupling rates κ_a and γ_b, respectively [32].

Such a system can be equivalently modeled as two coupled harmonic oscillators, as shown in Fig. 3b. $\delta {\widehat{q}}_{a,b}$and $\delta {\widehat{p}}_{a,b}$ are denoted as the position and momentum fluctuation operators around the mean-field values of the two oscillators, here a and b refer to the optical and mechanical oscillators, respectively. ω_a and ω_b are the frequencies of the two harmonic oscillators under the rotating frame approximation. Consequently, the equivalent linearized Hamiltonian can be written as follows [32]

$$\widehat{H}=\frac{\hslash {\omega }_a}{2}\left(\delta {\widehat{q}}_a^2+\delta {\widehat{p}}_a^2\right)+\frac{\hslash {\omega }_b}{2}\left(\delta {\widehat{q}}_b^2+\delta {\widehat{p}}_b^2\right)+\hslash g_{ab}\delta {\widehat{q}}_a\delta {\widehat{q}}_b$$ (2)

where g_ab is the coupling strength between the two modes.

In the NESS, all entropy produced in the system flows to the environments, and the entropy production rate can be obtained as [32]

$${\Pi }_s={\Phi }_s=4{\kappa }_a{n}_a+2{\gamma }_b(\frac{n_b+1/2}{n_{T_b}+1/2}-1)={\mu }_a+{\mu }_b$$ (3)

where $n_a={\langle \delta {\widehat{q}}_a^2+\delta {\widehat{p}}_a^2-1\rangle }_s/2$ and $n_b={\langle \delta {\widehat{q}}_b^2+\delta {\widehat{p}}_b^2-1\rangle }_s/2$ are the average occupation numbers of the cavity and mechanical modes in the NESS in excess of the zero-point motion of the respective harmonic oscillator. ${\kappa }_a$ and ${\gamma }_b$ are the corresponding decay rates. $n_{T_b}={(e^{\hslash {\omega }_b/k_B{T}_b}-1)}^{-1}$ is the average phonon number of mechanical oscillator at temperature $T_b$.

Eq. 3 quantifies the entropy production rate that the system has to pay to remain in its NESS. It comes from two parts, μ_a and μ_b, which are related to the optical and mechanical oscillators, respectively, implying that the entropy produced in the NESS is split into two separate fluxes with individual entropy flows to each environment. When the optomechanical coupling is disabled, the system reaches thermal equilibrium with $n_a=0$ and $n_b=n_{T_b}$, and the entropy production rate Π_s = 0.

Fig. 4a shows the density noise spectrum (DNS) of the cavity field quadratures [32]. In the experiment, the frequencies of two oscillators are in resonance, i.e., ω_a = ω_b, which is the most effective condition for the optomechanical cooling in the resolved-sideband regime. By measuring the light field leaking out of the cavity, both μ_a and μ_b can be reconstructed to determine Π_s quantitatively. The experimental assessment of Π_s at the NESS as a function of g_ab/κ_a is shown in Fig. 4b [32]. The inset depicts the behavior of μ_b. μ_a increases as the coupling becomes larger, which means that the stronger the coupling strength, the further the system operates away from thermal equilibrium and the more entropy is generated. Meanwhile, μ_b takes negative values, which increases in magnitude as g_ab increases. This is reasonable because μ_b is not the total entropy production rate, but rather an individual flux that can have negative values. It is worth mentioning that μ_a + μ_b has to be positive. The behavior of μ_b is a signature of optomechanical cooling. μ_b increases in absolute value with g_ab, which indicates a greater entropy flow from the mechanical resonator to the cavity field, corresponding to a lower effective temperature of the resonator.

Fig. 4.

(a) Density noise spectrum (DNS) of the phase quadrature of the output cavity field. The jagged blue and light-blue curves correspond to the rescaled coupling g_ab/κ_a = 0.49 and 2.29, respectively. The smooth lines are the fits of the DNS. (b) Experimental assessment of the irreversible entropy production rate Π_s at the NESS as a function of g_ab/κ_a. The vertical error bars are the statistical errors extracted from the fit, and the horizontal ones show the experimental error on the values of the parameter. The inset shows the behavior of μ_b[32].

2.2. Assessment of entropy production in a continuously measured optomechanical system

The fundamental connections between information and thermodynamics can be traced back to the Maxwell, Szilard, and Landauer's seminal contributions [46]. The acquisition of information can have an impact on the entropic balance of a physical process. As a result, when assessing a monitored system, it's important to distinguish between unconditional evolution and dynamics conditioned on measurement records. Acquiring information can make the process more reversible, resulting in the average entropy production of the conditional trajectories $\int {\Pi }_{c}d\tau$ being smaller than the unconditional ones $\int {\Pi }_{\mathit{uc}}d\tau$. The conditional and unconditional entropy production rates have the following relationship [20]:

$${\Pi }_c={\Pi }_{\mathit{uc}}+\dot{I}$$ (4)

Here $\dot{I}$ is the net information acquiring rate through measurement, which provides a valuable link between information and thermodynamics.

Recently, the impact of weak continuous measurements on the nonequilibrium thermodynamics of a mesoscopic mechanical resonator was studied experimentally by Rossi et al. [20]. A nanomechanical resonator is coupled to an optical cavity and the effects of both optical and phononic environments are considered in this system, as shown in Fig. 5a. Homodyne measurements on the output optical field are used to continuously monitor the position of mechanical resonator. Combining the phase-space formalism [47] with the state retrodiction methods [48], it is possible to characterize the entropy production at the level of individual quantum trajectories.

Fig. 5.

(a) A sketch of the experimental setup, which comprises a cryogenic optomechanical cavity resonantly driven by a coherent probe laser. The mechanical resonator is in thermal contact with two baths: a thermal, cryogenic bath and the optical bath. The output field is continuously monitored by means of a homodyne receiver. The photocurrent i is used to estimate the conditional mechanical state. (b) The dynamic behaviors of $\dot{I}$, $G\left(t\right)$, and $\dot{I}-G\left(t\right)$, which are represented by the solid, dashed, and dot-dashed curves, respectively [20].

Both conditional and unconditional entropy production rate can be experimentally obtained from the measured stochastic trajectories and the deduced conditional variance $V\left(t\right)=\langle {\widehat{X}}^2\rangle -{\langle \widehat{X}\rangle }^2=\langle {\widehat{Y}}^2\rangle -{\langle \widehat{Y}\rangle }^2$, where $\widehat{X}$ and $\widehat{Y}$ are the quadrature operators of mechanical mode. The difference between the conditional entropy production rate Π_c and the unconditional one Π_uc provides the influence of monitoring the system on the irreversibility of the dynamics, which is quantified by the net information acquiring rate through measuring:

$$\dot{I}={\Gamma }_m(V_{uc}/V\left(t\right)-1)-4{\eta }_{\det }{\Gamma }_{qba}V\left(t\right)$$ (5)

Here Γ_m is the energy dissipation rate of mechanical resonator, $V_{uc}={\langle {\widehat{X}}^2\rangle }_{uc}={\langle {\widehat{Y}}^2\rangle }_{uc}$ is the unconditional variance, ${\eta }_{\det }$ is the detection efficiency, and ${\Gamma }_{qba}$ is the decoherence rate due to the quantum measurement backaction.

Fig. 5b shows the dynamic behavior of $\dot{I}$reconstructed from the experimental data [20]. The system is initially prepared in the steady state of the unconditional dynamics, and evolves to the conditional steady state. One can see that $\dot{I}$ vanishes in the conditional steady state, and thus the quantity $-{\int }_0^{\infty }\dot{I}dt$ tends to a constant in the long-time limit. This is intuitively understood from the fact that monitoring the system adds no additional information in the steady state. The conditional steady state will not be maintained if the monitoring process suddenly stops, and the system will heat up back towards V_uc. To maintain the conditional steady state, continuous monitoring is required. In other words, information is constantly being acquired even in the steady state, but the phonon bath is constantly introducing noise. It is thus intriguing to identify which term in $\dot{I}$ (Eq. 5) accounts for the incremental information gains required to maintain the conditional steady state.

The differential gain of information is defined as $G\left(t\right)=-4{\eta }_{\det }{\Gamma }_{qba}V\left(t\right)$, i.e. the last term in Eq. 5. The behaviors of the differential gain $G\left(t\right)$ and the loss of information due to noise input by the phonon bath $\dot{I}-G\left(t\right)$ are also shown in Fig. 5b. The initial closeness of $G\left(t\right)$ to $\dot{I}$ suggests that the differential information gain has a significant impact on the early stages of the dynamics. However, as the dynamics approaches the steady state, the contribution from $G\left(t\right)$ become less important.

3. Fluctuation theorems

Fluctuation theorems are a generalization of thermodynamics that applies to small nonequilibrium systems [7,8]. They are the equalities regarding to the probability distribution functions of thermodynamic quantities that are essentially related to the entropy production, and they can be applied to a wide range of nonequilibrium situations. In the last two decades, different forms of fluctuation theorems have been developed, including Jarzynski equality [49] and Crooks relation [50], which reformulated the inequality of the second law into equalities and revealed the universal laws that the fluctuating thermodynamic variables must obey in processes which are arbitrarily far from equilibrium. Fluctuation theorems have been demonstrated in a variety of systems, including biomolecules, colloidal particles, and electric circuits. The majority of these experiments are described as an overdamped Langevin equation. With the advantages of optomechanical system, it is possible to extend fluctuation theorems to the underdamped regime, and in quantum systems.

Several groups have studied the fluctuation theorems in levitated optomechanical systems [51,52] under different situations, e.g. the transient fluctuation theorem [53], the differential fluctuation theorem [35], and the fluctuation theorem with fast control [54]. Although these experiments do not use a cavity, this is not an inherent limitation in the physics of optomechanics. In cavity optomechanical systems, the radiation pressure interaction is typically much stronger than in free-space devices. Recently, the levitated nanoparticle has been laser-cooled into its quantum ground state of motion within an optical cavity [55], which opens up the possibility of studying nonequilibrium thermodynamics with quantum motional states in cavity optomechanical systems with levitated particles in the future.

Fig. 6 shows a typical experimental setup of levitated optomechanics [53]. A silica nanoparticle of radius r ≈ 75 nm and mass m ≈ 3 × 10⁻¹⁸ kg is trapped in vacuum by the gradient force of a focused laser beam. The nanoparticle oscillates in all three spatial directions within the trap. In the experiment, the nanoparticle is initially cooled by parametric feedback. At time $\mathrm{t}={\mathrm{t}}_{\text{off}}$, the feedback is switched off and the nanoparticle trajectory is followed as it relaxes to equilibrium. After relaxation, the feedback is switched on again and the experiment is repeated. In this relaxation process, the dynamics satisfies the detailed balance with respect to the equilibrium distribution at reciprocal temperature β₀ = 1/k_BT₀. The time reversibility of the underlying dynamics implies the transient fluctuation theorem [53]:

$$p(-\Delta S)/p\left(\Delta S\right)=e^{-\Delta S}$$ (6)

Here $\Delta S={\beta }_{0}Q+\Delta \varphi$ is the relative entropy change, Q is the heat absorbed by the bath at temperature β₀ and Δϕ is the difference of the trajectory-dependent entropy.

Fig. 6.

Experimental setup of levitated optomechanics. A nanoparticle is trapped by a tightly focused laser beam in high vacuum. The nanoparticle can be cooled by parametric feedback or excited by a parametric drive [53].

Fig. 7a displays the probability density p(ΔS) for different times [53]. For long periods of time, p(ΔS) become increasingly asymmetric, with higher probabilities for positive ΔS and lower probabilities for negative ΔS. To test the transient fluctuation theorem, a time-independent function is defined as

$$\sum \left(\Delta S\right)=\ln \left[\frac{p\left(\Delta S\right)}{p(-\Delta S)}\right]=\Delta S$$ (7)

$\sum \left(\Delta S\right)$ is analyzed using the distributions of ΔS in Fig. 7a, and is shown in Fig. 7b, which agrees with the fluctuation theorem for ΔS.

Fig. 7.

(a) Probability density as a function of ΔS for different times after switching off the feedback. (b) Function Σ(ΔS) evaluated for the distributions shown in Fig. 7a [53].

4. Heat transfer

Heat transfer is an essential thermodynamic phenomenon, which is the transport of thermal energy from a warmer object to a cooler object. The convection, conduction, and radiation are the well-known basic ways to transfer thermal energy. Recently, the study of heat transfer in small nonequilibrium systems become an active research area. On the one hand, new types of heat transfer could exist at the nanoscale and atomic scale. Phonon heat transfer across a vacuum through quantum fluctuations has been observed by Fong et al. [56]. On the other hand, manipulation of heat transfer at small scales is significant to many problems, such as heat dissipation in electronic devices, energy harvesting, and communication with phonons. The ability to control heat flow results in various phononic devices [57], [58], [59], [60]. The reversal of heat flow has been demonstrated in quantum correlated spins by Micadei et al. [37]. Heat diffusion has been analyzed in optomechanical arrays by Xuereb et al. [61]. Optomechanical cooling has been treated as controlling the heat flow between the hot bath and cold bath with a heat valve [62]. Nonreciprocal manipulation of heat flow is proposed in an array of optomechanical cavities by breaking the time-reversal symmetry [63], which has also been demonstrated in a membrane-in-the-middle system by modulating the dynamical backaction [64].

Because heat flux is typically driven by a thermal gradient, it is natural to investigate heat transfer in a nonequilibrium system using multiple mechanical resonators contacting different thermal baths. Recently, a new mechanism of phonon heat transport has been realized in a two-membrane cavity optomechanical system [21]. The unique advantages of such a system are: (1) it consists of two nanomechanical resonators whose properties can be flexibly controlled, such as mechanical frequency and bath temperature; (2) the mechanical resonators are coupled via the cavity field, so the interacting range could be infinitely long in principle and the coupling strength can be tuned optomechanically; (3) the trajectories of two resonators can be real-time monitored independently at a very high sensitivity via an optical method [65].

Fig. 8a shows the experimental setup for a two-membrane cavity optomechanical system [21]. Two spatially separated silicon nitride nanomechanical membranes are placed inside a Fabry-Perot cavity independently with a distance of ∼ 60 mm. The membranes are 50 nm thick and a 1×1 mm² in size. The fundamental vibrational modes are utilized in the experiment, which are nearly degenerate with eigenfrequencies ω_m ≈ 2π×400 kHz (m = 1,2). Piezos are used to precisely control the frequencies of membranes [66]. One membrane is in contact with a room temperature thermal bath and the other is driven by a white noise to obtain the high temperature thermal bath. The motions of membranes are monitored by two weak probe laser fields. The optical cavity consists of two identical mirrors with a cavity length of 140 mm and a finesse ∼ 1000. The cavity is driven by a red-detuned laser field, which interacts with both membranes simultaneously due to the dynamical backaction, and consequently two membranes are effectively coupled by the cavity field.

Fig. 8.

(a) Two nanomechanical SiN membranes (M₁ and M₂) are placed inside an optical cavity separately and coupled to the cavity field via optomechanical interaction. (b) Equivalent model of two mutually coupled harmonic oscillators with a coupling strength Ʌ. Each oscillator is in contact with an independent thermal reservoir, i.e. R_H and R_L for high and low temperature reservoirs, respectively. γ_1,2 is the mechanical decay rate. T_i (i = 1, 2, H, L) represents the temperature [21].

As shown in Fig. 8b, such a system can be viewed as two mutually coupled harmonic oscillators, one in contact with a room temperature reservoir (T_L) and the other with a high temperature reservoir (T_H) [21]. Under the effective optomechanical coupling, the temperature gradient drives the system out of equilibrium, causing a mean heat flux flow from the hot reservoir to the cold reservoir. When the cavity field is turned off, the coupling between the membranes is zero, and the membranes have the same temperatures as their own reservoirs.

The total Hamiltonian of such a two-membrane cavity optomechanical system in the rotating frame of the driving laser frequency can be written as

$$\begin{array}{ccc}\hfill \widehat{H}& =& -\Delta {\widehat{a}}^{†}\widehat{a}+{\omega }_0{{\widehat{b}}_1}^{†}{\widehat{b}}_1+{\omega }_0{{\widehat{b}}_2}^{†}{\widehat{b}}_2-g_0{\widehat{a}}^{†}\widehat{a}\left({{\widehat{b}}_1}^{†}+{\widehat{b}}_1\right)-g_0{\widehat{a}}^{†}\widehat{a}\left({{\widehat{b}}_2}^{†}+{\widehat{b}}_2\right)\hfill \\ & & +i\epsilon \left({\widehat{a}}^{†}-\widehat{a}\right)\hfill \end{array}$$ (8)

Here $\widehat{a}$ and ${\widehat{b}}_{1,2}$ are the annihilation operators of the cavity mode and the mechanical oscillators, respectively. $\Delta ={\omega }_L-{\omega }_C$ is the frequency detuning between the driving laser and the cavity resonance. The two degenerate mechanical modes have the same frequency ${\omega }_0$. $g_0$ is the optomechanical coupling strength. $\epsilon =\sqrt{P{\kappa }_{in}/\hslash {\omega }_L}$ is the driving strength, P is the input laser power, and ${\kappa }_{in}$ is the loss of the input cavity mirror.

When the damping rate of the cavity mode is much larger than the mechanical damping rates, i.e., κ >> γ_1,2, the cavity field follows the dynamics of the mechanical modes and can be eliminated adiabatically. Consequently, the system can be described by an effective Hamiltonian:

$${\widehat{H}}_{eff}=\sum _{i=1,2}\left({\omega }_0+\Lambda -i{\gamma }_i/2\right){{\widehat{b}}^{†}}_i{\widehat{b}}_i+\Lambda \left({{\widehat{b}}^{†}}_1{\widehat{b}}_2+{{\widehat{b}}^{†}}_2{\widehat{b}}_1\right)$$ (9)

Here $\Lambda =g_0^2{\chi }_{eff}$ is the effective coupling strength between two membranes, and

$${\chi }_{eff}=\frac{P{\kappa }_{in}/\hslash {\omega }_L}{{\kappa }^2/4+{\Delta }^2}\left(\frac{\Delta +{\omega }_0}{{\kappa }^2/4+{\left(\Delta +{\omega }_0\right)}^2}+\frac{\Delta -{\omega }_0}{{\kappa }^2/4+{\left(\Delta -{\omega }_0\right)}^2}\right)$$ (10)

is the effective susceptibility introduced by the intracavity field. Generally, the effective coupling between membranes can be both dispersive and dissipative. Here, the laser frequency is detuned far-off the cavity resonance, and consequently the interacting of membranes is dominated by a conservative coupling.

In experiment, the thermal noise spectrum of two nanomembranes are measured in the nonequilibrium steady state [21]. Fig. 9a and b shows the noise spectra in the weak and strong coupling, respectively. In the strong coupling regime, the mechanical spectrum shows the normal mode splitting with the frequencies ${\omega }_{+}={\omega }_0$ and ${\omega }_{-}={\omega }_0+2\Lambda$. Due to the fact that the effective interaction is negative and the normal mode frequencies are also modified by the optomechanical coupling, the frequency of breathing mode ${\omega }_{+}$ is fixed and the frequency of center-of-mass mode ${\omega }_{-}$ decreases as the cavity photon number increases. Fig. 9c is the measured frequency shifts of normal modes as a function of the intracavity photon number.

Fig. 9.

(a, b) Thermomechanical noise spectra of membranes in the weak and strong coupling regimes, respectively. M₁ and M₂ represent the membranes with high and low temperature reservoirs, respectively. (c) Normal-mode splitting of the thermomechanical noise spectra as a function of cavity photon number. (d) Effective temperatures of membranes as a function of cavity photon number. Solid curves are the theoretical calculations and the dashed lines represent the reservoirs’ temperatures [21].

Integrating the measured thermal noise spectra, which are calibrated from the root mean square motions $\sqrt{k_{B}T/m{\omega }_0^2}$ at the room temperature, yields the effective temperature of mechanical mode. The effective temperatures of membranes as a function of cavity photon number are plotted in Fig. 9d. When the cavity photon number is low, there is a significant difference in the effective temperatures between two membranes. The effective temperatures gradually tend to equalize as the cavity photon number increases, indicating that the thermal phonon transport occurs between the membranes.

To fully characterize the process of heat transfer, the instant heat flux is investigated in the nonequilibrium steady-state. The heat flux from the membrane M₁ to M₂ is defined as

$$j_{\tau }=-\underset{t_0\to \infty }{\lim }\frac{2m}{\tau }\left({\omega }_0\Lambda +{\Lambda }^2\right){\int }_{t_0}^{t_0+\tau }{x}_1{\dot{x}}_{2}dt$$ (11)

where $x_{1,2}=\sqrt{\hslash /2m{\omega }_{1,2}}\left({\widehat{b}}_{1,2}^{†}+{\widehat{b}}_{1,2}\right)$ is the mechanical motion, and τ is the interval of integration windows. In a typical measurement, the membrane motions are recorded by the lock-in amplifier with τ ∼ 100 μs, which is much smaller than the relaxation time of membranes. As a result, it can be considered as an instant heat flux. Fig. 10 shows the instant heat flux as a function of time in the nonequilibrium steady state for weak and strong coupling regimes [21]. The instant heat flux fluctuates rather than remaining constant in the weak coupling regime, and the value of instant flux is less than zero at some points, implying that heat flows from the low temperature bath to the high temperature bath (see Fig. 10b). In the strong coupling regime, the instant heat flux oscillates back and forth between the membranes in the nonequilibrium steady-state (see Fig. 10d), which was not previously observed in the thermalization and is in contrast to the observations in the macroscopic world.

Fig. 10.

The instant heat flux as a function of time in the nonequilibrium steady state. (a, b) under the weak coupling and (c, d) strong coupling regimes, respectively [21].

Unlike most experimental platforms having been utilized for studying nonequilibrium thermodynamics, which use only one mechanical resonator, the two-membrane cavity optomechanical system allows researchers to investigate the situations where the nonequilibrium state is driven by thermal gradient, as well as nonequilibrium thermodynamic theories, such as fluctuation theorems and thermodynamics uncertainty relations [67], with interacting subsystems, particularly in the strong coupling regime. Although some nonequilibrium phenomena could be demonstrated using multiple modes of a single mechanical resonator [64], a two-membrane system not only offers more degrees of freedom to control, such as individual mechanical frequency and optomechanical coupling strength, but can also be straightforwardly extended to optomechanical arrays, which is crucial for investigating effects such as Fourier's law [68] and Frohlich condensate of phonons [69].

5. Heat engine

A heat engine is a thermodynamic device that converts thermal energy into mechanical power, and is the most successful application of classical thermodynamics. In small nonequilibrium systems, developing advanced and powerful synthetic heat engines becomes one of the most challenging goals of multiple disciplines. Thermodynamic machines operating at micro- and nanoscales can perform a variety of tasks that macroscopic engines can't access. Chemists and biologists have created artificial molecular systems that aim to accomplish complex and useful tasks at molecular level, including transporting cargo inside cells and selectively killing cancer cells, which has significant implications for medical research [70]. With the developments in nanotechnology and laser cooling, physicists have realized the stochastic and quantum heat engines at nano/microscales or single-atom levels with a single trapped ion [40,71], nano/microresonators [72], [73], [74], [75], [76], nitrogen-vacancy centers [77], cold atoms [78,79], and nuclear spins [80,81].

As an outstanding platform for studying nonequilibrium thermodynamics, cavity optomechanical systems have several appealing advantages, e.g. it is a truly mechanical system and has the ability to operate deep in the quantum regime. Various types of heat engines have been proposed based on optomechanical systems. Optomechanical heat engines based on Otto cycles are first investigated by Zhang et al. [82]. They propose a quantum optomechanical heat engine by using the polariton normal mode excitations via the optomechanical coupling between the cavity field and the oscillating end mirror [82], and a straight-twin engine in a hybrid microwave-optomechanical system [83]. The reversible work extraction of an Otto cycle is analyzed in a hybrid optomechanical system [84]. A quantum Otto heat engine is explored with coupled superconducting transmission line resonators via an optomechanical-like coupling [85] and is also studied based on a quadratically coupled optomechanical system [86]. A feedback loop is exploited to improve the performance of an optomechanical heat engine in an Otto cycle [87]. Stirling nanomechanical heat engines are theoretically investigated in a levitated optomechanical system [88], and driven by feedback-controlled light [89]. A continuous heat machine is proposed in an autonomous heat powered optomechanical setup [90]. The single cavity engine and the cascade engine are proposed in self-contained optomechanical setups [91]. An optomechanical heat pump is introduced based on polariton modes [92]. A thermodynamic machine that is controlled on timescales that are much faster than the oscillator period is discussed [93].

There could be various theoretical models for different kinds of optomechanical heat engines. The model of the optomechanical heat engine based on the polariton normal mode excitations is briefly introduced here [82]. A standard optomechanical setup with a cavity mode coupled to a mechanical resonator at frequency ω_m is considered. The linearized Hamiltonian of the entire system can be written as

$$H=-\hslash \Delta {\widehat{a}}^{†}\widehat{a}+\hslash {\omega }_m{\widehat{b}}^{†}\widehat{b}+\hslash G(\widehat{b}+{\widehat{b}}^{†})(\widehat{a}+{\widehat{a}}^{†})$$ (12)

where $\widehat{a}$ and $\widehat{b}$ account for the fluctuations of the photon and phonon mode annihilation operators. Δ is the frequency detuning and G is the optomechanical coupling strength. By introducing a normal mode representation of the system, after removing a constant term, Eq. 12 can be expressed in the diagonal form:

$$H=\hslash {\omega }_A{\widehat{A}}^{†}\widehat{A}+\hslash {\omega }_B{\widehat{B}}^{†}\widehat{B}$$ (13)

where $\widehat{A}$ and $\widehat{B}$ are the boson annihilation operators for the normal-mode excitations of the system with frequencies:

$${\omega }_{A,B}=\sqrt{\frac{{\Delta }^2+{\omega }_m^2\pm \sqrt{{\left({\Delta }^2-{\omega }_m^2\right)}^2-16G^2\Delta {\omega }_m}}{2}}$$ (14)

Since the photon bath is much warmer than the photon bath, the optomechanical heat engine can operate along a single polariton branch and extract work from the thermal energy of a mechanical resonator at finite temperature [82].

Despite the fact that various types of optomechanical heat engines have been proposed, the experimental realization has only recently been completed. Recently, a coupled-mode stochastic heat engine has been demonstrated in a two-membrane cavity optomechanical system [42]. Instead of the strong coupling between the mechanical resonator and cavity field as proposed in Ref. [82], such a coupled-mode heat engine utilizes the phonon-photon-phonon coupling. By using the normal mode of two nanomechanical resonators as the working medium, not only is a single cycle heat engine demonstrated, but also a straight-twin nanomechanical heat engine is realized for the first time.

The experimental setup of the coupled-mode nanomechanical heat engine is similar to the one used for the phonon heat transfer [21]. Two spatially-separated nanomechanical membranes are placed inside an optical cavity. One membrane is in contact with a room temperature thermal bath and the other is electrically driven by white noise (mimicking the high temperature thermal bath). The cavity is driven by a red-detuned laser field, which interacts with both membranes simultaneously due to the dynamical backaction, and as a result, two membranes are effectively coupled by the cavity field. In the strong coupling regime, two normal mode branches appear, as shown in Fig. 11.

Fig. 11.

(a) Measured thermomechanical noise spectra as a function of ∆ω, which exhibits the anti-crossing behavior in the strong coupling regime. (b) Corresponding theoretical plot of the normal mode frequencies. The black dashed lines represent the bare modes of membranes, and the red and blue solid curves are the upper and lower normal modes of the effective coupled system, i.e. ω₊ and ω_-. Four strokes of an Otto cycle operated on the upper normal mode are denoted. The strokes from status 1 to 2 and from 3 to 4 correspond to the adiabatic processes. The strokes from status 2 to 3 and from 4 to 1 correspond to the isochoric processes [42].

The basic operating principle of such a coupled-mode heat engine is similar to the previous theoretical proposal [82]. Here, the heat engine operates along the upper normal mode branch, which is utilized as the working medium. The frequency of the upper normal mode can be viewed as the volume of working medium. Thus, the expansion can be realized by reducing the upper normal mode frequency ω₊, which is controlled by the frequency mismatch of membranes ∆ω. The heat engine is based on an Otto cycle with two adiabatic and two isochoric processes, as shown in Fig. 12a [42]. W_1→2, Q_2→3, W_3→4, and Q_4→1 represent the energy changes of the four strokes for the upper normal mode, respectively. The total work done by the upper normal mode for an ideal Otto cycle can be obtained as W = W_1→2 + W_3→4 = ℏ(ω_+,i ‒ ω_+,f)(〈N₊〉_f ‒ 〈N₊〉_i), where 〈N₊〉_i = k_BT_H/ℏω_+,i and 〈N₊〉_f = k_BT_L/ℏω_+,f are the mean phonon numbers of upper normal mode at status 1 and 3, respectively. The heat of the upper normal mode consuming from the hot bath is Q_4→1 = ℏω_+,i(〈N₊〉_i ‒ 〈N₊〉_f). η=W/Q_4→1 is defined as the work efficiency of the heat engine for the upper normal mode.

Fig. 12.

(a) Schematic of the Otto cycle associated with the upper normal mode. The Otto cycle contains two adiabatic and two isochoric processes. (b) Experimental results of the corresponding thermodynamic cycle of the upper normal mode [42].

The thermodynamic cycle starts at the status 1. The process of the status 1 to 2 corresponds to the expansion stroke, with a decrease of the upper normal mode frequency ω₊. The heat engine performs work to the environment during this process, and this is the key stroke of the cycle. The isochoric stroke is implemented in constant frequencies and corresponds to the stroke from status 2 to 3. In this stroke, the upper normal mode is fully thermalized to a low phonon population at room temperature. The process of status 3 to 4 is the compression process. The environment does work on the heat engine during this process, but it is negligible because the upper normal mode has been thermalized at the room temperature. The high temperature thermal bath is switched on during the status 4 to 1 (isochoric stroke) process, and the upper normal mode is thermalized to a high phonon population state, which gets back to the origin of the thermodynamic cycle.

By measuring the phonon numbers of the membranes in real-time, the thermodynamic diagram of the Otto cycle can be obtained, as shown in Fig. 12b [42]. The phonon number of the upper normal mode as a function of ω₊ is plotted with a frequency sweep time of 15 ms. This is more close to an ideal Otto cycle compared to a longer sweep time. The thermodynamic cycle starts from the expansion stroke and the phonon number decays are determined by the rates of γ₁ and γ₂ (the red dots in Fig. 12b). The cycle then proceeds counterclockwise, as shown in Fig. 12b, through the processes of thermalization to a low phonon population (green dots), adiabatic compression (blue dots), and thermalization to a high phonon population state (orange dots). The red squares represent the total phonon numbers of the upper and lower normal modes and the red circles are the phonon numbers of the upper normal mode by ignoring the correlations [42].

A straight-twin nanomechanical heat engine can be realized in the same setup by utilizing the two-membrane cavity optomechanical system and properly engineering the normal mode branches, allowing both the upper and lower normal mode branches to operate alternatively in the same thermodynamic cycle, which is preferable in practice because multi-cylinder heat engines can provide more power and smoothness in each cycle. As in a single-cylinder heat engine, the thermodynamic diagrams of both normal modes can be obtained in a single cycle, as shown in Fig. 13a and b [42]. The initial time of a cycle is represented by the top right corner in Fig. 13a and the left bottom corner in Fig. 13b. The four strokes are depicted by the red, green, blue, and orange dots in order. In one thermodynamic cycle, the work done by the two normal modes is 2.6 × 10⁻²¹ J and 0.3 × 10⁻²¹ J, respectively. By using the eigenmodes of membranes with higher quality factors, the difference in work performed by two normal modes can be reduced.

Fig. 13.

Experimental results of the thermodynamic diagrams of the (a) upper and (b) lower normal mode branches in a straight-twin nanomechanical heat engine, respectively[42].

6. Conclusion and outlook

Nonequilibrium thermodynamics has emerged as an active field, attracting researches from various disciplines. This article mainly discusses the recent experimental progress of nonequilibrium thermodynamics in cavity optomechanical systems. Entropy production, fluctuation theorems, heat transport, and heat engines are introduced, as the examples of important thermodynamic quantities, theories, phenomena, and applications, respectively. Selective works have been discussed in each topic.

Among the various experimental platforms for studying nonequilibrium thermodynamics, cavity optomechanical systems have several distinct advantages, the most important two of which are the ability to operate in the quantum regime and scalability. We end this article by discussing some promising directions for further research regarding these two points.

Ability to operate in the quantum realm. Most systems for studying nonequilibrium thermodynamics can only operate in the thermal environment, and are difficult to reach the quantum regime, e.g. colloidal particles and biological molecules. Some of the systems can investigate quantum properties, while they are based on the internal states. Cavity optomechanical systems are real mechanical systems as well as being able to achieve the deep quantum regime. As a result, many open questions, such as the role of entanglement, the emergence of thermalization, and quantum thermal machines, especially in open quantum systems and finite-time thermodynamics, can be investigated with quantum motional states in the laboratory.

Scalability. Most experiments of nonequilibrium thermodynamics have so far relied on a single mechanical oscillator. Cold atomic ensembles are excellent platforms for studying many-body thermodynamics, though controlling every element on demand remains a challenge. Cavity optomechanical systems, such as multiple membrane systems [94], [95], [96] and photonic crystal cavity optomechanical systems [97,63], can be extended straightforwardly, and more importantly, individual mechanical resonator and optical field can be manipulated precisely. As a result, such multimode cavity optomechanical systems could significantly improve our understanding of nonequilibrium thermodynamics, and encourage researchers to study nonequilibrium thermodynamics in complex networks [98] and quantum many-body systems [99,100]. Moreover, cavity optomechanics can operate as an excellent interface for a hybrid thermal machine, which incorporates other quantum systems such as cold atomic ensembles, superconducting qubits, trapped ions, nitrogen-vacancy centers, and nuclear spins [101].

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Jiteng Sheng received his Ph. D degree from University of Arkansas in 2013. From 2014 to 2016, he worked in the University of Oklahoma as a postdoc fellow. Since September 2016, he is a full professor in East China Normal University. His research interests include cavity optomechanics and Rydberg atoms.

Haibin Wu obtained his Ph. D degree from University of Arkansas in 2009. From 2010 to 2012, he worked in Duke University as a research assistant. Since September 2012, he is a full professor in East China Normal University. His research group (http://www.whblab.ecnu.edu.cn/) focuses mainly on ultracold quantum gases and cavity optomechanics.