Introduction

Entropy holds fundamental significance across a wide range of scientific disciplines. In a standard thermodynamics textbook, it is described as a measure of system’s disorder, intimately connected to the second law of thermodynamics through the Clausius inequality, i.e., ΔS ≥ ΔQ/T1,2,3. Here, ΔS represents the change in system entropy, ΔQ denotes the heat absorbed by the system, and T signifies the temperature of the thermal reservoir. Traditional entropy assessments rely on macroscopic properties and typically occur within reversible processes where equality is maintained, ΔS = ΔQ/T. In small systems, novel techniques have emerged to measure entropy, addressing the challenge posed by unmeasurable bulk properties4,5,6,7,8,9,10,11. Nonetheless, most of these methods are still based on equilibrium or quasi-equilibrium states.

In finite-time thermodynamics, entropy production has gained considerable attention recently3. It provides a comprehensive framework for expressing the second law of thermodynamics in nonequilibrium scenarios, serves as a valuable metric for assessing the degree of irreversibility, and establishes crucial connections with fundamental principles of nonequilibrium thermodynamics, such as fluctuation theorems12,13,14,15,16,17,18 and thermodynamic uncertainty relations19,20,21. In this context, the second law is expressed as dS/dt = Π(t)-Φ(t), where Π(t) is the entropy production rate and Φ(t) is the entropy flux rate from the system to the environment. Currently, much of the research focuses on the nonequilibrium steady state (NESS), where the time variation of S vanishes (dS/dt = 0), and all entropy produced must be transferred to the environment (Π = Φ)22,23,24,25. However, in the nonstationary state, dS/dt ≠ 0, and Π ≠ Φ, indicate deviations from the steady-state conditions.

Quench dynamics explores unconventional phenomena resulting from a sudden change in system parameters26,27,28. For example, it investigates the nonstationary thermodynamic dynamics29,30, such as prethermalization in many-body systems with ultracold atoms31,32, as well as the irreversibility associated with the emergence of time’s arrow in microscopic quantum systems33. Despite tremendous advances in quench dynamics in both classical and quantum regimes, quench dynamics of entropy change in a finite-time thermodynamic process has yet to be thoroughly investigated, especially in strongly interacting composite systems.

In this work, we consider phonon heat transport in a mesoscopic system, a prototypical example of nonequilibrium thermodynamics. As shown in Fig. 1a, two subsystems are initially in equilibrium with their local thermal reservoirs. By switching on the interaction between two subsystems at a certain time, the phonon heat transport will be generated, and the total system will evenly approach a NESS. We address two central questions: (1) What is the dynamical evolution of system entropy, and (2) is it possible to manipulate it? To answer these questions, we experimentally investigate the nonequilibrium dynamics of phonon heat transport in a multimode cavity optomechanical system. The nonstationary dynamical behavior of the entropy production rate and entropy flux rate, and consequently the system entropy change in phonon heat transport, is investigated by repeatedly quenching the optomechanical coupling introduced by the intracavity field and real-time continuous monitoring of single trajectories. Furthermore, we find that the entropy change is strongly dependent on the system characteristics, revealing a rich entropic dynamic in mesoscopic composite systems.

Fig. 1: Phonon heat transport between two coupled nanomechanical resonators.
Fig. 1: Phonon heat transport between two coupled nanomechanical resonators.
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a The mesoscopic system consists of two coupled phonon modes, M1,2, as two subsystems, in contact with two thermal reservoirs R1,2, with temperatures T1,2, respectively. Λ is the effective optomechanical coupling strength between two phonon modes. γ1,2 is the dissipation rate of the phonon mode to its local reservoir. b Two nanomechanical membranes (M1,2) are placed inside an optical cavity with cavity mirrors CM1,2. Their displacements are measured by two weak probe fields (P1,2) independently. The driving laser is switched on and off with a period of 2 s.

Results

Experimental setup

Entropy measurement of nonequilibrium phonon heat transport is carried out between two silicon nitride nanomechanical membranes (M1,2) placed inside an optical cavity, where the interaction between M1 and M2 resulting from the optomechanical coupling can be dynamically controlled via the intracavity field25,34. The fundamental vibrational modes of mechanical resonators are used, which are frequency degenerate with ωm/2π = 400 kHz but have different dissipation rates γ1(2)/2π = 7(14) Hz. The trajectories of resonators are detected independently in real-time without crosstalk by two weak probe fields, as shown in Fig. 1b. A red-detuned laser field with a frequency detuning Δ/2π = − 10 MHz from the cavity resonance is used to drive the optical cavity. The intracavity field interacts with two mechanical resonators simultaneously via radiation pressure. The linearized interaction Hamiltonian of such a composite system is \({H}_{\mathrm{int}}/\hslash =-{\sum }_{i=1,2}{g}_{i}\sqrt{{n}_{a}}(\delta {a}^{\dagger }+\delta a)({b}_{i}^{\dagger }+{b}_{i})\)25,35. Here, δa and bi are the annihilation operators for the optical and phononic modes, respectively. gi is the single-photon optomechanical coupling rate. na is the mean intracavity photon number.

Hamiltonian description

After adiabatically eliminating the cavity field, an effective coupling between two mechanical resonators is achieved25. Such a multimode optomechanical system can be treated as a nonequilibrium system of two coupled phonon modes contacted with one cold and one hot thermal reservoirs R1,2 of different temperatures T1,2 with the average thermal phonon occupations \({\bar{n}}_{1,2}^{th}=1/({e}^{\hslash {\omega }_{m}/{k}_{B}{T}_{1,2}}-1)\approx {k}_{B}{T}_{1,2}/\hslash {\omega }_{m}\) (kB is the Boltzmann constant), as depicted in Fig. 1a. Therefore, the Hamiltonian of the composite system is described by (see the Supplementary Note 1)

$$H/{{\hbar}} =\frac{{\omega }_{m}}{2}({{q}_{1}}^{2}+{{p}_{1}}^{2})+\frac{{\omega }_{m}}{2}({{q}_{2}}^{2}+{{p}_{2}}^{2})-\frac{\varLambda }{2}{({q}_{1}-{q}_{2})}^{2}$$
(1)

Here \({q}_{1,2}=({b}_{1,2}+{b}_{1,2}^{\dagger })/\sqrt{2}\) and \({p}_{1,2}=-i({b}_{1,2}-{b}_{1,2}^{\dagger })/\sqrt{2}\) are the dimensionless displacement and momentum operators. In the experiment, the optomechanical coupling rates are controlled to be \({g}_{1}=-{g}_{2}={g}_{0}\), and the frequency detuning satisfies \(|\varDelta | > \!\!\! > (\kappa ,{\omega }_{m})\), which leads to the effective optomechanical coupling strength \(\varLambda \approx 4{g}_{0}^{2}{n}_{a}/|\varDelta |\).

Entropy formulation in phonon heat transport dynamics

To estimate the system’s entropy change for nonequilibrium phonon heat transport, we adopt the framework recently developed for inferring thermodynamic irreversibility in a bosonic quantum system based on the Wigner entropy formulation36,37. The study in ref.23, based on the Shannon entropy of the Wigner function, has successfully implemented the experimental determination of entropy production in two different physical systems by measuring the moments in NESS, where Π = Φ and dS/dt = 0. Our experiment focuses on the nonequilibrium nonstationary/transient processes, which is the first direct experimental inference of entropy production of interacting composite systems in the nonequilibrium nonstationary processes, where Π ≠ Φ, we need to measure Π, Φ, and dS/dt separately at the same time. In the following, we present the general idea of how to obtain time-dependent Π, Φ, and dS/dt from the experimentally measurable quantities. It is important to note that the emphasis of our study is on the entropy production in the transient heat transport between two phonon modes, excluding the optical mode. A comprehensive model for assessing the entropy production, including the optical mode, can be found in Supplementary Note 2.

The dynamics of the composite system presented in Fig. 1a evolves according to the Lyapunov equation \({\partial }_{t}\sigma =M\sigma +\sigma {M}^{T}+D\). Here \(\sigma =\langle u{u}^{T}+{(u{u}^{T})}^{T}\rangle /2\) is a \(4\times 4\) symmetric covariance matrix of quadratures with the zero-mean vector \(u={\{{q}_{1},{p}_{1},{q}_{2},{p}_{2}\}}^{T}\)(\(\langle \cdot \rangle\) represents the ensemble average). M is the drift matrix,

$$M=\left(\begin{array}{cccc}-{\gamma }_{1}/2 & {\omega }_{m} & 0 & 0\\ -({\omega }_{m}-\varLambda ) & -{\gamma }_{1}/2 & -\varLambda & 0\\ 0 & 0 & -{\gamma }_{2}/2 & {\omega }_{m}\\ -\varLambda & 0 & -({\omega }_{m}-\varLambda ) & -{\gamma }_{2}/2\end{array}\right)$$
(2)

and\(D={\rm{diag}}({\gamma }_{1}({\bar{n}}_{1}^{th}+1/2),{\gamma }_{1}({\bar{n}}_{1}^{th}+1/2),{\gamma }_{2}({\bar{n}}_{2}^{th}+1/2),{\gamma }_{2}({\bar{n}}_{2}^{th}+1/2))\) is the diffusion matrix. The linear dynamics of the coupled resonators in our experiment perfectly fit the theoretical proposals in refs. 36,37. In general, the entropy production derived from the Wigner function is not considered as the thermodynamic entropy production, especially for the general non-Gaussian state where the Wigner function allows for negativity3. However, for the bosonic systems characterized by Gaussian states, the entropy can be quantified through the Shannon entropy of the Wigner function3,23,36,37. The system’s entropy S can be formulated in terms of the Shannon entropy of the Wigner function \(S(t)=-\int W(u,t){\rm{ln}}W({u},{t}){du}\), where \(W(u,t)\) the Wigner function of the system with the phase space vector u at time t. The quadratic feature of Eq. (1) and the initial Gaussian thermal state of the phonon modes keep the Gaussian nature of the system at any time, ensure the nonnegativity of \(W(u,t)\) and allow for expressing it in terms of the covariance matrix \(\sigma\). Following the procedure of estimating the Shannon entropy of the Wigner function for an open system of two coupled bosonic modes prepared in Gaussian states, dS(t)/dt, Φ(t), and Π(t) at any time can be expressed as23,36,37,38,39,40,41

$$dS(t)/dt={\rm{Tr}}({\sigma }^{-1}(t)D)/2-({\gamma }_{1}+{\gamma }_{2})$$
(3)
$$\Phi (t)={\Phi }_{1}(t)+{\Phi }_{2}(t)$$
(4)
$$\varPi (t)=dS(t)/dt+\Phi (t)$$
(5)

where \({\Phi }_{1,2}(t)={\gamma }_{1,2}({N}_{1,2}(t)-{\bar{n}}_{1,2}^{th})/({\bar{n}}_{1,2}^{th}+1/2)\) with the phonon number N1,2(t) at time t is the local entropy flux rate of phonon mode M1,2 relative to thermal reservoir R1,2. According to Eqs. (3)–(5), the time revolution of the entropic property of the system can be fully extracted from σ(t), and all the elements σi,j(t) (i, j = 1, 2, 3, 4) can be obtained by averaging the fluctuating quantities over the measured individual trajectories of phonon modes. This needs to simultaneously measure real-time trajectories of all degrees of freedom and repeatedly implement quenching control at the common initial state in nonequilibrium nonstationary cases while keeping the long-term stability of the whole experimental system.

Experimental results

In the experiment, we use a lock-in amplifier to measure single trajectories of mechanical resonator quadratures and, consequently, obtain the second statistical moments σ(t). Before quantifying the time evolution of entropy, we first analyze the time-dependent average phonon numbers \({N}_{1,2}(t)=({\sigma }_{11,33}(t)+{\sigma }_{22,44}(t))/2\) and heat current \(J(t)=\hslash {\omega }_{m}\varLambda ({\sigma }_{12}(t)-{\sigma }_{23}(t))\), which are the basic properties of nonequilibrium phonon heat transport and could be of benefit for better understanding the entropic property. By setting T1 = 300 K (room temperature) and T2 = 11000 K (the effective high temperature is realized by driving the phonon mode piezoelectrically with white noise), and turning on and off the cavity field repeatedly, N1,2(t) and J(t) under different effective interaction strengths are obtained, as shown in Fig. 2, following the system quenched at t = 0. Since the physical observables fluctuate severely in such a mesoscopic system, each curve shown in Fig. 2 is averaged over 1000 single trajectories. Here, T1≪T2 indicates a significant temperature gradient, which breaks the detailed balance and plays the dominant role in the dynamics of entropy for photon heat transport.

Fig. 2: Observations of phonon heat transport in the transient regime.
Fig. 2: Observations of phonon heat transport in the transient regime.
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a–c The phonon numbers N1,2(t) as a function of time at the different coupling strengths Λ/2π = 8, 20, 140 Hz, respectively. The green and red dots represent the measured phonon numbers N1 and N2, and the solid curves are the theoretical simulations. Each curve is averaged over 1000 trajectories. The error bars are the standard deviations. d–f The heat current J(t) as a function of time for the cases corresponding to (a–c), respectively. The heat currents along individual trajectories are presented by the light lines. g–i J(t) as a function of ΔN(t) for the cases corresponding to (a–c), respectively.

The time evolutions of N1,2(t) and J(t) show the transitions from monotonic to oscillatory behaviors with the increase of Λ, as illustrated in Fig. 2a–c, d–f, respectively. The oscillatory behaviors found in the strong coupling limit (Λ≫γ1,2) are analogous to the Rabi oscillation of a two-level system coupled to a field and indicate that the energy goes back and forth between two phonon modes until the system relaxes into the NESS42. Correspondingly, the relationship between J(t) and ΔN(t) = N2(t)-N1(t) changes from monotonic to spiral characteristics as shown in Fig. 2g–i. In Fig. 2g, J(t) increases monotonically with the continuing decrease of ΔN(t) at Λ/2π = 8 Hz. For Λ/2π = 20 Hz in Fig. 2h, the oscillatory behavior starts to appear, and both J(t) and ΔN(t) eventually relax to a nonzero value. For a relatively strong coupling strength of Λ/2π = 140 Hz, multiple oscillations are observed and the signs of J(t) and ΔN(t) are reversed at certain times, as shown in Fig. 2i. The oscillations of J(t) and ΔN(t) have a phase shift of approximately π/2 and the period of oscillation equals 2π/Λ. Such behaviors between J(t) and ΔN(t) are the result of different responses of the system induced by competition between coherent interaction and dissipation under the driving of the nonequilibrium baths, which are akin to the relationship between the dipole moment and population inversion in two-level atom with radiative damping under different Rabi coupling strengths42.

According to Fig. 2, we expect that the entropy could have similar behaviors, since the entropy rates depend on J(t) and N1,2(t). The measured Π, Φ, and dS/dt for Λ/2π = 8 Hz and 80 Hz, are presented in Fig. 3a, c, respectively. Both Π and Φ increase from zero over time, with a monotonic increase at a relatively weak coupling strength (shown in Fig. 3a) and an oscillatory increase at a relatively strong coupling strength (shown in Fig. 3c), which eventually reach a positive constant value after the transient period. The positivity of Π manifests the time’s arrow and irreversibility in the nonequilibrium transport process and this irreversibility can be strengthened by enhancing the interaction strengths, e.g. Π(t) approaches ~75 Hz and 165 Hz in the NESS for Λ/2π = 8 Hz and 80 Hz, respectively (see Fig. 3a, c), which is consistent with the theoretical prediction43. During the transient time, we find that Π > Φ at most of time, which indicates that more entropy inside the system is produced than the entropy flowing into the environments until the NESS is reached with Π = Φ. Meanwhile, dS/dt first rises from zero to its maximum and then exponentially decreases to zero, which is the requirement for the NESS. However, dS/dt becomes negative at some time in Fig. 3c during the oscillation, which implies that dS/dt does not have to be positive as Π is. Moreover, according to Eq. (4), the total entropy flux is contributed from two local entropy fluxes. In Fig. 3b, d, one can see that more entropy from M1 flows into R1 (Φ1 > 0) than that from R2 flows into M2 (Φ2 < 0). This is because the ratio between N1 and \({\bar{n}}_{1}^{th}\) is much larger than that of N2 and \({\bar{n}}_{2}^{th}\)(for T1≪T2), i.e., \({N}_{1}/{\bar{n}}_{1}^{th} > \!\!\! > 1 > {N}_{2}/{\bar{n}}_{2}^{th}\), as shown in Fig. 2. The shaded regions of data curves of Fig. 3 indicate the corresponding standard deviation of the statistical fluctuations from numerous trajectories measured by repeatedly quenching control, which reveals the characteristic of inherent fluctuations of small system.

Fig. 3: Dynamical evolutions of the entropy rates.
Fig. 3: Dynamical evolutions of the entropy rates.
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a The entropy production rate Π, the entropy flux rate Φ, and the time variation of entropy dS/dt, and b Φ1,2 as a function of time for Λ/2π = 8 Hz. c Π, Φ, and dS/dt, and d Φ1,2 as a function of time for Λ/2π = 80 Hz. Each curve is averaged over 1000 trajectories. The effective temperatures of thermal reservoirs are the same as those of Fig. 2. The shaded regions represent the standard deviations. The solid curves are the theoretical simulations.

Figure 4 shows the time evolution of system entropy change, \(\varDelta S(t)={\int }_{0}^{t}(dS/dt)dt\), for different interaction strengths. In general, ΔS rises from zero to different saturation values depending on Λ. When Λ is relatively small, the dynamics of ΔS is close to Gaussian relaxation (the red circles in Fig. 4 are fitted by a Gaussian function \(g(t)=A-A{e}^{-{(t/\tau )}^{2}}\)), which is due to the dominantly nonunitary dynamics from the dissipation. For a larger Λ, the dynamics of ΔS becomes the mixture of exponential and Gaussian relaxations (the blue circles in Fig. 4 are fitted by a combined exponential and Gaussian function \(h(t)=A+B-(A{e}^{-{(t/{\tau }_{1})}^{2}}+B{e}^{-t/{\tau }_{2}})\) without considering the oscillatory tail) due to the competition between the unitary and nonunitary interactions3,44. Moreover, not only does an oscillatory tail appear, but also ΔS increases much faster for the early-time growth at a larger Λ. The inset of Fig. 4 shows the details of the oscillations, and it can be seen that the oscillation of ΔS has the same period as dS/dt, which is determined by 2π/Λ.

Fig. 4: Time evolution of the system entropy change.
Fig. 4: Time evolution of the system entropy change.
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The red circles, yellow triangles, purple squares, and blue circles present the time evolution of system entropy change ΔS for Λ/2π = 8, 20, 50, and 80 Hz, respectively. The inset shows the detailed oscillations. The effective temperatures of the two thermal reservoirs are the same as those of Fig. 2.

Lastly, we show that the entropy dynamics of phonon heat transport also strongly depend on the mechanical dissipation rates, besides the temperature gradient and the interaction strength. For this purpose, we set the temperature difference of reservoirs much smaller than the case of Fig. 3. In Fig. 5a, one can see that Φ becomes negative immediately after quenching instead of positive as shown in Fig. 3c. This is because less entropy from M1 flows into R1 (Φ1 > 0) than that from R2 flows into M2 (Φ2 < 0), as shown in Fig. 5b. The ratio between the local entropy fluxes depends on both the ratios of reservoir temperatures and dissipation rates, i.e. \(|\frac{{\varPhi }_{1}(t)}{{\varPhi }_{2}(t)}|\approx \frac{{\gamma }_{1}}{{\gamma }_{2}}\frac{{T}_{2}}{{T}_{1}}|\frac{{N}_{1}(t)-{\bar{n}}_{1}^{th}}{{N}_{2}(t)-{\bar{n}}_{2}^{th}}|\), therefore, the unequal dissipation rates (γ2 ≈ 2γ1) dominate the dynamics of entropy flux at T1 ≈ T2. Meanwhile, dS/dt exceeds Π during an initial time interval in contrast to the case shown in Fig. 3c, and relaxes to zero in an oscillatory way. The positivity of dS/dt indicates the system entropy increases in the transient regime, i.e., ΔS > 0, which is consistent with the situation of T1≪T2. Surprisingly, after reversing the temperature gradient, dS/dt approaches zero oscillatory from a negative value, indicating that the system entropy decreases, i.e., ΔS < 0, shown in Fig. 5c. Meanwhile, Φ exceeds Π in the transient regime and relaxes to Π = Φ in the NESS. This is due to the fact that more entropy from M2 flows into R2 (Φ2 > 0) than that from R1 flows into M1 (Φ1 < 0), as shown in Fig. 5d. The periodic oscillations of entropy rates in Fig. 5 are the result of the hybrid motion from the system’s normal mode splitting in a strong coupling regime25, which is also responsible for the oscillations in phonon number, heat current and entropy in Figs. 2–4. Additionally, the data curves of Fig. 5 after averaging abundant trajectories are still obviously fluctuant due to that the smaller temperature gradient of two reservoirs brings the weaker average heat current comparable to the inherent fluctuations, which is in contrast to the larger temperature gradient case in Fig. 3.

Fig. 5: Dynamical evolutions of the entropy rates by engineering thermal reservoirs with smaller temperature gradient.
Fig. 5: Dynamical evolutions of the entropy rates by engineering thermal reservoirs with smaller temperature gradient.
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a The entropy production rate Π, the entropy flux rate Φ, and the time variation of entropy dS/dt, and b Φ1,2 as a function of time for Λ/2π = 80 Ηz. The effective temperatures of two thermal reservoirs are T1 = 300 K and T2 = 400 K. c Π, Φ, and dS/dt, and d Φ1,2 as a function of time for Λ/2π = 80 Ηz. T1 = 400 K and T2 = 300 K. The fluctuating curves are the experimental measurements by averaging over 1500 trajectories. The solid lines are the theoretical simulations.

Conclusions

In conclusion, we have experimentally measured the nonstationary dynamics of entropic properties for nonequilibrium phonon heat transport in a multimode cavity optomechanical system. A rich variety of entropy dynamics has been observed. For example, the observations of the Gaussian, exponential and oscillatory relaxation dynamics, due to the competition between unitary and nonunitary processes3,44, indicate that the interaction between subsystems can strongly modify the entropy dynamics. The entropy production rate is always larger than zero, even in the transient regime, in accordance with the second law of thermodynamics; however, the system entropy change doesn’t have this requirement and can be controlled depending on the system parameters. Assessing entropy dynamics in such a nonequilibrium system could be extended to many-body and strongly interacting quantum systems within the realm of finite-time thermodynamics. It provides a profound platform for investigating fluctuation theorems and different versions of thermodynamic uncertainty relations in the transient regime21,45,46, as well as thermodynamic theories involving both unitary and nonunitary interactions. Our research provides further possibilities in the optimization of thermodynamic processes, such as the shortcut to equilibration and quantum control of open systems47.

Methods

Nanomechanical membranes

Two commercial stoichiometric silicon nitride membranes with size a \(1\times {{\rm{1mm}}}^{2}\) and a thickness of 50 nm are used in the experiment. Each membrane is attached to two cascaded ring piezo actuators so that its frequency and position can be controlled independently. The mechanical motions of two membranes are detected independently in real-time without crosstalk by two weak probe fields reflecting from two membranes. A lock-in amplifier is used to analyze the data of the membranes’ motions in real-time.

Two-membrane cavity optomechanical system

Two spatially separated nanomechanical membranes are placed inside an optical cavity, which constitutes a two-membrane cavity optomechanical system25,34. This system is placed inside a vacuum chamber with a vacuum of 10-8Torr. The Pound-Drever-Hall technique is used to stabilize the optical cavity frequency. A strong far-red-detuning pumping laser drives the optical cavity so that the intracavity field can enhance the optomechanical interaction between the cavity mode and nanomechanical membranes by the dynamical backactions. The power switching of the pumping laser can be controlled by the acousto-optic modulator to achieve the turning on and off of the cavity field repeatedly.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.