Accelerating the Heat Diffusion: Fast Thermal Relaxation of a Microcantilever

PHYSICAL REVIEW APPLIED 19, 034072 (2023) Accelerating the Heat Diffusion: Fast Thermal Relaxation of a Microcantilever Basile Pottier ,1Carlos A. Plata ,2Emmanuel Trizac ,3David Guéry-Odelin ,4and Ludovic Bellon 1,* 1Université de Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, Lyon F-69342, France 2Fisica Teórica, Universidad de Sevilla, Apartado de Correos 1065, Sevilla E-41080, Spain 3Université Paris-Saclay, CNRS, LPTMS, Orsay F-91405, France 4Université de Toulouse, CNRS, IRSAMC, Laboratoire de Collisions Agrégats Réactivité, Toulouse F-31062, France (Received 3 June 2022; revised 19 December 2022; accepted 24 February 2023; published 22 March 2023) In most systems, thermal diffusion is intrinsically slow with respect to mechanical relaxation. We devise here a generic approach to accelerate the relaxation of the temperature field of a one-dimensional object, in order to beat the mechanical time scales. This approach is applied to a micrometer-sized silicon cantilever, locally heated by a laser beam. A tailored driving protocol for the laser power is derived to quickly reach the thermal stationary state. The model is implemented experimentally yielding a significant acceleration of the thermal relaxation, up to a factor 30. An excellent agreement with the theoretical predictions is reported. This strategy allows a thermal steady state to be reached significantly faster than the natural mechanical relaxation. DOI: 10.1103/PhysRevApplied.19.034072 I. INTRODUCTION In condensed matter, the temperature fields evolve in general on much longer time scales than their mechanical counterparts (such as stress or strain). Indeed, combining the highest thermal diffusivity in solids [1](D∼ 10−4m2/s) and the order of magnitude of the speed of sound [1](c∼[103−104]m/s), one defines a typical length scale lth =D/c∼[10 −100] nm beyond which temperature is a slow phenomenon. Most systems imply larger length scales, thus a slow temperature evolution. In some devices or experiments, or for proof-of-concept demonstrations, it can however be desirable to accelerate the heat diffusion so as to impose temperature variations on time scales equivalent, or shorter, than those of the mechanical response of the system. One example is atomic force microscopy (AFM) [2], where a sharp tip attached to a cantilever scans a sample to map its topography and potentially more local properties of its surface. Most commonly, the measure of the interaction force through the deflection of the cantilever is performed with a laser beam [3], hence the photothermal response and thermal stability of AFM sensors have been extensively studied almost since the origin of AFM [4,5]. Photothermal excitation has been used in vacuum, air, or fluid, aiming at driving the cantilever efficiently while avoiding overheating of the system [6–11]. Let us compare the relaxation time τ=L2/Dof the temperature field *[email protected] along the cantilever length L, to the period of oscillation of the first resonance T0≃6.4L2/(cH)[12], with Hthe thickness of the cantilever: T0/τ ≃6.4lth/H=65 nm/H, where the numerical application has been done for silicon [13]. With Htypically in the few µm range, thermal diffusion is thus much slower than the oscillation period of the system, limiting the efficiency of photothermal excitation, or the speed of operation of scanning thermal microscopy [14–16]. Another example where fast temperature variations could be desirable is in stochastic thermodynamics experiments on micromechanical systems [17,18] [optically trapped particles, microcantilevers, microelectromechanical systems (MEMS)]. When constructing a stochastic Carnot heat engine, for instance [19–28], one needs to perform adiabatic heating or cooling, that is to say change the system temperature much faster than the time scale τrcorresponding to the heat exchanges with the thermostat. τris equivalent to the mechanical relaxation time, which writes for a cantilever τr=QT0/π, with Qthe quality factor of the resonance [17,29–34]. The situation is thus somewhat more favorable than previously for underdamped systems, but requires quality factors larger than a few hundred to start matching the natural thermal diffusion and mechanical relaxation time scales: τr/τ ≃Q×20 nm/H. Here again, an acceleration of the temperature dynamics would be welcome to perform efficient adiabatic transformations. The motivation of this work is to speed up the natural thermal relaxation, and we focus in this paper on the case of a cantilever, in relation to the previously 2331-7019/23/19(3)/034072(14) 034072-1 © 2023 American Physical Society BASILE POTTIER et al. PHYS. REV. APPLIED 19, 034072 (2023) mentioned examples. Engineered accelerated dynamics and shortcuts are protocols of the “shortcut to adiabaticity” type [35]. This class of ideas emerged in the quantum realm [36,37]. Yet, a number of techniques belonging to this family of accelerating methods has been successfully exported to classical and stochastic dynamics [28,38–41]. Some of these shortcuts are of inverse engineering type, requiring a careful monitoring of the time dependence of some control parameters, to impose a prescribed evolution of interest [39]. Such accelerating protocols have also been used fruitfully in systems such as cranes [42]and capacitors [43]. In the following, we address the question of the temperature-field control, so far untouched. The goal is to drive the thermal relaxation of a microcantilever in a time much smaller than its natural relaxation time. First, we model the thermal relaxation and derive theoretically the accelerating protocols allowing reduction of the relaxation time of the cantilever to reach thermal steady state. Second, the aforementioned protocols are experimentally implemented in a silicon cantilever. The efficacy of the driving is patent when comparing to a direct relaxation process. Moreover, experimental results are in full agreement with the predictions. Finally, we summarize the conclusions of our study and mention future perspectives. II. THEORY Our objective is to accelerate the relaxation towards a nonequilibrium steady state of a cantilever with a tunable punctual and well-localized heat source. To this end, we first identify the different eigenmodes (and corresponding time scales) of the heat equation that governs the dynamics of the system. We subsequently design the rate at which the deposit of heat should be carried out so as to cancel the contribution of the low-frequency eigenmodes that slow down the relaxation. This original trick enables one to benefit from a fast relaxation with the cancelation of just a few modes. A. Temperature rise of a cantilever irradiated by a laser beam We gather here the main ingredients useful for a quantitative description of the temperature field of a cantilever irradiated by a laser beam. As the cantilever is placed in vacuum, no heat transfer can occur by convection with its surroundings. Neglecting thermal radiation, the only possible heat-transfer mechanism is thus thermal conduction through the cantilever. Considering a cantilever having a length Lmuch larger than its transverse dimensions, the temperature θmay be assumed homogeneous across the cross section. Therefore, θdepends only on the longitudinal coordinate x∈[0,L] and time t, its dynamics being described by the one-dimensional heat-diffusion equation: ρcp ∂θ(x,t) ∂t=λ∂2θ(x,t) ∂x2+q(x,t),(1) where ρis the density, cpis the heat capacity, λis the thermal conductivity (all of them assumed constant), and qis the heat source or sink density. In vacuum, all cantilever surfaces are assumed to be thermally insulated, except at the location x0irradiated by the focused laser. We model the heating effect of the focused beam by the source term q(x,t)=aP(t) Sδ(x−x0),(2) where ais the fraction of light absorbed by the cantilever, P(t)is the incident power of the laser beam, which can be manipulated at wish, Sis the cantilever cross-section area, and δDirac’s distribution. Furthermore, we assume the following boundary conditions: θ(0, t)=0, (3a) ∂θ ∂x(L,t)=0. (3b) These conditions reflect, respectively, that the cantilever is in contact with the macroscopic chip acting as a thermostat at its clamp x=0, Eq. (3a), and is isolated on its free end x=L,Eq.(3b). Note that, for the sake of simplicity, θis not the absolute temperature, as we define the origin by the temperature of the chip. B. Stationary profile θs The stationary temperature profile θs(x)associated to a given power of the laser Pfis obtained solving Eq. (1) where ∂θ/∂t=0, with the boundary conditions Eq. (3). We get θs(x) θm s=x/x0,0<x<x0, 1, x0<x<L,(4) where θm s=aPfx0/(Sλ) corresponds to the maximum temperature elevation. Between the chip and the laser spot (x=x0), the stationary temperature increases linearly with the position x, while beyond the laser spot, it remains constant and equals θm s. From now on, we focus the analysis on the relaxation between two stationary states when driving the power P(t) from a constant initial value P(t<0)=P0to its constant final value P(t>tf)=Pf. Note that even if P(t) is constant after the final time tf, the temperature profile continues to evolve towards its final asymptotic stationary shape. Since the heat-diffusion equation [Eq. (1)] is linear, we make the simplifying assumption that P0=0: without loss of generality, P(t)or Pfrepresent the laser power 034072-2 ACCELERATING THE HEAT DIFFUSION... PHYS. REV. APPLIED 19, 034072 (2023) excess with respect to P0,andθ(x,t)or θs(x)stand for the temperature excess with respect to the initial stationary temperature profile. C. Transient solution through expansion The general solution of Eq. (1) can be expressed carrying out an expansion in eigenfunctions ψn(x)[44, 45], each satisfying the associated homogeneous problem ∂2 ˜xψn(˜x)=−k2 nψn(˜x)with ψn(0)=0andψ n(L)=0. Solving the homogeneous problem leads to ψn(˜x)=√2sin(kn˜x),˜x=x/L(5) with the eigenvalues kn=(n−1/2)π,n∈N∗. The temperature profile can be expressed as θ(x,t)=θs(x)+θm s ∞  n=1 Cn(˜ t)ψn(˜x),(6) where the components Cn(˜ t)depend solely on˜ t=t/τ, with τ=ρcpL2/λ, the characteristic diffusion time of the problem. These components are obtained projecting Eq. (1) on the eigenmodes basis ψn(x); they read as Cn(˜ t)=ψn(˜x0) ˜x0˜ t 0F(sτ)−1ek2 nsds −1 k2 ne−k2 n˜ t,(7) with ˜x0=x0/L,andFthe driving function defined as F(t)=P(t) Pf .(8) The temperature solution Eq. (6) is completely general for arbitrary F(t), even though here F(t<0)=0andF(t> tf)=1 by construction. For t>tf, the bracket term in Eq. (7) is constant, so each component in the expansion decays exponentially with its own decaying rate. Specifically, the nth component has a rate equal to k2 n, which increases with the mode number nas (2n−1)2: the second mode decays k2 2/k2 1=9 times faster than the fundamental mode, the third mode decays k2 3/k2 1=25 times faster than the fundamental one, etc. The higher the mode n, the less time it takes for that component of the temperature to relax. D. Accelerating the dynamics We explain hereafter how the driving function can be designed to remove the contribution of the slow relaxation modes, thereby speeding up the relaxation dynamics in a given arbitrary small (chosen) time lapse tf. Our strategy is somewhat similar to that of Ref. [46] on a different thermal acceleration problem. For this purpose, we use an ansatz for F(t)that involves as many parameters as modes to be canceled. Let us assume that we want to cancel the Nfirst modes. We can take the following convenient polynomial ansatz: F(t)= N  m=1 γmt/tfm−1,0<t<tf.(9) Canceling the Nfirst modes corresponds to impose Cn(˜ tf)=0forn=1toN, where ˜ tf=tf/τ. This implies for n∈[1,N] N  m=1 γm˜ tf 0s ˜ tfm−1 ek2 nsds =ek2 n˜ tf k2 n , (10) which is a system of Nlinear equations for Nunknown variables γn, analytically solvable. When tfis reached, we change the laser power abruptly to Pf[i.e., F(t>tf)= 1] and the relaxation will occur only through the modes higher than the Nth one (since we already force the cancelation of the first Nmodes). Examining Eq. (10), the coefficients γndepend only on the normalized protocol duration ˜ tfand the number of canceled modes N; the protocol to apply thus does not depend on the laser-beam position ˜x0. For the sake of concreteness, we illustrate our protocols with ˜x0=0.95 (we exemplify the independence of the protocol on ˜x0by working out the solution for ˜x0=0.5 in Appendix B). If one wants only to cancel the relaxation of the fundamental mode (N=1), the function F(t)to apply during the protocol is constant and equals 1/(1−e−k2 1˜ tf);for˜ tf=0.1, we have F(t)=4.57. In Fig. 1, we show the functions F(t)that allow cancelation of up to the third mode for a protocol duration ˜ tf=1/10 and ˜ tf=1/30. Further details on the computation and properties of F(t)are given in Appendix A.The needed range of F(t)(thus the laser power) increases with both the number of canceled modes Nand the acceleration factor τ/tf. Note that for N>1, F(t)requires both positive and negative values. In practice, the cantilever can only absorb the power from the laser beam. However, P(t) corresponds to the power excess with respect to the initial value P0, so that the protocol will be valid as long as P0is larger than −min[P(t)]. A priori, this constraint implies a limitation for our protocols. However, we prove below that our method succeeds in accelerating thermal relaxation substantially in a wide variety of experimentally realizable situations. III. EXPERIMENTAL RESULTS A. Experimental setup For the experiments presented below, a silicon cantilever placed in vacuum is heated with a laser and the evolution of its temperature at various positions xalong its length Lis measured. 034072-3 BASILE POTTIER et al. PHYS. REV. APPLIED 19, 034072 (2023) FIG. 1. Driving function F(t)to apply in order to cancel successively the first thermal modes for an acceleration (towards the stationary profile) such that tf=τ/10 (left) and tf=τ/30 (right). The coefficients γndefining F(t)are found solving Eq. (10); they are listed in Table Iof Appendix A. At a thickness of a few micrometers, silicon is semitransparent for visible light, the light experiences multiple reflections within the thickness of the cantilever, which acts as a lossy Fabry-Perot [47]. As a result, the fraction of light reflected Rdepends on the thickness and the silicon refractive index, both varying with temperature θ.By measuring the local change in reflectivity, one can thus infer the local temperature variation with respect to the reference temperature field. For small variations, we have R(x)/R(x)=βθ(x), where the sensitivity βdepends on the local properties (refractive index and thickness). The calibration procedure for βis detailed in Appendix C. The experimental setup comprises two laser beams (Fig. 2): (i) a heating laser beam (at 532 nm) of variable power P(t)focused at a fixed position close to the cantilever free end (x0=0.95L) that serves to heat the cantilever, and (ii) a probe laser beam (at 633 nm) of constant power (350 µW) that allows us to measure the local temperature rise taking benefit of the thermo-optical effect described above. The radius of the heating and probe beam at the cantilever surface are, respectively, 14 and 7µm. The power of the heating beam is modulated with an acousto-optic modulator, with a rise time of 100 ns. Three photodiodes P1, P2, and P3 measure the incident heating beam power P(t)and the reflectivity Rof the probe beam, respectively, as shown in Fig. 2. By tilting the probe laser, one scans the cantilever length facilitating temperature measurements at various locations x. During the experiment, the cantilever is placed in vacuum at 2 ×10−2mBar. At this pressure level, the contribution of convective heat transfer is negligible compared to thermal conduction [8]. All the measurements are performed using a single raw silicon cantilever OCTO-1000D from Micromotive, with the following size: length L= 1 mm, width W=90 µm, and thickness H=5µm. AOM Probe laser @633 nm Heating laser @532 nm Dichroic Focusin g lens QWP NPBS Vacuum Data acquisition t P1 P2 P3 AOM: acousto-optic modulator NPBS: nonpolarizing beam splitter PBS: polarizing beam splitter QWP: quarter-wave plate P: photodiode Probe beam spot (various positions) Heating beam spot (fixed) L Chip Cantilever xH Function generator PBS W FIG. 2. Sketch of the experimental setup. The heating laser beam (at 532 nm) is fixed and focused at the extremity of a cantilever placed in vacuum. The power of the heating beam is modulated with an acousto-optic modulator. The probe laser beam (at 633 nm) is used to measure the local temperature change by analyzing the cantilever reflectivity variations. By tilting the probe laser, the probe beam spot scans the cantilever length allowing measurement of the temperature variation at different positions x. To record the spatial dependence of the temperature profile, we choose 20 evenly spaced positions xof the probe beam along the cantilever. At each position, we measure the time evolution of the temperature θ(t)induced by the applied incident power P(t)of the heating beam. To increase the signal-to-noise ratio, the displayed temperature variations are obtained averaging over 700 heating procedures at each x. The spatial resolution is approximately the size of the laser spot, 7 µm, hence smaller that 1% of the cantilever length. B. Relaxation for a step function [F(t)=1] Before testing the performance of the accelerating protocols worked out in Sec. II, we first present the results obtained imposing a jump in power, P(t)=Pffor t>0. The analysis of the temperature relaxation allows measuring the characteristic diffusion time τneeded to determine the function F(t), to later successfully accelerate the dynamic given a protocol duration tf. Moreover, this is the simplest reference that can be thought of and it defines the time scale that we would like to beat. In Fig. 3and in the Movies within the Supplemental Material [48], we display the cantilever temperature rise measured imposing a sudden increase in the power of Pf= 034072-4 ACCELERATING THE HEAT DIFFUSION... PHYS. REV. APPLIED 19, 034072 (2023) 5.6 µW. The heating laser spot is focused close to the free end of the cantilever at x0=0.95L. The convergence of the temperature profile towards the expected linear stationary profile given by Eq. (4) confirms that for the temperature variations explored, the silicon conductivity λcan be assumed constant. The maximum temperature elevation is θm s=38 mK, this value is consistent with the theoretical one θm s=aPfx0/(WHλ) with λ=156Wm −1K−1[49] and a=0.5, an absorption coefficient also consistent with the H=5µm thickness of the cantilever [47]. In the case of a step function [F(t)=1], the temperature is predicted by Eq. (6) where the components Cn(˜ t) are reduced to Cn(˜ t)=−ψn(˜x0) ˜x0 1 k2 n e−k2 n˜ t. (11) Using Eq. (11), all experimental data are properly described by theory for a characteristic time τ=14.2 ms. From the thermal diffusivity of bulk silicon given in the literature [49], D=λ/ρcp=86 mm2/s, we expect τ= L2/D=11.6 ms. The 18% difference could be explained by the reduced diffusivity of silicon due to the phonon confinement effect. Indeed, for a 5-µm-thick film, the silicon conductivity is expected to be approximately 15% smaller than the value for bulk silicon [50]. C. Relaxation for an acceleration factor τ/tf=10 In Fig. 4, and in the Movies within the Supplemental Material [48], we display the evolution of the measured cantilever temperature when applying the driving function F(t)given by Eq. (9) for a protocol duration tf=1.4 ms, corresponding to ˜ tf=0.1. We test our protocol varying the number of canceled modes Nfrom 1 up to 4. Note that the imposed functions F(t)correspond to the ones already presented in Fig. 1(left panel). In order to be able to impose the negative variations in F(t)(for N>1), we choose a reference incident power of P0=210 µW. As the number of canceled modes Nincreases, the temperature at the end of the protocol (t=tf) clearly converges towards the stationary profile θs(x)(Fig. 5). Contrary to the step function (Fig. 3), the temperature takes intermediate values higher than the stationary target θs(x). The maximum transient temperature is obtained for N=4 and corresponds to θ=4.7 ×θm sat x=0.95L. For each protocol, we compare the experiment to theory computing the temperature from Eq. (6) limiting the infinite sum to 60 terms; all experimental data remarkably coincide with theory. To highlight the success of our tailored acceleration, we compare in Fig. 6the difference of convergence towards the stationary profile θsbetween the step function power (displayed in Fig. 3) and the accelerating protocol for N=4 (displayed in the last panel of Fig. 4). Using the step function, the duration needed to converge within 2 % (at the positions ˜x=0.3,0.67, and 0.95) is 25 ms (1.8τ), while in the accelerating protocol this milestone is reached at the end of the program tf=1.4 ms (0.1τ), beating remarkably the natural relaxation time. Actually, having gotten rid of the first four modes, the fifth and higher orders do remain and are responsible for a slight mismatch with respect to the target profile. They decay at least 92=81 times faster than the power step-forcing relaxation, with time scales smaller or equal to τ/(4.5π)2≃0.07 ms, much smaller than tfitself. Hence, since the characteristic evolution time scale (after tf)isk−2 N+1τ, high values of Nare needed for extremely low values of tf/τ only. (a) (b) (c) FIG. 3. Transient temperature profiles (a) of a cantilever heated by a laser whose incident power P(t)is the step function (b). The heating laser is located near the cantilever free end (˜x0=0.95). The 5.6-µW increase in power induces a maximum temperature elevation of θm s=38 mK. As time goes on, the temperature profile converges towards the expected linear stationary profile given by Eq. (4). Graph (c) shows the normalized evolution of temperature at the positions ˜x=0.3, 0.67, and 0.95, corresponding to the vertical dashed lines in (a). All experimental data are perfectly described by the model of Eq. (6), using the characteristic time τ=14.2 ms. 034072-5 BASILE POTTIER et al. PHYS. REV. APPLIED 19, 034072 (2023) (c) (b) (a) (b) (a) (c) (b) (a) (c) (b) (a) (c) FIG. 4. Transient temperature profiles measured when applying the protocol function F(t)given by Eq. (9) with a protocol duration tf=0.1τdesigned to cancel the Nfirst thermal modes. For each tested protocol (N=1uptoN=4) we display (a) the temperature profiles, (b) the imposed power P(t), and (c) the temperature variations with time at the positions ˜x=0.3, 0.67, and 0.95. All experimental data coincide with the expected theoretical temperature variations. 034072-6 ACCELERATING THE HEAT DIFFUSION... PHYS. REV. APPLIED 19, 034072 (2023) FIG. 5. Relative temperature difference from the targeted stationary profile θs(˜x)at the end of the protocol t=tf=0.1τ. As the number of canceled modes Nincreases, the temperature profile converges towards the stationary profile. For N=4, the relative temperature difference remains below 10% along the cantilever length. D. Relaxation for an acceleration factor τ/tf=30 In Fig. 7(a), we compare the temperature convergence towards the stationary state varying Nfor a shorter protocol duration of 0.47 ms corresponding to τ/tf=30. The driving functions applied, F(t), correspond to the ones displayed in Fig. 1(right panel). To be able to impose the large variations in F(t), a reduced target power change Pf=0.6 µW, implying a maximum temperature elevation FIG. 6. Comparison of the temperature convergence towards the stationary profile θswhen heating the cantilever with a step function power (black) or with the accelerating protocol function of Eq. (9) for N=4 (orange). The red shaded area corresponds to the protocol duration, tf=0.1τ=1.4 ms. (a) (b) FIG. 7. (a) Temperature relaxation towards the stationary state (measured close to the cantilever free end ˜x=0.95) varying N for a protocol duration tf=τ/30 =0.47 ms. Inset: temperature variations during the protocol. (b) Comparison of the temperature decay. The horizontal dashed lines correspond to a relative difference (θ −θs)/θsof 4%. of θm s=5 mK, is considered. To maintain the signal-to- noise ratio, each curve displayed is now obtained averaging over 3000 measurements. As observed previously for the slower acceleration of tf=τ/10, the temperature variations during the protocol duration (see inset) increase with the number of canceled modes N. Note that the transient temperature variations are now much larger and can take negative values. In that case, the temperature at the end of the protocol is notably different from the stationary value θs. The relative difference (θ(tf)−θs)/θs(measured at ˜x=0.95) is, respectively, 100 %, −150 %, and 170 % for Nincreasing from 1 to 3. It may seem counterintuitive that the difference from the target value obtained at the end of the 034072-7 BASILE POTTIER et al. PHYS. REV. APPLIED 19, 034072 (2023) protocol increases with the number of canceled modes. We recall that our protocol allows converging towards the stationary profile at the end of the protocol only in the limit N→∞. For a finite value Nthe use of the protocol guarantees that starting from tfonly the modes higher than N will relax. Because the rate of relaxation (equal to k2 nτ) increases with the mode number n, increasing Nshould allow a faster convergence towards the stationary profile. It cannot be guaranteed that the coefficients of the modes higher than the Nth one take small values at t=tf, but its relaxation is ensured to be exponential, faster and faster for increasing n. Numerical resolution of the heat equation with our protocols are provided in Appendix D, quantifying the limitation caused by these modes surviving after tf. In Fig. 7(b), we verify that the temperature starting from tfrelaxes faster as Nincreases, as predicted. For N=3, for example, the relaxation times are at most τ/(3.5π2)≃ 0.12 ms, smaller than tfitself, and in agreement with the observation (green curve). IV. DISCUSSION AND CONCLUSIONS In this work, we tailor in time the power of a localized irradiating laser, in order to speed up the thermal relaxation of a microcantilever in a duration much smaller than its natural relaxation time. The protocol duration can be reduced, in principle, as much as desired. In practice, however, a smaller duration unavoidably translates into a larger dynamics of the heating and cooling power (minimum and maximum power with respect to final value, faster time control of the imposed drive), and of the transient values of the temperature field. Experimentally, acceleration factors up to a factor 30 are demonstrated. The theoretical approach relies on canceling the first N eigenmodes of the thermal diffusion equation in a finite time. Of course, the system is getting closer to the exact steady states as more Nare considered. Nevertheless, we prove that for an experimentally accessible range of parameters, relatively small values of Nmay suffice, N= 4 providing a final state already very close to the stationary one. We emphasize that such a strategy to speed up a given dynamics described by partial differential equations is generic, and can be applied to a wide variety of mesoscopic systems [43]. It complements the methods that have been recently developed in stochastic thermodynamics to speed up a relaxation process [28]. Our approach requires that local equilibrium holds at all times, which allows us to define the temperature field and ensure the validity of Eq. (1). The specific case worked out here also requires that a one-dimensional (1D) description of the system is valid, thus that the accelerated dynamics is still slow with respect to the transverse directions of the cantilever. With a width- to-length ratio of W/L=0.09 in our experiments, the transverse relaxation time is (W/L)2τ=8.10−3τ, smaller than the smallest tfwe probe, but it could be an issue for more ambitious accelerations! Some driving protocols require a heat sink (negative power), which is technically impossible with laser absorption. From the experimental point of view, this represents a technical problem that is bypassed by choosing a heated initial state. This trick allows redefinition of the driving function with an offset that gives access to effective negative powers. With our notation, P0+P(t)needs to remain positive. Our solution is proven to work in all the considered processes. Let us point out that our approach is quite different from classic feedback-control methods. Those would typically rely on measuring the temperature in one position, say x0, and adjusting the heating power to reach the setpoint, θm s. A perfect feedback loop would therefore achieve θ(x0,t>0)=θm s, acting as an effective boundary condition (a temperature step at the specific point x0)for the heat Eq. (1). The typical time scale to reach the stationary state would then be x2 0/(π2D)(see Appendix E), close to one fourth of the power-step slowest decay time 4L2/(π2D): only a 4 times acceleration of the global dynamics would consequently be achieved. Note that reaching the stationary value in one point is very different from reaching the final profile at every position. As a further illustration, we superimpose in the Movies within the Supplemental Material [48] the time evolution of the temperature field in response to a power step, to a temperature step (perfect feedback loop), and to our protocols. The acceleration provided by our strategy is clear, at the expense of noticeable local temperature oscillations and overshoots. The order of magnitude of the temperature jumps that we experimentally perform is modest: 38 mK for a 10 times acceleration, and 5 mK for a 30 times acceleration. This choice hinges upon a practical consideration: linearity and dynamic range. The thermal conductivity of silicon depends on temperature in a noticeable way (roughly as λ∝1/T[49]), so that we would like to avoid large temperature excursions to match the linearity hypotheses of the model. Moreover, we need an initial power P0to allow negative values for P(t), hence the initial temperature profile is actually not flat. For the set of data presented in Figs. 3to 6, the maximum temperature due to P0= 210 µW and the 350 µW of the red laser probe beam is 4 K, corresponding to a variation of λbelow 1.3%. Larger P0, allowing larger Pf, would imply nonuniform values for λ, again deviating from the model. Our goal in this paper is to demonstrate the excellent agreement between theory and experiment, so we restrict ourselves to small temperature jumps. We also test larger steps corresponding to θm s=1 K, and as illustrated in Fig. 8, the protocols works fine with only tiny adjustments to the parameter τ to compensate for emerging nonlinearities. In this case, the maximum temperature due to P0=400 µW and the 034072-8 ACCELERATING THE HEAT DIFFUSION... PHYS. REV. APPLIED 19, 034072 (2023) FIG. 8. Transient temperature profiles measured when applying a step function and the accelerating protocol (N=1upto N=3) for a maximum temperature elevation θm s=1K.The small deviation from theory is attributed to the thermal conductivity dependence of silicon with temperature: as the initial profile corresponds to a maximum temperature rise of 50 K, λis up to 16% smaller near the cantilever’s free end. 600 µW of the red laser probe beam is 50 K, corresponding to a nonuniformity of λof 16 %. Larger jumps would require the simple form of the dynamic evolution that facilitates the theoretical derivation of the driving protocol to be reconsidered. The introduction of linear corrections or even nonlinearities in λconstitutes a challenging future perspective. One way to partially avoid the need for an extended power range is the freedom gained when the laser position x0(t)is no longer fixed but controlled in time: it would allow the required amount of heat to be deposited at each location of the cantilever, bypassing the slow diffusion process to reach the stationary state of the temperature field. As a final consideration, let us rewind to our initial motivation: get closer or beat the time scales of the mechanical system. With an acceleration of a factor 30, the ratio of the mechanical and thermal time writes T0/tf=2µm/H. For our H=5-µm-thick cantilever, we are thus able to stabilize the temperature field in only 2.5 oscillations, and we would reach half an oscillation with H=1µm. As for the mechanical relaxation time τr, which is Q/π times longer than the period T0, with a quality factor of the order of Q=3000 in our experiment in vacuum, we actually reach thermal steady state in τr/400. Such an acceleration even allows stabilizing temperature faster than the “equilibration” of the first 12 mechanical resonant modes of the cantilever. Our strategy to accelerate the heat diffusion thus achieves its goal, and could be useful in numerous applications once we reach a meaningful temperature step amplitude. The data that support the findings of this study are openly available in Zenodo [51]. ACKNOWLEDGMENTS This work is financially supported by the Agence Nationale de la Recherche through grant ANR-18- CE30-0013. C.A.P. acknowledges financial support from Grant No. PID2021-122588NB-I00 funded by MCIN/AEI/ TABLE I. Coefficients γnof the polynomial function F(t) found solving Eq. (10) that allow us to cancel successively the four first thermal modes for a protocol duration tfsuch that τ/tf=10, τ/tf=30, and τ/tf=100. N=1N=2N=3N=4 τ/tf=10 γ14.57 16.5 42 88.1 γ2−22.9 −155 −585 γ3122 1040 γ4−548 τ/tf=30 γ112.7 113 641 2749 γ2−198 −3204 −26156 γ32926 57017 γ4−34647 τ/tf=100 γ141 1141 19361 238632 γ22191 −109748 −2662356 γ3106680 6372575 γ4−4126391 034072-9