Linking fluctuation and dissipation in spatially extended out-of-equilibrium systems

For systems in equilibrium at a temperature $T$, thermal noise and energy damping are related to $T$ through the fluctuation-dissipation theorem (FDT). We study here an extension of the FDT to an out-of-equilibrium steady state: a microcantilever subject to a constant heat flux. The resulting thermal profile in this spatially extended system interplays with the local energy dissipation field to prescribe the amplitude of mechanical fluctuations. Using three samples with different damping profiles (localized or distributed), we probe this approach and experimentally demonstrate the link between fluctuations and dissipation. The thermal noise can therefore be predicted a priori from the measurement of the dissipation as a function of the maximum temperature of the micro-oscillator.

Figure 1

Experimental setup: The deflection and torsion of a cantilever are captured thanks to the optical lever technique. The red laser beam ($1\mathrm{mW}$ at $633\mathrm{nm}$), focused on the cantilever tip, is reflected towards the four-quadrant photodiode. This sensor records the temporal signals of deflection, $\delta \left(t\right)$, and torsion, $\theta \left(t\right)$. A green laser beam (0–12 mW at $532\mathrm{nm}$) focused close to the tip of the triangular end of the cantilever acts as the heater. A camera is used to visualize the position of both lasers on the sample. The cantilever, in vacuum at $5\times {10}^{-6}\mathrm{mbar}$, is monolithically clamped to its macroscopic chip, which is thermalized at room temperature, $T^{\min }$.

Figure 2

PSDs of the (a) thermal-noise-induced deflection and (b) torsion of the cantilever, where each resonance is identified as a sharp peak with a quality factor in the range of tens of thousands. The modes can safely be considered decoupled and each can be treated as a simple harmonic oscillator. In the inset, a zoom-in around the second flexural resonance shows how the resonance is redshifted with the laser power increasing. The shapes of the modes are simulated in comsol [23] and shown as snippets corresponding to each peak, yielding resonance frequencies very close to the ones found in our experiment and in agreement with the Euler-Bernoulli description.

Figure 3

Cantilever C100, flexural modes. (a) The thermal noise amplitude $T_n^{\mathrm{fluc}}$ is shown with respect to $T^{\mathrm{avg}}$. The black solid line represents the equilibrium temperature, i.e., the fluctuations an object would show had it been in thermal equilibrium with a thermal bath at $T^{\mathrm{avg}}$. All the modes lie below this line, showing a dearth of thermal noise. Furthermore, we note how they are also much lower than the maximal temperature of the system, represented by the black dashed line. The modes shown span from 2 to 8, excluding mode 5 because of the laser probe being on a mechanical node. (b) The measured loss angles ${\phi }_n$ show little change with the average temperature. We note also how the loss angle of the mode $n=6$ is roughly twice that of the other modes. (c) The PSD of the same mode at two different powers alongside the fit with Eq. (6).

Figure 4

Cantilever C100, torsional modes. (a) The thermal noise content of the first eight torsional modes (except the fifth due to the sensitivity being too low) is shown to be roughly independent of the temperature profile imposed on the system. (b) The loss angle also shows little changes at different temperatures. (c) The resonance $m=1$ at two powers alongside the fit with Eq. (6).

Figure 5

Cantilever C30, flexural modes. (a) Dissipation ${\phi }_n$ versus $T^{\mathrm{avg}}$ for flexural modes $n=1$–5, alongside a quadratic fit of each. [(b)–(e)] Thermal noise amplitude $T_n^{\mathrm{fluc}}$ alongside its theoretical prediction (dotted line) from Eq. (5) versus $T^{\mathrm{avg}}$ for the same modes $n$ (excluding $n=3$, shown in Fig. 6). The model nicely predicts the thermal noise evolution, except for modes 2 and 4 around $T=400\mathrm{K}$ where fluctuations are somewhat below the expected value.

Figure 6

Cantilever C30, flexural mode $n=3$. (a) The thermal noise of the third flexural mode shows little to no dependence on the temperature profile, which is the opposite behavior expected looking at the dissipation (b). In this case, the simple quadratic approximation of the dissipation fails, and the model cannot predict the thermal fluctuations.

Figure 7

Cantilever C30, torsional mode $m=1$. (a) The thermal noise amplitude of the first torsional mode stays roughly below the average temperature line. In this case, the model qualitatively predicts this behavior. (b) The evolution of the dissipation with the temperature and its fit with a quadratic function.

Figure 8

Cantilever C30C, flexural modes $n=1$–4. (a) Dissipation ${\phi }_n$ versus $T^{\mathrm{avg}}$, alongside a quadratic fit of each. We note how the coating increases the magnitude of the dissipation at least ten times with respect to the bare C30 sample. [(b)–(e)] Thermal noise amplitude $T_n^{\mathrm{fluc}}$ alongside its theoretical prediction (dotted line) from Eq. (5) versus $T^{\mathrm{avg}}$. The theoretical framework and the simple hypotheses formulated for the C30 sample allow us to approximately predict the nonequilibrium thermal content of the C30C sample. We nevertheless note how mode 3 deviates slightly from the prediction.

Figure 9

Cantilever C30C, torsional mode $m=1$. (a) The theoretical prediction of the thermal noise thanks to the evolution of the dissipation [(b), quadratic fit] would predict an increase of the fluctuations below the average temperature of the system. This is not observed in the measurement. A possible explanation is discussed in the text.

Figure 10

Top: Cross section of samples C100 and C30, to scale. The red lines highlight the slanted faces of the cantilever. Bottom: Scanning electron microscopy image of the samples, with dimensions superposed. The green lines highlight the clamp length across the cantilever.