Low thermal fluctuations in a system heated out of equilibrium

We study the mechanical fluctuations of a micrometer sized silicon cantilever subjected to a strong heat flow, thus having a highly nonuniform local temperature. In this nonequilibrium steady state, we show that fluctuations are equivalent to the thermal noise of a cantilever at equilibrium around room temperature, while its mean local temperature is several hundred degrees higher. Changing the mechanical dissipation by adding a coating to the cantilever, we recover the expected rise of fluctuations with the mean temperature. Our work demonstrates that inhomogeneous dissipation mechanisms can decouple the amplitude of thermal fluctuations from the average temperature. This property could be useful to understand out-of-equilibrium fluctuating systems, or to engineer low noise instruments.

Figure 1

Experiment principle: the deflection $d$ of a microcantilever (typically $500\mu \mathrm{m}\times 100\mu \mathrm{m}\times 1\mu \mathrm{m})$ is recorded through the interference of two laser beams, one reflected on the cantilever free end, the other on the chip holding the cantilever [26]. The sensing beam heats the cantilever and a steady temperature profile driven by the absorbed light power is reached, setting the system in a nonequilibrium steady state. In a first approximation, the temperature $T$ grows linearly with $x$, the coordinate along the cantilever length [27].

Figure 2

Power spectrum density (PSD) of thermal noise induced deflection in vacuum as a function of frequency. (a) Each resonant mode is clearly identified by a sharp peak, associated to the normal mode shape pictured in the insets for the first three modes. These resonances can be modeled as uncoupled simple harmonic oscillators. The area under the PSD curve directly gives the mean square deflection of each of these degrees of freedom, and defines their effective temperatures. (b) When the light power grows, all the resonance frequencies of the cantilever decrease, as illustrated here with a zoom on mode 2. This frequency shift, due to the softening of the cantilever with temperature, is used to track the amplitude of the temperature profile, and thus the average temperature [27].

Figure 3

Effective temperatures of the first 3 flexural modes $\left(T_n^{\mathrm{eff}}\right)$ and average temperature (stars, $T^{\mathrm{avg}})$ measured on two different cantilevers as a function of the impinging light power $P$. (top) Raw silicon cantilever: $T_n^{\mathrm{eff}}$ is mode independent and almost constant (equal to room temperature), when $T^{\mathrm{avg}}$ raises by $400\mathrm{K}$. (bottom) tantala coated silicon cantilever: $T_n^{\mathrm{eff}}$ is mode dependent, and increases with $P$ similarly to $T^{\mathrm{avg}}$ for $n>1$. Error bars (one standard deviation) correspond to the statistical uncertainty for $T_n^{\mathrm{eff}}$ (at most $15\mathrm{K})$, and to the systematic error for $T^{\mathrm{avg}}$ (discrepancy between the evaluation on the first 3 modes). For both cantilevers, measurable higher order modes present the same behavior, but their amplitude being smaller uncertainty increases and we do not present them for sake of clarity.

Figure 4

Average temperatures: $T^{\mathrm{fluc}}$, deduced from fluctuations, versus $T^{\mathrm{avg}}$, deduced from the frequency shift of the resonances. $T^{\mathrm{fluc}}$ quantifies the thermal fluctuations amplitude: it is the average on the first three modes of the effective temperatures $T_n^{\mathrm{eff}}$ rescaled for the mode shape through coefficient ${\kappa }_n$, thus assuming a uniform mechanical damping along the cantilever. The raw silicon cantilever presents a strong deficit of fluctuations with respect to equilibrium, while the tantala coated one presents a small excess of fluctuations. Error bars (one standard deviation) account for the statistical uncertainty and systematic error (discrepancy between the evaluation on the first three modes).