Nonequilibrium Control of Thermal and Mechanical Changes in a Levitated System

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small nonequilibrium systems. While work and heat are equally important forms of energy exchange, fluctuation relations have not been experimentally assessed for the generic situation of simultaneous mechanical and thermal changes. Thermal driving is indeed generally slow and more difficult to realize than mechanical driving. Here, we use feedback cooling techniques to implement fast and controlled temperature variations of an underdamped levitated microparticle that are 1 order of magnitude faster than the equilibration time. Combining mechanical and thermal control, we verify the validity of a fluctuation theorem that accounts for both contributions, well beyond the range of linear response theory. Our results allow the investigation of general far-from-equilibrium processes in microscopic systems that involve fast mechanical and thermal changes at the same time.

Figure 1

Nonequilibrium processes with fast thermal and mechanical changes. We realize changes of the center-of-mass temperature $T$ and of the spring constant $\kappa$ (corresponding to frequency $\omega$) of a harmonically trapped levitated microparticle by varying the feedback gain and the laser trap power (inset). Protocol $P_1$ (purple) corresponds to a thermal change with fixed spring constant, while protocol $P_2$ (turquoise) refers to simultaneous thermal and mechanical changes. Full circles are data, with error bars smaller than the symbol size. Shaded areas represent the temperatures set experimentally, taking into account uncertainty due to laser power drifts. In the inset, the gray lines show position distributions during protocol $P_2$ when the temperature is decreased from $T_1$ to $T_2$ and the frequency is simultaneously increased from ${\omega }_1$ to ${\omega }_2$ during a time $\tau$.

Figure 2

Experimental setup and single trajectories. (a) Schematic of the experimental setup. Two counterpropagating laser beams (red arrows) of wavelength 1064 nm trap a 969 nm silica particle at the intensity maximum of the standing wave formed inside a hollow-core photonic crystal fiber (HCPCF). The scattered light (red lines) along the HCPCF provides the center-of-mass motion $x(t)$ of the particle. An additional feedback laser beam (blue arrow) cools the center-of-mass motion of the particle. Here, the velocity contribution of the feedback force $F_{\mathrm{fb}}=g{\gamma }_{p}m\sin (\mathrm{\Delta }\varphi )\dot{x}$ depends on the feedback gain $g$ and the trapping laser power $P$ via the feedback phase $\mathrm{\Delta }\varphi$ (see the Supplemental Material [40]). Protocol $P_1$ (thermal change, purple) is implemented by a linear increase of the feedback gain $g$ at constant optical trap power. Protocol $P_2$ (thermal and mechanical change, turquoise) is realized by changing the optical trap power $P$. (b)–(e) Measured single-particle trajectories (solid lines) and ensemble variances (dashed lines, based on 15 000 trajectories) of the particle center-of-mass motion are plotted together with the corresponding change of the respective control parameter (gray lines). Panels (b),(d) and (c),(e) correspond to the slowest equilibrium (fastest nonequilibrium) protocols, 1 order of magnitude slower (faster) than the relaxation time of the system. (f)–(g) Experimental distributions of the dimensionless thermal work $W_{\mathrm{ther}}$ (black) and dimensionless heat $\int \beta \delta Q$ (blue) in the case of protocol $P_2$, for slow (f) as well as fast (g) drivings.

Figure 3

Fluctuation theorem for thermal and mechanical modulations. Normalized free energy difference $\mathrm{\Delta }(\beta F)$ versus inverse driving time $\tau$ for (a) thermal protocol $P_1$ and (b) thermal and mechanical protocol $P_2$. Dots represent experimental data evaluated using Eq. (1). Shaded areas are theoretical predictions with an uncertainty that incorporates long term laser drifts [40]. Errors bars are determined by the standard deviation over 15 000 runs. Equilibrium, $\mathrm{\Delta }(\beta F{)}_{\mathrm{eq}}$, (triangle up) and linear response, $\mathrm{\Delta }(\beta F{)}_{\mathrm{lr}}$, (triangle down) results only hold for slow driving. The horizontal dashed line in (b) indicates the contribution of thermal change only.

Figure 4

Clausius inequality for thermal and mechanical changes. Comparison between the measured (dimensionless) heat exchanged with the environment $⟨\int \beta \delta Q⟩$ (triangles) and the (dimensionless) entropy variation of the system $\mathrm{\Delta }S$ (gray full circles), as a function of the inverse protocol time for thermal and mechanical changes (protocol $P_2$) followed by an isothermal thermalization. The Clausius inequality, $\mathrm{\Delta }S-⟨\int \beta \delta Q⟩\ge 0$, is verified for any protocol speed. Shaded areas are theoretical predictions that include the uncertainty on the underlying experimental parameters. Error bars are determined via the standard deviation of the respective value over 15 000 runs and are smaller than the symbol size.