Non-Markovian Feedback Control and Acausality: An Experimental Study

Causality is an important assumption underlying nonequilibrium generalizations of the second law of thermodynamics known as fluctuation relations. We here experimentally study the nonequilibrium statistical properties of the work and of the entropy production for an optically trapped, underdamped nanoparticle continuously subjected to a time-delayed feedback control. Whereas the non-Markovian feedback depends on the past position of the particle for a forward trajectory, it depends on its future position for a time-reversed path, and is therefore acausal. In the steady-state regime, we show that the corresponding fluctuation relations in the long-time limit exhibit a clear signature of this acausality, even though the time-reversed dynamics is not physically realizable.

Figure 1

Schematic representation of a path $\mathbf{X}=\{x_t{\}}_{0\le t\le \mathcal{T}}$ and its time-reversal ${\mathbf{X}}^{†}=\{x_{\mathcal{T}-t}{\}}_{0\le t\le \mathcal{T}}$. While the time-delayed feedback control depends on the past position of the particle $x_{t-\tau }$ (causal feedback) for the forward trajectory, it depends on the future position of the particle $x_{t+\tau }^{†}$ (acausal feedback) for the time-reversed path.

Figure 2

Experimental setup. A nanoparticle is levitated in a harmonic optical trap formed by two counterpropagating beams (red), inside a hollow-core photonic crystal fiber. The nanoparticle is subjected to a delayed feedback force $F_{\mathrm{fb}}\propto x_{t-\tau }$ implemented with an acousto-optic modulator using radiation pressure from an additional laser beam (green). The feedback loop is characterized by a gain $g$ and time delay $\tau$.

Figure 3

Steady-state kinetic temperatures versus time delay in the second stability region of the oscillator (in units of ${\mathrm{\Omega }}_0^{-1}$). The black solid line is the theoretical prediction for the (causal) temperature ratio $T_v/T$ and symbols are data obtained by averaging $v_t^2$ over a trajectory of length $1000Q_0$. The red solid line is the prediction for the (acausal) temperature ratio ${\tilde{T}}_v/T$ in the region (labeled I) where the acausal dynamics admits a stationary solution.

Figure 4

(a)–(d) Estimates of ${\mu }_W(1,\mathcal{T})$ (squares) and ${\mu }_{\mathrm{\Sigma }}(1,\mathcal{T})$ (stars) as a function of the observation time $\mathcal{T}$ (in unit of ${\mathrm{\Omega }}_0^{-1}$), for various delays $\tau$. Panels (a) and (c): from bottom to top, $\tau =6.13$, 6.32, 6.51, 6.70 (red) and $\tau =6.88$, 7.07, 7.26, 7.44, 7.63, 7.82 (black). Panels (b) and (d): from top to bottom, $\tau =8.00$, 8.19, 8.38, 8.57, 8.75 (black) and $\tau =8.94$, 9.13, 9.31, 9.50 (red). Solid lines are a guide to the eye. (e) Values of ${\mu }_{\mathrm{\Sigma }}(1)$ (stars) and ${\mu }_{\mathcal{W}}(1)$ (squares) deduced from the data plotted in panels (a)–(d). The solid line is the theoretical expression of the acausal contribution ${\dot{S}}_J$ [19, 23]. In region I (red symbols), the acausal dynamics has a stationary solution and ${\mu }_{\mathrm{\Sigma }}(1)={\mu }_W(1)={\dot{S}}_J$. In region II (black symbols), ${\mu }_W(1)=1/Q_0$ and ${\mu }_{\mathrm{\Sigma }}(1)={\dot{S}}_J$. (f) Ratio of the preexponential factors $g_W(1)/g_{\mathrm{\Sigma }}(1)$ in region I. The red solid line is the conjectured formula (5). Error bars are discussed in Supplemental Material [23].

Figure 5

(a) Experimental distributions of the entropy production $P({\mathrm{\Sigma }}_{\mathcal{T}}=\sigma \mathcal{T})$ for $\tau =8.75$ and $\mathcal{T}/Q_0=(0.4,1,10)$. The horizontal dashed line indicates the threshold $\theta$ (here fixed at ${10}^{-3}$). (b) Estimates of ${\mu }_{\mathrm{\Sigma }}(1,\mathcal{T})$ versus time for different thresholds. Solid lines are a guide to the eye. The plateau disappears as $\theta$ is increased from ${10}^{-5}$ to ${10}^{-2}$.