Extracting work from hidden degrees of freedom

Abstract

Thermodynamics establishes that information acquired through measurement can be converted into work, as exemplified by Maxwell's demon and Szilard engines. Most experimental realizations of information engines, however, implicitly assume Markovian environments, in which information exchanged with the surroundings is irreversibly lost. Many physical systems instead exhibit environmental memory, with hidden degrees of freedom retaining correlations with the system's past and giving rise to non Markovian dynamics. Whether and how such concealed memory can be harnessed as a thermodynamic resource has remained an open question. Here we experimentally demonstrate work extraction from environmental memory. Using time resolved measurements on an optically trapped Brownian particle in equilibrium, we implement a time delayed double measurement protocol that retrieves information via backflow from hidden bath degrees of freedom. We show that this information backflow alters relaxation dynamics, can be quantified independently of initial state effects, and when appropriately exploited enhances work extraction. Notably, we identify regimes in which the extracted work exceeds the energy stored in the observable degree of freedom alone. Our results establish environmental memory as an experimentally accessible thermodynamic resource and reveal how non Markovian dynamics can be systematically explored to improve the performance of information engines operating in time-correlated environments.

Figures (4)

• Figure 1: Two-measurement protocol and state-conditioned particle distributions.a, Measurement protocol illustrated using an experimental particle trajectory $x(t)$ (gray) corresponding to the state $|01\rangle$. At $t=t_i$ and $t_j=t_i+\delta t$, the particle position is compared to a fixed threshold $x_{\mathrm{th}}$; positions below and above the threshold are indicated by the blue and orange symbols and define the state $|0\rangle$ and $|1\rangle$, respectively. b, (left) Schematic equilibrium position distribution (gray) in the harmonic potential $V(x)$ (solid black); the vertical dashed line marks the threshold $x_{\mathrm{th}}$. (middle) The first measurement truncates the equilibrium distribution and prepares the conditional states $\rho_{|0\rangle}(x,t_i)$ (blue) and $\rho_{|1\rangle}(x,t_i)$ (orange), shown relative to the equilibrium distribution (black dashed). (right) A second measurement at $t_j=t_i+\delta t$ prepares a further refined conditional ensemble with the four initial states $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$ and corresponding distributions $\rho_{|ij\rangle}(x,t_j)$ (dark blue, light orange, light blue, and red), which subsequently relax back to equilibrium (gray).
• Figure 2: Memory modifies relaxation: State-conditioned relaxation of the averaged particle potential energy.a, Relaxation curves following a two-measurement protocol with $\delta t=1s$ for all possible conditional binary states which show marked differences. b, With increasing $\delta t$, the influence of the first measurement progressively vanishes and the relaxation dynamics become determined solely by the second measurement, as demonstrated by the data obtained for $\delta t = 200s$. This delay corresponds to the regime $\delta t \gg \tau_r$, for which the relaxation curves become indistinguishable from those following a single measurement (inset). Independent of $\delta t$, all curves asymptotically relax to the equilibrium energy $0.5\,k_{\mathrm{B}} T$ (dashed lines). The shaded bands correspond to the SEM calculated using a bootstrapping procedure (see Methods).
• Figure 3: Information backflow: Information decay and non-Markovian order parameter.a, Time evolution of the KL divergence $I_{|ij\rangle}(t)$ (solid lines) and $\tilde{I}_{|ij\rangle}(t)$ (dashed lines) for $\delta t = 1s$, conditioned on the second measurement outcome $j = 1$. The shaded regions denote enhanced persistence (accelerated decay) of information relative to the adjusted state. Notably, for the state $|01\rangle$, the difference $I_{|10\rangle}(t)-\tilde{I}_{|10\rangle}(t)$ is negative, corresponding to accelerated decay of information. b, Same as a, but for states conditioned on $j = 0$. c, State-averaged information $I(t)$ and $\tilde{I}(t)$, obtained by weighting individual contributions with their empirical probabilities $\mathrm{P}_{ij}$. The shaded region highlights the net excess information retained at $\delta t = 1s$. d, Integrated order parameters weighted by their relative probabilities $\mathrm{P}_{ij} \Phi_{|ij\rangle}(\delta t)$ for individual measurement-conditioned states, together with the state-averaged quantity $\Phi(\delta t)$, showing a pronounced maximum at $\delta t \approx 1s$.
• Figure 4: Work extraction from hDoF through two-measurement protocols.a, Sketch of the work extraction protocol from a $|11\rangle$ state within a single well potential. After measuring a $|11\rangle$ state, the trap is translated at constant velocity $\dot{\lambda}$ during $t_\mathrm{f}=3s$ over a total distance $2x_\mathrm{th}$. b, Averaged normalized work $W/\bar{V}_m(0)$ accumulated over time for increasing $\delta t$ (purple to yellow). Markers denote the maximal normalized work for each interval, ($\eta^*_{|11\rangle}$), which peaks at $\delta t = 1s$ (see inset) but remains below unity, confirming that $W$ never exceeds $\bar{V}_m(0)$. Shaded bands represent the SEM from 50 events determined via bootstrap resampling (Methods). c, Corresponding work extraction protocol for a double well potential with states $|0\rangle$ and $|1\rangle$ corresponding to the particle in the left and right well. After a barrier crossing event, i.e. a $|01\rangle$ state, the trap is translated at constant velocity during $t_\mathrm{f}=1.2s$ over a total distance of $360n\m$. d, Averaged normalized work $W/\bar{V}_m(0)$ extracted from a $|01\rangle$ state within a double well geometry. The maximum extracted work clearly exceeds $\bar{V}_m(0)$, demonstrating the extraction of energy stored within the hDoF. The average has been obtained from 100 extraction protocols and the shaded area corresponds to the SEM obtained from bootstrapping (Methods).