Information Thermodynamics in a Quantum Dot Szilard Engine - Experimentally Investigating Fluctuation Theorems and Thermodynamic Uncertainty Relations

TL;DR

The paper experimentally investigates information thermodynamics in a quantum-dot Szilard engine by implementing two distinct backward experiments and comparing their fluctuation theorems and thermodynamic uncertainty relations. It demonstrates that both mutual information and inferable entropy provide bounds on the extracted work, with the inferable entropy often yielding tighter constraints and being robust to finite driving speeds. The work combines stochastic thermodynamics, quasistatic and finite-speed theory, and precise quantum-dot experiments to reveal that information quantified from measurement outcomes can be a more relevant predictor of dissipation than mutual information in many regimes. The findings advance understanding of information-processing thermodynamics in mesoscopic systems and offer practical insights for designing feedback-controlled quantum devices.

Abstract

In Szilard's engine, measurement and feedback allows to extract work from an equilibrium environment, a process otherwise forbidden by the laws of thermodynamics. Recent theoretical developments have established fluctuation theorems and thermodynamic uncertainty relations that constrain the fluctuations in Szilard's engine. These relations rely on auxiliary experimental protocols known as backward experiments. Here, we experimentally investigate the thermodynamics of Szilard's engine by implementing two distinct types of backward experiments. We verify and compare the corresponding fluctuation theorems and thermodynamic uncertainty relations associated with each protocol. Our results reveal that the entropy production inferable from measurement may serve as a more relevant quantifier of information than the widely used mutual information.

Figures (7)

• Figure 1: Szilard engine operation. a) Initially, the spin-degenerate energy level $E$ is occupied by one or two electrons with probability one half, requiring an offset from the chemical potential by $E_0 = -k_BT\ln2$. During the operation, the level is either raised by $\Delta E_0$ or lowered by $\Delta E_1$, depending on $y$, the outcome of an occupation measurement. b) The heat deposited to the reservoir when an electron tunnels into it from the quantum dot is $Q$. c) Example trajectories for $y=0$. Top panel: The charge detector current $I_d$ is used to measure the occupation $n$ of the quantum dot. Second panel: The energy level is increased by $\Delta E_0$ and then ramped back. Third panel: While $n=1$, work is extracted when $E$ is lowered. Bottom panel: Every tunneling event increases or reduces the total heat $Q$ deposited in the reservoir.
• Figure 2: Fluctuation theorems. a) Example histograms of work extracted in the forward experiment (top panel) and the two different backward experiments with drive amplitudes $\Delta E_0 = \Delta E_1 = 0.75k_BT$ and error rate $\epsilon = 0.3$. The fluctuation theorem is verified for the backward experiment related to the mutual information in b) and the inferable entropy in c). The error bars are estimates of the standard error of the mean.
• Figure 3: Generalized second law and thermodynamic uncertainty relation. Symbols are experimental data and lines correspond to theory curves. a) Top panel: The mutual information $\langle I \rangle$ (green squares) and the inferable entropy $\langle\mathcal{E}\rangle$ (orange diamonds) provide bounds on the extracted work $\langle W\rangle$ (blue circles). Bottom panel: The SNR (blue circles) are plotted against a variety of TUR quantities. Pink triangles are the RHS of the original TUR in Eq. \ref{['eq:basicTUR']}. Green squares and orange diamonds are the RHS of Eq. \ref{['eq:GTUR']}. Inset: Zooming in at the low-SNR area reveals that when $\epsilon = 0.5$, Eq. \ref{['eq:basicTUR']} is valid. Parameters: $\Delta E_0 = \Delta E_1 = 0.75k_BT$. b) Shows the same quantities as a), but now plotted against $\Delta E_1$ while $\epsilon = 0.2$ and $\Delta E_0 = 0.75k_BT$. The top horizontal axis denotes the equilibrium occupation probability at the lowest energy point. The error bars in a) and b) all represent the statistical scatter of the mean which was estimated through subsampling in the cases of the RHS of Eq. \ref{['eq:GTUR']} and $\langle\!\langle W^2\rangle\!\rangle$.
• Figure 4: Thermodynamic uncertainty relations. The signal-to-noise ratio of the Szilard engine is plotted against the right-hand sides of Eqs \ref{['eq:basicTUR']} (green points) and \ref{['eq:GTUR']} for two different backward experiments (blue and orange points) for the same data as in Fig. \ref{['fig:backwards']}. The dashed diagonal indicates when equality is achieved. Points situated below the diagonal show the TUR in question is violated and points close to the diagonal indicate the bound is tight. All error bars show the standard error of the mean, which in the case of the blue and orange points are estimated through subsampling.
• Figure S1: The same experimental data and full theory curves from the bottom panels of Figure \ref{['fig:backwards']} (a) and (b) in the main text. The additional black lines are theory curves that neglect the effect of the finite driving speed of the protocol.