Optimal Probabilistic Work Extraction beyond the Free Energy Difference with a Single-Electron Device

We experimentally realize protocols that allow us to extract work beyond the free energy difference from a single-electron transistor at the single thermodynamic trajectory level. With two carefully designed out-of-equilibrium driving cycles featuring kicks of the control parameter, we demonstrate work extraction up to large fractions of $k_{B}T$ or with probabilities substantially greater than $1/2$, despite the zero free energy difference over the cycle. Our results are explained in the framework of nonequilibrium fluctuation relations. We thus show that irreversibility can be used as a resource for optimal work extraction even in the absence of feedback from an external operator.

Figure 1

(a) Scanning electron micrograph of the single-electron transistor (SET) capacitively coupled to a voltage-biased detector SET. Leads (blue) made of superconducting aluminum are coupled through oxide (tunnel) barriers to the copper (red) island. (b) Electrical circuit representation. (c) Protocol used to maximize work extraction, with a zoom on the detector SET output current under system driving, around the quench event.

Figure 2

(a),(b) Work histograms obtained for (a) $n_g^{*}=0.608$ and (b) $n_g^{*}=0.698$, with the same ramp time $t_1=1.25\mathrm{s}$. (c) Probability for $W=-\mathrm{\Delta }E(n_g^{*})$ (orange dots, mind the sign) and $W=\mathrm{\Delta }E(n_g^{*})$ (blue dots) events as a function of the quench amplitude. Solid lines are fits of Fermi functions (see text) with $E_C=110\mu \mathrm{eV}$ and $T=670\mathrm{mK}$. Error bars are calculated from the number of protocol realizations. (d) Work performed on the system averaged over all outcomes as a function of the quench peak amplitude $\mathrm{\Delta }{n}_g=n_g^{*}-1/2$. The solid line is obtained from Eq. (5). Inset: Verification of Eq. (1) for all values of $n_g$. (e),(f) Work histograms obtained for the same quench amplitude $\mathrm{\Delta }{n}_g=0.048$, but with ramp times (e) $t_1=0.1\mathrm{s}$ and (f) $t_1=0.025\mathrm{s}$, much shorter than in (a) and (b). In (a), (b), (e), and (f), solid lines are obtained by numerically solving the master equation [27]. All work values are normalized to $E_C$.

Figure 3

(a) Protocol used to extract work with high probability (see text). (b),(c) Work histograms and experimental values of work and exponentiated work averages obtained for (b) $n_{g,a}=0.656$, $n_{g,b}=0.698$ and (c) $n_{g,a}=0.752$, $n_{g,b}=0.863$, with ramp time $t_1=1.25\mathrm{s}$. The vertical dashed line sets the zero free energy difference to guide the eye. Solid lines are obtained by numerically solving the master equation [27].