Experimental Verification of the Work Fluctuation-Dissipation Relation for Information-to-Work Conversion

We study experimentally work fluctuations in a Szilard engine that extracts work from information encoded as the occupancy of an electron level in a semiconductor quantum dot. We show that as the average work extracted per bit of information increases toward the Landauer limit $k_{B}T\ln 2$, the work fluctuations decrease in accordance with the work fluctuation-dissipation relation. We compare the results to a protocol without measurement and feedback and show that when no information is used, the work output and fluctuations vanish simultaneously, contrasting the information-to-energy conversion case where increasing amount of work is produced with decreasing fluctuations. Our study highlights the importance of fluctuations in the design of information-to-work conversion processes.

Figure 1

(a) Scanning electron microscope image of the nanowire device. Embedded in the nanowire are three QDs, each aligned to one of the plunger gates $V_{g1}$,$V_{g2}$, or $V_{gd}$. Contacts separate the device into one part with two QDs and one part with a single QD. The coupler couples the two systems together, allowing the current $I_d$ through the lone QD to provide a measure of the charge state of the other system. Here, the QD involved in the experiment is marked in blue (close to the plunger gate with $V_{g1}$) while the quantum dot marked in red is put in Coulomb blockade. The sensor QD is marked in purple and the tunnel barriers are colored orange. (b) An example of the sensor current $I_d$ at the start of the experiment. $V_{gd}$ is set so that a high current represents the state $n=1$ and $V_{g1}$ is set so that the average $⟨n⟩=0.5$.

Figure 2

(a) The middle panel illustrates how the energy level is driven during the information-to-work conversion. Originally, it is offset from the reservoir Fermi level by $E=-k_{B}T\ln 2$. At time 0, a measurement is made. Depending on the outcome, the energy is rapidly driven either up or down by $6k_{B}T$ before slowly being driven back to the starting point. In this example, the protocol time $\tau =2\mathrm{s}$. The upper (lower) panel shows the detector signal during one realization of the protocol where the energy was initially driven up (down). (b) The distributions of extracted work for the Szilard engine operation for the protocol lengths $\tau =0.25$ s (top), 2 s (middle), and 5 s (bottom) together with the theoretically expected distribution (purple). The dotted black line indicates the Landauer limit $k_{B}T\ln 2$ and the orange dashed line indicates the mean extracted work. (c) The two quantities from Eq. (1) plotted against each other. Here, $W_{\mathrm{diss}}=k_{B}T\ln 2-⟨W_{\mathrm{ex}}⟩$. The FDR line is Eq. (1), whereas the simulations are obtained by a full counting statistics method with Eq. (3). (d) Same quantities as in (c), but plotted against $\tau$. For details on the computation of error bars in (c) and (d), see the Supplemental Material [49].

Figure 3

(a) Top panel shows the detector signal $I_d$ during the dissipative drive protocol, which is shown in the bottom panel. In the dissipative drive, the energy level is initially offset from the reservoir Fermi energy by $E=-k_{B}T\ln 2$. Then, it is driven down by $2.5k_{B}T$ and back up again during a time $\tau$. (b) The work distribution of the $2.5k_{B}T$ dissipative drive for total protocol lengths 0.5 s (top), 2 s, and 4 s (bottom) together with what is expected theoretically (purple). The orange line is the mean of the distribution, which in this case is equal to $W_{\mathrm{diss}}$. The black line is drawn at $W=0$ to help guide the eye. (c) The two sides of Eq. (1) plotted against each other, for protocol lengths between 0.5 s and 4 s. The orange diagonal is given by Eq. (4), which is obtained analytically in the slow-driving limit, and from which the FDR directly follows (see details in the Supplemental Material [49]). (d) The same quantities as in (c), but now plotted against the protocol length. For details on the computation of error bars in (c) and (d), see the Supplemental Material [49].

Figure 4

The quantities in the work FDR of Eq. (1) plotted against each other for all protocols tested. This summarizes the data from the information conversion protocol (purple squares) and the dissipative drive with no information involved (blue triangles) plotted in Figs. 2 and 3, respectively. Blue diamonds represent a dissipative drive with an amplitude of $6k_{B}T$ but otherwise identical to the $2.5k_{B}T$ dissipative drive. For all protocols implemented, the points follow the orange line given by Eq. (1) up to the long time saturation of ${\sigma }_W$.