Probing the power of an electronic Maxwell's demon: Single-electron transistor monitored by a quantum point contact

We suggest that a single-electron transistor continuously monitored by a quantum point contact may function as Maxwell's demon when closed-loop feedback operations are applied as time-dependent modifications of the tunneling rates across its junctions. The device may induce a current across the single-electron transistor even when no bias voltage or thermal gradient is applied. For different feedback schemes, we derive effective master equations and compare the induced feedback current and its fluctuations as well as the generated power. Provided that tunneling rates can be modified without changing the transistor level, the device may be implemented with current technology.

Figure 1

Sketch of the setup: an SET with two attached leads (sketched are their Fermi functions $f_L$ and $f_R$) is capacitively coupled to a nearby QPC. Its current $I_{\mathrm{QPC}}$ yields information on the instantaneous occupation of the SET. One may either apply Liouvillians ${\mathcal{L}}_E$ and ${\mathcal{L}}_F$ with different tunneling rates conditioned on the present state of the SET (scheme I; compare different tunneling barriers in SET sketch) or control operations ${\kappa }_I$ and ${\kappa }_O$ conditioned on the change of the SET state (schemes IIa/b; not shown in SET sketch). During the control operations, electrons may also tunnel into or out of the SET (dashed shorter spikes), which may not (IIa) or may (IIb) recursively trigger further control operations.

Figure 2

Current-voltage characteristics under feedback strength ${\delta }_L^E={\delta }_R^F=-{\delta }_R^E=-{\delta }_L^F=1$ in scheme I (thin solid curve, red), maximum feedback in schemes IIa and IIb (dashed and dotted, respectively), and no feedback (bold black) for symmetric tunneling rates ${\Gamma }_L=\Gamma ={\Gamma }_R$ and $\beta \epsilon =2$. The current may point in the other direction than the voltage leading to a positive power generated by the device. The inset (for $\beta e/C=1$) shows the mean-field evolution of the voltage from an initial value (empty circles) to the stable fixed points (filled circle) for symmetric capacitances of the two conductors.

Figure 3

Comparison of a single (thin red curve with jumps; same realization in all panels) and the average of 100 (medium thickness, green) and 10000 (bold smooth curve, turquoise) trajectories with the solution from the effective feedback master equation [Eq. (3), thin black] for the dot occupation (top), the number of particles on the left (middle), and the number of particles on the right (bottom). The average of the trajectories converges to the effective feedback master equation result. The reference curve without feedback (dashed orange) may be obtained from Eq. (2) or by using vanishing feedback parameters and demonstrates that the direction of the current may actually be reversed via sufficiently strong feedback. Parameters: ${\Gamma }_L={\Gamma }_R\equiv \Gamma$, $f_L=0.45$, $f_R=0.55$, ${\delta }_L^E={\delta }_R^F=1.0$, ${\delta }_R^E={\delta }_L^F=-10.0$, and $\Gamma \Delta t=0.01$.

Figure 4

Comparison of a single and the average of 100 and 10 000 trajectories with the solution from the effective feedback master equation (6) for the dot occupation (top), the number of particles on the left (middle), and the number of particles on the right (bottom). The average of the trajectories converges to the effective feedback master equation result. The single trajectory only records the state after all control operations have been performed, i.e., jumps with no net effect are not visible. Therefore, control operations (bold kinks in top graph) are not always correlated with a net change in the occupation or the particle number left and right. The second last top kink, for example, stands for an out-control operation $e^{{\kappa }_O}$ that was triggered by an electron jumping to the left (information that is not provided by the QPC) contact. During the control operation, an electron jumped back from the left contact to the system, such that no net change in $n_{\mathrm{dot}}$, $n_L$, nor $n_R$ occurred. Color coding and parameters are as in Fig. 3, where applicable. Feedback parameters were chosen as ${\delta }_L^I={\delta }_R^O=0$ and ${\delta }_R^I={\delta }_L^O=1.0$.

Figure 5

Comparison of a single (only net changes as in Fig. 4) and the average of 100 and 10 000 trajectories with the solution from the effective feedback master equation (7) for the dot occupation (top), the number of particles on the left (middle), and the number of particles on the right (bottom). The average of the trajectories converges to the effective feedback master equation result. Electrons jumping during control operations may now recursively trigger further control operations—marked by the height of the bold kinks in the top graph. Color coding and parameters are as in Fig. 4.