Autonomous conversion of information to work in quantum dots

We consider an autonomous implementation of Maxwell's demon in a quantum dot architecture acting on a system without changing its number of particles or its energy. As in the original thought experiment, only the second law of thermodynamics is seemingly violated when disregarding the demon. The autonomous architecture allows us to compare descriptions in terms of information to a more traditional, thermoelectric characterization. Our detailed investigation of information-to-work conversion is based on fluctuation relations and second-law-like inequalities in addition to the average heat and charge currents. By introducing a time reversal on the level of individual electrons, we find a fluctuation relation that is not connected to any symmetry of the moment-generating function of heat and particle flows. Furthermore, we show how an effective Markovian master equation with broken detailed balance for the system alone can emerge from a full description, allowing for an investigation of the entropic cost associated with breaking detailed balance. Interestingly, while the entropic cost of performing a perfect measurement diverges, the entropic cost of breaking detailed balance does not. Our results connect various approaches and idealized scenarios found in the literature and can be tested experimentally with present-day technology.

Figure 1

Composite system under investigation. (a) Illustration of the demon, measuring the occupation of a quantum dot and manipulating the transition probabilities such that electrons enter the dot from the left reservoir and leave the dot toward the right reservoir. Within the system alone, the first law of thermodynamics holds while the second law is violated. (b) Real-space schematics and (c) spectral properties of an implementation with quantum dots. Both the system as well as the demon can host at most one electron. The demon includes a temperature gradient which is used to overcome the voltage bias within the system. Capacitive couplings (with equal charging energy $U)$ between the system dot and the demon dots serve for measuring the occupation of the system dot. Capacitive couplings between the demon dots and the tunnel barriers within the system result in the desired modulation of transition rates.

Figure 2

Working principle and graph representation. Inner part: Graph representation of Eq. (4). Each node corresponds to a state $(s,d)$ and each line corresponds to possible transitions between states. The transitions illustrated in dashed red are suppressed by the demon while the transitions illustrated in green are not. Outer part: Desired behavior of the composite system. Starting with three empty dots (left-hand side), the system dot cannot be filled due to the superconducting gaps. The cold dot can be filled, lifting the energy level in the system above the superconducting gaps. When the cold dot is filled, transitions involving the right reservoir are suppressed. The system dot can be filled from the left. This raises the energy level of the cold dot above ${\mu }_{\mathrm{C}}$, such that the electron can leave. The electron in the system dot is now trapped due to the superconducting gaps until the hot dot is being filled. When the hot dot is full, transitions involving the left reservoir are suppressed and the system dot is emptied to the right. The cycle is closed when the hot dot is emptied. In one cycle, one electron is transported against the voltage bias and one quantum of heat of size $U$ is transported from hot to cold.

Figure 3

Performance of the demon. The generated power and the heat flow out of terminal L are compared for the cases where the demon is (a), (b) in equilibrium $\left(T_{\text{C}}=T_{\text{H}}=T\right)$ and (c), (d) out of equilibrium $\left(T_{\text{C}}<T<T_{\text{H}}\right)$, as functions of the applied voltage $V$ and the tunneling asymmetry $\delta ={\delta }_{\text{L}}={\delta }_{\text{R}}$. For positive voltages, the nonequilibrium demon generates power while cooling the left reservoir at the same time. (e) The power generation reduces the entropy of the system by $P/T$, which is always smaller than the entropy production in the demon, ${\dot{S}}_{\text{d}}$, except for the case at the stall voltage with $\delta \to \infty$, where they are equal. At this point, (f) the demon performs in the reversible limit and the maximal efficiency ${\eta }_0$ is achieved. Here we explicitly included the effect of a finite superconducting gap, $\mathrm{\Delta }=0.8k_{\text{B}}T$, which does not influence the shown plots. Parameters: $T_{\text{H}}=1.2T,T_{\text{C}}=0.8T,U=1.5k_{\text{B}}T,{\varepsilon }_{\text{S}}={\xi }_{\text{H}}=0,{\xi }_{\text{C}}=-0.4k_{\text{B}}T,{\mu }_{\text{L}}=-eV/2,{\mu }_{\text{R}}=eV/2,{\mathrm{\Gamma }}_{\alpha }=0.1k_{\text{B}}T$, and $\gamma ={10}^{-8}{k}_{\text{B}}T$.

Figure 4

Performance of the strong demon. (a) Generated power and (b) heat flow out of terminal L as functions of the applied voltage $V$ and the temperature of the cold reservoir. $P\le 0$ for $T_{\text{C}}=T_{\text{H}}$. The stall voltage increases for lower $T_{\text{C}}$, but is cut off by the gap. (c) The generated power also increases when lowering $T_{\text{C}}$. (d) The superconducting gap however avoids reaching the highest efficiencies. (e) By increasing the ratio ${\mathrm{\Gamma }}_{\text{d}}/\mathrm{\Gamma }$ for the case $T_{\text{C}}=0.5T$, the power increases toward the fast-demon limit, (f) but the efficiency is not affected. Close to the gap, a small peak appears in $P$ reflecting the superconducting density of states (8). Same parameters as in Fig. 3 except for those explicitly indicated.

Figure 5

Second-law-like inequalities relating the rates for system and demon entropy change ${\dot{S}}_{\text{s}}$ and ${\dot{S}}_{\text{d}}$, the information flow $\mathcal{I}$, and the fast information flow ${\mathcal{I}}_{\mathrm{f}}$ for different strengths of the demon: (a) ${\delta }_{\text{L}}={\delta }_{\text{R}}=2$ and (b) ${\delta }_{\text{L}}={\delta }_{\text{R}}=10$. In the strong-demon limit, ${\dot{S}}_{\mathrm{d}}={\mathcal{I}}_{\mathrm{f}}$, and all inequalities are saturated at the stall voltage. Away from the fast-demon limit, ${\mathcal{I}}_{\mathrm{f}}$ is defined as the right-hand side of Eq. (44). Other parameters are as in Fig. 3.

Figure 6

Illustration of time-reversed trajectories. Left panel: Desired trajectory, where an electron enters the system dot from the left through an open (green) barrier and leaves the system dot to the right through an open barrier. Completing this loop once results in a single charge transported against the voltage bias $w=1$, a single quantum of heat transported from hot to cold $q=1$, and a single electron contributing positively to the feedback-assisted current $n$. Middle panel: Time-reversing the composite system results in changing the direction of all arrows. While $w$ and $q$ change sign, the contribution to the feedback-assisted current remains positive because the electron passes through two open barriers. Right panel: Time-reversing only the system exchanges origin and destination of the electron that passes through the system dot. This inverts $w$ and $n$ while $q$ remains invariant since the demon is not affected by the transformation.

Figure 7

Information-related efficiencies, ${\eta }_j$, with $j=\{\mathrm{T},\mathrm{I},\mathrm{f},\mathrm{F}\}$, calculated from the right-hand side of Eqs. (54, 55, 56, 57), for (a) a weak, (b) a weak and fast, (c) a strong, and (d) a strong and fast demon. The cases with $T_{\text{C}}=0.5T$ (black) and $T_{\text{C}}=0.3T$ (red lines) are compared. We set ${\delta }_{\text{L}}={\delta }_{\text{R}}=2$ for weak, ${\delta }_{\text{L}}={\delta }_{\text{R}}=10$ for strong, ${\mathrm{\Gamma }}_{\text{d}}=50\mathrm{\Gamma }$ for fast, and ${\mathrm{\Gamma }}_{\text{d}}=\mathrm{\Gamma }$ otherwise. For the fast demon (b) and (d), ${\eta }_{\mathrm{T}}$ and ${\eta }_{\mathrm{I}}$ overlap. For a strong demon, ${\eta }_{\mathrm{T}}$ coincides with ${\eta }_{\mathrm{f}}$, panels (c) and (d), except for being limited by the gap.

Figure 8

Effect of unequal charging energies. Generated power (black lines) and heat current absorbed by the demon (red lines) for different factors of asymmetry, $\theta$, having (a) $U_{\text{C}}>U_{\text{H}}$ and (b) $U_{\text{C}}<U_{\text{H}}$ (in both cases fixing $U_{\text{H}}=1.5k_{\text{B}}T)$. Same parameters as in Fig. 3 for the strong-demon limit with ${\delta }_{\text{L}}={\delta }_{\text{R}}=\infty$, except for those explicitly indicated.

Figure 9

Sketch of an analog device formed by metallic islands. (a) The pale gray islands are used to avoid or permit transport into the system island (dark gray) depending on the occupation of the demon islands. Labels indicate superconducting leads (S), insulating tunneling barriers (I), and normal-metal islands (N). The transitions in (b) and (c) picture the sequence of the demon action. Dashed arrows mark the change in the electrochemical potentials due to a tunneling transition in a different island (numbered full arrows).