Autonomous quantum Maxwell's demon based on two exchange-coupled quantum dots

I study an autonomous quantum Maxwell's demon based on two exchange-coupled quantum dots attached to the spin-polarized leads. The principle of operation of the demon is based on the coherent oscillations between the spin states of the system which act as a quantum iSWAP gate. Due to the operation of the iSWAP gate, one of the dots acts as a feedback controller which blocks the transport with the bias in the other dot, thus inducing the electron pumping against the bias; this leads to the locally negative entropy production. Operation of the demon is associated with the information transfer between the dots, which is studied quantitatively by mapping the analyzed setup onto the thermodynamically equivalent auxiliary system. The calculated entropy production in a single subsystem and information flow between the subsystems are shown to obey a local form of the second law of thermodynamics, similar to the one previously derived for classical bipartite systems.

Figure 1

(a) Scheme of the studied system consisting of two exchange-coupled quantum dots attached to the spin-polarized leads. $J$ denotes the exchange interaction, ${\mathrm{\Gamma }}_{i\nu }^{\sigma }$ the tunneling rates, and $f_{i\nu }$ the Fermi distribution functions. (b) Scheme of the dynamics of the system. Black straight arrows denote transitions associated with the electrons tunneling while the violet curved double-headed arrow denotes the coherent oscillations between the states $|\uparrow \downarrow \rangle$ and $|\downarrow \uparrow \rangle$. Notation as at the end of Sec. 2.

Figure 2

Phase diagram of the system over $f_1=f(V_1/k_{B}T)$ and $f_2=f(V_2/k_{B}T)$ for arbitrary nonzero values of ${\mathrm{\Gamma }}_{1L}^{\uparrow },{\mathrm{\Gamma }}_{1R}^{\downarrow },{\mathrm{\Gamma }}_{2L}^{\downarrow },{\mathrm{\Gamma }}_{2R}^{\uparrow },{\epsilon }_i$, and $J$ showing the different operation modes: blue and red regions correspond to the Maxwell demon modes, with red (violet) regions denoted with a number 1 (2) corresponding to the negative entropy production in the first (second) dot; the green region denoted with a number 3 corresponds to the mode in which entropy production in both dots is positive.

Figure 3

(a) Scheme of the dynamics of the auxiliary system $A$. Notation of the tunneling rates defined at the end of Sec. 2. Blue numbers (1–18) below the circles representing quantum dots denote the states $|i\rangle$ of the auxiliary system. Red underlined numbers on the left and top sides of the scheme denote rows and columns. (b) Alternative arrangement with transitions in the second dot placed in columns, referred to as the auxiliary system $B$.

Figure 4

Entropy production and the information flow as a function of $f(V_2/k_{B}T)$ for $f(V_1/k_{B}T)=0.2,{\mu }_{iL}=V_i/2,{\mu }_{iR}=-V_i/2,{\epsilon }_1={\epsilon }_2=0,{\mathrm{\Gamma }}_{1L}^{\uparrow }={\mathrm{\Gamma }}_{1R}^{\downarrow }={\mathrm{\Gamma }}_{2L}^{\downarrow }={\mathrm{\Gamma }}_{2R}^{\uparrow }=\mathrm{\Gamma },J=1.5\mathrm{\Gamma }$. For line identification see attached labels in (a) and attached legend in (b).

Figure 5

Entropy production and the information flow as a function of $J$ for ${\mu }_{iL}=V_i/2,{\mu }_{iR}=-V_i/2,{\epsilon }_1={\epsilon }_2=0,{\mathrm{\Gamma }}_{1L}^{\uparrow }={\mathrm{\Gamma }}_{1R}^{\downarrow }={\mathrm{\Gamma }}_{2L}^{\downarrow }={\mathrm{\Gamma }}_{2R}^{\uparrow }=\mathrm{\Gamma },f(V_1/k_{B}T)=0.2,f(V_2/k_{B}T)=0.6$. For line identification see attached labels.

Figure 6

Work inputs ${\dot{N}}_i{V}_i$, heat dissipated in the $i\mathrm{th}$ dot ${\dot{Q}}_i$ and the energy exchange between the first and the second dot ${\dot{E}}_{12}$ as a function of $\left|J\right|/V_1$ for $V_1=k_{B}T$ and $V_2=-k_{B}T/2,{\mu }_{iL}=V_i/2,{\mu }_{iR}=-V_i/2,{\epsilon }_1={\epsilon }_2=0,{\mathrm{\Gamma }}_{1L}^{\uparrow }={\mathrm{\Gamma }}_{1R}^{\downarrow }={\mathrm{\Gamma }}_{2L}^{\downarrow }={\mathrm{\Gamma }}_{2R}^{\uparrow }=\mathrm{\Gamma },{\mathrm{\Gamma }}_{1L}^{\downarrow }={\mathrm{\Gamma }}_{1R}^{\uparrow }={\mathrm{\Gamma }}_{2L}^{\uparrow }={\mathrm{\Gamma }}_{2R}^{\downarrow }=0$. For line identification see attached labels.