Second laws for an information driven current through a spin valve

We propose a physically realizable Maxwell's demon device using a spin valve interacting unitarily for a short time with electrons placed on a tape of quantum dots, which is thermodynamically equivalent to the device introduced by Mandal and Jarzynski [D. Mandal and C. Jarzynski, Proc. Natl. Acad. Sci. USA 109, 11641 (2012)]. The model is exactly solvable and we show that it can be equivalently interpreted as a Brownian ratchet demon. We then consider a measurement-based discrete feedback scheme, which produces identical system dynamics, but possesses a different second law inequality. We show that the second law for discrete feedback control can provide a smaller, equal, or larger bound on the maximum extractable work as compared to the second law involving the tape of bits. Finally, we derive an effective master equation governing the system evolution for Poisson distributed bits on the tape (or measurement times, respectively) and we show that its associated entropy production rate contains the same physical statement as the second law involving the tape of bits.

Figure 1

(a) Simple sketch of the system showing the tape of bits coupled to the quantum dot spin valve. (b) Equivalent model of the Brownian ratchet showing the two different modes, which can be regarded as shifted versions of an infinite lattice. Transitions between the two different modes are due to the interaction with a bit or due to the feedback.

Figure 2

Sketch of the possible transitions during a single cycle, starting from the left with the initial state ${\rho }_S^{-}\otimes {\rho }_B^{-}$, followed by a swap operation, and ending after a time $\tau$ in the final state, which becomes the new initial state for the next cycle if we replace it with a new bit. The arrows indicate possible transitions and the color code symbolizes the net number of exchanged particles with the left and right reservoir contributing to Eqs. (10) and (11).

Figure 3

Phase diagram of the system over ${\delta }^{-}$ and $V$ showing the different working modes: information eraser mode (regions 1 and $1^{\prime }$, red color), Maxwell demon mode (regions 2 and $2^{\prime }$, green color), and the “third mode” (regions 3 and $3^{\prime }$, blue color). The primed regions are connected to the unprimed regions by the symmetry relation. For clarity, we also plot the line ${\delta }^{-}=0$ (dashed). Note that the line ${\delta }^{+}=-{\delta }^{-}$ changes with $\tau$, but the overall structure remains the same.

Figure 4

Probabilities to find the system and the memory in their respective states prior to the measurement, after the measurement, and after the feedback. The possible transitions are marked with shaded gray lines.

Figure 5

Plot of the delivered work $-\Delta W=V\Delta N^L$ (black thin line), the change in the Shannon entropy of the bits $\Delta H_B$ (black thick line), and the maximum $I_{\max }$ (green dash-dotted line), the minimum $I_{\min }$ (red dashed line), and the error-free mutual information $I_{\mathrm{ef}}$ (blue dotted line) over the excess parameter ${\delta }^{-}$. We see that the sum of $V\Delta N^L$ and any of the information theoretic quantities $\Delta H_B,I_{\max },I_{\min }$, or $I_{\mathrm{ef}}$ is always positive, as imposed by Eqs. (22) and (26). We further chose $V=1$ and $\tau =10$.

Figure 6

Plot of the entropy production rates ${\dot{\stackrel{̃}{S}}}_i$ [Eq. (42), black solid line] as well as ${\dot{S}}_i$ [Eq. (39), blue lines] and ${\Delta }_i{S}_{\mathrm{tape}}/\tau$ [Eq. (22), red lines] for $\gamma =0.1,\tau =10$ (dotted lines), $\gamma =1,\tau =1$ (dashed lines), and $\gamma =10,\tau =0.1$ (dash-dotted lines) over the bias voltage $V$. We chose ${\delta }^{-}=0.75$.