Maxwell's demon for quantum transport

Abstract

While most of the existing quantum information engines assisted by Maxwell's demon harness thermal fluctuations, those that rectify only quantum fluctuations have recently been constructed. We propose an alternative type of quantum information engine that harnesses only quantum fluctuations to achieve cumulative energy storage and unidirectional transport of a particle. This unidirectional transport makes a stark contrast with the case without Maxwell's demon where the motion of a particle is confined to a finite region due to Bloch oscillations. We find a trade-off relationship between the maximum power and the maximum velocity. With an improved definition of efficiency that includes all possible energy flows in the engine cycle, we numerically demonstrate the absence of a trade-off relationship among power, efficiency, and power fluctuations that is present for classical heat engines and classical information engines. We also evaluate the influence of experimentally unavoidable measurement imprecision on the performance of the quantum Maxwell's demon.

Figures (10)

• Figure 1: (a) Setup. A particle hops on a one-dimensional tilted lattice with lattice constant $d$, hopping amplitude $J$, and step height $\Delta$. If a projective position measurement finds the particle at site $j$, the potential at site $j-1$ is instantly raised by $V\gg \Delta, J$ to prevent the particle from hopping downwards. The particle then evolves according to Eq. \ref{['eq: our limiting H']} during time $t$. Then we perform the projective position measurement again which may find a particle at a higher position $j'>j$. Through repetitive measurement and feedback, the particle climbs up the stairs to gain potential energy. (b) Numerical simulations of the traveled distance for 100 realizations (thin colored), each of which runs 1000 engine cycles with $\alpha:=\Delta/J =10$ and $\tilde{V}=10^3$. The single-cycle optimization (black dotted) agrees well with the trajectory ensemble average (thick red).
• Figure 2: (a) Maximal power $\tilde{p}_{\rm max}(\alpha)$ and (b) maximal velocity $\tilde{v}_{\rm max}(\alpha)$ as functions of gradient $\alpha$. The blue curves are numerical results and the orange dashed curves show the asymptotic results for large $\alpha$ (see Sec. \ref{['sec: p and v def']}). The inset in (a) shows $\tilde{p}(1,\tau)$ as a function of $\tau$, indicating a well-defined optimal time. (c) Optimal time $\tau^*_p(\alpha)$ for $\tilde{p}_{\rm max}$ and $\tilde{v}_{\rm max}$. The orange-dashed curve is predicted by the large-$\alpha$ theory (see Sec. \ref{['sec: p and v def']}). Data fitting (red) gives $\tau^* \propto\alpha^{-0.64}$ for small $\alpha$. (d) Velocity for $\alpha=0$ with $N=10000$ sites. The inset shows the displacement $\langle n(\tau) \rangle$. Both insets are plotted in linear scales. After a negligible transient stage, the velocity reaches about 1.7 until the wave packet reaches the right boundary as plotted in the inset.
• Figure 3: (a) Energy flow diagram for our engine. The probe $M$ plays the role of an agent because it is cyclic. The total energy input $E_{\rm cost}$ consists of the costs of measurement and information erasure, whose resource is not a heat bath. Heat $q:= E_{\rm cost} - E$ is dissipated into a heat bath at temperature $T$ to erase the information in $M$. The FB stores energy $E$ in the potential energy of the particle $S$. (b) The maximal efficiency $\eta_{\rm max}(\alpha)$ for $\tilde{T} = 0.1$ (blue), $1$ (orange), and $10$ (green). The dashed curves show the large-$\alpha$ approximation. (c) Optimal time $\tau^*_{\eta}(\alpha)$ with $\tilde{T}=1$. The orange dashed line is the large-$\alpha$ calculation and data fitting (red) for the small-$\alpha$ region gives an almost inverse linear relation.
• Figure 4: (a) Numerical calculation of $Q(\alpha,\tau)-2$ defined in inequality \ref{['eq: TUR to test']} for $\alpha\in[10^{-2},10^2]$ and $\tau\in [0,500]$. Data points representing $Q-2<0$ are plotted in red and nonnegative points are shown in gray, where whiter colors represent larger values. A clear boundary near $\alpha=3.6$ that separates the two regions is indicated by the dashed line; to the left of it, violation of the inequality is allowed. (b) $Q_{\rm min}$ (red curve) versus $\alpha$. The gray dashed line shows the bound \ref{['eq: TUR to test']} to be tested (see the main text). (c)-(f) Numerical results of $\tilde{p}$, ${\rm Var}[\tilde{p}]$, $H$, and $\Delta_{\tilde{p}}$, where $\tau = \tau^*_Q(\alpha)$. The orange dashed curves show the results obtained with the large-$\alpha$ approximation.
• Figure 5: Influence of the error $\varepsilon$ on the performance with $\alpha =1$ for the $j-2$ FB-contolled engine. We take 9 values of $\varepsilon$ from 0 to 0.5 with a logarithmic interval as shown in the legend. We plot the trajectory ensemble average (colored curve) of (a) the traveled distance and (b) the accumulated energy gain for (a) $M=30$ cycles and (b) $M=10$ cycles where each engine cycle evolves for (a) $\tau=\tau^*_v = 2.65$ and (b) $\tau=\tau^*_E(\alpha=1) = 45.42$ for different $\varepsilon$'s. The black dotted lines show the single-cycle optimized results. (c) Change $\delta \tilde{v}$ in the average transport velocity from the error-free velocity $\tilde{v}_{\rm max}(\alpha=1) = 0.49$ as a function of $\varepsilon$. (d) Change in the average energy gain per cycle compared with the error-free value $\tilde{E}_{\rm max}(\alpha=1) = 1.52$. (e) Change in the efficiency with $\tau^*_{\eta}(\alpha=1) = 94.0$ compared with the error-free value $\eta_{\rm max} = 0.51$.