Converting heat into directed transport on a tilted lattice

We present a self-contained engine, which is made of one or more two-level systems, each of which is coupled to a single bath, as well as to a common load composed of a particle on a tilted lattice. We show that an increase in time of energy and entropy in the system composed of the spins and the particle, due to the interaction with the bath, can set the particle into upward motion at an average constant speed, even when driven by a single spin connected to a single bath. When considering an ensemble of different spins, the velocity of the particle is larger when the tilt is on resonance with any of the spins' energy splitting. Interestingly, we find regimes where the spins' polarization enters periodic cycles with the oscillation period being determined by the tilt of the lattice.

Figure 1

Pictorial representation of the system studied. An ensemble of spins, each coupled to a single bath (depicted by clouds around the spins), and to a particle on a tilted lattice.

Figure 2

(a) Average and (b) variance of the position of the particle vs time for different values of ${\lambda }^{+}-{\lambda }^{-}$. In (a) and (b) we only consider a single spin and keep $({\lambda }_1^{+}+{\lambda }_1^{-})/2={\lambda }_1$ constant. The parameters used for (a) are ${\lambda }_1^{+}/\nu =\{0.01,0.02,0.03,0.07,0.08,0.09\}$ while for (b), ${\lambda }_1^{+}/\nu =\{0.045,0.07,0.09\}$. In both (a,b) the arrow indicates the direction of increasing ${\lambda }_1^{+}$. (c) Density plot of $P_n$ versus position and time. (d) Total entropy $S_t$ (circles) and particle entropy $S_p$ (triangles) vs time. The continuous lines are fit with the function $y\left(t\right)=a+[ln(t\left)\right]/2$ predicted by classical normal diffusion. In (a)–(d), we used ${\lambda }_j=0.05\nu , {\omega }_1=\nu$ and $g_1=0.15\nu$. In (c) and (d) ${\lambda }^{+}=0.07\nu$. In (e) and (f) we show the particle velocity vs $\omega -\nu$. (e) shows two identical spins coupled by $g_1=g_2$ while (f) shows the results for three different spins. Plotted are both the asymptotic results of a full numerical simulation as well as fits using a sum of Lorentzian lineshapes, fitted using a least squares algorithm. The parameters in (e) are ${\omega }_1={\omega }_2$ and $g_1=g_2=0.15{\omega }_2$. In (f) instead ${\omega }_1={\omega }_2/4, {\omega }_3=2{\omega }_2, g_1/{\omega }_2=0.15, g_2/{\omega }_2=0.2$ and $g_3/{\omega }_2=0.1$. In both (e) and (f) ${\lambda }_j^{+}/{\omega }_2=0.071$ and ${\lambda }_j/{\omega }_2=0.05$ for every $j$.

Figure 3

Asymptotic velocity of the particle vs coupling to the spin $g_1$. The filled circles (joined by a dashed line) represent the numerical results, the continuous line represent the asymptotic value $({\lambda }^{-}-{\lambda }^{+})/2$ while the dashed line represent the prediction of the adiabatic elimination Eq. (13). The inset shows a detail of Fig. 3 which shows the accuracy of the prediction of Eq. (13). In the plot, we have considered only a single spin with ${\omega }_1=\nu , {\lambda }_1^{+}=0.071\nu$, and ${\lambda }_1=0.05\nu$.

Figure 4

Asymptotic behavior of $\langle {\widehat{\sigma }}_{x,j}\rangle , \langle {\widehat{\sigma }}_{y,j}\rangle$, and $\langle {\widehat{\sigma }}_{z,j}\rangle$ for different values of ${\lambda }_1^{+}-{\lambda }_1^{-}$. From bottom to top we have used ${\lambda }_1^{+}/\nu =\{0.01,0.02,0.03,0.07,0.08,0.09\}$ while keeping ${\lambda }_1/\nu =0.05$. The other parameters are ${\omega }_1/\nu =1$ and $g/\nu =0.15$.