Landauer’s Erasure Principle in a Squeezed Thermal Memory

Landauer’s erasure principle states that the irreversible erasure of a one-bit memory, embedded in a thermal environment, is accompanied with a work input of at least $k_{B}T\ln 2$. Fundamental to that principle is the assumption that the physical states representing the two possible logical states are close to thermal equilibrium. Here, we propose and theoretically analyze a minimalist mechanical model of a one-bit memory operating with squeezed thermal states. It is shown that the Landauer energy bound is exponentially lowered with increasing squeezing factor. Squeezed thermal states, which may naturally arise in digital electronic circuits operating in a pulse-driven fashion, thus can be exploited to reduce the fundamental energy costs of an erasure operation.

Figure 1

(a) Stroboscopic phase space probability density of a squeezed thermal state (squeezing factor $r=0.5$, relative timing $t_0=0.24{\nu }^{-1}$). The thermal fluctuations in one quadrature are reduced, while the orthogonal quadrature shows increased fluctuations. The variances of the line integrated distributions (shown in blue) correspond to two temperatures $T_x$, $T_p$ that govern the thermal fluctuations of the system. (b) Scheme to erase one bit of information in a squeezed thermal memory: (i) removal of partition and free expansion, (ii) stroboscopic compression to half of the volume, (iii) insertion of partition. During the process, the single-particle gas is assumed to be in contact with the squeezed thermal reservoir at all times.

Figure 2

(a) Stroboscopic phase space densities of a harmonically confined particle subject to squeezed thermal noise for three different damping regimes: underdamped motion (damping ratio $\zeta =c/2m{\omega }_0=0.05$), critically damped motion ($\zeta =0.5$), and overdamped motion ($\zeta =5$). The diagrams represent position-momentum histograms at equidistant points in time $t_0,{\nu }^{-1}+t_0,2{\nu }^{-1}+t_0,\dots$, where $t_0=0.24{\nu }^{-1}$ sets the relative timing of the observations with respect to the stochastic force. (b) Collisions with the piston (indicated by the red bar) induce phase jumps in the particle motion, which cancel the squeezing effect in the case of underdamped motion. In the case of overdamped motion, the particle tends to “stick” to the piston, which leads to a strong enhancement of the probability density in this region. (c) Typical trajectories $x(t)$ of the trapped particle.

Figure 3

Required work $W$ to compress the single-particle gas to half of its initial volume as a function of the relative timing $t_0$, which defines the points in time, namely, $t_0,{\nu }^{-1}+t_0,2{\nu }^{-1}+t_0,\dots$, at which the compression steps are executed. The given values of $W_r$ (for different squeezing parameters $r$) are normalized to the work at vanishing squeezing $W_{r=0}$. The temperature $T$ is kept fixed in all simulations. (a) In the critically damped regime ($\zeta =0.5$), the work is observed to exponentially decrease with the squeezing parameter $r$ close to $t_0=0.33{\nu }^{-1}$ and $t_0=0.83{\nu }^{-1}$. (b) Work at constant squeezing $r=0.5$ for various damping ratios $\zeta$. The squeezing effect is observed to vanish for underdamped systems.