High-Precision Test of Landauer’s Principle in a Feedback Trap

We confirm Landauer’s 1961 hypothesis that reducing the number of possible macroscopic states in a system by a factor of 2 requires work of at least $kT\ln 2$. Our experiment uses a colloidal particle in a time-dependent, virtual potential created by a feedback trap to implement Landauer’s erasure operation. In a control experiment, similar manipulations that do not reduce the number of system states can be done reversibly. Erasing information thus requires work. In individual cycles, the work to erase can be below the Landauer limit, consistent with the Jarzynski equality.

Figure 1

Schematic of feedback trap operation. (a) Acquisition of an image of a fluorescent particle. (b) Determination of particle position from that image using a centroid algorithm. (c) Evaluation of feedback force $F_x=-{\partial }_{x}U(x,t)$ at the observed position $\overline{x}$. (d) Application of electric force, with voltage set by electrodes (light blue), held constant during the update time $t_s=10\mathrm{ms}$. The long gray arrow indicates repetition of the cycle.

Figure 2

Erasure protocol and trajectories for full erasure ($p=1$) and no erasure ($p=0.5$). Full erasure requires a strong tilt of the potential towards the desired well ($A=0.5$). In the no-erasure protocol, the potential is symmetric at every time step ($A=0$), implying that a particle ends up in a random final state. The image intensity $I(x,t)\propto P(x,t)$ the occupation probability for a particle in a time-dependent, double-well potential and was generated from 30 trajectories for each case using kernel density estimation. We used a Gaussian kernel with standard deviation equal to 0.1 in time and $0.15\mu \mathrm{m}$ in space, evaluated on a $500\times 160$ grid. Scale bar at lower left measures $5\mu \mathrm{m}$.

Figure 3

(a) Histograms of work series for individual cycles of duration $\tau =0.5$, 1, and 2.5, with Gaussian fits shown as solid curves. The gray shaded area shows the part of the probability distribution that is below the Landauer limit. (b) Variance versus mean for the work distribution, in units of $kT$. Solid lines are plotted from the Jarzynski relation, Eq. (4), and have slope 2.

Figure 4

Mean work measured in the full erasure ($p=1$) and no-erasure ($p=0.5$) protocols. (a) Mean work approaches the Landauer limits for each protocol. Solid line shows fit to asymptotic ${\tau }^{-1}$ correction. (b) Mean work as a function of inverse time. The dimensionless cycle times $\tau$ are in units of ${\tau }_0=(2x_m{)}^2/D$.