Information and Thermodynamics: Fast and Precise Approach to Landauer’s Bound in an Underdamped Micromechanical Oscillator

The Landauer principle states that at least $k_{B}T\ln 2$ of energy is required to erase a 1-bit memory, with $k_{B}T$ the thermal energy of the system. We study the effects of inertia on this bound using as one-bit memory an underdamped micromechanical oscillator confined in a double-well potential created by a feedback loop. The potential barrier is precisely tunable in the few $k_{B}T$ range. We measure, within the stochastic thermodynamic framework, the work and the heat of the erasure protocol. We demonstrate experimentally and theoretically that, in this underdamped system, the Landauer bound is reached with a 1% uncertainty, with protocols as short as 100 ms.

Figure 1

(a) Schematic diagram of the experiment. The conductive cantilever is sketched in yellow. Its deflection $x$ is measured with a differential interferometer [28], through two laser beams focused respectively on the cantilever and on its base. The cantilever at voltage $V=\pm V_1$ is facing an electrode at $V_0$. The voltage difference $V-V_0$ between them creates an attractive electrostatic force $F\propto (V-V_0{)}^2$. The dashed box encloses the feedback controller made of a comparator and a multiplier, which create the double-well potential. (b) Measured power spectrum density (PSD) of the cantilever deflection thermal noise with no feedback ($V_1=0$, blue), and best fit by the theoretical thermal noise spectrum of a simple harmonic oscillator (SHO, red dashed line). Up to 10 kHz, the cantilever behaves like a resonator at $f_0=1270\mathrm{Hz}$, with a quality factor $Q=10$. We infer from this measurement the variance ${\sigma }^2=⟨x^2⟩=k_{B}T/k$, used to normalize all measured quantities. (c) Measured double-well potential energy (blue line) obtained with the feedback on, with $x_0=0$ and $V_1$ adjusted to have a $5k_{B}T$ barrier height. The potential is inferred from the measured probability density function (PDF) of $x$ during a 10 s acquisition and the Boltzmann distribution. The fit using Eq. (1) is excellent (red dashed line).

Figure 2

Description of the erasure protocol. (a) Evolution of the potential $U(z,z_0,z_1)$. (b) Experimental static PDF $P(z,z_0,z_1)$ (blue), best fit to the Boltzmann distribution (red line), and position of the threshold $z_0$ (green dashed line).

Figure 3

Time recording of the cantilever deflection $z$ on a single trajectory (blue, starting in state 1 in this example), superposed with the two wells’ centers $+z_1$ (black line) and $-z_1$ (red line), and the threshold $z_0$ (green dashed line). Stage 1 (red background) and 2 (green background) both last $\tau =200\mathrm{ms}$. $\tau$ is very long compared to the natural relaxation time ${\tau }_r$ of the cantilever: $f_0\tau \sim 250\gg f_0{\tau }_r=Q/\pi \sim 3$. The equilibrium periods around stages 1 and 2 are chosen freely as long as they allow the cantilever to relax, in this example ${\tau }_0=50\mathrm{ms}\gg {\tau }_r$.

Figure 4

Time evolution of the mean power over 2000 trajectories following the slow protocol of Fig. 3. The work takes off when the cantilever starts switching between the wells, during stage 1 (red background). Since the process is quasistatic, stage 2 (green background) doesn’t contribute on average. The red dashed line is the analytical prediction in the quasistatic regime [31]. (Inset) Work distributions for the 2000 trajectories during stage 1 (${\mathcal{W}}_1$, red) and 2 (${\mathcal{W}}_2$, green) fitted by gaussians (plain lines). As theoretically predicted for a slow erasure, stage 1 reaches the LB of 0.693 (dashed red) while stage 2 requires negligible work (dashed green). A PDF of $\mathcal{Q}$ is available in the Supplemental Material [31].

Figure 5

Energy cost from erasure protocols at different speeds ${\dot{z}}_1=Z_1/\tau$, with $Z_1\sim 5$. Experimental data (blue) are fitted by $L_{\infty }+B^{\prime }{Z}_1/\tau$ (red dashed line), with $B^{\prime }=(437\pm 27)\mu \mathrm{s}$ and $L_{\infty }=0.695\pm 0.012$. (Inset) Experimental mean work and heat during stage 2 (green) at different speeds. The theoretical prediction $B_2^{\prime }{Z}_1/\tau$ (red dashed line) with $B_2^{\prime }=Z_1/Q{\omega }_0$ works perfectly (no adjustable parameters).