Heat production and error probability relation in Landauer reset at effective temperature

Abstract

The erasure of a classical bit of information is a dissipative process. The minimum heat produced during this operation has been theorized by Rolf Landauer in 1961 to be equal to k_BT ln2 and takes the name of Landauer limit, Landauer reset or Landauer principle. Despite its fundamental importance, the Landauer limit remained untested experimentally for more than fifty years until recently when it has been tested using colloidal particles and magnetic dots. Experimental measurements on different devices, like micro-mechanical systems or nano-electronic devices are still missing. Here we show the results obtained in performing the Landauer reset operation in a micro-mechanical system, operated at an effective temperature. The measured heat exchange is in accordance with the theory reaching values close to the expected limit. The data obtained for the heat production is then correlated to the probability of error in accomplishing the reset operation.

Figure 1. Schematic of the whole system and measurement setup.

Lateral view of the whole system and measurement setup. Two magnets with opposite magnetic orientations are used to induce bistability in the system. Two electrodes are used to apply electrostatic forces on the mechanical structure: V_L and V_R to force the cantilever to bend to the left (negative x) and to the right (positive x) respectively. The magnetic interaction can be engineered by changing geometric parameters such as d and Δx. (b) Color-map of the reconstructed potential energy as function of the distance between the magnets, d. The distance is expressed in arbitrary units proportional to the voltage applied to the piezoelectric stage. Decreasing the distance between the magnets the potential energy softens and eventually two stable states appear. (c) Dependence of the effective temperature, T_eff, with the root mean square of the white Gaussian voltage applied to the piezoelectric shaker. The red dot represents the condition accounted for the experimental data presented.

Figure 2. Reset protocol.

(a) Protocol used to perform the reset operation. In order to account for all the possible transitions we considered the reset to 1 (first two columns) and reset to 0 (last two columns) starting from both 0 and 1 states. The first row depict the protocol used for removing the barrier. Once the barrier is removed a lateral force is applied (second row). The resulting displacement of the cantilever tip is represented in the third row. Once all the forces are removed and the barrier is restored the cantilever tip encodes the final state. (b) Schematic time evolution of the potential energy and state of the system for the case presented in the first column of panel (a). The operation starts from a double well potential and in the first step of the protocol the barrier is removed. During the next step the potential is tilted and the barrier is restored to its initial value. Finally, the lateral force is removed recovering the initial condition where the barrier between wells is at its maximum and the electrostatic forces are equal to zero.

Figure 3. Produced heat and probability of success for the reset operation.

(a) Average heat produced during the reset operation as function of the lateral alignment Δx. For Δx < 0 the counter magnet is moved to the right and the 0 state (x < 0) is favorable. Accordingly, for Δx > 0 the 1 state is more favorable. Introducing an asymmetry on the potential Q decreases, which is accounted to the probability of success, P_s, that tends to decrease (P_s is encoded in the color map). (b) Success rate of the reset operation as function of lateral alignment. Solid violet circles represent the overall success rate while black and red symbols account for the success rate resetting to 0 and 1 state respectively. The maximum overall success rate is present when the system is almost symmetric, Δx ≈ 0. (c) Relation between success rate and heat dissipated. Red circles correspond to the resetting to 1 case while black ones correspond to the resetting to 0. (d) Dependence of Q with the protocol time duration, τ_p. As τ_p is increased the effects of frictional phenomena becomes negligible and the produced heat should approach the thermodynamic limit. However, for large τ_p the reset operation fails giving a wrong logic output. In these cases, where the error probability is high, the produced heat is clearly below the Landauer limit. Inset shows the obtained relation between error probability (1 − P_s) and produced heat. The data are compatible with the minimum energy required for a given error probability as predicted by Eq. 1, represented by dashed line.