Nonequilibrium thermodynamics of erasure with superconducting flux logic

We implement a thermal-fluctuation-driven logical bit reset on a superconducting flux logic cell. We show that the logical state of the system can be continuously monitored with only a small perturbation to the thermally activated dynamics at 500 mK. We use the trajectory information to derive a single-shot estimate of the work performed on the system per logical cycle. We acquire a sample of ${10}^5$ erasure trajectories per protocol and show that the work histograms agree with both microscopic theory and global fluctuation theorems. The results demonstrate how to design and diagnose complex, high-speed, and thermodynamically efficient computing using superconducting technology.

Figure 1

Gradiometric flux logic cell. (a) False-color electron micrograph of the device, realized as a two-layer superconducting circuit on an insulating silicon substrate. (b) Simplified circuit schematic of the information-bearing subsystem. We take the dynamical coordinates to be the total magnetic fluxes $\phi$ and ${\phi }_{dc}$ threading the loops. (c) Contour plot of the potential calculated for the component values of the studied device and external bias fluxes $({\phi }_x,{\phi }_{x,dc})=(-0.1018,-2.5887)$ coinciding with the start of the bit erasure protocols studied later. Two local metastable minima ($L$ and $R$, red dots) and the unique saddle point ($B$, black dot) are marked.

Figure 2

Quasistatic response of the flux logic cell. (a) and (b) Two-dimensional scan of tilt and barrier controls (${\mathrm{\Phi }}_x$ and ${\mathrm{\Phi }}_{x,dc}$, respectively). Experimental plots (left) show the unprocessed phase of the local magnetometer readout. Theoretical plots (right), from Eqs. (1) and (2), show the coordinate $\phi$ at a local minimum of the potential. A bidirectional sweep of tilt was performed for each value of the barrier control. Top panels (a) show the mean response to the two sweep directions. Bottom panels (b) show the response to the positive-direction sweep subtracted from the response to the negative-direction sweep, revealing metastability. (c) Detailed features of the flux response for a few different values of the barrier control [indicated by arrows on top of the (a) panels] in the nonhysteretic regime. Results of data are shown by markers and of theory by solid lines. The nonlinear flux-to-phase transfer function of the magnetometer has been inverted.

Figure 3

Backaction-free escape dynamics. (a) Pulse sequence used for determining the escape rates. The sequence begins with a deterministic reset. The fine tilt pulse is superimposed with the tilt waveform with a large attenuation. (b) Escape rate as a function of the theoretical barrier height during the fine tilt pulse at different sample temperatures (markers). The experiment was performed at barrier setting ${\phi }_{x,dc}/2\pi =-0.3778$ using both positive and negative polarity tilt pulses (upward and downward triangles, respectively). Lines are fits to data (solid and dashed for $+$ and $-$ polarity, respectively). (c) Escape energy, extracted as the inverse negative slope of the escape rate data, as a function of temperature, separately for $+$ and $-$ polarity data (upward and downward triangles, respectively). The solid line is a zero-intercept fit to data at $T>250$ mK and the dashed horizontal line indicates the low-temperature saturation level.

Figure 4

Magnetometer readout with different probe powers and backaction on system dynamics at $T=500$ mK. (a) Flux modulation curves for $-79$ to $-72$ dBm incident power, in 1-dBm steps. Pickup from magnetometer bias to tilt and barrier fluxes has not been compensated. (b) Phase response for the three lowest powers. Data have been offset by ${20}^{\circ }$ per dBm for clarity. (c) and (d) Escape rates at ${\phi }_{x,dc}/2\pi =-0.3778$ under continuous readout for the three lowest powers. (e) and (f) Backaction-free pulsed readout under otherwise identical conditions. The operation point has been chosen so that the logical state $R$ is at the peak of the modulation curve. Consequently, the magnetometer is (close to) a zero-voltage state in the logical state $L$. The dash-dotted line is the same in all panels and serves as a guide for the eye.

Figure 5

Bit erasure at $T=500$ mK. (a) and (b) Time-domain waveforms applied to tilt and barrier controls. The reverse protocol is obtained by reversing both control channels in time. The reset protocol targeting state $L$ has the polarity of the tilt waveform inverted. (c) Two randomly chosen magnetometer traces. (d) Average occupation of the state $R$ throughout the protocol when the initial state is $R$ (blue), $L$ (red), or an equilibrium mixture of the idling potential (black). The average is calculated from ${10}^5$ experimental trajectories (solid lines) and theory (dashed lines). (e) Distribution of work $W$ from ${10}^5$ experimental trajectories (markers) and from Markovian theory (lines). Full distribution (black) and conditional distributions based on initial and final states (magenta, green, red, and blue for trajectories of $R\to L, L\to L, R\to R$, and $L\to R$ type, respectively) are shown. Here PDF denotes probability density function.

Figure 6

One-dimensional projection of the potential at key stages of the reset-to-$R$ (blue solid line) and reset-to-$L$ (red dashed line) protocols. The plotted quantity is the potential as a function of the $\phi$ coordinate evaluated along a curve in $(\phi ,{\phi }_{dc})$ space that passes through the left minima, the saddle point, and the right minima (in this order) and is parallel to the local principal axis of curvature at these points. A constant offset of 71.4 K has been subtracted for clarity.

Figure 7

Crooks relation ratio of the probability densities of per-trajectory work in the forward and reverse directions (see the text for definitions). The experimental ratio is for bit reset targeting the $R$ and $L$ states (blue and red markers, respectively) and the fluctuation theory prediction is $exp(W/k_{\mathrm{B}}T)$ with $T=500$ mK (solid line).