Rapid Flipping of Parametric Phase States

We experimentally demonstrate flipping the phase state of a parametron within a single period of its oscillation. A parametron is a binary logic element based on a driven nonlinear resonator. It features two stable phase states that define an artificial spin. The most basic operation performed on a parametron is a bit flip between these two states. Thus far, this operation involved changing the energetic population of the resonator and therefore required a number of oscillations on the order of the quality factor $Q$. Our technique takes a radically different approach and relies on rapid control of the underlying potential. Our work represents a paradigm shift for phase-encoded logic operations by boosting the speed of a parametron bit flip to its ultimate limit.

Figure 1

Parametron phase states and basic idea of rapid phase flipping. (a) Parametric driving corresponds to a harmonic modulation of the resonator’s natural frequency $f_0$. Solid (dashed) lines represent the modulated (original) potential. If the drive is sufficiently strong, the resonator locks to $f_d$ and settles into one of two stable phase states that are separated by $\pi$, illustrated as oscillating red and blue spheres. (b) In phase space, the parametrically driven resonator experiences an effective double-well potential, which is the key signature of the parametron. The phase states now appear as stationary red and blue spheres in the quasipotential minima. (c) Simplified illustration of the parametron in phase space. In the lab frame, the two states rotate around the origin at frequency $f_d$. (d) Illustration of rapid phase flip. The parametron is initialized in the red phase state ($\varphi =\pi$). At time $t=0$, the phase evolution of the system is paused for half an oscillation period by freezing the resonator’s position. Upon release, the parametron resumes oscillation in the blue phase state ($\varphi =0$).

Figure 2

Experimental demonstration of phase flip via potential deformation. (a) Experimental setup. A silica nanoparticle (diameter 136 nm) is trapped in a focused laser beam (wavelength 1064 nm) inside a vacuum chamber (not shown). The stiffness of the optical potential can be modulated with an electro-optic modulator (EOM). The particle displacement is detected with a quadrant photo diode (QPD). (b) Thermally driven power spectral density $S_x$ of the particle displacement. From the red line fit, we extract a quality factor $Q=1970$. (c) Schematic illustration of the phase-flip protocol. The parametron is initialized in the red phase state. When the particle reaches its maximum displacement, we reduce the resonator frequency from $f_0$ (potential sketched as solid line) to $f_0/2$ (dashed line) by attenuating the laser intensity with an acousto-optic modulator (AOM). We then let the particle evolve for the pulse length ${\tau }_{\mathrm{def}}=1/f_0$, such that the phase states of the parametron undergo a full oscillation, while the particle only traverses the trap and acquires a phase delay of $\pi$. (d) Same as (c) but illustrated in phase space. (e) Measured phase of the parametron as a function of time. A switch of the potential as outlined in (c) and (d) is applied at a rate of 2 Hz with ${\tau }_{\mathrm{def}}=8.1\mu \mathrm{s}$, periodically flipping the parametron phase state. Note the failed flip around 8 s. (f) Flipping probability $P_{\mathrm{flip}}$ for varying pulse length ${\tau }_{\mathrm{def}}$. Our model (black line) takes into account the finite thermal population of the resonator (see the Supplemental Material [32]). Error bars represent statistical uncertainty.

Figure 3

Experimental demonstration of phase flips via potential displacement. (a) Schematic of the electrical LC resonator circuit with a varicap diode to provide a nonlinear capacitance $C$. (b) Linear response of the resonator to a small external driving voltage ($U_{\text{drive}}=50\mathrm{mV}$). From the red line fit we extract $f_0=3.3\mathrm{MHz}$ and $Q=245$. (c) Illustration of the phase-flip protocol. The parametron is initialized in the red phase state. When the resonator reaches its maximum displacement at $t=0$, the potential is displaced by an external force (from dashed to solid lines) such that the resonator is momentarily at rest. At $t=1/(2f_d)$, the force is turned off and the parametron resumes its evolution, now in the blue phase state. (d) Same as (c) but illustrated in phase space. (e) Demonstration of two different phase flips, performed with (i) ${\tau }_{\mathrm{dis}}=153\mathrm{ns}$, the ideal pulse duration for flipping, and with (ii) ${\tau }_{\mathrm{dis}}=230\mathrm{ns}$. The signal was demodulated by a digital lock-in amplifier and filtered for clarity (see the Supplemental Material [32]). (f) Results of flipping experiments with varying ${\tau }_{\mathrm{dis}}$. Each datapoint represents the state of the resonator directly after a pulse. Here, $u=X\cos (2\pi f_{d}t)-Y\sin (2\pi f_{d}t)$ and $v=Y\cos (2\pi f_{d}t)+X\sin (2\pi f_{d}t)$ are the phase-space quadratures in a frame rotating at the drive frequency $f_d$. A black circle serves as a guide to the eye.