Gate-controlled phase switching in a parametron

The parametron, a resonator-based logic device, is a promising physical platform for emerging computational paradigms. When the parametron is subject to both parametric pumping and external driving, complex phenomena arise that can be harvested for applications. In this paper, we experimentally demonstrate deterministic phase switching of a parametron by applying frequency tuning pulses. To our surprise, we find different regimes of phase switching due to the interplay between a parametric pump and an external drive. We provide full modeling of our device with numerical simulations and find excellent agreement between model and measurements. Our result opens up new possibilities for fast and robust logic operations within large-scale parametron architectures.

Figure 1

(a) Phase states of a parametron in the rotating frame (open dots). In-phase amplitude is denoted by $X$, out-of-phase amplitude by $Y$. A resonant force with relative phase $\theta$ can break the symmetry of the parametron and shift the states (solid dots). (b) In our experiment, an electrical resonator is driven and measured inductively. The varactor diode with capacitance $C_1$ is biased with $U_{\mathrm{tune}}$ and $U_{\mathrm{gate}}$ for dc and ac tuning of $f_0$, respectively. (c) Response to external driving with $U_d=50\mathrm{mV}$ ($U_p=0$) and (d) parametric response to $U_p=5\mathrm{V}$ ($U_d=0$) with $U_{\mathrm{tune}}=2.2\mathrm{V}$. Bright dots represent data, fits are shown as lines. The dashed line in panel (d) indicates an unstable solution branch. From the fits, we obtain $f_0=3.37\mathrm{MHz}$, $Q=243$, $F_d/U_d=2.65\times {10}^{10}{\mathrm{s}}^{-2}$, $\alpha =3.2\times {10}^{17}{\mathrm{V}}^{-2}{\mathrm{s}}^{-2}$, and $\eta =5.8\times {10}^8{\mathrm{V}}^{-2}{\mathrm{s}}^{-1}$.

Figure 2

(a) Amplitude and (b) phase response during sweeps of $f$ with parametric pumping (at $f_p=2f$) and external driving (at $f_d=f$) applied simultaneously. A characteristic double hysteresis is observed, with two jumps when sweeping from low to high frequency and one jump in the opposite direction. In all graphs, bright squares are measurements and dark lines are simulations. Black dots in panel (b) indicate the position of the phase states in frequency. $U_{\mathrm{tune}}=2.2\mathrm{V}$, $U_{\mathrm{gate}}=0$, $U_p=5\mathrm{V}$, $U_d=50\mathrm{mV}$, $\theta =\pi /4$. (c) Response to a periodic voltage $U_{\mathrm{gate}}$ that shifts $f_0$ relative to the fixed $f_{p,d}$. Black dots indicate the approximate positions of the two phase states within one $U_{\mathrm{gate}}$ period. $f_p=2f_d=6.74\mathrm{MHz}$, $U_{\mathrm{mod}}=0.2\mathrm{V}$, and $T_{\mathrm{mod}}=10\mathrm{ms}$.

Figure 3

Periodic response of the parametron to $U_{\mathrm{gate}}$ modulations (a) in the rotating frame of the lock-in amplifer (i.e., the in-phase and out-of-phase quadratures $U_X$ and $U_Y$ at $f_p/2$) and (b) in terms of amplitude and phase as a function of time. Bright squares are measurements, solid lines are simulations. The starting phase of the gate voltage modulation is a free parameter in the simulation. $U_{\mathrm{tune}}=2.2\mathrm{V}$, $U_p=5\mathrm{V}$, $U_d=50\mathrm{mV}$, $\theta =\pi /4$, $T_{\mathrm{mod}}=67\mathrm{ms}$, and the pulse has an amplitude of $U_{\mathrm{mod}}=0.45\mathrm{V}$. In panels (c) and (d) we show the same for $T_{\mathrm{mod}}=2\mathrm{ms}$.

Figure 4

Hysteresis in the phase response to $U_{\mathrm{gate}}$ modulations. Bright squares are measurements, solid lines are simulations, and curves are offset for visibility. (a) Response for $T_{\mathrm{mod}}=67\mathrm{ms}$, $12.5\mathrm{ms}$, $6.7\mathrm{ms}$, $2\mathrm{ms}$, and $0.33\mathrm{ms}$ from bottom to top, with $U_{\mathrm{tune}}=2.2\mathrm{V}$, $U_p=5\mathrm{V}$, $U_d=50\mathrm{mV}$, $U_{\mathrm{mod}}=0.45\mathrm{V}$, and $\theta =\pi /4$. (b) The same study for $U_p=0$. Note that the color coding for $T_{\mathrm{mod}}$ is consistent with Fig. 3.

Figure 5

(a) $C_1$ as a function of $U_{\mathrm{tune}}$ calculated from measurements of $f_0$ with Eq. (B1). (b) Frequency-dependent gain $G$ of the voltage buffer used for $U_{\mathrm{gate}}$. In the main text, we compensated for this frequency dependence.

Figure 6

(a) Peak amplitude $U_{\mathrm{amp}}$ of the Lorentzian response of the resonator measured in frequency sweeps with small driving voltage amplitudes $U_d$. (b) Width of the Arnold tongue $\mathrm{\Delta }f$ for varying parametric driving voltage amplitude $U_p$.

Figure 7

(a) Maximum amplitude $U_{\mathrm{jump}}$ measured in parametric sweeps for different $U_{\mathrm{tune}}$. (b) Extracted maximum voltage applied to the varactor diode before the jump.

Figure 8

(a) Schematic representation of a logic unit. $P_1$ and $P_2$ are two gate parametron devices that are acting on a third gate parametron, $P_3$. The two driving voltages at frequencies $f_d$ and $2f_d=f_p$ are labeled $U_{\omega }$ and $U_{2\omega }$, respectively. Gate operations are achieved by applying voltage pulses via $U_{\mathrm{gate}}$. The output of the logic operation is encoded in the phase of $U_{\mathrm{out}}$. (b) Gate operations can be performed by applying a negative tuning voltage to $U_{\mathrm{gate}}$, such that the device is effectively driven below its resonance frequency for a short time. The phase that is imprinted onto $P_3$ in this case is the (majority) phase of the combined drives $U_{\omega }$, $P_1$, and $P_2$. Note that $f_d$ is always constant. (c) For positive $U_{\mathrm{gate}}$, the phase imprint is inverted and the phase that is imprinted onto $P_3$ is opposite to that of the combined drives. (d) Predicted outcome of logic operations. Here we have defined “0” as the phase of $U_{\omega }$ and “1” as the phase shifted by $\pi$.