Phase-Locking Transition in a Chirped Superconducting Josephson Resonator

We observe a sharp threshold for dynamic phase locking in a high-$Q$ transmission line resonator embedded with a Josephson tunnel junction, and driven with a purely ac, chirped microwave signal. When the drive amplitude is below a critical value, which depends on the chirp rate and is sensitive to the junction critical current $I_0$, the resonator is only excited near its linear resonance frequency. For a larger amplitude, the resonator phase locks to the chirped drive and its amplitude grows until a deterministic maximum is reached. Near threshold, the oscillator evolves smoothly in one of two diverging trajectories, providing a way to discriminate small changes in $I_0$ with a nonswitching detector, with potential applications in quantum state measurement.

Figure 1

(a) Schematic of the experimental setup. Input microwave lines are attenuated and filtered at the indicated temperature stages of the cryostat. Isolators provide a total of 80 dB isolation in the relevant frequency band. (b) Series $RLC$ model of the resonator. (c) Scanning electron micrograph of the junction.

Figure 2

(a) Transmitted power $P_{\mathrm{out}}$ in steady state, measured with a network analyzer. Input powers are $-155.5$ to $-123\mathrm{dBm}$ at 2.5 dB step. (b) Simulated in-phase and quadrature components of current oscillations for detuning $(\omega -{\omega }_0)/2\pi =0.5\mathrm{MHz}$ and drive power ramped in time from $-126$ to $-120\mathrm{dBm}$. $A$ and $B$ label the low- and high-amplitude attractors, and $\Theta$ is the phase mismatch. (c) Response of the resonator to a chirped drive, $\alpha /2\pi =5\times {10}^{11}\mathrm{Hz}/\mathrm{s}$, at $P_{\mathrm{in}}=-126\mathrm{dBm}$ (blue, dark) and $-120\mathrm{dBm}$ (red, light). (d) Simulated quadratures of the current in a chirped resonator with the same parameters as the data in (c). The simulation was terminated at $\omega /2\pi =1.6106\mathrm{GHz}$.

Figure 3

Normalized amplitude of current oscillations in the resonator as a function of drive power and frequency, with the frequency swept downwards at a rate of $5\times {10}^{11}\mathrm{Hz}/\mathrm{s}$. The arrow indicates the critical power $P_c$ for phase locking.

Figure 4

Comparison of the experimental (□) critical drive voltage versus chirp rate to Eq. (3), with (solid line) and without (dashed line) dissipation. Also shown are values for the critical drive from simulations of the fully nonlinear equation of motion for the equivalent $RLC$ circuit, for a lossless resonator (×) and for $Q=27500$ (+). Inset: Probability of phase locking versus drive amplitudes near threshold at $\alpha /2\pi =2\times {10}^{12}\mathrm{Hz}/\mathrm{s}$, evaluated over a 0.5 MHz frequency band centered at 1.6095 GHz. Solid line is a sigmoidal fit.