Transient Dynamics of a Superconducting Nonlinear Oscillator

We investigate the transient dynamics of a lumped-element oscillator based on a dc superconducting quantum interference device (SQUID). The SQUID is shunted with a capacitor, forming a nonlinear oscillator with a resonance frequency in the range of several gigahertz. The resonance frequency is varied by tuning the Josephson inductance of the SQUID with on-chip flux lines. We report measurements of decaying oscillations in the time domain following a brief excitation with a microwave pulse. The nonlinearity of the SQUID oscillator is probed by observing the ringdown response for different excitation amplitudes while the SQUID potential is varied by adjusting the flux bias. Simulations are performed on a model circuit by numerically solving the corresponding Langevin equations incorporating the SQUID potential at the experimental temperature and using parameters obtained from separate measurements characterizing the SQUID oscillator. Simulations are in good agreement with the experimental observations of the ringdowns as a function of applied magnetic flux and pulse amplitude. We observe a crossover between the occurrence of ringdowns close to resonance and adiabatic following at a larger detuning from the resonance. We also discuss the occurrence of phase jumps at a large amplitude drive. Finally, we briefly outline prospects for a readout scheme for superconducting flux qubits based on the discrimination between ringdown signals for different levels of magnetic flux coupled to the SQUID.

Figure 1

(a) Circuit schematic showing the input and output coupling capacitors and the SQUID oscillator. (b) Optical image of the fabricated circuit with an enlarged view of the SQUID with on-chip flux lines.

Figure 2

(a) Schematic of the general measurement setup. The network analyzer is used to measure the frequency response, while the pulse-generation setup [expanded in (b)] is used to measure the ringdowns in the time domain. The two types of measurement are separate, and, as indicated by dashed lines in (a), the network analyzer is never connected simultaneously with the pulse-generation setup. The SQUID oscillator device shown as a blue box is displayed in Fig. 1.

Figure 3

Density plot of $|S_{21}|$ vs flux and frequency of the SQUID oscillator as measured from a network analyzer at $-125\text{dBm}$ power at the input of the SQUID oscillator chip at 300 mK. The dashed line and magenta symbols are from fits to the SQUID modulation as described in the text. The marker in black indicates the bias point where the pulsed measurements are taken.

Figure 4

(a) The input pulse as measured at the top of the cryostat. The red dots represent the measured data, while the blue curve is the fit used as the input voltage in the simulations. (b) Example measured output voltage ringdown from the SQUID oscillator. Once again, the red dots are obtained directly from experimental data, while the gray curves in the background are a subset of the realizations obtained from the simulations of the circuit. The average of these realizations is shown in black.

Figure 5

(a) Full circuit model of the SQUID oscillator of Fig. 1 and 1 reduced circuit, valid in the limit of small amplitudes when the Josephson junction responds linearly. In this regime, the SQUID is treated as an effective, flux-dependent inductance $L_t$. In both (a) and (b), the blue numbers represent node labels.

Figure 6

A comparison of the amplitude dependence of ringdowns at a noninteger flux bias of $f_s=0.30$, for the input signal of Fig. 4 at 2.4 GHz. The plot in (a) shows experimental data, while in (b) the corresponding simulations. The attenuation on the drive pulse in decibels is shown on the $y$ axis, with decreasing numbers implying an increasing amplitude of the input pulse. The ringdown time is shown on the $x$ axis, while the ringdown amplitude is represented by the color scale. The purple dashed line in (a) indicates the amplitude corresponding to the ringdown trace shown in Fig. 4.

Figure 7

Low and high drive response. Each row shows plots of ${\phi }_2(t)$, ${\varphi }_0{\dot{\phi }}_2(t)$, and ${\varphi }_0{\dot{\phi }}_3(t)$ (the output voltage), respectively. In plots (a)–(c) the amplitude of the input pulse is low with respect to the critical current of the Josephson junctions, whereas in (d)–(f) it is high. The gray curves show a small subset of individual realizations used to calculate the averages (blue curves). In the case of the third column, we also show the experimental data for the same parameters (red dots) and how it compares to the simulation results. From the plots, one clearly sees how, when the drive is low, all the realizations stay within the same potential well [see plot (a)], and their relative phase shift varies only slightly. In the case of a high drive, we observe that different realizations tend to end up in different wells at different times [as shown in plot (d)], which introduces a relative phase shift between them. This in turn leads to faster decay of the average voltage, which is reflected in plots (e) and (f).

Figure 8

Fixed-amplitude flux scans with the applied flux bias $f_s$ between 0 and 1 and with the input signal of Fig. 4 at 2.4 GHz. The top row (a)–(c) shows experimental data, while the bottom row (d)–(f) uses data obtained by running stochastic simulations. The amplitude of the input pulse increases from left to right, with the leftmost column showing results for 20-dB attenuation pulses, the middle column for 15-dB attenuation pulses, and finally the rightmost column for 10-dB attenuation pulses. As the amplitude increases, one clearly observes the effects of the nonlinearity of the system. See the main text for a more detailed discussion.

Figure 9

Root-mean-square output voltage $V_{\mathrm{rms}}$ as a function of the applied flux bias ${\mathrm{\Phi }}_s$, calculated over a time range between 2.1 and 3.4 ns. The plot uses data obtained with the 20-dB attenuation input pulses and is the same as in the left column of the flux scans from Fig. 8. The dots represent results calculated from the experimental data (top row, leftmost column), while the solid line is produced using the simulations (bottom row, leftmost column). By biasing the flux through the SQUID near a point where the slope is high, for example, at ${\mathrm{\Phi }}_s\sim 0.36{\mathrm{\Phi }}_0$, one can have a means of distinguishing between different flux signals—see the main text for more details.