Resonant and off-resonant microwave signal manipulation in coupled superconducting resonators

We present an experimental demonstration as well as a theoretical model of an integrated circuit designed for the manipulation of a microwave field down to the single-photon level. The device is made of a superconducting resonator coupled to a transmission line via a second frequency-tunable resonator. The tunable resonator can be used as a tunable coupler between the fixed resonator and the transmission line. Moreover, the manipulation of the microwave field between the two resonators is possible. In particular, we demonstrate the swapping of the field from one resonator to the other by pulsing the frequency detuning between the two resonators. The behavior of the system, which determines how the device can be operated, is analyzed as a function of one key parameter of the system, the damping ratio of the coupled resonators. We show a good agreement between experiments and simulations, realized by solving a set of coupled differential equations.

Figure 1

(a) Model of the system under study. A storage resonator with frequency ${\omega }_a/\left(2\pi \right)$ is coupled to a transmission line via a coupling resonator with tunable frequency ${\omega }_b/\left(2\pi \right)$. (b) Optical microscope image of the corresponding superconducting integrated circuit (sample I).

Figure 2

Schematics of the cryogenic microwave measurement setup. The chip is located at the coldest stage of a dilution refrigerator equipped with coaxial lines. The input of the device is probed with a reflectometry setup: a microwave signal is routed to the device via a circulator and the reflected signal is amplified with low noise amplifiers, both at 4 K and room temperature. The output signal is down-converted and then numerically demodulated and sampled with a vector signal analyzer. The control port of the chip is driven with an arbitrary waveform generator (AWG). The input lines are equipped with attenuators and filters in order to prevent room temperature thermal noise from heating the device. The output line is equipped with a circulator acting as an isolator.

Figure 3

Spectroscopy measurements on sample I. (a) Magnitude and phase of the reflection coefficient for $\mathrm{\Phi }=0.3{\mathrm{\Phi }}_0$, i.e., when the two resonators are on resonance. Two resonance lines are observed, corresponding to the coupled modes of the two resonators. They are fitted separately with the model given by Eq. (16). (b) Magnitude of the reflection coefficient as a function of the frequency and the flux in the SQUID loop. The frequency of the two resonance lines evolve periodically with the flux. The dashed line indicates the cut shown in panel (a).

Figure 4

Effective coupling time of the storage resonator as a function of the detuning for both samples. The data points come from the fit with Eq. (16) of the reflection coefficient. The plain lines show the model given by Eq. (13) without adjustable parameters. The horizontal dashed lines indicate the intrinsic lifetime $1/{\kappa }_{ia}$ of the storage resonators. In both samples the storage resonator can be set to the over- or undercoupled regimes.

Figure 5

Direct release of microwaves from the storage resonator. (a) Output voltage measured for sample I at several detunings. As the detuning is reduced, the decay becomes faster. Beating is observed for low detunings. (b) Same experiment for sample II. (c) Magnitude and quadrature signals for sample I for a detuning of $41\mathrm{MHz}$ at $-0.266{\mathrm{\Phi }}_0$. Blue line: fit; black line and dots: measurement. The model contains two decaying and oscillating components (see text).

Figure 6

Capture and release through field swapping for sample I. (a) Principle of the experiment. A microwave pulse on resonance with the initially detuned (zero flux) coupling resonator is sent to the input port. The detuning is then quickly reduced to zero, leading to a transfer (swapping) of the energy to the storage resonator. After a delay, the swap is repeated, such that the energy is transferred to the coupling resonator from which the microwaves can leak out to the transmission line. (b) Measurement sequence showing the RF input and the detuning. Note that the detuning pulses are 10 times shorter than sketched, for clarity reasons. (c) Magnitude, phase and quadratures of the measured output voltage for an excitation at the input of the resonator of $-139$ dBm. Red lines: simulation. The phase of the measured trace, which has a random reference, has been compensated to match the phase obtained with the simulation. (d) Same experiment and simulation for an excitation of $-119$ dBm, driving the coupling resonator to a nonlinear regime. The signal power is a hundred times higher, hence the reduced noise. (e) Color map of the output voltage traces for different duration of the swapping pulses. The swap is effective only for certain pulse widths, where the signal is minimum after the first swap pulse, and maximum after the second pulse. For other values, the energy transfer is only partial.