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arXiv:1801.04566 · Physical review. B./Physical review. B

Nondegenerate parametric oscillations in a tunable superconducting resonator

Andreas Bengtsson, Philip Krantz, Michaël Simoen, Ida-Maria Svensson, Ben H. Schneider, Vitaly S. Shumeiko, Per Delsing, Jonas Bylander

Published Apr 4, 2018• 18 citations
arXivHTMLPDF

Abstract

We investigate nondegenerate parametric oscillations in a superconducting microwave multimode resonator that is terminated by a superconducting quantum interference device (SQUID). The parametric effect is achieved by modulating magnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the oscillation spectral linewidths. These linewidths are significantly reduced by applying a weak on-resonant tone, which also suppresses the phase fluctuations. When the weak tone is detuned, we observe synchronization of the oscillation frequency with the frequency of the input. For the detuned input, we also observe an emergence of three idlers in the output. This observation is in agreement with theory indicating four-mode amplification and squeezing of a coherent input.

DOI
10.1103/physrevb.97.144502

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We investigate nondegenerate parametric oscillations in a superconducting microwave multimode resonator that is terminated by a superconducting quantum interference device (SQUID). The parametric effect is achieved by modulating magnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the oscillation spectral linewidths. These linewidths are significantly reduced by applying a weak on-resonant tone, which also suppresses the phase fluctuations. When the weak tone is detuned, we observe synchronization of the oscillation frequency with the frequency of the input. For the detuned input, we also observe an emergence of three idlers in the output. This observation is in agreement with theory indicating four-mode amplification and squeezing of a coherent input.
We develop a theory for nondegenerate parametric resonance in a tunable superconducting cavity. We focus on nonlinear effects that are caused by nonlinear Josephson elements connected to the cavity. We analyze parametric amplification in a strong nonlinear regime at the parametric-instability threshold, and we calculate maximum gain values. Above the threshold, in the parametric-oscillator regime, the cavity linear response diverges at the oscillator frequency at all pump strengths. We show that this divergence is related to the continuous degeneracy of the free oscillator state with respect to the phase. Applying on-resonance input lifts the degeneracy and removes the divergence. We also investigate quantum noise squeezing. It is shown that in the strong amplification regime, the noise undergoes four-mode squeezing, and that, in this regime, the output signal-to-noise ratio can significantly exceed the input value. We also analyze the intermode frequency conversion and identify the parameters at which full conversion is achieved.
We experimentally demonstrate a coupled superconducting resonator system in which a superconducting quantum interference device (SQUID) modulated nonlinear resonator is capacitively coupled to a fixed-frequency linear resonator. This configuration enables flux-tunable mode separation and parametric interactions between spatially separated modes. By exploiting the SQUID-induced nonlinearity, we achieve both single-mode (degenerate) and two-mode (nondegenerate) parametric amplification via three-wave mixing processes. Gains exceeding 15 dB are obtained in both the tunable resonator and the fixed frequency linear resonator with near quantum limit noise performance. Our device may provide an attractive solution for storing and releasing squeezed photons.
In this licentiate thesis, I present the design, fabrication, and characterization of superconducting parametric resonators, for use in quantum information processing. These devices are quarter-wavelength coplanar waveguide resonators (~5 GHz), terminated at one end by a non-linear inductance provided by a SQUID (superconducting quantum interference device). The SQUID acts as a flux-controlled boundary condition, which effectively changes the electrical length of the resonator. This enables the modulation of the resonant frequency by coupling microwave magnetic flux into the SQUID, using on-chip transmission lines. The modulation occurs on a timescale much faster than the photon loss out of the resonator. The non-linearity provides several different regimes for operating this device. I focused on the parametric regime, in which the non-linear element was being modulated (pumped) at or near twice the resonant frequency. This pumping can lead to amplification and frequency conversion of an incoming signal. It can also give rise to instabilities and self-oscillations "parametric oscillations" - that is, the generation of an intense electric field in the resonator (the creation of photons). Parametric oscillations set in when the pumping strength exceeds a threshold value, and also depends on the pump-frequency detuning from twice the resonance and on the static flux through the SQUID. I characterized the devices over a wide parameter range by doing homodyne detection of the reflected signal or of the emitted field oscillations. In particular, I investigated the damping and two leading nonlinearities, dominating in different operating regimes, which influence the dynamics of the parametric resonator. First, I extracted the Duffing nonlinearity by studying the line width of the resonator as a function of the input signal (without pumping the SQUID). Second, with high pump amplitude, I extracted the pump-induced nonlinearity by determining the threshold parameters (pump amplitude and detuning) for the onset of parametric oscillations; this nonlinearity leads to the generation of higher-order pump terms that complicate the system dynamics, but can be mitigated now that they have been understood. My results validate a recent theoretical model for the classical, non-linear dynamics of parametric resonance, and helps determine workable design parameters. Finally, I helped develop a linearized model −-− the "pumpistor" - describing this device in terms of a flux-controlled impedance. This will be useful when designing more complex circuits. One goal of this work is to demonstrate single-shot measurements on superconducting qubits (quantum bits of information). We will use the parametric resonator as a threshold discriminator, associating the qubit's two energy eigenstates with the parametric oscillations either turning on or not. This would be a very useful device in the quantum engineer's toolbox when designing systems for quantum information processing and communication.
We study self-sustained oscillations in a Nb superconducting stripline resonator integrated with a dc superconducting quantum interference device (SQUID). We find that both the power threshold where these oscillations start and the oscillation frequency are periodic in the applied magnetic flux threading the SQUID loop. A theoretical model which attributes the self-sustained oscillations to a thermal instability in the dc-SQUID yields a good agreement with the experimental results. This flux dependant nonlinearity may be used for quantum state reading of a qubit-superconducting resonator integrated device.
We have created a doubly tunable resonator, with the intention to simulate relativistic motion of the resonator boundaries in real space. Our device is a superconducting coplanar-waveguide microwave resonator, with fundamental resonant frequency ω 1 / (2 π ) ~ 5 GHz. Both of its ends are terminated to ground via dc-SQUIDs, which serve as magnetic-flux-controlled inductances. Applying a flux to either SQUID allows the tuning of ω 1 / (2 π ) by approximately 700 MHz. Using two separate on-chip magnetic-flux lines, we modulate the SQUIDs with two tones of equal frequency, close to 2 ω 1 . We observe photon generation, at ω 1 , above a certain pump amplitude threshold. By varying the relative phase of the two pumps we are able to control this threshold, in good agreement with a theoretical model. At the same time, some of our observations deviate from the theoretical predictions, which we attribute to parasitic couplings resulting in current driving of the SQUIDs.
We have observed period-tripling subharmonic oscillations in a driven superconducting coplanar waveguide resonator operated in the quantum regime, ${k}_{B}T\ensuremath{\ll}\ensuremath{\hbar}\ensuremath{\omega}$. The resonator is terminated by a tunable inductance that provides a Kerr-type nonlinearity. We detected the output field quadratures at frequencies near the fundamental mode, $\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\sim}5\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$, when driving the resonator with a current at $3\ensuremath{\omega}$, with amplitude exceeding an instability threshold. We observed three stable radiative states with equal amplitudes, phase shifted by $2\ensuremath{\pi}/3$ rad, red detuned from the fundamental mode. The down-conversion from $3\ensuremath{\omega}$ to $\ensuremath{\omega}$ is strongly enhanced by near-resonant excitation of the second mode of the resonator and the cross-Kerr effect. Our experimental results are in quantitative agreement with a model for the driven dynamics of two coupled modes.