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EPJ Quantum Technology

Quantum and classical nonlinear dynamics in a microwave cavity

Charles P. Meaney, Hyunchul Nha, T. Duty, Gerard J Milburn

Published Jun 10, 2014• 26 citations
Publisher

Abstract

We consider a quarter wave coplanar microwave cavity terminated to ground via asuperconducting quantum interference device. By modulating the flux through the loop,the cavity frequency is modulated. The flux is varied at twice the cavity frequencyimplementing a parametric driving of the cavity field. The cavity field also exhibitsa large effective nonlinear susceptibility modelled as an effective Kerrnonlinearity, and is also driven by a detuned linear drive. We show that thesemi-classical model corresponding to this system exhibits a fixed point bifurcationat a particular threshold of parametric pumping power. We show the quantum signatureof this bifurcation in the dissipative quantum system. We further linearise about thebelow threshold classical steady state and consider it to act as a bifurcationamplifier, calculating gain and noise spectra for the corresponding small signalregime. Furthermore, we use a phase space technique to analytically solve for theexact quantum steady state. We use this solution to calculate the exact small signalgain of the amplifier.

DOI
10.1140/epjqt7

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We consider a quarter wave coplanar microwave cavity terminated to ground via asuperconducting quantum interference device. By modulating the flux through the loop,the cavity frequency is modulated. The flux is varied at twice the cavity frequencyimplementing a parametric driving of the cavity field. The cavity field also exhibitsa large effective nonlinear susceptibility modelled as an effective Kerrnonlinearity, and is also driven by a detuned linear drive. We show that thesemi-classical model corresponding to this system exhibits a fixed point bifurcationat a particular threshold of parametric pumping power. We show the quantum signatureof this bifurcation in the dissipative quantum system. We further linearise about thebelow threshold classical steady state and consider it to act as a bifurcationamplifier, calculating gain and noise spectra for the corresponding small signalregime. Furthermore, we use a phase space technique to analytically solve for theexact quantum steady state. We use this solution to calculate the exact small signalgain of the amplifier.
We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semiclassical approximation of this system has an interesting fixed-point bifurcation structure. In the semiclassical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positive to negative detuning. We show that signatures of this bifurcation structure are also present in the full dissipative quantum system and further show that the bifurcation structure leads to mixed-state entanglement between the nanomechanical resonator and the microwave cavity in the dissipative quantum system that is a maximum close to the semiclassical bifurcation. Quantum signatures of the semiclassical limit cycles are presented.
We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the superconducting quantum interference device (SQUID) attached to the cavity and damping due to connection of the cavity to a transmission line are taken into consideration. We study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance, featuring regimes of multistability and parametric radiation. We investigate the phase-sensitive amplification of external signals on resonance, as well as amplification of detuned signals, and relate the amplifier performance to that of linear parametric amplifiers. We also discuss applications of the device for dispersive qubit readout. Beyond the classical response of the cavity, we investigate small quantum fluctuations around the amplified classical signals. We evaluate the noise power spectrum both for the internal field in the cavity and the output field. Other quantum-statistical properties of the noise are addressed such as squeezing spectra, second-order coherence, and two-mode entanglement.
Nonlinear Josephson circuits play a crucial role in the growing landscape of quantum information and technologies. The typical circuits studied in this field consist of qubits, whose anharmonicity is much larger than their linewidth, and also of parametric amplifiers, which are engineered with linewidths of tens of MHz or more. The regime of small anharmonicity but also narrow linewidth, corresponding to the dynamics of a high-Q Duffing oscillator, has not been extensively explored using Josephson cavities. Here, we use two-tone spectroscopy to study the susceptibility of a strongly driven high-Q Josephson microwave cavity. Under blue-detuned driving, we observe a shift of the cavity susceptibility, analogous to the AC Stark effect in atomic physics. When applying a strong red-detuned drive, we observe the appearance of an additional idler mode above the bifurcation threshold with net external gain. Strong driving of the circuit leads to the appearance of two exceptional points and a level attraction between the quasimodes of the driven cavity. Our results provide insights on the physics of driven nonlinear Josephson resonators and form a starting point for exploring topological physics in strongly-driven Kerr oscillators.
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.
Nonlinear Josephson circuits play a crucial role in the growing landscape of quantum information and technologies. The typical circuits studied in this field consist of qubits, whose anharmonicity is much larger than their linewidth, and also of parametric amplifiers, which are engineered with linewidths of tens of MHz or more. The regime of small anharmonicity but also narrow linewidth, corresponding to the dynamics of a high-QQQ Duffing oscillator, has not been extensively explored using Josephson cavities. Here, we use two-tone spectroscopy to study the susceptibility of a strongly driven high-QQQ Josephson microwave cavity. Under blue-detuned driving, we observe a shift of the cavity susceptibility, analogous to the AC Stark effect in atomic physics. When applying a strong red-detuned drive, we observe the appearance of an additional idler mode above the bifurcation threshold with net external gain. Strong driving of the circuit leads to the appearance of two exceptional points and a level attraction between the quasi-modes of the driven cavity. Our results provide insights on the physics of driven nonlinear Josephson resonators and form a starting point for exploring topological physics in strongly-driven Kerr oscillators.
We give a semiclassical analysis of the average photon number as well as photon number variance (Fano factor F)F)F) for a Josephson junction (JJ) embedded microwave cavity system, where the JJ is subject to a fluctuating (i.e., noisy) bias voltage with finite dc average. Through the ac Josephson effect, the dc voltage bias drives the effectively nonlinear microwave cavity mode into an amplitude squeezed state (F<1)(F<1)(F<1), as has been established previously [Armour et al., Phys. Rev. Lett. 111, 247001 (2013)], but bias noise acts to degrade this squeezing. We find that the sensitivity of the Fano factor to bias voltage noise depends qualitatively on which stable fixed point regime the system is in for the corresponding classical nonlinear steady-state dynamics. Furthermore, we show that the impact of voltage bias noise is most significant when the cavity is excited to states with large average photon number.