Injection locking replicates the frequency and phase of a driven oscillator in a target oscillator, with applications in electronic, mechanical, and optical systems, and lately in mesoscopic circuits. The technique requires the driven oscillator to be almost resonant with the target oscillator. This study uses a superconducting circuit to demonstrate injection locking in a well-controlled parametric microwave oscillator. The authors furthermore present another technique, $p\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c$ $l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g$, that removes the constraint on detuning between the oscillators. This should impact any device that should not or cannot be close to resonant driving, as in quantum information processing.
Injection locking replicates the frequency and phase of a driven oscillator in a target oscillator, with applications in electronic, mechanical, and optical systems, and lately in mesoscopic circuits. The technique requires the driven oscillator to be almost resonant with the target oscillator. This study uses a superconducting circuit to demonstrate injection locking in a well-controlled parametric microwave oscillator. The authors furthermore present another technique, p\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}cl\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}k\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g, that removes the constraint on detuning between the oscillators. This should impact any device that should not or cannot be close to resonant driving, as in quantum information processing.
Superconducting optomechanical circuits enable frequency mixing of optical and mechanical modes, facilitating the generation of microwave frequency combs. However, such optomechanical combs suffer from frequency fluctuations, requiring their stabilization for applications in precision sensing and signal processing. Here, we investigate the sideband injection locking of microwave frequency combs in a niobium-based superconducting optomechanical circuit. By strongly driving the device with a blue-detuned pump to induce parametric instability and introducing an additional tone near individual comb peaks, we study how the locking range varies with the power, the frequency position, and the sweep direction of the injection tone. The locking responses show interesting features such as injection hysteresis, which cannot be explained by existing models. Numerical simulations of the classical optomechanical equations implementing a cubic mechanical nonlinearity show that the nonlinearity contributes to broadening the locking range. We also characterize the Allan deviations and phase noise of the injection-locked combs for different injection frequencies, demonstrating enhanced stability performance. Our results lay the foundation for the utilization of optomechanical combs for applications in nanomechanical sensing and cryogenic microwave pulse generation.
A parametric oscillator is an oscillating system in which one of the parameters, typically either the resonance frequency or damping, can be modulated by an external pump. Parametric oscillations can be found in a wide variety of systems including radiofrequency circuits, optical and mechanical systems, and even single electrons in a Penning trap. In recent years, interest in parametric oscillators has revived in many areas of physics, ranging from basic physics to applications. For instance, they are being used as quantum-limited amplifiers in an increasingly large number of experiments in quantum information and computing. At the same time, interest in their basic physics in the quantum regime, in which they are a model system for driven, nonlinear systems, has grown commensurately. This chapter gives a largely self-contained introduction to the theoretical description of the dynamics of parametric oscillators, both classical and quantum, and reviews some of the recent experimental work in superconducting circuits.
Oscillators designed to function at cryogenic temperatures play a critical role in superconducting electronics and quantum computing by providing stable, low-noise signals with minimal energy loss. Here, we present a comprehensive numerical study of injection locking and mutual coupling dynamics in superconducting nanowire (ScNW)-based cryogenic oscillators. Using the design space of a standalone ScNW-based oscillator, we investigate two critical mechanisms that govern frequency synchronization and signal coordination in cryogenic computing architectures: (1) injection locking induced by an external AC signal with a frequency near the oscillator's natural frequency, and (2) the mutual coupling dynamics between two ScNW oscillators under varying coupling strengths. We identify key design parameters—such as shunt resistance, nanowire inductance, and coupling strength—that govern the locking range. Additionally, we examine how the amplitude of the injected signal affects the amplitude of the locked oscillation, offering valuable insights for power-aware oscillator synchronization. Furthermore, we analyze mutual synchronization between coupled ScNW oscillators using capacitive and resistive coupling elements. Our results reveal that the phase difference between oscillators can be controlled by tuning the coupling strength, enabling programmable phase-encoded information processing. These findings could enable building ScNW-based oscillatory neural networks, synchronized cryogenic logic blocks, and on-chip cryogenic resonator arrays.
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 propose a superconducting microwave relaxation oscillator based on a nanowire shunted by a resistor and an inductor controlled by quasiparticle injection from a tunnel junction positioned on it: the QUISTRON. This device exhibits relaxation oscillator behavior with dc voltage--controlled frequency tuning and dc bias. The device frequency is modulated via the tunnel junction, which induces localized heating by injecting quasiparticles. This heating mechanism modulates the nanowire switching current, enabling relaxation oscillations when it falls below the bias current. We simulate the device operating principles and its performance across various parameters, including different choices of shunt resistor, shunt inductance, and bath temperature ranging from 20 mK to 1 K. The device showed oscillation with a frequency range approximately between 1 and 10 GHz and total energy dissipation per cycle of approximately 100 zJ. Our results suggest that this design offers a promising platform for compact, tunable superconducting oscillators in the microwave spectrum with potential applications in quantum information processing, microwave technology, and ultralow-power electronics. The straightforward frequency control mechanism and integration potential make this device an attractive candidate for superconducting microwave local oscillators.
Injection locking can stabilize a source of radiation, leading to an efficient suppression of noise-induced spectral broadening and therefore, to a narrow spectrum. The technique is well established in laser physics, where a phenomenological description due to Adler is usually sufficient. Recently, locking experiments were performed in Josephson photonics devices, where microwave radiation is created by inelastic Cooper pair tunneling across a dc-biased Josephson junction connected in-series with a microwave resonator. An in-depth theory of locking for such devices, accounting for the Josephson nonlinearity and the specific engineered environments, is lacking. Here, we study injection locking in a typical Josephson photonics device where the environment consists of a single mode cavity, operated in the classical regime. We show that an in-series resistance, however small, is an important ingredient in describing self-sustained Josephson oscillations and enables the locking region. We derive a dynamical equation describing locking, similar to an Adler equation, from the specific circuit equations. The effect of noise on the locked Josephson phase is described in terms of phase slips in a modified washboard potential. For weak noise, the spectral broadening is reduced exponentially with the injection signal. When this signal is provided from a second Josephson device, the two devices synchronize. In the linearized limit, we recover the Kuramoto model of synchronized oscillators. The picture of classical phase slips established here suggests a natural extension towards a theory of locking in the quantum regime.