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IEEE Transactions on Applied Superconductivity

Novel Coupling Scheme for the Dynamics of Non-Uniform Coupled SQUID

Patrick Longhini, Anna Leese de Escobar, F. Escobar, Visarath In, Adi R. Bulsara

Published May 27, 2009• 5 citations
Publisher

Abstract

Recently, there is a growing interest in non-uniform coupled SQUID devices in the science community. Similar to how antenna array technology has moved from uniform arrays to non-uniform, sparsely populated arrays, the traditional uniform SQUID arrays evolves to an unconventional structure making these mesoscopic quantum devices (for RF reception) a feasible concept. The focus of this work covers theoretical and numerical computations which exploits well-designed coupling schemes to modeling the dynamics of the device where the behavior of the non-periodic voltage response related to the external flux exhibits an anti-peak which is present only around the zero applied magnetic field. It is know that many nonlinear dynamical systems are highly sensitive to small external perturbations, especially when they are near the onset of a bifurcation. Exploring parameter variations, such as; inductive coupling between neighboring elements, unconventional grating and fabrication parameters may lead to a significant performance enhancement. There are theoretical indications that variations in coupling could provide very high dynamic range. Application of these devices with the ability to detect and amplify will result in a new generation of electrically small antennas.

DOI
10.1109/tasc.2009.2017852

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Recently, there is a growing interest in non-uniform coupled SQUID devices in the science community. Similar to how antenna array technology has moved from uniform arrays to non-uniform, sparsely populated arrays, the traditional uniform SQUID arrays evolves to an unconventional structure making these mesoscopic quantum devices (for RF reception) a feasible concept. The focus of this work covers theoretical and numerical computations which exploits well-designed coupling schemes to modeling the dynamics of the device where the behavior of the non-periodic voltage response related to the external flux exhibits an anti-peak which is present only around the zero applied magnetic field. It is know that many nonlinear dynamical systems are highly sensitive to small external perturbations, especially when they are near the onset of a bifurcation. Exploring parameter variations, such as; inductive coupling between neighboring elements, unconventional grating and fabrication parameters may lead to a significant performance enhancement. There are theoretical indications that variations in coupling could provide very high dynamic range. Application of these devices with the ability to detect and amplify will result in a new generation of electrically small antennas.
In this work we numerical model series and parallel non‐uniform superconducting quantum interference device (SQUID) array. Previous work has shown that series SQUID array constructed with a random distribution of loop sizes, (i.e. different areas for each SQUID loop) there exists a unique ‘anti‐peak’ at the zero magnetic field for the voltage versus applied magnetic field (V‐B). Similar results extend to a parallel SQUID array where the difference lies in the arrangement of the Josephson junctions. Other system parameter such as bias current, the number of loops, and mutual inductances are varied to demonstrate the change in dynamic range and linearity of the V‐B response. Application of the SQUID array as a low noise amplifier (LNA) would increase link margins and affect the entire communication system. For unmanned aerial vehicles (UAVs), size, weight and power are limited, the SQUID array would allow use of practical ‘electrically small’ antennas that provide acceptable gain.
The dynamics of a small Josephson junction, driven either by a current source or by an external flux when the junction is the weak link element of a SQUID ring, were studied. Approximate analytical arguments and numerical simulations were used for the lumped circuit model of the junction. The various characteristic times of these devices were systematically obtained as functions of the parameters G and K of the junction and of the SQUID, respectively. The forms of the flux transitions and the times governing them are obtained for linear and periodic time variations of the external flux on the SQUID. A coupling of the SQUID to an LC circuit was treated. Limitations on the device parameters consistent with fast response, stability, and sensitivity were obtained both for the current driven junction and for the rf SQUID.
Abstract We consider, for the first time, the effects of strong capacitive and inductive coupling between radio frequency superconducting quantum interference devices (rf SQUIDs) in an overlapping metamaterial geometry when driven by rf flux at and near their self-resonant frequencies. The equations of motion for the gauge-invariant phases on the Josephson junctions in each SQUID are set up and solved. Our model accounts for the high-frequency displacement currents through capacitive overlap between the wiring of SQUID loops. We begin by modeling two overlapping SQUIDs and studying the response in both the linear and nonlinear high-frequency driving limits. By exploring a sequence of more and more complicated arrays, the formalism is eventually extended to the N × N × 2 overlapping metamaterial array, where we develop an understanding of the many ( 8 N 2 − 8 N + 3 ) resulting resonant modes in terms of three classes of resonances. The capacitive coupling gives rise to qualitatively new self-resonant responses of rf SQUID metamaterials, and is demonstrated through analytical theory, numerical modeling, and experiment in the 10–30 GHz range on capacitively and inductively coupled rf SQUID metamaterials.
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.
Superconducting quantum interference devices (SQUIDs) are used across physics and engineering, from amplification to detecting tiny magnetic fields to quantum computing. In many applications the SQUID's steady-state response is of interest, but here the authors measure and model the $t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t$ response to short bursts of radiation. The resulting ringdown oscillations in voltage decay over time, with a rich structure that may be particularly useful in discriminating weak magnetic fluxes---in qubits, for example.