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Physical Review

Coupled Superconducting Quantum Oscillators

Arnold H. Silver, James Edward Zimmerman

Published Jun 10, 1967• 15 citations
Publisher

Abstract

Voltage biasing of a single superconducting weak link results in coherent electromagnetic oscillation at the frequency $\ensuremath{\nu}=\frac{V}{{\ensuremath{\Phi}}_{0}}$; simultaneous voltage biasing of two weak links in a superconducting ring produces two strongly coupled oscillators at the same frequency. The electromagnetic phase difference is the difference in the gauge-invariant quantum phases across the two contacts. Because of the quantization of the superconducting ring, this phase difference is $\frac{2\ensuremath{\pi}〈\ensuremath{\Phi}〉}{{\ensuremath{\Phi}}_{0}}$, where $〈\ensuremath{\Phi}〉$ is the time-averaged magnetic flux in the superconducting circuit and ${\ensuremath{\Phi}}_{0}$ is the flux quantum. We can characterize the coupled system in terms of normal modes, and we have experimentally observed these modes of oscillation at 30 MHz. This provides a means of understanding the earlier observations of quantum interference in the resistive state.

DOI
10.1103/physrev.158.423

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Voltage biasing of a single superconducting weak link results in coherent electromagnetic oscillation at the frequency \ensuremathν=V\ensuremathΦ0\ensuremath{\nu}=\frac{V}{{\ensuremath{\Phi}}_{0}}\ensuremathν=\ensuremathΦ0​V​; simultaneous voltage biasing of two weak links in a superconducting ring produces two strongly coupled oscillators at the same frequency. The electromagnetic phase difference is the difference in the gauge-invariant quantum phases across the two contacts. Because of the quantization of the superconducting ring, this phase difference is 2\ensuremathπ〈\ensuremathΦ〉\ensuremathΦ0\frac{2\ensuremath{\pi}〈\ensuremath{\Phi}〉}{{\ensuremath{\Phi}}_{0}}\ensuremathΦ0​2\ensuremathπ〈\ensuremathΦ〉​, where 〈\ensuremathΦ〉〈\ensuremath{\Phi}〉〈\ensuremathΦ〉 is the time-averaged magnetic flux in the superconducting circuit and \ensuremathΦ0{\ensuremath{\Phi}}_{0}\ensuremathΦ0​ is the flux quantum. We can characterize the coupled system in terms of normal modes, and we have experimentally observed these modes of oscillation at 30 MHz. This provides a means of understanding the earlier observations of quantum interference in the resistive state.
Abstract Superconducting circuits represent a versatile playground to explore novel quantum phenomena and a platform to realize quantum bits (qubits). Superconducting qubits hinge on the coherent tunneling of Cooper pairs across the insulating barrier of a Josephson junction. This qubit implementation is, however, plagued with the variability of the fabricated junction, along with the need for magnetic tunability of the qubit transition frequency. Alternatively, a dual to the Josephson effect—referred to as quantum phase slips—allows for a coherent coupling of flux states and has been used to build phase-slip qubits. In this paper, we propose coupling flux states of a superconducting ring using a bias voltage. By solving the time-dependent Ginzburg–Landau equations, we demonstrate that the bias voltage locally suppresses the density of Cooper pairs, thereby forming a weak link where phase slips occur. Moreover, we show that a voltage-biased superconducting ring serves as an electrically-controlled magnetic memory. The work presented in this paper is an important step towards realizing electrically-tunable superconducting devices, such as qubits and SQUIDs.
Abstract We demonstrate the frequency synchronization of multiple single-flux quantum (SFQ) oscillators with different oscillation frequencies. To synchronize these SFQ oscillators, a common constant bias current is supplied to the SFQ oscillators without any bias resistors. When an SFQ oscillator oscillates at a frequency of f, the average voltage across the Josephson junction comprising the SFQ oscillator is f Φ 0 , where Φ 0 is the flux quantum in the superconductor. The bias currents supplied to the SFQ oscillators are redistributed to eliminate the average voltage difference output from the SFQ oscillators. As a result, the oscillation frequencies of all the SFQ oscillators are synchronized. Simulation results indicate that SFQ oscillators with an oscillation frequency difference of more than 50 GHz can be synchronized. We experimentally demonstrate the frequency synchronization of two SFQ oscillators composed of circular Josephson transmission lines. Frequency synchronization is expected to contribute toward the development of a low-power stable clock source stabilizing SFQ circuit operation.
New experiments demonstrating macroscopic coherence and quantization in partially resistive superconducting rings with point contacts are described. These include: (1) the observation of quantized magnetic transitions for frequencies $\ensuremath{\omega}>{\ensuremath{\omega}}_{1}=\frac{R}{L}$, where RRR is the resistance and LLL the inductance of the ring, and (2) the measurement of coherent oscillations at frequencies limited by thermal noise fluctuations, $\frac{8{k}_{B}\mathrm{TR}}{{{\ensuremath{\Phi}}_{0}}^{2}}$, where kB{k}_{B}kB​ is Boltzmann's constant, TTT the absolute temperature, and ${\ensuremath{\Phi}}_{0}$ the flux quantum. These results are interpreted by introducing into the phenomenological model of weakly connected superconducting rings a continuous time dependence of the quantum states.
We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113 ) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the qubit. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux qubit. In particular, when the number of Josephson junctions is increased, several low-energy modes can have frequencies lower than the qubit frequency. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of oscillations between the two counterpropagating current states at short times, followed by oscillation-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime.
Weakly connected superconductors can function as very low power sensors by utilizing the coherence properties of superconductors. One configuration which has produced significant success incorporates a superconducting point contact and a small resistance element in a low-inductance superconducting circuit. When the critical supercurrent (ic) of the point contact is of the order of φ0/L, (φ0=h/2e, L=inductance of the ring), that circuit exhibits coherent quantum behavior. Used as a parametric amplifier, it has been used to measure ultralow voltages, limited by the thermal noise fluctuations in the resistance element R. The voltage sensitivity is approximated by 8kTR/φ0, where k is Boltzmann's constant and T the absolute temperature of the resistance R. We have measured 4×10−16 V across 1.7×10−10 Ω at 4.2°K with a signal-to-noise ratio approaching 10. The operation of this sensor follows from the Josephson oscillation of voltage-biased superconducting point contacts. A dc voltage V0 across the resistance of the circuit produces an oscillating current at the angular frequency ωJ=2πV0/φ0 and an oscillating voltage across the point contact at ωJ. Upon supplying an rf or microwave current at frequency ωD, the point contact mixes ωJ and ωD. Using a homodyne detector at ωD of bandwidth larger than ωJ, we can retrieve the oscillation at ωJ. Direct measurement of this frequency ωJ yields V0. Such superconducting circuits may be useful as cryogenic thermometers, as thermal radiation sensors, and in low-level spectroscopy.
Frequency locking effects in two interacting Josephson junctions are investigated. The junctions are described in the usual resistively-shunted-junction (RSJ) model and coupling between junctions is provided by a resonant circuit. The system can be used to simulate interactions between superconducting weak links in a resonant cavity or via a general impedance shunt. Various types of v-i curves in the voltage locking region are obtained in this calculation by using a first harmonic approximation in the high-frequency regime. Stability of order parameter phase oscillations for two identical junctions in the voltage-locked region is examined. We also show that dissimilar junctions can be synchronized in a resonant cavity using a common series bias current.