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Journal of Physics Condensed Matter

Flux and voltage periodic behaviour in a superconducting weak-link ring

Jason F. Ralph, Terrence D. Clark, J. Diggins, Robert J. Prance, Helen Prance, A. Widom

Published Sep 29, 1997• 7 citations
Publisher

Abstract

Starting with a simple Lagrangian for the electromagnetic field with broken gauge symmetry, we derive an effective circuit Hamiltonian for a superconducting weak-link ring. The energy eigenstates of this Hamiltonian exhibit sensitivity to both external magnetic flux and an applied Faraday law voltage. We show that the flux periodicity of the rf SQUID ring, and the voltage periodic behaviour found in ultra-small-capacitance weak-link rings, can be seen as limiting cases of a more general phenomenon.

DOI
10.1088/0953-8984/9/39/012

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Review of Scientific Instruments•Sep 1, 2010•2 citations
Starting with a simple Lagrangian for the electromagnetic field with broken gauge symmetry, we derive an effective circuit Hamiltonian for a superconducting weak-link ring. The energy eigenstates of this Hamiltonian exhibit sensitivity to both external magnetic flux and an applied Faraday law voltage. We show that the flux periodicity of the rf SQUID ring, and the voltage periodic behaviour found in ultra-small-capacitance weak-link rings, can be seen as limiting cases of a more general phenomenon.
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.
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.
The response of a double hole RF SQUID to modulation by magnetic flux phi e introduced into one of the holes is first considered. The output voltage v 0 is a periodic function of phi e but it is shown that one would expect the period not to be equal to the flux quantum phi 0 itself but to some multiple of it. The flux in either hole of the SQUID may be calculated by imagining the weak link broken and the RF drive removed: one period of the v 0 variation is produced by a change in phi e sufficient to change the flux in either hole (averaged over one RF cycle), due both to current in the coil and to current induced on the inside wall of the SQUID, by phi 0 . This principle is applied to a double hole SQUID modified by the replacement of part of one wall by a block of normal metal (an R SQUID). Explicit expressions for the response to modulation, which is frequency dependent, are derived in the limit of low frequency, and more generally, at any frequency, for an R SQUID in which both the holes and the modulation coil are long in comparison with their diameter. The size of the superconducting screen which must surround the R SQUID, if only to avoid interference from other magnetic fields in the laboratory, is shown to play an important part in determining the response at low frequencies. The spectrum of noise currents is also calculated.
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.
It is theoretically explained that a response of a superconducting quantum interference device (SQUID) is periodically dependent on total magnetic flux coupling to the SQUID ring (Φ) and its period is a flux quantum (Φ(o)=h/2e, where h and e, respectively, express Planck's constant and elementary charge). For example, the voltage of an electromagnetically oscillated rf-SQUID or a current biased dc-SQUID is thought to be periodically dependent on Φ with a period of Φ(o). In this paper, we propose an accurate method to check the periodicity of a SQUID response by using a set of sensing coils covered with a superconducting sheath. As a demonstration, we measured periodicity of a commercially available thin-film type rf-SQUID response in magnetic flux ranging up to approximately 4300Φ(o). Its flux dependence was periodic below about 3400Φ(o).