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Physical Review

Coherence and Quantization in Nearly Superconducting Rings

James Edward Zimmerman, Arnold H. Silver

Published Mar 10, 1968• 23 citations
Publisher

Abstract

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 $R$ is the resistance and $L$ 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 ${k}_{B}$ is Boltzmann's constant, $T$ 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.

DOI
10.1103/physrev.167.418

Related papers

Quantum States and Transitions in Weakly Connected Superconducting Rings

Arnold H. Silver, James Edward Zimmerman

This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance $L$ incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ${i}_{c}$. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ${i}_{c}$ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.

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Superconducting quantum electronics

Arnold H. Silver

The quantum electronic properties of superconductors have been vividly demonstrated by means of experiments on Josephson tunneling junctions and small area superconducting contacts. Emphasis is placed on ring configurations where quantum state transitions corresponding to changes of magnetic flux of2.07 \times 10^{15}webers are directly observed. Such systems can be used to make ultrasensitive measurements of magnetic fields, electric currents and voltages, cryogenic temperatures, and electro-magnetic radiation.

IEEE Journal of Quantum Electronics•Nov 1, 1968•6 citations
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 \ensuremathω>\ensuremathω1=RL\ensuremath{\omega}>{\ensuremath{\omega}}_{1}=\frac{R}{L}\ensuremathω>\ensuremathω1​=LR​, 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, 8kBTR\ensuremathΦ02\frac{8{k}_{B}\mathrm{TR}}{{{\ensuremath{\Phi}}_{0}}^{2}}\ensuremathΦ0​28kB​TR​, where kB{k}_{B}kB​ is Boltzmann's constant, TTT the absolute temperature, and \ensuremathΦ0{\ensuremath{\Phi}}_{0}\ensuremathΦ0​ 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.
This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance LLL incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ic{i}_{c}ic​. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ic{i}_{c}ic​ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.
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.
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.
Thermodynamic fluctuations limit the ultimate performance of superconducting devices of practical interest, devices such as the quantum interference magnetometer and the Josephson junction radiation detector. We report direct observations of thermodynamically driven quantum phase fluctuations near the superconducting critical temperature. The experiments used point-contact quantum interference techniques with niobium points on a single-crystal tin substrate. Near the superconducting transition of tin, TC=3.72°K, the niobium points served as probes to measure fluctuations in the phase of the superconducting order parameter within the tin crystal. The temperature was slowly increased, beginning well below TC for tin, where the quantum phase was stable. Quantum phase fluctuations were first observed approximately 5 mdeg below TC. As the temperature continued to increase through this 5 mdeg range near TC, the characteristic frequency of the phase fluctuations increased 6 orders of magnitude. At TC a discontinuous transition to a resistive state gave rise to a finite voltage difference in the tin with the associated driven quantum phase precession. Thermodynamic Johnson noise voltages developed across this resistance of order 10−10Ω observably broadened the precession line. We will present the detailed temperature dependence of the frequency power spectrum of quantum phase fluctuations below and above TC.
The quantum electronic properties of superconductors have been vividly demonstrated by means of experiments on Josephson tunneling junctions and small area superconducting contacts. Emphasis is placed on ring configurations where quantum state transitions corresponding to changes of magnetic flux of2.07 \times 10^{15}webers are directly observed. Such systems can be used to make ultrasensitive measurements of magnetic fields, electric currents and voltages, cryogenic temperatures, and electro-magnetic radiation.