Quantum States and Transitions in Weakly Connected Superconducting Rings

Abstract

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.