Quantum phase slips in Josephson junction rings

Abstract

We study quantum phase-slip (QPS) processes in a superconducting ring containing $N$ Josephson junctions and threaded by an external static magnetic flux ${\ensuremath{\Phi}}_{B}$. In such a system, a QPS consists of a quantum tunneling event connecting two distinct classical states of the phases with different persistent currents [Matveev et al., Phys. Rev. Lett. 89, 096802 (2002)]. When the Josephson coupling energy ${E}_{J}$ of the junctions is larger than the charging energy ${E}_{C}={e}^{2}/2C$, where $C$ is the junction capacitance, the quantum amplitude for the QPS process is exponentially small in the ratio ${E}_{J}/{E}_{C}$. At given magnetic flux, each QPS can be described as the tunneling of the phase difference of a single junction of almost $2\ensuremath{\pi}$, accompanied by a small harmonic displacement of the phase difference of the other $N\ensuremath{-}1$ junctions. As a consequence, the total QPS amplitude ${\ensuremath{\nu}}_{\mathrm{ring}}$ is a global property of the ring. Here, we study the dependence of ${\ensuremath{\nu}}_{\mathrm{ring}}$ on the ring size $N$, taking into account the effect of a finite capacitance ${C}_{0}$ to ground, which leads to the appearance of low-frequency dispersive modes. Josephson and charging effects compete and lead to a nonmonotonic dependence of the ring's critical current on $N$. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the system converges either towards a superconducting or an insulating state, depending on the ratio between the charging energy ${E}_{0}={e}^{2}/2{C}_{0}$ and the Josephson coupling energy ${E}_{J}$.