QUROPE Quantum Information Processing and Communication in Europe

We study quantum phase-slip processes in a one-dimensional phase-biased chain containing $N$ Josephson junctions. 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 phase-slip processes is exponentially small in the ratio $E_J/E_C$; hence they can occur one by one. Then, each phase-slip consists of a single tunneling of the phase difference by $2 \pi$ in one of the junctions, accompanied by a small harmonic displacement of the phase difference of the other $N-1$ junctions such as to satisfy the phase bias. As a consequence the total phase-slip amplitude $\nu_\mathrm{chain}$ is a global property of the chain. Here we study for the first time the dependence of $\nu_\mathrm{chain}$ on the chain length $N$ taking into account the effect of a finite capacitance $C_0$ to ground which leads to the appearance of low-frequency propagating modes. Josephson and charging effects compete and lead to a non-monotonic dependence of the chain's critical current on $N$. For $N \rightarrow \infty$, the system converges either towards a superconducting or an insulating state, depending on the ratio $E_J/E_0$, where $E_0=e^2/2C_0$.