Theory of coherent quantum phase slips in Josephson junction chains with periodic spatial modulations

We study coherent quantum phase slips which lift the ground state degeneracy in a Josephson junction ring, pierced by a magnetic flux of the magnitude equal to half of a flux quantum. The quantum phase-slip amplitude is sensitive to the normal mode structure of superconducting phase oscillations in the ring (Mooij-Schön modes). These, in turn, are affected by spatial inhomogeneities in the ring. We analyze the case of weak periodic modulations of the system parameters and calculate the corresponding modification of the quantum phase-slip amplitude.

Figure 1

A schematic representation of a superconducting ring threaded by a magnetic flux $\mathrm{\Phi }$ and containing $N+1$ Josephson junctions with a capacitance $C$ between the neighboring islands and a capacitance $C_{\text{g}}$ to the ground. $E_J$ is the Josephson energy. $a$ is the size of each superconducting island. ${\varphi }_n$ is the condensate phase of the $n\mathrm{th}$ superconducting island.

Figure 2

A schematic representation of the Josephson junction chain in the continuum limit with the boundary junction shown explicitly.

Figure 3

The dispersion curve of the phase oscillations (plasmons, Mooij-Schön modes), determined by Eq. (7).

Figure 4

Flux dependence of the ground state energy (upper panel) and the persistent current (lower panel), shown schematically in the purely classical approximation (gray dashed line) and taking into account quantum tunneling (red solid line).

Figure 5

A schematic representation of the classical trajectory $\varphi (x,\tau )$ going from the static configuration with $\vartheta =\pi$ (solid line on upper panel) to $\vartheta =-\pi$ (solid line on the lower panel) for a spatially homogeneous ring. Straight arrows correspond to the slow adjustment of the phase in the whole ring, the round arrows show the fast flip of the phase in the vicinity of the ${\tilde{E}}_J$ junction. In a ring with spatially modulated parameters, some modulation will also be superimposed on $\varphi (x,\tau )$.

Figure 6

Deformation of the integration contour in Eq. (48) from the real axis (solid red line) into the upper complex half-plane (dashed red line). The dots represent branching points $z_j$ of $s\left(z\right)$.

Figure 7

The two potentials, $V\left(\varphi \right)=-{\tilde{E}}_J(1+cos\varphi )$ (left panel) and $V\left(\varphi \right)=-{\tilde{E}}_J{\left(\right|\varphi |-\pi )}^2/2$ (right panel), for which the slow part of the instanton trajectory (solid arrows) should be similar.