Anomalous Josephson Effect between Even- and Odd-Frequency Superconductors

We demonstrate that, contrary to standard wisdom, the lowest-order Josephson coupling is possible between odd- and even-frequency superconductors. The origin of this effect is the induced odd- (even-)frequency pairing component at the interface of bulk even- (odd-)frequency superconductors. The resulting current-phase relation is found to be proportional to $\cos \phi$, where $\phi$ is the macroscopic phase difference between the two superconductors.

Figure 1

Spatial dependence of the normalized pair potential (solid line), even-frequency pair amplitudes $E_s(x)$ and $E_{px}(x)$ (dotted lines) and odd-frequency pair amplitudes $O_s(x)$ and $O_{px}(x)$ (dashed lines). (a) $s$-wave bulk state, full-transparency junction with $Z=0$; (b) ETO $p_x$-wave bulk state, low-transparency junction with $Z=5$. Distance $x$ is normalized by $\xi =v_F/\Delta$.

Figure 2

Schematic illustration of $\pi$-SQUID made of ESE and OSO superconductor junctions. $f_{1L}=-f_{1R}$.