Controllable odd-frequency Cooper pairs in multisuperconductor Josephson junctions

We consider Josephson junctions formed by multiple superconductors with distinct phases and explore the formation of nonlocal or intersuperconductor pair correlations. We find that the multiple superconductor nature offers an additional degree of freedom that broadens the classification of pair symmetries, enabling nonlocal even- and odd-frequency pairings that can be highly controlled by the superconducting phases and the energy of the superconductors. Specially, when the phase difference between two superconductors is $\pi$, their associated nonlocal odd-frequency pairing is the only type of intersuperconductor pair correlations. Finally, we show that these nonlocal odd-frequency Cooper pairs dominate the nonlocal conductance via crossed Andreev reflections, which constitutes a direct evidence of odd-frequency pairing.

Figure 1

JJs formed by coupling superconductors $S_i$ with distinct phases ${\varphi }_i$, and same induced pair potential $\mathrm{\Delta }$. In each $S_i$ local pairs are depicted in gray ellipses containing two electrons (black filled circles), referred to as intrasuperconductor (local) pairs. Due to the tunneling between superconductors, intersuperconductor (nonlocal) pair correlations emerge (cyan) which can be controlled by ${\varphi }_i$. Normal leads (green) are attached to two $S_i$ for exploring nonlocal transport and detecting intersuperconductor Cooper pairs.

Figure 2

(a), (b) Symmetric even-$\omega$ and antisymmetric odd-$\omega$ nonlocal pair amplitudes ($F_{12}^{\pm }$) in a JJ with two superconductors coupled directly as a function of $\omega$ and $\varphi$ at $\epsilon =0.1$, with the color scale cut off at 10 for visualization. White arrows in (a) indicate that $|F_{12}^{+}|$ vanishes at $\varphi =\pi$. (c) Ratio $R^{\pm }=|F_{12}^{+}|/|F_{12}^{-}|$ as a function of $\epsilon$ and $\varphi$ at $\omega =0.2$. Cyan arrows indicate that $R$ vanishes either at $\epsilon =0$ or $\varphi =\pi$. (d) Line cuts of (a), (b) at fixed $\omega$. Parameters: $\mathrm{\Delta }=0.5$ and $t_0=0.3$.

Figure 3

(a), (b) Symmetric even-$\omega$ ($F_{12}^{+}$) and antisymmetric odd-$\omega$ ($F_{12}^{-}$) nonlocal pair amplitudes in a JJ with three superconductors coupled directly as a function of ${\varphi }_2$ and ${\varphi }_3$ at $\omega =0.1$ and $t_0=0.3$, with the color scale cut off at 5 for visualization. (c), (d) Same as in (a),(b) but at $\omega =1$ and $t_0=0.5$. Parameters: $\mathrm{\Delta }=0.5$, $\epsilon =0$, and ${\varphi }_1=0$.

Figure 4

Electron tunneling (top row) and crossed Andreev reflection (bottom) processes as a function of $\omega$ and $\varphi$. Parameters: $\mathrm{\Delta }=0.5$, $\epsilon =0$, ${\mathrm{\Gamma }}_j=0.1$, and $t_0=0.5$.