Nonlocal Josephson diode effect in minimal Kitaev chains

Abstract

We study the emergence of the nonlocal Josephson effect in a system composed of three laterally coupled minimal Kitaev chains and exploit it to realize the nonlocal Josephson diode effect. We find that an imbalance between crossed Andreev reflections and electron cotunneling in the middle Kitaev chain gives rise to an asymmetric $2π$-periodic phase-dependent Andreev spectrum, controlled by the superconducting phases across the left and right junctions. We then show that the asymmetric Andreev spectrum, formed by hybridized Andreev bound states at the left and right junctions, enables a supercurrent across one junction via the phase difference at the other junction, thereby signaling the nonlocal Josephson effect. Notably, these nonlocal Josephson supercurrents exhibit distinct positive and negative critical currents, demonstrating the realization of the nonlocal Josephson diode effect with highly tunable polarity and efficiencies exceeding $50\%$. The nonlocal Josephson diode effect requires breaking local time-reversal and local charge-conjugation symmetries, with the latter being unique to minimal Kitaev chains. Our results establish minimal Kitaev chains as a highly controllable platform for engineering nonlocal Josephson phenomena.

Figures (7)

• Figure 1: Three minimal Kitaev chains laterally coupled to form a double Josephson junction, where the phase of the left (right) junction is controlled by an external magnetic flux $\Phi_{\rm L (R)}$. Here, a minimal Kitaev chain is formed by connecting two quantum dots (QDs, green) by a superconductor (S, blue). The onsite energy of the QDs is controlled by gates (bars, brown), while the coupling between minimal Kitaev chains can be tuned by an insulating barrier (red).
• Figure 2: Energy spectrum as a function of $\phi_{\rm L}$ at distinct $\phi_{\rm R}$ for $t_{\rm M}=\Delta_{\rm M}$ (a-d) and $t_{\rm M}<\Delta_{\rm M}$ (e-h). The vertical dotted line marks $\phi_{\rm L}=\pi$, while the red vertical dashed line marks $\phi_{\rm L}$ at which the dispersing lowest ABSs reach zero energy. Parameters: $\varepsilon_{\alpha}=0$, $t_{\rm L,R}=\Delta_{\rm L,R}\equiv\Delta=1$, $\tau_{\rm L}=1$, $\tau_{\rm R}=0.9$.
• Figure 3: (a,c,e) Supercurrent as a function of $\phi_{\rm L}$ and $\phi_{\rm R}$ for $t_{\rm M}<\Delta_{\rm M}$ (a), $t_{\rm M}=\Delta_{\rm M}$ (c), and $t_{\rm M}>\Delta_{\rm M}$ (e). (b,d,f) Line cuts as a function of $\phi_{\rm L (R)}$ for distinct $\phi_{\rm R (L)}$. In (d,f), the magenta arrow indicates variations of $\phi_{\rm L,R}$ within $(0,\pi)$ in steps of $0.1\pi$. Parameters: $I_{0}=e\Delta/\hbar$, the rest as in Fig. \ref{['Fig2']}.
• Figure 4: (a-c) Critical currents $I_{\rm cL}^{\pm}$ as functions of $\phi_{\rm R}$ for $t_{\rm M}<\Delta_{\rm M}$ (a), $t_{\rm M}=\Delta_{\rm M}$ (b), and $t_{\rm M}>\Delta_{\rm M}$ (c). The dashed and solid lines in (a) correspond to $t_{\rm M}=0.2\Delta_{\rm M}$ and $t_{\rm M}=0.5\Delta_{\rm M}$, while to $t_{\rm M}=1.5\Delta_{\rm M}$ and $t_{\rm M}=1.8\Delta_{\rm M}$ in (c). (d,e) Diode's quality factor $\eta_{\rm L}$ as a function of $\phi_{\rm R}$ for distinct $t_{\rm M}$ in steps of $0.2\Delta_{\rm M}$, marked by colored bars in (e). (f) $\eta_{\rm L}$ as a function of $\phi_{\rm R}$ for distinct $\varepsilon_{\rm M}$ for $t_{\rm M}<\Delta_{\rm M}$ and $t_{\rm M}>\Delta_{\rm M}$. (g) The same as in (f) but for distinct $\tau_{\rm R}$. Parameters: $I_{0}=e\Delta/\hbar$, the rest as in Fig. \ref{['Fig2']}.
• Figure 5: Energy spectrum as a function of $\phi_{\rm L}$ at distinct values of $\phi_{\rm R}$ for $t_{\rm M}=1.8\Delta_{\rm M}$. The vertical dotted line marks $\phi_{\rm L}=\pi$, while the red vertical dashed line marks the value of $\phi_{\rm L}$ at which the dispersing lowest ABSs reach zero energy. Parameters: $\varepsilon_{\alpha}=0$, $t_{\rm L,R}=\Delta_{\alpha}\equiv\Delta=1$, $\tau_{\rm L}=1$, $\tau_{\rm R}=0.9$.