Supercurrent from the imaginary part of the Andreev levels in non-Hermitian Josephson junctions

TL;DR

The paper addresses how non-Hermiticity in open superconducting junctions manifests in transport, specifically through a current component J_Im tied to the phase derivative of the imaginary part of Andreev levels. By deriving a non-Hermitian effective Hamiltonian for a QD-based JJ coupled to a ferromagnetic bath and applying a current formula that includes both real- and imaginary-part contributions, the authors analytically and numerically map regimes with EPs and global zero-energy states. They identify parameter windows (1G-ZES, 2G-ZES, and quartet regimes) where J_Im is large and accessible, and they propose an experimental protocol combining dI/dV spectroscopy and CPR to detect J_Im, including off-resonant regimes that enhance the effect. The work establishes a practical route to probe non-Hermitian physics in solid-state devices beyond EP signatures, linking symmetry properties of the shifted Hamiltonian to observable transport phenomena and guiding future NH-Josephson experiments.

Abstract

We investigate the electronic transport properties of a superconductor-quantum dot-superconductor Josephson junction coupled to a ferromagnetic metal reservoir in the presence of an external magnetic field. The device is described by an effective non-Hermitian Hamiltonian, whose complex eigenvalues encode the energy (real part) and the broadening (imaginary part) of the Andreev quasi-bound states. When extending the Andreev current formula to the non-Hermitian case, a novel contribution arises that is proportional to the phase derivative of the levels broadening. This term becomes particularly relevant in the presence of exceptional points (EPs) in the spectrum, but its experimental detection is not straightforward. We identify optimal Andreev spectrum configurations where this novel current contribution can be clearly highlighted, and we outline an experimental protocol for its detection. We point out that the phase dependence in the levels imaginary part originates from the breaking of a time-reversal-like symmetry. In particular, spectral configurations in the broken phase of the symmetry and without EPs can be obtained, where this novel contribution can be easily resolved. The proposed protocol would allow to probe for the first time a fingerprint of non-Hermiticity in open junctions that is not strictly related to the presence of EPs.

Figures (10)

• Figure 1: Regions of the parameters space, magnetic field amplitude $B$ (x-axis) and its angle $\theta$ with the ferromagnet magnetization (z-axis), hosting exceptional points (EPs), one pair of global zero-energy states (G-ZES) and without ZES (noZES). In the insets, the corresponding Andreev spectra along with the corresponding CPR. Here, we also show the CPR components coming from the phase dispersion in the real and imaginary parts of the levels, i.e. $J_{\Re}$ and $J_{\Im}$. Between two zero-energy EPs (ZE-EPs), e.g. phase interval among the green-dashed lines, the Andreev levels are pinned at zero energy giving rise to ZES. For a JJ hosting EPs (a) $J_{\Im}$ is piecewise defined while in the G-ZES regime (b) it is continue and its detection should be easier.
• Figure 2: Scheme of the Quantum Dot Josephson junction (QD JJ) connected to the ferromagnetic lead. We show the relevant parameters: the dot energy level $\varepsilon_{d}$; the superconducting gap $\Delta$ and the phase in the left/right lead $\phi_{L/R}$; the chemical potential $\mu_{L/R/F}$ respectively of the left/right superconductors and ferromagnet; the F lead magnetization $\vec{M}_{F}$; the external magnetic field $\vec{B}_{0}$ inducing a Zeeman splitting of order $B=g\mu_{B}|\vec{B}_{0}|$ on the dot; the hybridization parameter of the dot with the left/right S leads $\Gamma_{L/R}$ and the spin-dependent hibridizations with the F lead $\Gamma_{\uparrow/\downarrow}$.
• Figure 3: Schematic representation of the emergence of the EPs in a NH JJ at $\varepsilon_{d}=0$ (a), along with the complex Andreev spectrum (b). In (a) ABS spectrum of the Hermitian system, i.e. $\Gamma_{N}=0$ (gray lines), and the real part of the quasi-ABS of its NH counterpart (red-dashed lines). ZE-EPs and FE-EPs are shown.
• Figure 4: The position of the external FE-EP (a) and the ZE-EPs (b) vs. the dissipation imbalance for the two spins in F, $\gamma_N/\Gamma_N$, for different values of the coupling asymmetry $\gamma$. The more asymmetric is the junction the higher is the $\gamma_N$ value at which FE-EPs manifest. External ZE-EPs, in the absence of finite energy ones, can annihilate in pairs at $\phi=\pm\pi$, while internal ones tend to annihilate at $\phi=0$. The system parameters are $\Gamma=1$, $\Gamma_N=2$, $B=0.6$, $\theta=1$.
• Figure 5: Emergence of regions with spectral configurations without EPs and with phase-dependent level broadening, along with the corresponding CPRs. In the upper panel: phase positions of all EPs, ZE-EPs ( thick lines) and FE-EPs (dashed lines) vs. $\gamma_N/\Gamma_N$ for different values of the S-leads coupling asymmetry, $\gamma=0.5,\,0.8$. The $\gamma_N$-windows with no EPs in the spectrum for the JJ with $\gamma=0.8$ are shaded in brown. The system parameters are the same of Fig. \ref{['fig: 14 External_EPs_diff_gamma']}. In the lower panels: typical spectra in each of the three shaded regions of the parameters space, 1G-ZES (a) 2G-ZES (b) and no ZES quartet (c), along with their corresponding CPR contributions, $J_{\Re}$ and $J_{\Im}$ (d-f). Here all parameters stay the same, save $\gamma=0.8$ and $\gamma_N/\Gamma_{N}=0.25,\,0.73,\,0.95$ respectively in (a,d), (b,e) and (c,f).