Quartet Tomography in Multiterminal Josephson Junctions

Abstract

We investigate the detection of quartets in hybrid multiterminal Josephson junctions. Using simple models of quantum dots coupled to superconducting leads, we find that quartets are ubiquitous and show how to rigorously extract their contribution to the current-phase relation (CPR). We also demonstrate that quartets are closely related to the hybridization of Andreev bound states (ABSs) in these systems and propose a method to identify quartets directly in ABS spectra. We illustrate our method by analyzing the spectroscopic measurements of the ABS spectrum in a three-terminal Josephson junction realized in an InAs/Al heterostructure. Our analysis strongly suggests the existence of quartets in the studied hybrid system.

Figures (2)

• Figure 1: (a) Schematics of the single-dot model. A single-level quantum dot is coupled to $N$ superconducting terminals with phases $\varphi_j$. The parameter $\Gamma_j$ describes the strength of the coupling between the dot and lead $j$. (b) Example of the current-phase relation $I_1(\varphi_{12},\varphi_{23})$ for the model in (a) with three identical leads with energy gap $\Delta$. The different parameters are: $\Gamma_{1} = \Gamma_{2} = \Gamma_{3} = 5\Delta$ and $\epsilon_0 = 0$. (c) The corresponding $c_{nm}$ coefficients in the expansion of Eq. \ref{['eq_I1_1D']} in units of $e\Delta / \hbar$. The phase dependence associated to some of these coefficients is indicated in the graph. (d) Andreev bound state spectrum $E_{\rm ABS}(\varphi_{12},\varphi_{23})$ for the example shown in panel (b). (e) The corresponding $d_{nm}$ coefficients in the expansion of Eq. \ref{['eq-ABS-phase']} in units of $\Delta$.
• Figure 2: (a) Schematics of the double-dot model. (b) Example of the energy $E_{\rm ABS}^{(1)}(\varphi_{12}, \varphi_{23})$ of the highest occupied ABS in the model of panel (a). The different parameters are: $\Delta_{1,2,3} = \Delta$, $\Gamma_{1} = 5.5\Delta$, $\Gamma_{2,1} = 6\Delta$, $\Gamma_{2,2} = 5\Delta$, $\Gamma_{3} = 6\Delta$, $t = 5\Delta$, and $\epsilon_1 = \epsilon_2 = 0$. (c) The corresponding $d^{(1)}_{nm}$ coefficients in the expansion of Eq. \ref{['eq-ABS-phase']} measured in units of $\Delta$. The phase dependence associated to some of these coefficients is indicated in the graph. (d) False-colored scanning electron micrograph of the device measured in Ref. Coraiola2023, near to the three-terminal Josephson junction region. Exposed semiconductor is shown in orange, aluminum in blue and gate electrodes in gold. (e) The upper ABS extracted from the tunneling spectroscopy data of Ref. Coraiola2023, where $V_{\rm SD}$ is the bias voltage applied to the superconducting probe. (f) The corresponding $d^{(1)}_{nm}$ coefficients of Eq. \ref{['eq-ABS-phase']} in $\mu$eV that show the presence of quartets in the studied device.