We propose the "Andreev molecule," an artificial quantum system composed of two closely spaced Josephson junctions. The coupling between Josephson junctions in an Andreev molecule occurs through the overlap and hybridization of the junction's "atomic" orbitals, Andreev Bound States. A striking consequence is that the supercurrent flowing through one junction depends on the superconducting phase difference across the other junction. Using the Bogolubiov-de-Gennes formalism, we derive the energy spectrum and non-local current-phase relation for arbitrary separation. We demonstrate the possibility of creating a ϕ-junction and propose experiments to verify our predictions. Andreev molecules may have potential applications in quantum information, metrology, sensing, and molecular simulation. 1 arXiv:1809.11011v3 [cond-mat.mes-hall] 4 Sep 2019 Keywords superconductivity, Josephson junction, Andreev bound states, superconducting circuits, quantum informationUnderstanding and exploiting the interaction between Josephson junctions is paramount for superconducting device applications in quantum information 1 , magnetometry 2 , metrology 3 , and quantum simulation 4 . In typical superconducting circuits, junctions interact indirectly via electromagnetic coupling to inductors, capacitors, transmission lines, and microwave resonators. In addition to this well understood long-range interaction 5 , there is a short range interaction via quasiparticle diffusion which can modify superconducting energy gaps and critical currents, but is only important close to T c , the superconducting transition temperature, or at large bias voltages 6 .A second short-range interaction, mediated by Cooper pairs, is relevant to the majority of applications where characteristic energies are much smaller than the gap, but is still poorly understood. It becomes significant when the distance between Josephson junctions is comparable to ξ 0 , the superconducting coherence length, and can modify the electrical properties in a dramatic way.Initially, minor effects resulting from this "order-parameter interaction" were calculated for temperatures near T c using the Ginzburg-Landau equations 7 . More recently, theorists have investigated this problem at arbitrary temperature using Green's function techniques.In the two-electrode geometry, where it is not possible to independently apply a phase difference to each junction, the overall current-phase relation and dc current were obtained 8,9 .For the more relevant three-electrode geometry, non-local out-of-equilibrium supercurrents were calculated and the existence of π shifts in the current-phase relation were demonstrated 10-15 . A remarkable phase-locking similar to Shapiro steps was predicted and subsequently measured experimentally in superconducting bi-junctions biased with commensurate voltages 16,17 . The authors attribute these phenomena to the formation of entangled Cooper pairs called "quartets."

The authors report on parametric luminescence in a single pillar semiconductor microcavity. Nonresonant photoluminescence measurements at 10K show that micropillars operate in the exciton-photon strong coupling regime. Under resonant excitation, polariton parametric scattering is observed: signal and idler beams are of comparable intensities, a promising result in the framework of quantum optics with correlated photon pairs.

An Andreev molecule is a system of closely spaced superconducting weak links accommodating overlapping Andreev Bound States. Recent theoretical proposals have considered one-dimensional Andreev molecules with a single conduction channel. Here we apply the scattering formalism and extend the analysis to multiple conduction channels, a situation encountered in epitaxial superconductor/semiconductor weak links. We obtain the multi-channel bound state energy spectrum and quantify the contribution of the microscopic non-local transport processes leading to the formation of Andreev molecules.