Andreev Molecule in Parallel InAs Nanowires

Coupling individual atoms fundamentally changes the state of matter: electrons bound to atomic cores become delocalized turning an insulating state to a metallic one. A chain of atoms could lead to more exotic states if the tunneling takes place via the superconducting vacuum and can induce topologically protected excitations like Majorana or parafermions. Although coupling a single atom to a superconductor is well studied, the hybridization of two sites with individual tunability was not reported yet. The peculiar vacuum of the Bardeen–Cooper–Schrieffer (BCS) condensate opens the way to annihilate or generate two electrons from the bulk resulting in a so-called Andreev molecular state. By employing parallel nanowires with an Al shell, two artificial atoms were created at a minimal distance with an epitaxial superconducting link between. Hybridization via the BCS vacuum was observed and the spectrum of an Andreev molecule as a function of level positions was explored for the first time.


YSR states in an intermediate limit
In superconducting QDs, sub-gap states can be formed by the hybridization of a QD level with the SC. [1][2][3][4][5][6][7][8][9] There are commonly used schemes to describe the formation of bound states: the two simplest of them is the superconducting atomic limit (or Andreev limit), where the superconducting gap is considered to be much larger than the charging energy of the QD (∆ U ), and the opposite limit, the so-called Yu-Shiba-Rusinov limit (∆ U ).
In the Andreev picture, quasi-particles are not allowed to appear in the SC. However, the superconductivity can couple the empty and doubly occupied QD states via Andreev reflection (AR) as shown in Fig. 1a leading to a singlet superposition of them. Depending on the coupling strength, the ground state is either the doublet (singly occupied QD) or the singlet, and transitions can be induced between them by changing the on-site potential exhibiting dispersively evolving excitation lines.
In the YSR limit, we first assume that the double occupation of the QD is forbidden as a result of the large U . Therefore the relevant number of electrons in the QD and quasiparticles in the SC is restricted to 0 or 1, as shown in Supp. Fig. 1b. In the following we use the notation of |n, m for |n QD ⊗ |m SC and σ for a single spin. Due to the spin degeneracy, the |0 QD , | ↑ QD , | ↓ QD and |0 SC , | ↑ SC , | ↓ SC configurations in the QD and in the SC define 9 different states. When the total electron number parity of the system is even, the The blue rectangles denote the SC electrode, whereas the red circle the QD. Different configurations if different rows are coupled by AR and tunneling processes (arrows). In the Andreev limit (a), the empty and double occupied QD states can hybridize via AR to form a singlet state (marked by "S"), while the doublet (marked by "D") remains intact. In the YSR limit (b), the double occupation of the QD is penalized. States with the same parity can hybridize in the presence of a finite tunnel coupling. In the intermediate limit (c), double occupation of the QD is allowed. Triplet states of parallel spins are excluded. d Simulated spectrum of a YSR state in the intermediate limit exhibiting an "eye-shaped" curve. The coupling strength was chosen as t/U = 0.15 and ∆/U = 0.2 was used. Transport takes place when a ground state transition is induced.
altogether. Analogously to the YSR limit, the tunneling implies hybridization within the singlet and doublet subspaces, as depicted in Supp. Fig. 1c. The states |0, 0 , | ↑, ↓ − | ↓, ↑ , | ↑↓, 0 in the singlet sector and states |σ, 0 , |0, σ , | ↑↓, σ in the doublet sector couple (by tunneling or AR) providing level-repulsions in the energy diagram. In transport experiments, one can probe the excitation energies via finite-bias spectroscopy leading to an "eye-shaped" curve as a function of a plunger gate voltage, as outlined above. A simulation showing such a curve is depicted in Supp. Fig. 1d, where the singlet and doublet grounds states are addressed with "S" and "D", respectively. We note that all of the calculated spectra introduced in the main text and here were derived in the framework of this intermediate limit with the only difference of allowing 2 quasi-particles in the system.

Uncoupled YSR states (device A)
In the main text, we showed the SEM micrograph and the transport model of the sample exhibiting interacting YSR states, called device B. Here we provide the same for the uncoupled one, device A, in Supp. Fig. 2. Figure 2: Device hosting the uncoupled YSR states. a False-color SEM micrograph of device A. Compared to device B (see Fig. 1a in the main text), the nanowires originally merged by the epitaxial Al got disconnected and moved further away from each other. b Schematic illustration of device A, similarly to Fig. 1d in the main text. As the nanowire branches (green) are only linked by the common SC (blue), CAR is strongly suppressed by the long distance. Due to the larger spatial separation, the interdot capacitance is also weaker.
A small gap visible between the wires along the segment covered by epitaxial Al is the indication of the wires being fallen apart during the manipulation process, thus the epitaxial SC link between the wires is missing. On one hand, as the probability of CAR decays with increasing spatial separation of the conducting channels, 10 it is strongly suppressed in device A as it can take place between the QDs only via the ex-situ evaporated common SC. This results in an effective distance of ∼800 nm between the QDs. On the other hand, the interdot capacitance of the two InAs branches is also reduced.  to the bottom QD resonances, therefore bias slices along the pink and blue cuts are taken.
In panels b and c, the spectra along line cuts are presented off and close to resonance, respectively, with the "eye-shaped" YSR T (red) and the movement of YSR B (green) depicted.
We emphasize that YSR B is weakly coupled, thus its excitation energy is expected to be at

Coupled YSR states (device B)
In the main text, a pair of bias slices were introduced from the measurements performed on device B revealing the hybridization of the YSR states (see Fig. 3e-f ), which were compared to numerical simulations of the fully interacting system introduced. We present the spectra with the same gate settings with excluding the superconducting coupling and only allowing interdot Coulomb repulsion between the QDs. The results are depicted in Supp. Fig. 4.   As further sub-gap states appeared in the spectra, the single YSR state picture can not be applied anymore to describe the system. Albeit the spectra became more complex due to the multiple excitation lines and their broadenings dropping the visibility of the hybridization the signatures of the Andreev molecular state are still present.

Modeling
In this Supplementary Note we outline the framework used in the main text to simulate the transport spectrum of the Andreev molecule. First, the Hamiltonian is introduced, then we discuss the transport calculation.
The system consists of two, parallel-coupled quantum QDs and two superconducting electrodes as depicted in Fig. 1e. One of the SC is strongly coupled to the QDs, their hybridization forms the YSR states. The other SC is weakly coupled.
The total Hamiltonian of the system is The first term describes the double QD, where n ασ = d † ασ d ασ is the number of electrons with spin σ on QD α , with d where c SC1σ is the annihilation (creation) operator of an electron with spin σ in the SC and ∆ 1 is the superconducting gap. This Hamiltonian can be diagonalized by a Bogoljubov- The tunnel coupling between superconductor SC1 and the QDs writes as where t α is tunneling amplitude. Using the Bogoljubov-transformation above this Hamiltonian translates to The already defined three Hamiltonian terms, H DQD + H SC1 + H T1 are numerically diagonalized to obtain the energy spectrum and wavefunction of the Andreev molecular state.
In Eq. S1 H SC2 and H T2 describes the second superconducting lead and its weak tunnel coupling to the QDs, respectively. The superconductor SC2 is described by the BCS Hamiltonian, where c SC2kσ is the annihilation (creation) operator for electrons with momentum k and spin σ in the SC2 superconductor, ε SC2k is normal state dispersion and ∆ 2 is superconducting gap. Note that here both superconducting gaps, ∆ 1 and ∆ 2 are assumed to be real, the possible effects originating from the superconducting phase difference are neglected. The tunnel coupling Hamiltonian is where t SC2 is tunneling amplitude, for the simplicity, assumed to be the same for the two QDs.
The tunnel coupling to SC2 is assumed to be weak and treated perturbatively using Fermi's golden rule. In this description the tunnel coupling induces transitions between eigenstates of the QD T -SC1-QD B system. The time evolution of the occupation of the eigenstates |χ , P χ is governed by a master equation together with the normalization condi- Here W χχ the total transition rate from |χ state to |χ induced by the tunnel coupling to SC2. The rates are the sum of two processes, when an electron is added to the Andreev molecule and when one is removed, i.e. W χχ = W χ χ d † ασ + W χ χ (d ασ ). These two contributions are expressed as where ρ S (E) = ρ 0 Re is the Dynes-like density of states (DOS) in SC2 superconductor, with γ being the Dynes-parameter and ρ 0 is the normal state DOS, assumed to be constant, f (E) is the Fermi function, E χ denote the energy of the |χ state and µ SC2 = eV SD is chemical potential difference of the two superconducting leads, due to the applied bias voltage, V SD .
To derive the current through the device the master equation, Eq. S9 is solved in the stationary limit, dP χ /dt = 0 to obtain the occupations. The current writes as (S11) The differential conductance is obtained as the derivative of the current, i.e. G = e dI dµ SC2 . For the case, when only capacitive coupling is assumed between the QDs, one has to eliminate the SC mediated tunneling processes that couples the states of the two QDs. An example for such a process is when a Cooper pair from the SC1 electrode splits between the QDs. As the QD T -SC1-QD B subsystem is treated up to all orders in the tunneling, such processes are necessarily present in the description above. One can formally remove them by coupling the QDs to two separate SCs. This can be formulated in the following Hamiltonians: The difference compared to Eqs. S3&S5 is that the SC1 superconductor is split into two parts, which are only coupled to one of the QDs, This way the tunnel coupling hybridize the QD and SC states to local YSR states, but the further hybridization of the YSR state into molecular states are prevented. The rest of the transport calculation is the same as above.