Andreev Molecules: Hybridized Quantum States

• Andreev molecules are artificial quantum systems formed by hybridizing spatially separated Andreev bound states via direct tunneling or photon-mediated coupling.
• They exhibit unique nonlocal effects such as modulated supercurrents and a Josephson diode effect with efficiencies up to 45%, driven by phase shifts and asymmetric processes.
• Experimental platforms including quantum dot arrays, Josephson junctions, and nanowires demonstrate their potential for scalable quantum devices and topological superconductivity.

Andreev molecules are artificial quantum systems in which two or more Andreev bound states (ABSs), localized in distinct physical regions such as superconducting quantum dots or Josephson junctions, interact coherently via hybridization. The resulting molecular states inherit properties from the parent ABSs but display new spectra, nonlocal transport features, and susceptibility to symmetry breaking mechanisms, distinguishing them fundamentally from isolated ABSs in conventional superconducting devices. Andreev molecules have emerged as a key platform for investigating nonlocal superconducting correlations, quantum spin and charge states, topological superconductivity, parity phenomena, and advanced superconducting device functionalities.

1. Fundamental Concepts and Formation

Andreev molecules originate from the hybridization of ABSs confined to spatially separated regions—typically realized in closely spaced Josephson junctions, semiconductor quantum dots proximitized by superconductors, or molecular-scale systems on superconducting substrates. In a minimal setup, two ABSs are coupled such that their wavefunctions overlap, forming bonding and antibonding molecular states. The resulting hybridization splits the original energy degeneracies and introduces nonlocal dependencies: observable quantities such as the supercurrent in one junction become explicit functions of the phase and state of the other.

The formation is not limited to direct electronic overlap at short distances (on the order of the superconducting coherence length, ξ₀); recent developments demonstrate that photon-mediated coupling via an embedding electric circuit allows for long-distance Andreev molecules, in which hybridization occurs even if direct electron transfer is suppressed (Samuelsen et al., 21 Jul 2025). In these designs, alignment of ABS energies and phase matching are crucial to achieve strong molecular coupling.

2. Theoretical Models and Multichannel Generalization

The physics of Andreev molecules can be described with models extending the Anderson impurity model, Bogoliubov–de Gennes (BdG) formalism, and scattering-matrix approaches:

• For ABSs in quantum dots, the Hamiltonian is typically

$$H = \sum_{i=L,R} [\epsilon_i n_i + U n_{i\uparrow} n_{i\downarrow} + \Delta (c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger + c_{i\downarrow} c_{i\uparrow}) ] + t \sum_\sigma (c_{L\sigma}^\dagger c_{R\sigma} + h.c.)$$

describing dot levels (ε₁,ε₂), Coulomb interaction (U), induced pairing (Δ), and interdot coupling (t) (Su et al., 2016).

• In Josephson junction geometries with hybridization, the effective two-level Hamiltonian is

$$H_{\text{eff}} = \begin{bmatrix} E_1(\varphi_1) & t(\varphi_1,\varphi_2) \ t^*(\varphi_1,\varphi_2) & E_2(\varphi_2) \end{bmatrix}$$

where $t(\varphi_1,\varphi_2)$ is determined by overlapping Andreev states' matrix elements and $E_{1,2}$ are the bare ABS energies (Haxell et al., 2023).

• The scattering-matrix approach generalizes to multi-channel Andreev molecules by constructing large-dimensional scattering matrices (size $8N\times 8N$ for $N$ channels per contact) and solving for the resulting subgap spectrum via the secular equation

$\det(\mathbb{I} - S_n S_S) = 0$

where $S_n$ models normal scattering at the weak links, and $S_S$ captures energy-dependent Andreev reflections and circuit-transmitted amplitudes (Pillet et al., 2020).

These models encompass direct elastic cotunneling (dEC), crossed Andreev reflection (CAR), multiple Andreev reflections (MAR), and continuum effects (including leaky Andreev states in the presence of strong hybridization). The inclusion of many-body effects such as electron-phonon coupling (with Morse potentials for molecular vibrations) further enriches the accessible physics (Golez et al., 2012).

3. Nonlocal Effects and Diode Phenomena

Andreev molecules are defined by their nonlocal properties. In a conventional Josephson junction, the supercurrent $I(\delta)$ is a function of the local superconducting phase difference $\delta$, and often obeys $I(-\delta) = -I(\delta)$. Hybridization in Andreev molecules breaks this locality: the current-phase relation (CPR) in one junction becomes

$$I_1(\varphi_1, \varphi_2) = I_0 \sin(\varphi_1 + \Delta\varphi_1(\varphi_2))$$

where $\Delta\varphi_1(\varphi_2)$ is a phase shift generated by the state of the other junction (Haxell et al., 2023, Pillet et al., 2018). This nonlocal Josephson effect enables $\varphi$-junctions and tunable superconducting phase sources.

A critical manifestation of nonlocality is the Josephson diode effect (JDE): the critical current becomes direction-dependent, and even the sign of the current can be reversed depending on external control parameters (Pillet et al., 2023, Zhu et al., 19 Aug 2025). The diode efficiency

$\eta = \frac{I_{c+} - |I_{c-}|}{I_{c+} + |I_{c-}|}$

can reach considerable values (up to 45%) and be modulated by local and nonlocal gate voltages and phases. In addition to phase control, local magnetic configurations (spin valves with tunable barrier magnetization) enable switching and reversal of the diode effect (Hodt et al., 2023).

The microscopic origin of JDE in Andreev molecules lies in the asymmetric weighting of double elastic cotunneling and double crossed Andreev reflection processes, further enhanced by the contribution of leaky ABSs merging with the continuum.

4. Experimental Realizations and Spectroscopic Signatures

Andreev molecules have been experimentally realized in various platforms:

1. Quantum dot arrays: Hybridized ABSs in double quantum dots (DQDs) proximitized by superconductors were observed via bias and tunnel spectroscopy, with clear gate control of singlet, doublet, and triplet ground states (Su et al., 2016, Zalom et al., 29 Jul 2024, Kürtössy et al., 30 Jun 2024).
2. Josephson junction arrays: Planar JJ devices with shared electrodes at separations $\lesssim\xi_0$ demonstrate modulated CPRs and anomalous phase shifts (Matsuo et al., 2023, Haxell et al., 2023, Shi et al., 6 Aug 2025).
3. Nanowire devices: InAs/Al nanowire-based Andreev molecules exhibit nonlocal JDE, gate-controlled critical current modulation, and periodic oscillations of diode efficiency with flux (Zhu et al., 19 Aug 2025).
4. Long‐distance coupling: Experiments have shown photon-mediated hybridization and coherent dynamics between Josephson junctions separated by millimeters, using engineered microwave circuits and two-tone spectroscopy to probe molecular states via mutual inductance (Samuelsen et al., 21 Jul 2025).
5. Single-molecule junctions: STM measurements of phthalocyanine molecules on Pb(111) demonstrate AR controlled by aligning a molecular orbital to the Fermi energy, indicating that single-molecule Andreev molecules are realizable and their AR response is tunable via tip-molecule distance (Meyer et al., 2 Apr 2025).

Key spectroscopic signatures of Andreev molecules include avoided crossings in the ABS spectrum, side peaks corresponding to vibrational (vibronic) transitions, phase-dependent superconducting gap closing, and dispersive shifts in coupled oscillator readouts.

5. Gate Control, Scalability, and Sensing

A pivotal recent advance is the demonstration of electrostatic (gate) control over Andreev molecule behavior. In gate-controlled Andreev molecules (“Type II”), local tuning of the density of states (DOS) in one junction nonlocally modulates the critical current and phase behavior of a neighboring junction without the need for global magnetic flux control (Shi et al., 6 Aug 2025). This approach facilitates:

• Superior architectural scalability (elimination of mutual flux loops and cross-talk)
• Independent site control enabling extended chains and emulation of Kitaev models
• Integration of noninvasive parity/charge sensors capable of single Cooper pair sensitivity

In these systems, the nonlocal enhancement of critical current, high gate sensitivity, and ability to resolve parity states via charge sensing provide essential capabilities for future quantum computing and sensing platforms.

6. Topological and Correlated Phenomena

Andreev molecules serve as versatile building blocks for engineered topological superconductors. Double quantum dot Andreev molecules can be extended to Kitaev chains, with hybridization enabling the realization and manipulation of Majorana zero modes (Su et al., 2016, Shi et al., 6 Aug 2025). The ability to drive quantum phase transitions (e.g., from antiferromagnetic to ferromagnetic ground states in heteroatomic Andreev molecules) through gate control or occupation tuning has been quantitatively analyzed via advanced numerical methods including density matrix renormalization group (DMRG) and numerical renormalization group (NRG) (Kürtössy et al., 30 Jun 2024, Zalom et al., 29 Jul 2024).

In parallel DQD Andreev molecules, the emergence of triplet ground states is a haLLMark of strong lead-mediated interactions and can only be captured by full NRG or extended zero-bandwidth models; standard atomic-limit approximations are inadequate in these regimes (Zalom et al., 29 Jul 2024).

7. Implications, Applications, and Future Directions

The demonstrated control, nonlocal phenomena, and tunable ground state properties of Andreev molecules have direct impact for:

• Design of quantum devices and qubits exploiting ABS/Andreev molecular degrees of freedom
• Parity-to-charge conversion and high-fidelity parity readout, essential for topological qubits (Driel et al., 2023)
• Robust and scalable quantum circuit architectures for metrology, logic, and sensing
• Investigation of nonreciprocal superconductivity and Josephson diode devices with tunable, local control (including spin-dependent configurations)
• Realization of long-range coherent coupling between spatially separated superconducting elements, crucial for modular quantum information networks

Andreev molecules thus offer a unified platform for exploring correlations, quantum transport, nonequilibrium dynamics, symmetry-breaking phenomena, and topological effects in engineered superconducting nanostructures. The continued development of gate-controlled, tunable, and scalable Andreev molecule devices is poised to shape future progress in quantum information processing, quantum simulation, and superconducting electronics.