🔬 mesoscale physics

Quasiparticle-resolved variational theory of Andreev spin qubits

This paper develops a continuum variational theory for Andreev spin qubits that reveals how coherent two-quasiparticle configurations dominate spin-dependent Josephson couplings and provides closed-form expressions for key parameters, bridging strongly correlated quantum-dot models with effective circuit Hamiltonians.

Original authors: Teodor Iličin, Rok Žitko

Published 2026-10-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Teodor Iličin, Rok Žitko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum computing, information is stored in the most fragile of states: the spin of a single electron. This spin acts like a tiny compass needle, pointing either up or down to represent a zero or a one. To protect this delicate information from the noisy environment, scientists often trap the electron inside a superconducting circuit, a material that conducts electricity with zero resistance. When an electron is confined in such a circuit, it forms a special kind of bound state known as an Andreev bound state. These states are the heart of a new type of quantum bit, or qubit, called an Andreev spin qubit. The challenge for researchers is to understand exactly how this trapped electron interacts with the superconducting sea of electrons around it. While the electron carries the information, it does not exist in isolation; it is constantly exchanging energy and virtual particles with the superconductor, creating a complex dance of quantum mechanics that determines how the qubit behaves and how it can be controlled.

A team of physicists at the Jožef Stefan Institute in Slovenia has developed a new way to map this interaction with high precision. They focused on a specific setup where a quantum dot, acting as a tiny island for electrons, is connected to two superconducting leads. In this system, the electron on the island can interact with the superconducting leads through two different paths: a direct path through the island itself, and a background path that bypasses the island. The researchers wanted to know how these interactions create the energy differences between the qubit's spin-up and spin-down states, a property essential for reading and writing quantum information. To do this, they created a detailed mathematical model that tracks the behavior of the electron and the "quasiparticles"—excitations in the superconductor that act like particles—surrounding it.

The core of their work involves a method called variational theory, which is essentially a way of guessing the most likely shape of the quantum wavefunction and then refining that guess until it fits the laws of physics as closely as possible. In this case, the researchers built a model that includes the possibility of the system having zero, one, or two quasiparticles in the superconducting leads at any given moment. While the probability of finding two quasiparticles is very small, the team discovered that these rare configurations are surprisingly important. Even though they appear infrequently, they interfere with the more common zero-quasiparticle state in a way that significantly alters the energy of the qubit. Specifically, these two-quasiparticle configurations contribute exactly half of the leading effect that makes the spin-up and spin-down states have different energies, even when the electron-electron repulsion is weak. As the repulsion between electrons grows stronger, these two-quasiparticle states become the dominant factor in determining the qubit's behavior.

The researchers tested their model against powerful computer simulations known as numerical renormalization group calculations, which are considered a gold standard for solving these types of problems. Their results matched the simulations very closely, confirming that their simplified model captures the essential physics. They found that the strength of the interaction between the qubit and the superconductor depends heavily on the Coulomb repulsion, which is the force that pushes electrons apart. When this repulsion is strong, the conventional energy that drives the supercurrent is suppressed much more quickly than the spin-dependent energy. This means that in systems with strong electron repulsion, the spin-dependent effects become relatively more important, even though the overall coupling to the superconductor gets weaker. This finding helps explain why certain experimental setups show strong spin-dependent signals despite having strong electron interactions.

Beyond just calculating energies, the team mapped out where the virtual quasiparticles actually live in space. They found that these particles are not confined to the tiny quantum dot but are spread out over a distance comparable to the coherence length of the superconductor, which is a measure of how far the superconducting order extends. Furthermore, they discovered that the spin of the electron is not entirely localized on the dot; a portion of it is shared with the superconducting leads. This sharing is influenced by a property called spin-orbit coupling, which links the electron's spin to its motion. When a magnetic field is applied, this spin sharing creates a unique effect: a supercurrent that flows even when there is no voltage difference across the junction. This current is the same for both spin-up and spin-down states, meaning it does not depend on the qubit's information but rather on the physical redistribution of the spin itself.

The study provides a clear bridge between the complex, microscopic reality of interacting electrons and the simpler, effective models used to design quantum circuits. By showing exactly how the wavefunction structure determines the couplings, the researchers have offered a tool for engineers to predict how these qubits will behave in real devices. They demonstrated that ignoring the rare two-quasiparticle events would lead to significant errors in predicting the qubit's performance. The work confirms that to truly understand and control these quantum bits, one must account for the subtle, coherent interplay between the trapped electron and the virtual excitations of the superconductor, a relationship that is far more intricate and influential than previously assumed.

Technical Summary: Quasiparticle-resolved variational theory of Andreev spin qubits

Problem Statement
Andreev spin qubits (ASQs) encode quantum information in the spin of a single quasiparticle confined within an Andreev bound state of a superconducting point contact or hybrid heterostructure. While phenomenological Hamiltonians successfully describe ASQs in many regimes, a microscopic understanding of the spin-dependent Josephson coupling ( $E_S$ ) and the role of virtual quasiparticle (QP) configurations remains incomplete. Specifically, the interplay between Coulomb interactions, spin-orbit coupling (SOC), and the continuum of superconducting states requires a description that goes beyond simple effective models or low-order perturbation theory. Open questions include the spatial extent of the spin-carrying bound state, the onset of partial Kondo screening, and the precise microscopic origin of spin splitting in the presence of continuum states.

Methodology
The authors develop a continuum variational description for the odd-parity doublet in a quantum-dot Josephson junction modeled by an Anderson impurity model (AIM) coupled to two superconducting leads. The system includes:

Hamiltonian: An interacting impurity orbital (Coulomb repulsion $U$ , detuning $\delta$ ) coupled to BCS leads via spin-conserving hopping ( $V$ ) and a direct interlead tunneling term ( $W_\sigma$ ) that incorporates spin-dependent amplitudes and SOC.
Variational Ansatz: The wavefunction is constructed as a superposition of configurations containing up to two Bogoliubov quasiparticles in the leads. This includes the vacuum state (0-QP), single-QP states (1-QP), and double-QP states (2-QP), which are further decomposed into singlet and triplet bath pairs.
Symmetry and Channels: The authors utilize a unitary symmetry ( $\eta$ ) and introduce symmetric/antisymmetric QP channels ( $\gamma_{s,k,\sigma}$ ) to simplify the variational equations.
Approximation Schemes: The coupled integral equations derived from the variational principle are solved using three levels of kernel truncation (Level 1: 1-QP only; Level 2: 2-QP without feedback; Level 3: 2-QP with mixed-kernel feedback).
Validation: The variational results are benchmarked against Numerical Renormalization Group (NRG) and Density Matrix Renormalization Group (DMRG) calculations using the open-source qdjj solver.

Key Contributions and Results

Analytical Expressions for Josephson Couplings:
The authors derive closed-form analytical expressions for both the conventional Josephson energy ( $E_J$ ) and the spin-dependent Josephson energy ( $E_S$ ) at finite superconducting gap $\Delta$ .
- Conventional Coupling ( $E_J$ ): Recovered at second order in hybridization ( $\Gamma^2$ ), consistent with fourth-order perturbation theory results for multilevel dots.
- Spin-Dependent Coupling ( $E_S$ ): Derived at leading order in the direct tunneling amplitude ( $W$ ) and hybridization ( $\Gamma$ ). The expression shows that $E_S$ is suppressed by the factor $\Delta/(\Delta + U/2)$ relative to the atomic limit.
Role of Two-Quasiparticle Configurations:
A central finding is that 2-QP configurations, despite having small probability weights, are essential for the magnitude of the spin splitting.
- Intersector Coherence: The spin-dependent coupling arises from the coherence between the 0-QP (vacuum) and 2-QP sectors, as well as the 1-QP sector.
- Quantitative Contribution: At zero Coulomb repulsion ( $U=0$ ), the 0–2-QP interference contributes exactly half of the leading spin-dependent coupling. As $U$ increases, this interference term dominates the 1-QP contribution, supplying the leading $1/U$ behavior of the coupling.
Accuracy and Convergence:
- Benchmarking: The Level-3 variational approximation (including 2-QP feedback) shows excellent agreement with NRG and DMRG for energies and wavefunction components, with discrepancies primarily arising from omitted 3-QP and higher sectors.
- Truncation Errors: The error introduced by kernel truncation within the 2-QP space is found to be minor (less than 0.255% for energy at $\phi=0$ ). The larger deviations from NRG at strong hybridization are attributed to the exclusion of higher-QP sectors rather than the kernel approximation itself.
Spin Redistribution and Magnetic Response:
- SOC-Induced Transfer: The inclusion of SOC induces a transfer of spin between the dot and the leads. This redistribution is captured by the change in the local-moment fraction $q_d$ .
- Common Josephson Current: When the Zeeman energy on the dot differs from that in the leads ( $b_{rel} \neq 0$ ), the SOC-induced spin transfer generates a Josephson current contribution common to both qubit states. This current is finite even at zero phase bias, representing an anomalous Josephson effect ( $\phi_0$ -junction behavior) driven by the interplay of SOC and magnetic field gradients.
Spatial Structure:
The authors map the localization of virtual QPs and the physical spin density in the leads.
- Localization: Virtual QPs extend over the BCS coherence length $\xi_0$ . The 2-QP component is dominated by a single natural orbital per channel, with occupation probabilities exceeding 95%.
- Spin Density: The physical spin density in the leads is positive, but the dot-lead spin correlation function changes sign with distance. The 0–2-QP interference term dominates the spatial redistribution of spin without altering the integrated spin value.

Significance
The paper establishes a rigorous microscopic link between interacting quantum-dot models and the effective $E_J$ – $E_S$ Hamiltonians used to design ASQ circuits. It clarifies that the wavefunction structure, specifically the coherence between different QP sectors, dictates the couplings. The results provide quantitative constraints for circuit design, demonstrating that Coulomb repulsion suppresses the conventional and spin-dependent couplings differently ( $E_J \propto U^{-2}$ vs. $E_S \propto U^{-1}$ in the large- $U$ limit). This implies that while the relative spin dependence ( $E_S/E_J$ ) may increase with interaction, the absolute magnitude of the spin-dependent coupling decreases. The work also identifies the specific mechanism (0–2-QP coherence) responsible for the spin splitting, offering a more accurate predictive framework than previous phenomenological or low-order perturbative approaches.

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