[Paper Review] Strong-coupling theory of quantum dot Josephson junctions: role of the residual quasiparticle

Luka Pavešič, Ramón Aguado+1 more|arXiv (Cornell University)|Apr 24, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper develops a strong-coupling effective Hamiltonian for quantum dot Josephson junctions by introducing symmetry-adapted orbitals that include both proximal and distal quasiparticles in superconducting leads. It reveals that a residual quasiparticle bound near the dot at φ ≈ π stabilizes the doublet ground state, explaining the 'doublet chimney' phase diagram feature, while at φ ≈ 0 the quasiparticle delocalizes and plays no active role. The model captures partial screening of the dot spin across all coupling regimes and provides a minimal, finite-bandwidth description beyond standard approximations like ZBA and SAL.

ABSTRACT

We consider an interacting quantum dot strongly coupled to two superconducting leads in a Josephson junction geometry. By defining symmetry-adapted superpositions of states from the leads, we formulate an effective Hamiltonian for the strong-hybridisation regime with a single orbital directly coupled to the dot and three additional indirectly coupled orbitals. This minimal basis set allows to account for the quasiparticles in the vicinity of the dot as well as those further away in the leads, and to describe how their role evolves as a function of coupling strength and phase bias $ϕ$. This formulation also reveals the changing nature of the spin-doublet state for the experimentally relevant coupling strengths. The binding of a nearly decoupled quasiparticle in the vicinity of the QD explains the "doublet chimney" in the phase diagram for $ϕ\sim π$, in contrast to $ϕ\sim 0$ where the residual quasiparticle escapes to infinity and plays no active role.

Motivation & Objective

• To develop a minimal, symmetry-adapted effective Hamiltonian that captures strong hybridization effects in quantum dot Josephson junctions beyond zero-bandwidth and superconducting atomic limits.
• To clarify the role of quasiparticles in the superconducting leads—both near the dot and further away—in determining subgap state properties and phase diagrams.
• To explain the origin of the 'doublet chimney' in the phase diagram at φ ≈ π, which is absent at φ ≈ 0, by identifying a bound residual quasiparticle near the dot.
• To characterize the evolution of the doublet state wavefunction and spin-screening behavior across the full coupling-strength and phase-bias range.
• To provide a framework for understanding and designing Andreev spin qubits by accounting for non-local quasiparticle contributions and partial spin screening.

Proposed method

• Formulate an effective Hamiltonian using symmetry-adapted superpositions of lead orbitals, including one directly coupled orbital and three indirectly coupled ones (proximal and distal symmetric/antisymmetric channels).
• Introduce a finite bandwidth via a hopping term t between lead orbitals, allowing for non-local quasiparticle effects beyond the zero-bandwidth approximation (ZBA).
• Use the resulting minimal basis to describe the full coupling regime—from weak to strong coupling—by incorporating phase-dependent anomalous hopping that models superconducting pairing.
• Map the system's low-energy spectrum and ground state properties using numerical methods, with analytical insights from perturbation theory for small t.
• Analyze the spin-screening character of the ground state via the degree of Kondo-like screening, distinguishing between fully screened, partially screened, and unscreened regimes.
• Construct phase diagrams and addition amplitudes to probe the stability and composition of the doublet state under varying coupling strength V and phase bias φ.

Figure 1: (a) Sketch of the quantum dot Josephson junction. (b, d) Phase diagrams for $\phi=0$ and $\phi=\pi$ within the minimal model, Eqs. ( 1 ) and ( 2 ), for a left-right symmetric system. $V$ is the coupling strength, $\nu=1/2-\epsilon/U$ is the dot filling in units of charge. Parameters are $U

Experimental results

Research questions

• RQ1 Why does the doublet ground state form a 'chimney'-shaped region in the phase diagram only at φ ≈ π and not at φ ≈ 0?
• RQ2 How do quasiparticles in the superconducting leads—especially those not at the dot—contribute to the formation and stability of subgap states?
• RQ3 What is the role of finite bandwidth in the superconducting leads in modifying the effective coupling and quasiparticle localization?
• RQ4 How does the degree of spin screening of the quantum dot evolve with coupling strength and phase bias, and what is its impact on the doublet state?
• RQ5 To what extent can the doublet state be manipulated locally if part of its spin character is delocalized into the superconducting leads?

Key findings

• The 'doublet chimney' at φ ≈ π arises from a residual quasiparticle bound near the quantum dot, which stabilizes the doublet state through a blocking effect, unlike at φ ≈ 0 where the quasiparticle delocalizes and becomes inert.
• At φ ≈ π, the doublet state is stabilized by a nearly localized quasiparticle in the vicinity of the dot, while at φ ≈ 0, the quasiparticle escapes to infinity and does not contribute to the state.
• The quantum dot spin is partially screened in the doublet ground state for all non-zero coupling strengths, with full screening occurring only at very strong coupling.
• The inclusion of finite bandwidth via the hopping term t leads to a splitting of the doublet at φ = 0 due to inter-site singlet formation between distant orbitals, with an energy gain ∼t².
• The doublet chimney is highly sensitive to system symmetry: it vanishes for even small left-right asymmetry (η > 0.05), indicating it requires high symmetry to emerge.
• The addition amplitude χ shows distinct φ- and V-dependent behavior, with the doublet phase stability strongly modulated by both coupling strength and phase bias, especially near φ = π.

Figure 2: Low-energy spectra, singlet (red, $S=0$ ) and doublet (blue, $S=1/2$ ) states. (a) $\phi=0$ and (b) $\phi=\pi$ eigenenergies vs coupling strength $V$ . The top row shows the spectra at small to intermediate $V/\Delta$ and the bottom row shows the spectra up to large $V/\Delta$ . The black