Current precision in interacting hybrid Normal-Superconducting systems

We study Andreev-mediated transport and current fluctuations in interacting normal-superconducting quantum-dot systems. Using a generalized master equation based on real-time diagrammatics and full counting statistics, we compute the steady-state current, zero-frequency noise, and rate of entropy production in the large superconducting-gap limit. We show how Coulomb interactions modify Andreev-mediated transport by renormalizing resonant conditions and suppressing superconducting coherence, leading to a pronounced reduction of current precision even when average currents are only weakly affected. These effects are particularly evident at high temperatures, where conventional Coulomb-blockade features are thermally smeared while fluctuation properties remain highly sensitive. By analyzing thermodynamic uncertainty relations, we demonstrate that violations of the quantum bound present in the noninteracting regime are progressively reduced and eventually suppressed as interactions increase, whereas the recently proposed hybrid bound remains satisfied. Our results clarify how Coulomb interactions, and nonequilibrium fluctuations jointly determine transport properties in hybrid superconducting devices, and establish current precision as a robust benchmark for interacting Andreev transport beyond the noninteracting limit.

💡 Research Summary

In this work the authors investigate how electron‑electron interactions affect current precision and thermodynamic uncertainty relations (TURs) in hybrid normal‑superconducting nanostructures. The study focuses on two prototypical setups in the limit of an infinitely large superconducting gap (Δ→∞): (i) a single quantum dot described by the Anderson model and (ii) a Cooper‑pair splitter consisting of two quantum dots coupled to a common superconductor. Both systems are attached to one (or two) normal metallic leads. The normal leads are treated within the wide‑band approximation with a constant tunneling rate Γ_N, while the coupling to the superconductor is split into a local Andreev tunneling amplitude Γ_S (pair creation/annihilation on a single dot) and a non‑local amplitude Γ_C (pair splitting between two dots).

The theoretical framework combines real‑time diagrammatic perturbation theory with full counting statistics (FCS). Transition rates are evaluated to first order in the normal‑lead coupling Γ_N and to linear order in the superconducting couplings Γ_S and Γ_C, while the Coulomb interaction U is treated exactly within the reduced density‑matrix formalism. By introducing a counting field χ for the normal leads, the χ‑dependent kernel W(χ) is constructed; its derivatives at χ=0 give the stationary current I, the zero‑frequency noise S, and, via the Drazin pseudoinverse, the current‑current correlations. The entropy production rate follows from σ = V I/(e T), where V = –μ_N/e is the bias applied to the normal leads.

Key findings can be summarized as follows:

1. Interaction‑induced renormalization of Andreev resonances – At particle‑hole symmetry (ε = –U/2) the local Andreev reflection (LAR) current exhibits a resonance when the empty‑ and doubly‑occupied dot energies coincide (E_0 = E_D). Increasing U shifts this resonance and introduces a Coulomb‑blockade threshold at |μ_N| = U/2. While the average current remains relatively robust against U (especially at high temperature where thermal smearing dominates), the noise S is strongly affected: the Fano factor F = S/(e|I|) grows with U, indicating that interactions suppress the coherent Andreev contribution and enhance incoherent fluctuations.

2. Thermodynamic uncertainty relations – Three TURs are examined: the classical TUR (F – 1 ≥ 0), the quantum TUR for non‑interacting phase‑coherent conductors (Q = F sinh(eσ/2|I|) – 1 ≥ 0), and the recently proposed hybrid TUR (Q_H = F² sinh(eσ|I|) – 1 ≥ 0). In the non‑interacting limit (U = 0) the system can violate both the classical and quantum TURs when Γ_S ≈ 5/3 Γ_N and the temperature is comparable to the tunnel broadenings, reproducing earlier results obtained with exact nonequilibrium Green’s functions. As U is increased, the quantum TUR violation diminishes and eventually disappears: the interaction‑induced decoherence raises the noise relative to the current, while the entropy production (proportional to V I) is reduced, driving Q back to positive values. The hybrid TUR, however, remains satisfied for all parameter sets, confirming its robustness in Andreev‑dominated transport even when interactions are strong.

3. Effect of non‑local Andreev processes (Cooper‑pair splitter) – Introducing Γ_C adds a second transport channel that splits Cooper pairs between the two dots. This enhances the sensitivity of the noise to U and can amplify TUR violations in the non‑interacting case. Nevertheless, the same trend as in the single‑dot case is observed: stronger Coulomb repulsion suppresses the quantum TUR violation, while the hybrid TUR stays intact.

4. Temperature dependence – At high temperatures (k_B T ≫ Γ_N, Γ_S) the average current is thermally broadened and shows only a weak dependence on U, but the noise retains a pronounced U‑dependence. Consequently, the current precision (quantified by the TURs) is governed more by interaction effects than by thermal smearing.

Overall, the paper demonstrates that Coulomb interactions, by renormalizing Andreev resonances and diminishing superconducting coherence, lead to a substantial reduction of current precision even when the mean current is barely altered. This highlights current precision as a sensitive benchmark for the interplay of superconducting proximity effects, electron‑electron interactions, and nonequilibrium fluctuations. The authors also show that while the quantum TUR can be violated in ideal, non‑interacting hybrid devices, realistic interacting nanostructures naturally restore the bound, whereas the hybrid TUR provides a universally valid constraint.

Methodologically, the combination of real‑time diagrammatics with full counting statistics proves capable of treating strong local interactions exactly while retaining a systematic expansion in the tunnel couplings. The approach can be extended to multi‑level dots, finite superconducting gaps, and temperature gradients, opening avenues for future studies of precision thermodynamics in more complex hybrid superconducting circuits.