Current precision in interacting hybrid Normal-Superconducting systems

TL;DR

This work analyzes how Coulomb interactions alter Andreev-mediated transport and current fluctuations in interacting normal–superconducting quantum-dot devices, using a generalized master equation with real-time diagrammatics and full counting statistics in the $\Delta\to\infty$ limit. By treating Coulomb repulsion exactly within a reduced density-matrix framework, the authors compute steady-state current, zero-frequency noise, and entropy production for a single quantum dot and a Cooper-pair splitter, revealing that interactions renormalize Andreev resonances and suppress coherence, thereby reducing current precision even when average currents are weakly affected. Through thermodynamic uncertainty relations (TURs), they show that violations of the quantum TUR observed in the noninteracting regime shrink and eventually vanish as interactions grow, while the hybrid quantum bound remains satisfied, providing a robust diagnostic of coherent transport in hybrid superconducting systems. Overall, current precision emerges as a powerful benchmark for interaction-induced decoherence in Andreev-dominated transport, clarifying the joint roles of superconducting coherence, nonequilibrium fluctuations, and Coulomb effects in these nanoscale devices.

Abstract

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.

Figures (12)

• Figure 1: Schematic setup representation of the two interacting hybrid normal-superconducting systems studied: (a) Single quantum dot, and (b) Cooper-pair splitter. The blue dashed line corresponds to the local tunneling rate $\Gamma_S$, and the red dashed line corresponds to the nonlocal tunneling rate $\Gamma_C$.
• Figure 2: (a) Superconducting current, (b) Noise, and (c) Rate of entropy production in the single dot setup as a function of the Coulomb interaction $U$ and the chemical potential of the normal lead $\mu_N$. The dashed black line represents the Coulomb blockade threshold occurring at $\mu_N=-U/2$. The parameters are $\varepsilon=-U/2$, $k_B T=10\Gamma_N$ and $\Gamma_S=\sqrt{5/3}\Gamma_N$. Energies are in units of $\Gamma_N$.
• Figure 3: Violation of the quantum TUR in the single dot as a function of the Coulomb interaction and the chemical potential of the normal lead. The dashed blue line corresponds to the saturation of the quantum bound $\mathcal{Q}=0$, and the gray area represents the region where the classical TUR is violated. The hybrid quantum bound $\mathcal{Q}_H$ is never violated. Same parameters as in Fig. \ref{['figure2']}.
• Figure 4: (Up) Superconducting current, (Center) Noise, and (Down) Rate of entropy production in the single quantum dot as a function of the gate level $\varepsilon$ and the chemical potential of the normal lead. Each column corresponds to a different Coulomb interaction, $U/\Gamma_N=0,30,60$ from left to right. The rest of parameters are $\Gamma_S=\sqrt{5/3}\Gamma_N$, and $k_BT=10\Gamma_N$.
• Figure 5: Violation of the quantum TUR in the single dot as a function of the gate level $\varepsilon$, centered at the LAR resonance condition $\varepsilon+U/2=0$, and the chemical potential of the normal lead. The outer colored region corresponds to $U=0$. The inner colored region corresponds to $U/\Gamma_N=2$. The dashed blue and white lines correspond to the saturation of the quantum bound $\mathcal{Q}=0$, for $U/\Gamma_N=0,2$, respectively. The dark and light gray areas represents the regions where the classical TUR is violated for $U/\Gamma_N=0,2$, respectively. Same parameters as in Fig. \ref{['figure4']}.