Quantum Kinetic Uncertainty Relations in Mesoscopic Conductors at Strong Coupling

Abstract

Kinetic Uncertainty Relations (KURs) establish quantum transport precision limits by linking signal-to-noise ratio (SNR) to the system's dynamical activity, valid in the weak-coupling regime where particle-like transport dominates. At strong coupling, quantum coherence challenges the validity of KURs and questions the concept of activity itself. In this Letter, we achieve two distinct, yet complementary main results. First, we introduce a general definition of dynamical activity valid at arbitrary coupling, which reveals the breakdown of standard KURs at strong coupling. Second, we prove a novel uncertainty relation valid at arbitrary coupling strength, which we denote Quantum KUR (QKUR). This QKUR corresponds to a nontrivial quantum extension of KUR, involving fundamental contributions of the generalized dynamical activity. These two achievements provide a general framework for out-of-equilibrium quantum transport precision analysis, in close analogy with the transition from TURs to QTURs [Phys. Rev. Lett. 135, 046302]. Explicit steady-state expressions are obtained within Green's-function and Landauer-Büttiker formalisms. We illustrate these concepts for paradigmatic quantum-coherent mesoscopic devices: a single quantum channel pinched by a quantum point contact and open single- and double-quantum dot systems.

Figures (3)

• Figure 1: Top panel -- Scheme of a generic multiterminal setup. A quantum system ($\hat{H}_S$) couples to reservoirs ($\hat{H}_\alpha$) via interaction Hamiltonians ($\hat{V}_\alpha$). The generalized dynamical activity is defined through two-time exchange-rate fluctuations $\langle\langle \hat{V}_\alpha(t)\hat{V}_\alpha(t+\tau)\rangle\rangle$. Bottom panel — SNR, i.e.$I^2_L/S_{LL}=I^2_R/S_{RR}$ (solid black), for a two-terminal SQD with equal-temperature reservoirs, as a function of coupling strength $\Gamma/k_B T$, characterized by the transmission probability $\mathcal{T}_{RL}(\epsilon)=\Gamma_L\Gamma_R/\left[\Gamma^2/4+(\epsilon-\epsilon_d)^2\right]$Blasi2024. The KUR bound $\xi_{\rm KUR}$ (blue dotted) and the QKUR bound $\xi_{\rm QKUR}$ (red dashed), calculated from Eq. \ref{['eq:QKUR']}, are shown. KUR is valid only at small $\Gamma$, whereas QKUR holds for any coupling. Parameters: $\epsilon_d/k_BT=3$, $\Delta\mu/k_BT=2$.
• Figure 2: Uncertainty relations for a perfectly transmitting QPC ($\tau = 1$). The ratios $\text{SNR}/\xi_{\mathrm{QKUR}}$ (solid) and $\text{SNR}/\xi_{\mathrm{QTUR}}$ (dashed) Brandner2025 are shown as functions of voltage bias $\Delta\mu/k_B T$ for different thermal biases $\Delta T/k_B T$. Both QKUR and QTUR remain valid at arbitrary coupling strength. Notably, QKUR provides a tight bound far from equilibrium, with $\text{SNR}/\xi_{\mathrm{QKUR}} \rightarrow 1$ at large $\Delta\mu/k_B T$ for all $\Delta T$.
• Figure 3: Contour plots of the ratio ${\rm SNR}/\xi_\mathrm{QKUR}$ for the DQD as a function of $k_B T$ and $\Delta\mu$ (panel (a)), and as a function of the coupling strength $\Gamma \equiv \Gamma_L/2 = \Gamma_R/2$ and interdot tunneling $g$ (panel (b)). White stars mark the parameter values used in the complementary panel. An asymmetric voltage bias is applied such that $\Delta\mu = \mu_L - \mu_R$ with $\mu_R = 0$. No temperature bias is applied, and $\epsilon_1 = \epsilon_2 = 0$.