Thermodynamic bound on current fluctuations in coherent conductors

TL;DR

This work addresses how to bound the full distribution of particle currents in coherent quantum conductors. It develops a universal large-deviation bound I(j) <= I_qu(j) by using scattering theory and the Levitov-Lesovik scaled cumulant generating function, with the bound depending only on the mean current J and the total entropy production sigma under symmetry conditions. The bound recovers the quantum thermodynamic uncertainty relation for typical fluctuations and can be saturated in simple two-terminal boxcar transmission models, as demonstrated in a chain of quantum dots. The results provide a universal, experimentally accessible benchmark for current fluctuations in nanoscale devices and extend thermodynamic uncertainty principles to coherent transport, with implications for thermodynamic inference and device optimization; they also discuss robustness to dephasing and potential extensions to broken time-reversal symmetry and to other currents.

Abstract

We derive a universal bound on the large-deviation functions of particle currents in coherent conductors. This bound depends only on the mean value of the relevant current and the total rate of entropy production required to maintain a non-equilibrium steady state, thus showing that both typical and rare current fluctuations are ultimately constrained by dissipation. Our analysis relies on the scattering approach to quantum transport and applies to any multi-terminal setup with arbitrary chemical potential and temperature gradients, provided the transmission coefficients between reservoirs are symmetric. This condition is satisfied for any two-terminal system and, more generally, when the dynamics of particles within the conductor are symmetric under time-reversal. For typical current fluctuations, we recover a recently derived thermodynamic uncertainty relation for coherent transport. To illustrate our theory, we analyze a specific model comprising two reservoirs connected by a chain of quantum dots, which shows that our bound can be saturated asymptotically.

Figures (2)

• Figure 1: Current fluctuations in a single quantum dot. Left: The sketch shows the considered system, which consists of $n$ quantum dots with nearest-neighbor tunneling, coupled to two reservoirs with the same temperature $T$ and chemical potentials $\mu_1 = E_0 + k_\mathrm{B} T\mathcal{F}/2$ and $\mu_2 = E_0 -k_\mathrm{B} T\mathcal{F}/2$. The blue line shows the rate function $I(j)$ in units of $I_0 = k_\mathrm{B} T/h$ as a function of the normalized fluctuating current $j/J$; the orange line indicates the upper bound $\hat{I}_\mathrm{qu}(j)$. Right: Behavior of $I(j)$ and $\hat{I}_\mathrm{qu}(j)$ around the mean current $J$. For comparison, the dashed line shows the classical bound $\hat{I}_\mathrm{cl}(j)$. For all plots, we have used the transmission function (\ref{['eq:exp:TransFun']}) with $w=k_\mathrm{B} T/4$ and $n=1$, and set $\mathcal{F}=4$ such that $\sigma/4k_\mathrm{B} J =\mathcal{F}/4 = 1$ and $R\simeq 1.31$.
• Figure 2: Quantum suppressed current fluctuations in chains of quantum dots. The plots show the same quantities as the right panel of Fig. \ref{['fig:exp:1']} for an increasing number $n$ of dots under otherwise identical conditions. The classical upper bound $\hat{I}_\mathrm{cl}(j)$ on $I(j)$ is violated for $n\geq 2$, while the quantum bound $\hat{I}_\mathrm{qu}(j)$ is almost saturated for $n=20$.