Universal departure from Johnson-Nyquist relation caused by limited resolution

Abstract

Exploiting the two-point measurement statistics, we propose a quantum measurement scheme of current with limited resolution of electron counting. Our scheme is equivalent to the full counting statistics in the long-time measurement with the ideal resolution, but is theoretically extended to take into account the resolution limit of actual measurement devices. Applying our scheme to a resonant level model, we show that the limited resolution of current measurement gives rise to a positive excess noise, which leads to a deviation from the Johnson-Nyquist relation. The deviation exhibits universal single-parameter scaling with the scaling variable $Q\ensuremath{\equiv}{S}_{\mathrm{M}}/{S}_{0}$, which represents the degree of the insufficiency of the resolution. Here, ${S}_{0}$ is the intrinsic noise, and ${S}_{\mathrm{M}}$ is the positive quantity that has the same dimension as ${S}_{0}$ and is defined solely by the measurement scheme. For the lack of the ideal resolution, the deviation emerges for $Q<1$ as $2exp[\ensuremath{-}{(2\ensuremath{\pi})}^{2}/Q]$ having an essential singularity at $Q=0$, which followed by the square root dependence $\sqrt{Q/4\ensuremath{\pi}}$ for $Q\ensuremath{\gg}1$. Our findings offer an explanation for the anomalous enhancement of noise temperature observed in Johnson noise thermometry.