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 $Q\equiv S_{ \rm{ }M}/S_0$, which represents the degree of the insufficiency of the resolution. Here, $S_0$ is the intrinsic noise, and $S_{ \rm{ }M}$ 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 $2\exp[-(2\pi)^2/Q]$ having an essential singularity at $Q=0$, which followed by the square root dependence $\sqrt{ Q/4\pi}$ for $Q\gg1$. Our findings offer an explanation for the anomalous enhancement of noise temperature observed in Johnson noise thermometry.
Categories: cond-mat.mes-hall cond-mat.stat-mech quant-ph