Numerical evaluation of shot noise using real-time simulations

Abstract

We present a method to determine the shot noise in quantum systems from knowledge of their time evolution -the latter being obtained using numerical simulation techniques. While our ultimate goal is the study of interacting systems, the main issues for the numerical determination of the noise do not depend on the interactions. To discuss them, we concentrate on the single resonant level model, which consists in a single impurity attached to non-interacting leads, with spinless fermions. We use exact diagonalisations (ED) to obtain time evolution, and are able to use known analytic results as benchmarks. We obtain a complete characterization of finite size effects at zero frequency, where we find that the finite size corrections scale ∝ G 2 , G the differential conductance. We also discuss finite frequency noise, as well as the effects of damping in the leads. PACS numbers: U U J J' J' J J J J J V g FIG . 1: Sketch of the IRLM model. J ′ denotes the coupling of the impurity to the left and right lead, J is the hopping element in the leads, U is the interaction on the contact link, and Vg is a gate voltage. In this work we set Vg = 0 (the dot level is on resonance), and U = 0 (non-interacting case, which allows to treat the problem via ED).

Shot Noise in the Self-Dual Interacting Resonant Level Model

“…We use this idea to study the noise in the IRLM at the self dual point. Extending and adapting the ideas in [15] we are able to obtain this noise analytically in the scaling limit, while we are also able to carry out the proposal of [16] to determine this noise numerically. These two are in excellent agreement.…”

“…− S analyt. ∝ G 2 /M with G the differential conductance that has been given for the free case in [16]. The strong deviations in the high voltage regime from this relation may be traced back to different sources: the approximative td-DMRG scheme introduces a cutoff error that gets especially pronounced for values of the voltage of the order of the bandwidth.…”

“…Nevertheless, it has been realized very recently that the shot noise could indeed be reliably extracted from real time simulations, the finite size effects being controllable partly by analytical arguments, and partly by linear finite size extrapolations [16]. We use this idea to study the noise in the IRLM at the self dual point.…”

“…1, where we show zero-frequency shot noise as well as the finite size error of the numerical data, rescaled by the system size. In the low frequency limit, strong finite size effects have to be expected, that get mostly pronounced for small values of the voltage [16]. Since the rescaled finite size error happens to collapse on a single curve in the low voltage regime, the numerical data can be linearly extrapolated to infinite system size in order to obtain results for the thermodynamic limit.…”

“…Third, after choosing a starting point t ′ ≫ t S we have to simulate until t max < t R , where t max −t ′ has to be large enough to allow for a faithful calculation of the noise power spectrum. A detailed analysis of the finite size correction in the non-interacting case is provided in [16].…”