Finite frequency current noise in the Holstein model

We investigate the effects of local vibrational excitations in the nonsymmetrized current noise $S(\omega)$ of a nanojunction. For this purpose, we analyze a simple model - the Holstein model - in which the junction is described by a single electronic level that is coupled to two metallic leads and to a single vibrational mode. Using the Keldysh Green's function technique, we calculate the nonsymmetrized current noise to the leading order in the charge-vibration interaction. For the noise associated to the latter, we identify distinct terms corresponding to the mean-field noise and the vertex correction. The mean-field result can be further divided into an elastic correction to the noise and in an inelastic correction, the second one being related to energy exchange with the vibration. To illustrate the general behavior of the noise induced by the charge-vibration interaction, we consider two limit cases. In the first case, we assume a strong coupling of the dot to the leads with an energy-independent transmission whereas in the second case we assume a weak tunneling coupling between the dot and the leads such that the transport occurs through a sharp resonant level. We find that the noise associated to the vibration-charge interaction shows a complex pattern as a function of the frequency $\omega$ and of the transmission function or of the dot's energy level. Several transitions from enhancement to suppression of the noise occurs in different regions, which are determined, in particular, by the vibrational frequency. Remarkably, in the regime of an energy-independent transmission, the zero order elastic noise vanishes at perfect transmission and at positive frequency whereas the noise related to the charge-vibration interaction remains finite enabling the analysis of the pure vibrational-induced current noise.