Detection of finite frequency photo-assisted shot noise wit h a resonant circuit D. Chevallier, 1,2 T. Jonckheere, 1 E. Paladino, 3,4 G. Falci, 3,4 and T. Martin 1,2 $^{1}$Centre de Physique Th´eorique, UMR6207, Case 907, Luminy, 13288 Marseille Cedex 9, France $^{2}$Universit´e de la M´editerran´ee, 13288 Marseille Cedex 9, France $^{3}$Centre de Physique Th´eorique, Universit´e de la M´editerr an´ee, Case 907, 13288 Marseille, France $^{4}$MATIS CNR-INFM, and D.M.F.C.I., Universit´a di Catania, 95125 Catania, Italy Photo-assisted transport through a mesoscopic conductor occurs when an oscillatory (AC) voltage is super- posed to the constant (DC) bias which is imposed on this conductor. Of particular interest is the photo assisted shot noise, which has been investigated theoretically and experimentally for several types of samples. For DC biased conductors, a detection scheme for finite frequency n oise using a dissipative resonant circuit, which is inductively coupled to the mesoscopic device, was developped recently. We argue that the detection of the finite frequency photo-assisted shot noise can be achieved with the same setup, despite the fact that time translational invariance is absent here. We show that a measure of the photo-assisted shot noise can be obtained through the charge correlator associated with the resonant circuit, where the latter is averaged over the AC drive frequency. We test our predictions for a point contact placed in the fractional quantum Hall effect regime, for the case of weak backscattering. The Keldysh elements of the photo-assisted noise correlator are computed. For simple Laughlin fractions, the measured photo-assisted shot noise displays peaks at the frequency corresponding to the DC bias voltage, as well as satellite peaks separated by the AC drive frequency. PACS numbers: 73.23.-b, 72.70.+m, 73.63.-b, The understanding of the transport properties of nanoscale conductors at low temperatures has known tremendous successes via experiments in a wide range of systems performed for the most past in the stationary regime. Correspondingly, theoretical modelling has allowed the description of these transport processes via scattering theory approaches as well as Hamiltonian formulations, in a fruitful dialogue with experimental investigations. Transport is first characterized by the averag e current flowing through conductors. But further information can be g ained via the measurement and analysis of the current fluctua tions 1,2 and more generally via the higher current moments. 3 Early investigations of quantum transport focused almost exclusively on the low frequency regime. Few recent experiments have probed quantum system on timescales comparable with the electron correlation time, where new physical effects are expected. The present work deals with the detection of quantum noise at such high frequencies, when both a DC and a AC bias is imposed between the source and the drain of the mesoscopic system. Indeed, high frequency measurements can mean several things. First, if only a DC bias is imposed on the sample, a stationary current is generated and high frequencies refer to the Fourier component of the current-current correlation function in time. 4–7 Second, high frequencies can be injected as a drive on the mesoscopic circuit,$^{8–11}$for instance when an additional AC drive is superposed to the DC bias. The later effect is called photo-assisted (PA) transport: electrons undergoing transmission from one lead to another are able to absorb/emit “photons” during thi s process. PA transport, and in particular PA noise has been studied theoretically and experimentally on several occasions for diffusive metals, 5 tunnel junctions, 12 normal metal/ superconductor junctions 13,14 as well as quantum point contacts. 6 The noise caracteristics then displays some structure at values of the DC bias which are multiple of the AC drive frequency. However, high requency noise detection requires special care: conventional (low) frequency noise detection setups are often inadequate for such measurements, and one must often resort to on-chip detection schemes, or alternatively/equivalently to schemes where a good connection to the measurement circuit is achieved through adapted impedence lines. 15 On chip detectors have allowed the detection of single electrons travelling through quantum dots. Such detectors and the device they probe are parts of the same quantum system and must be treated on the same footing. They bear the peculiarity that the noise which is measured is a non trivial combination of non-symmetrized noise correlators. For DC driven systems there are existing proposals to detect high frequency noise using either capacitive or inductive coupling with an on-chip circuit. 16 In a recent theoretical work, a LC resonant circuit, which was coupled inductively to the mesoscopic device circuitry, was employed as a detector of both noise and higher current moments (third moment). 17 The description of this generic detector included its electromagnetic environement, described at a bath of harmonic oscillators with the Caldeira Legett model$^{18}$. Pre- dictions were made on the role of such a dissipative environment and on the relevance of this harmonic detector to capture on high frequency current moments. However, this study considered the case of a mesoscopic device in a stationary regime (with a DC bias only). The hypothesis of a stationary regime greatly simplies the analysis of the detection process because of time translational invariance. The presence of an additional AC voltage drive breaks such a property. Given the interest in the study of time driven mesoscopic systems and in particular PA noise, it seems necessary to address how detection with an auxiliary circuit can be achieved in such situations. The purpose of this work is to present a high frequency detection scheme for photoassisted noise, and to illustrate it with a calculation of photoassisted noise in a specific si tuation arXiv:1001.1910v1[cond-mat.mes-hall]12 Jan 2010 I. INTRODUCTION