Frequency-dependent counting statistics in interacting nanoscale conductors

Abstract

We present a formalism to calculate finite-frequency current correlations in interacting nanoscopic conductors. We work within the n-resolved density matrix approach and obtain a multi-time cumulant generating function that provides the fluctuation statistics solely from the spectral decomposition of the Liouvillian. We apply the method to the frequency-dependent third cumulant of the current through a single resonant level and through a double quantum dot. Our results, which show that deviations from Poissonian behaviour strongly depend on frequency, demonstrate the importance of finite-frequency higher-order cumulants in fully characterizing transport.PACS numbers: 73.23. Hk,72.70.+m,03.65.Yz Following the considerable success of shot-noise in the understanding of transport through mesoscopic systems [1], attention is now turning towards the higher-order statistics of electron current. The so-called Full Counting Statistics (FCS) of electron transport yields all moments (or cumulants) of the probability distribution P (n, t) of the number of transferred electrons during time t. Despite their difficulty, measurements of the third moment of voltage fluctuations have been made [2,3], and recent developments in single electron detection [4,5,6] promise to open new horizons on the experimental side.The theory of FCS is now well established in the zerofrequency limit [7,8,9]. However, this is by no means the full picture, since the higher-order current correlators at finite frequencies contain much more information than their zero-frequency counterparts. Already at second order (shot-noise), one can extract valuable information about transport time scales and correlations. When the conductor has various intrinsic time scales like, for example, the charge relaxation time and the dwelling time of a chaotic cavity [10], one needs to go beyond second-order in order to fully characterize electronic transport. Apart from this example, and some other notable exceptions [11,12,13], the behaviour of finite-frequency correlators beyond shot-noise is still largely unexplored.In this Rapid Communication, we develop a theory of frequency-dependent current correlators of arbitrary order in the context of the n-resolved density matrix (DM) approach, -a Quantum Optics technique [14] that has recently found application in mesoscopic transport [15]. Within this approach, the DM of the system, ρ(t), is unravelled into components ρ (n) (t) in which n = n(t) = 0, 1, . . . electrons have been transferred to the collector. Considering a generic mesoscopic system with Hamiltonian H = H S + H L + H T , where H S and H L refer to the system and leads respectively, and provided that the Born-Markov approximation with respect to the tunnelling term H T is fulfilled, the time-evolution of this n-resolved DM can be written quite generally aṡwhere the vector ρ (n) (t) contains the nonzero elements of the DM, written in a suitable many-body basis. The Liouvillian L 0 describes the 'continuous' evolution of the system, whereas L J describes the ...

Factorial cumulants reveal interactions in counting statistics

“…13,14 A powerful and elegant formulation in this semiclassical regime is the stochastic path integral approach, which was introduced for FCS in the pioneering works by Pilgram et al [15][16][17] For interacting systems, the FCS can be related to a generalized master equation describing the charge transport [18][19][20][21] or obtained using Keldysh Greens functions.…”

Waiting time distribution of quantum electronic transport in the transient regime

“…9 Theoretically, finite frequency FCS has been studied using quantum master equation and scattering matrix approach. [10][11][12][13] Less attention has been paid on FCS of transient transferred charge. 14,15 Besides FCS, another complementary quantity to characterize the stochastic processes is the waiting time distribution (WTD) which is the distribution of time interval between two successive events.…”

Waiting Time Distributions for the Transport through a Quantum-Dot Tunnel Coupled to One Normal and One Superconducting Lead

“…A well established theoretical tool is the full counting statistics (FCS) [3][4][5][6][7][8][9][10][11][12][13]. The FCS is usually defined in the long-time limit, when the time interval during which transport events are counted is long enough that many particles have passed through the system.…”