Influence of a Random Telegraph Process on the Transport through a Point Contact Fabian Hassler, 1 Gordey B. Lesovik, 2 and Gianni Blatter 1 $^{1}$Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerland $^{2}$L.D. Landau Institute for Theoretical Physics RAS, 117940 Moscow, Russia (Dated: November 18, 2018) We describe the transport properties of a point contact under the influence of a classical two-level fluctuator. We employ a transfer matrix formalism allowing us to calculate arbitrary correlation functions of the stochastic process by mapping them on matrix products. The result is used to obtain the generating function of the full counting statistics of a classical point contact subject to a classical fluctuator, including extensions to a pair of two-level fluctuators as well as to a quantum point contact. We show that the noise in the quantum point contact is a sum of the (quantum) partitioning noise and the (classical) noise due to the two-level fluctuator. As a side result, we obtain the full counting statistics of a quantum point contact with time-dependent transmission probabilities. PACS numbers: 73.23.-b, 73.63.Nm, 73.50.Bk, 05.60.Gg The impact of mobile impurities on transport through a quantum conductor attracted a lot of attention soon after it was realized that the conductance of a dirty co- herent sample is sensitive to the position of a single im- purity; this discovery formed the basis for the explana- tion of flicker noise as it appears due to the presence of bistable mobile impurities. 1,2 A detailed characteriza- tion of charge transport through a quantum conductor is provided by the full counting statistics (FCS). 3 Dur- ing the past decade, this description has been applied to numerous systems 4 and a first attempt to describe the influence of a two-level fluctuator on the FCS of quan- tum transport has been given in Ref. 5. In the present paper, we calculate the full counting statistics of charge transport through a (classical or quantum) point contact coupled to a classical two-level fluctuator; we go beyond previous studies by considering the combined effects of the presence of one or many mobile impurities, as well as quantum-partitioning and the Fermi statistics on the FCS. Also, we reconsider carefully the case when parti- tioning is neglected and correct previous findings which are flawed when calculating the fourth or higher cumu- lants. The influence of a fluctuating (uncontrollable) envi- ronment on a (controllable) device is a generic problem 6 and our work is related to other studies, e.g., the trans- port statistics through a quantum dot in the Coulomb- blockade regime 7,8 or the effect of a bistability on the transport through a quantum point contact $^{9,10}$. Another example is the dephasing of qubits due to a classical 11 or quantum 12 two-level system, or the studies on 1 /f -noise originating from telegraph noise due to classical 13,14 or quantum 11,15–17 two-level fluctuators. The two problems, full counting statistics of charge transport and dephasing of a quantum system (qubits) are related through the equivalence of fidelity 18 and full counting statistics $^{3}$, as has been pointed out recently. 19 In our analysis below, we describe the time evolution Consider a classical particle which can be trapped in an external potential at two positions denoted by x$_{1}$ and x$_{2}$ , cf. Fig. 1. The potential is characterized by the energies E$_{1}$$_{,}$$_{2}$ associated with the two valleys, which are shifted by the amount Δ = E$_{2}$ − E$_{1}$ with respect to each other, and the height of the barrier U . We assume the dynam- ics to be given by thermally activated hopping over the barrier. 22 The particle performs a random (Brownian) of the two-level fluctuator by rate equations which can be solved explicitly. We then study the full counting statistics of a wire with a conductance depending on the state of the fluctuator. The fluctuator induces noise in the transport current through the wire which we evaluate using a mapping of correlation functions on matrix prod- ucts. Using this mapping, we are able to calculate the full counting statistics of a classical wire coupled to a two- level fluctuator. Furthermore, we discuss the situation of a second (independent) fluctuator and show that a non- linear interaction with the wire can lead to correlations in the noise even though the fluctuators evolve indepen- dent of each other. Finally, we apply our method to the case of a quantum wire which exhibits intrinsic partition- ing (shot) noise. 20,21 We derive a formula which incorpo- rates both classical- (due to the two-level fluctuator) and quantum- (due to the point contact) noise. Thereby, we give an explicit expression for the full counting statistics of a quantum point contact whose transmission probabil- ities change with time. As two-level fluctuators seem to be a major obstacle for achieving solid state implemen- tations of qubits with long coherence times, being able to characterize the influence of a two-level fluctuator on transport through a point contact offers the possibility to learn about the fluctuating environment by measuring the full counting statistics through a nearby quantum point contact. The (partial) overlap of our results with previous work 7,9,11 will be discussed below. II. SINGLE TWO-LEVEL FLUCTUATOR I. INTRODUCTION arXiv:1001.0578v2 [cond-mat.mes-hall] 10 Aug 2011