Frequency-dependent current correlation functions from scattering theory

Abstract

We present a general formalism based on scattering theory to calculate quantum correlation functions involving several time-dependent current operators. A key ingredient is the causality of the scattering matrix, which allows one to deal with arbitrary correlation functions. The formalism proves useful, e.g., in view of recent developments in full counting statistics of charge transfer, where detecting schemes have been proposed for measurement of frequency dependent spectra of higher moments. Some of these schemes are different from the well-known fictitious spin detector and therefore generally involve calculation of non-Keldysh-contourordered correlation functions. As an illustration of the approach we consider various third order correlation functions of current, including the usual third cumulant of current statistics. We investigate the frequency dependence of these correlation functions explicitly in the case of energy-independent scattering. The results can easily be generalized to the calculation of arbitrary nth order correlation functions, or to include the effect of interactions.

Time Ordering and Counting Statistics

“…In [7] the discrepancy was claimed to be entirely due to the difference existing between unordered and T -ordered correlators [9,4]. Our explanation is different, as we spell out momentarily.…”

“…Second, we will present a model of energy independent scattering, where that difference matters. Though the model is implicit in previous works [2,5,7], its formalization allows to establish the equivalence between the in/out-and the T * -ordering of currents and hence between the two approaches mentioned at the beginning.…”

“…This remark being made, the scattering amplitude from |t 1 , α 1 to |t 2 , α 2 is denoted by S α 2 α 1 (t 2 − t 1 ). Equivalence between in/outand T -ordering has been shown [10,7] under the assumption that the scattering matrix satisfies S(t) = 0 , (t ≤ 0) .…”

“…An application of these findings is the clarification of a discrepancy between [8] and [7] concerning the third cumulant of charge transfer through a tunnel junction. In [7] the discrepancy was claimed to be entirely due to the difference existing between unordered and T -ordered correlators [9,4].…”

Model of qubits as devices to detect the third moment of current fluctuations

“…27 within the framework of scattering theory, 2 we can obtain an explicit expression of ⌳ ͑3͒ in terms of the transmission eigenvalues T n and of the voltage bias V across the junction. In particular, in the case of energy-independent scattering and in the limit of zero temperature of the noise source, we obtain…”