Scattering approach to counting statistics in quantum pumps

Abstract

We consider the Fermi gas in a non-equilibrium state obtained by applying an arbitrary timedependent potential to the Fermi gas in the ground state. We present a general method that gives the quantum statistics of any single-particle quantity, such as the charge, total energy or momentum, in this non-equilibrium state. We show that the quantum statistics may be found from the solution of a matrix Riemann-Hilbert problem. We use the method to study how the finite measuring time modifies the distribution of the charge transferred through a biased quantum point contact.

Time Ordering and Counting Statistics

“…The first proposal [9] is where I(t) is the current operator at the junction and at time t, and · has become the expectation in the initial quantum state of the system. The second one [4] reads, as recast by [14], χ(λ, t) = e iλQ(t) e −iλQ = e iHt e iλQ e −iHt e −iλQ ,…”

Fredholm Determinants and the Statistics of Charge Transport

“…We illustrate the hypotheses (14,15) by presenting a model in which they can be verified. The left and right portions of the single lead mentioned in Sect.…”

“…Finally, in (59, 63) we may, by (15), replace N and N ′ by N U and N ′ U in those places where the subscript is not already present. By passing to a unitary conjugate and adjoint, they reduce to (61, 62).…”

“…Effectively, this means that dP (n)Ω N is the state relevant for a second measurement. The charge transfer is inferred from two measurements [15] of the charge Q, one before and one after the evolution of the system by U . The joint probability for measurements n and m is ( U dP (n)Ω N , dP (m) UdP (n)Ω N ) and the generating function appropriately defined as…”

“…Let us mention a few connections to other works. A related regularization of the LevitovLesovik determinant at zero temperature was used in [15], where the relation of counting statistics to a Riemann-Hilbert problem was studied. Another one, exhibiting the symmetry (10), was proposed in [16].…”