One-channel conductor coupled to a quantum of resistance: Exact ac conductance and finite-frequency noise

Abstract

We consider a one-channel coherent conductor with a good transmission embedded into an Ohmic environment, the impedance of which is equal to the quantum of resistance ${R}_{q}=h/{e}^{2}$ below the $RC$ frequency. This choice is motivated by the mapping of this problem to a Tomonaga-Luttinger liquid with one impurity, the interaction parameter of which corresponds to the specific value $K=1/2$, allowing for a refermionization procedure. The ``new'' fermions have an energy-dependent transmission amplitude, which incorporates the strong correlation effects and yields the exact dc current and zero-frequency noise through expressions similar to those of the scattering approach. We recall and discuss these results for our present purpose. Then, we compute the finite-frequency differential conductance and the finite-frequency nonsymmetrized noise. Contrary to intuitive expectation, both can not be expressed within the scattering approach for the new fermions, even though they are still determined by the transmission amplitude. Even more, the finite-frequency conductance obeys an exact relation in terms of the dc current, which is similar to that derived perturbatively with respect to weak tunneling within the Tien-Gordon theory, and extended recently to arbitrary strongly interacting systems coupled eventually to an environment and/or with a fractional charge. We also show that the emission excess noise vanishes exactly above $eV$, even though the underlying Tomonaga-Luttinger liquid model corresponds to a many-body correlated system. Our results apply for all ranges of temperature, voltages, and frequencies below the $RC$ frequency, and they allow us to explore fully the quantum regime.