One-Channel Conductor in an Ohmic Environment: Mapping to a Tomonaga-Luttinger Liquid and Full Counting Statistics

It is shown that a one-channel coherent conductor in an Ohmic environment can be mapped to the impurity problem in a Tomonaga-Luttinger liquid. This allows one to determine nonperturbatively the effect of the environment on $I-V$ curves, and to find an exact relationship between dynamic Coulomb blockade and shot noise. We investigate critically how this relationship compares to recent proposals in the literature. The full counting statistics is determined at zero temperature.

Figure 1

A one-channel coherent conductor with transmission coefficient $\mathcal{T}$ in series with an impedance $Z\left(\omega \right)=R$ for $\omega <{\omega }_R=1/RC$. It is mapped to a TLL with parameter $K^{\prime }=1/(1+r)$, where $r=e^{2}R/h$. The strong (weak) backscattering limit is characterized by a dimensionless amplitude ${\Gamma }_T^{\prime }$ ($v_B^{\prime }$).

Figure 2

Limit of a weak resistance, $r=0.05$: The ratio $x$ of the exact differential noise $G_2(V,r)$ to that at $r=0$ (continuous curve), and to a “renormalized” noise given by $\mathcal{T}(V,r)[1-\mathcal{T}(V,r\left)\right]$ (dashed curve) is plotted against $\log V$ where $V$ is implicitly divided by an arbitrary voltage scale. The bare transmission is taken to be $\mathcal{T}=0.65$, while $\mathcal{T}(V,r)=(h/e^2)(1+r)dI/dV$.