Transport in a Dissipative Luttinger Liquid

We study theoretically the transport through a single impurity in a one-channel Luttinger liquid coupled to a dissipative (Ohmic) bath. For nonzero dissipation, the single impurity is always a relevant perturbation which suppresses transport strongly. At zero temperature, the current voltage relation of the link is $I\sim \exp (-E_0/eV)$, where $E_0\sim \eta /\kappa$ and $\kappa$ denotes the compressibility. At nonzero temperature $T$, the linear conductance is proportional to $\exp (-\sqrt{\mathcal{C}{E}_0/k_{B}T})$. The decay of Friedel oscillation saturates for a distance larger than $L_{\eta }\sim 1/\eta$ from the impurity.