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Dynamical Coulomb Blockade in an interacting 1D system coupled to an arbitrary environment

We study the out-of-equilibrium transport in a Tomonaga-Luttinger liquid containing a weak or a tunneling barrier coupled to an arbitrary electromagnetic environment. This applies as well to a coherent one-channel non-interacting conductor with a transmission coefficient close to one or to zero. We derive formal expressions for the current and finite-frequency (FF) noise at arbitrary voltages, temperatures and frequency-dependent impedance $Z(ω)$ in the regimes of weak and strong backscattering. We show that these two regimes are no longer related by duality at finite frequency. We then carry explicit computations of the nonlinear conductance and FF noise when $Z(ω)$ describes an harmonic oscillator such as a LC circuit or a cavity.