Tunneling and quantum noise in one-dimensional Luttinger liquids

We study nonequilibrium noise in the transmission current through barriers in one-dimensional Luttinger liquids and in the tunneling current between edges of fractional quantum Hall liquids. The distribution of tunneling events through narrow barriers can be described by a Coulomb gas lying in the time axis along a Keldysh (or nonequilibrium) contour. We show that the charges tend to reorganize as a dipole gas,which we use to describe the tunneling statistics. The dipole-gas picture allows us to have a unified description of the low-frequency shot noise and the high-frequency Josephson noise. The correlation between the charges within a dipole (intradipole) contributes to the high-frequency Josephson noise, which has an algebraic singularity at ω=${\mathit{e}}^{\mathrm{*}}$V/ħ, whereas the correlations between dipoles (interdipole) are responsible for the low-frequency noise. We show that an independent or noninteracting dipole approximation gives a Poisson distribution for the locations of the dipole centers of mass, which gives a flat noise spectrum at low frequencies and corresponds to uncorrelated shot noise. Including interdipole interactions gives an additional 1/${\mathit{t}}^2$ correlation between the tunneling events that results in an ‖ω‖ singularity in the noise spectrum. We present a diagrammatic technique to calculate the correlations in perturbation theory, and show that contributions from terms of order higher than the dipole-dipole interaction should only affect the strength of the ‖ω‖ singularity, but its form should remain ∼‖ω‖ to all orders in perturbation theory. A counting argument also suggests that the leading algebraic singularity at ${\mathrm{\omega }}_{\mathit{J}}$ should be ∝‖ω-${\mathrm{\omega }}_{\mathit{J}}$${\mathrm{\Vert }}^{2\mathit{g}\mathrm{-}1}$ to all orders in perturbation theory.