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Time resolved transport properties of a $Y$-junction of Tomonaga-Luttinger liquids

We study time resolved transport properties of a $Y$-junction composed of interacting one-dimensional quantum wires using a bosonization approach. In particular, we investigate the AC conductivity of the $Y$-junction formed from finite length Tomonaga-Luttinger liquid wires based on a plasmon scattering approach for injected charge pulses of arbitrary shapes. In addition, we calculate the tunneling current and quantum noise of the $Y$-junction arising from point-like tunneling impurities at the junction, including finite temperature effects. Our results will be useful for designing nano-electronic quantum circuits, and for interpreting time-resolved experiments [H. Kamata et. al. in Nat. Nanotechnol. 9, 177 (2014)] in interacting wires and their junctions.