  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.
