Energy Reflection and Transmission of Interfaces in $T\bar{T}$-deformed CFT

Conformal interfaces gluing a pair of two-dimensional conformal field theories enjoy a large degree of universality in terms of the coefficients of reflection and transmission of energy, that describe the scattering of conformal matter at the interface. In this article, we study these coefficients beyond conformality, by gluing a pair of $T\bar T$-deformed 2D CFTs across an interface, which requires the condition $c_L \mu_L = c_R \mu_R$ to be obeyed. We show that, at least when the interface admits a holographic description, the $T\bar T$ deformation of the CFTs can be extended to the interface. We propose a generalization of the linear matching condition in the universal sector of the undeformed ICFT to a non-linear one, which is captured by a universal antisymmetric \emph{transmission function} of the incoming fluxes. We employ the flow equations of the $T\bar T$-deformed CFTs to compute this function in two special classes of states, namely the non-equilibrium steady state (NESS) and scattering state. We show that the results can also be reproduced using holographic techniques in the bulk dual of these states.