Efficient variational approach to the Fermi polaron problem in two dimensions, both in and out of equilibrium

We develop a non-Gaussian variational approach that enables us to study both equilibrium and far-from-equilibrium physics of the two-dimensional Fermi polaron. This method provides an unbiased analysis of the polaron-to-molecule phase transition without relying on truncations in the total number of particle-hole excitations. Our results -- which include the ground state energy and quasiparticle residue -- are in qualitative agreement with the known Monte Carlo calculations. The main advantage of the non-Gaussian states compared to conventional numerical methods is that they enable us to explore long-time polaron evolution and, in particular, study various spectral properties accessible to both solid-state and ultracold atom experiments. We design two types of radiofrequency spectroscopies to measure polaronic and molecular spectral functions. Depending on the parameter regime, we find that these spectral functions and fermionic density profiles near the impurity display either long-lived oscillations between the repulsive and attractive polaron branches or exhibit fast relaxational dynamics to the molecular state.