  Quantum simulations with ultracold fermions in triangular optical lattices
have recently emerged as a new platform for studying magnetism in frustrated
systems. Experimental realizations of the Fermi Hubbard model revealed striking
contrast between magnetism in bipartite and triangular lattices. In bipartite
lattices magnetism peaks at half filling, and doped charge carriers tend to
suppress magnetic correlations. In triangular lattices for large $U/t$,
magnetism is enhanced by doping away from $n=1$ because kinetic energy of
dopants can be lowered through developing magnetic correlations. This
corresponds to formation of magnetic polarons, with hole and doublon doping
resulting in antiferro- and ferromagnetic polarons respectively. Snapshots
obtained with quantum gas microscopes revealed formation of magnetic polarons
around dopants at temperatures exceeding the superexchange energy scale. In
this work we discuss theoretically that additional insight into properties of
magnetic polarons can be achieved using spectroscopic experiments with
ultracold atoms. We consider starting from a spin polarized state with small
hole doping and applying a two-photon Raman photoexcitation, which transfers
atoms into a different spin state. We show that such magnon injection spectra
exhibit a separate peak corresponding to formation of a bound state between a
hole and a magnon. This polaron peak is separated from the simple magnon
spectrum by energy proportional to single particle tunneling and can be easily
resolved with currently available experimental techniques. For some momentum
transfer there is an additional peak corresponding to photoexciting a bound
state between two holes and a magnon. We point out that in two component Bose
mixtures in triangular lattices one can also create dynamical magnetic
polarons, with one hole and one magnon forming a repulsive bound state.
