  Quantum gas microscopes are a promising tool to study interacting quantum
many-body systems and bridge the gap between theoretical models and real
materials. So far they were limited to measurements of instantaneous
correlation functions of the form $\langle \hat{O}(t) \rangle$, even though
extensions to frequency-resolved response functions $\langle \hat{O}(t)
\hat{O}(0) \rangle$ would provide important information about the elementary
excitations in a many-body system. For example, single particle spectral
functions, which are usually measured using photoemission experiments in
electron systems, contain direct information about fractionalization and the
quasiparticle excitation spectrum. Here, we propose a measurement scheme to
experimentally access the momentum and energy resolved spectral function in a
quantum gas microscope with currently available techniques. As an example for
possible applications, we numerically calculate the spectrum of a single hole
excitation in one-dimensional $t-J$ models with isotropic and anisotropic
antiferromagnetic couplings. A sharp asymmetry in the distribution of spectral
weight appears when a hole is created in an isotropic Heisenberg spin chain.
This effect slowly vanishes for anisotropic spin interactions and disappears
completely in the case of pure Ising interactions. The asymmetry strongly
depends on the total magnetization of the spin chain, which can be tuned in
experiments with quantum gas microscopes. An intuitive picture for the observed
behavior is provided by a slave-fermion mean field theory. The key properties
of the spectra are visible at currently accessible temperatures.
