  Motivated by recent experimental progress we revisit the theory of pump-probe
time- and angle-resolved photoemission spectroscopy (trARPES), which is one of
the most powerful techniques to trace transient pump-driven modifications of
the electronic properties. The pump-induced dynamics can be described in
different gauges for the light-matter interaction. Standard minimal coupling
leads to the velocity gauge, defined by linear coupling to the vector
potential. In the context of tight-binding (TB) models, the Peierls
substitution is the commonly employed scheme for single-band models.
Multi-orbital extensions -- including the coupling of the dipole moments to the
electric field -- have been introduced and tested recently. In this work, we
derive the theory of time-resolved photoemission within both gauges from the
perspective of nonequilibrium Green's functions. This approach naturally
incorporates the photoelectron continuum, which allows for a direct calculation
of the observable photocurrent. Following this route we introduce
gauge-invariant expressions for the time-resolved photoemission signal. The
theory is applied to graphene pumped with short terahertz pulses, which we
treat within a first-principles TB model. We investigate the gauge invariance
and discuss typical effects observed in subcycle time-resolved photoemission.
Our formalism is an ideal starting point for realistic trARPES simulations
including scattering effects.
