  Poynting flux is the flux of magnetic energy, which is responsible for
chromospheric and coronal heating in the solar atmosphere. It is defined as a
cross product of electric and magnetic fields, and in ideal MHD conditions it
can be expressed in terms of magnetic field and plasma velocity. Poynting flux
has been computed for active regions and plages, but estimating it in the quiet
Sun (QS) remains challenging due to resolution effects and polarimetric noise.
However, with upcoming DKIST capabilities, these estimates will become more
feasible than ever before. Here, we study QS Poynting flux in Sunrise/IMaX
observations and MURaM simulations. We explore two methods for inferring
transverse velocities from observations - FLCT and a neural network based
method DeepVel - and show DeepVel to be the more suitable method in the context
of small-scale QS flows. We investigate the effect of azimuthal ambiguity on
Poynting flux estimates, and we describe a new method for azimuth
disambiguation. Finally, we use two methods for obtaining the electric field.
The first method relies on idealized Ohm's law, whereas the second is a
state-of-the-art inductive electric field inversion method PDFI SS. We compare
the resulting Poynting flux values with theoretical estimates for chromospheric
and coronal energy losses and find that some of Poynting flux estimates are
sufficient to match the losses. Using MURaM simulations, we show that
photospheric Poynting fluxes vary significantly with optical depth, and that
there is an observational bias that results in underestimated Poynting fluxes
due to unaccounted shear term contribution.
