  The elastic production of top quark pairs in $pp$ collisions at low and high
luminosity regimes is investigated in detail. We extend the study performed in
Phys. Rev. D102, 074014 (2020) which has demonstrated that the sum of two
semi-exclusive $t\bar{t}$ production modes, namely in photon-Pomeron and
Pomeron-Pomeron interactions, can be experimentally measured when the
$t\bar{t}$ system decays semi-leptonically, $t\bar{t}\rightarrow
jjbl\nu_l\bar{b}$, both forward protons are tagged and a low amount of pile-up
is present. In this study we focus on separating individual channels and a
special attention is paid to the situation at high-luminosity LHC environment.
We observe that the separation of the pomeron-pomeron and photon-photon events
is a challenging task, especially at high amounts of pile-up even with an
optimistic 10~ps resolution of timing detectors. In contrast, the
photon-Pomeron signal is relatively well separable from all backgrounds at low
levels of pile-up, allowing us to discover the elastic $t\bar{t}$ production
and probe, for the first time, the production of a top quark pair in the
photon-Pomeron interactions. The diffractive photoproduction of such a complex
system as the $t\bar{t}$ pair hence can be used not only to study diffractive
properties of the scattering amplitude but also to search for new physics
beyond the Standard Model, and consequently to be a solid part of the physics
programme of forward proton detectors at LHC.
