  Massive neutrinos demand to ask whether they are Dirac or Majorana particles.
Majorana neutrinos are an irrefutable proof of physics beyond the Standard
Model. Neutrinoless double electron capture is not a process but a virtual
$\Delta L = 2$ mixing between a parent $^AZ$ atom and a daughter $^A(Z-2)$
excited atom with two electron holes. As a mixing between two neutral atoms and
the observable signal in terms of emitted two-hole X-rays, the strategy,
experimental signature and background are different from neutrinoless double
beta decay. The mixing is resonantly enhanced for almost degeneracy and, under
these conditions, there is no irreducible background from the standard
two-neutrino channel. We reconstruct the natural time history of a nominally
stable parent atom since its production either by nature or in the laboratory.
After the time periods of atom oscillations and the decay of the short-lived
daughter atom, at observable times the relevant "stationary" states are the
mixed metastable long-lived state and the non-orthogonal short-lived excited
state, as well as the ground state of the daughter atom. We find that they have
a natural population inversion which is most appropriate for exploiting the
bosonic nature of the observed atomic transitions radiation. Among different
observables of the atom Majorana mixing, we include the enhanced rate of
stimulated X-ray emission from the long-lived metastable state by a
high-intensity X-ray beam: a gain factor of 100 can be envisaged at current
XFEL facilities. On the other hand, the historical population of the daughter
atom ground state can be probed by exciting it with a current pulsed optical
laser, showing the characteristic absorption lines: the whole population can be
excited in a shorter time than typical pulse duration.
