  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 short-lived excited state, as well as
the ground state of the daughter atom. Their natural population inversion is
most appropriate for exploiting the bosonic nature of the observed X-rays by
means of stimulating X-ray beams. 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 in a facility like European XFEL.
