  One-atom laser based on electromagnetically-induced transparency, suggested
recently [Phys. Rev. Lett. 124, 093603 (2020)], is capable of generating
Schr\"odinger cat states in the regime of strong ground-state coupling. In this
regime, we find the exact solution for the Schr\"odinger equation with a
time-dependent effective Hamiltonian by considering the Magnus expansion of the
time-ordered exponential and calculating analytically the time-ordering terms,
omitted in the previous study. We show that the time-ordering term affects the
relative phase of two coherent components of the generated Schr\"odinger cat
state. We show this influence by calculating various nonclassicality indicators
for the cavity field, such as total noise, average parity and relative total
noise. We find, that time-ordering becomes important at the average photon
number in the cavity below 1, in striking contrast to the case of single-pass
parametric downconversion, where it becomes important at average photon number
in one optical mode above 4.
