The CREMA collaboration is pursuing a measurement of the ground-state
hyperfine splitting (HFS) in muonic hydrogen
(\muμp)
with 1 ppm accuracy by means of pulsed laser spectroscopy to determine
the two-photon-exchange contribution with 2\times10^{-4}2×10−4
relative accuracy. In the proposed experiment, the
\muμp
atom undergoes a laser excitation from the singlet hyperfine state to
the triplet hyperfine state, {then} is quenched back to the singlet
state by an inelastic collision with a H_22
molecule. The resulting increase of kinetic energy after the collisional
deexcitation is used as a signature of a successful laser transition
between hyperfine states. In this paper, we calculate the combined
probability that a \muμp
atom initially in the singlet hyperfine state undergoes a laser
excitation to the triplet state followed by a collisional-induced
deexcitation back to the singlet state. This combined probability has
been computed using the optical Bloch equations including the inelastic
and elastic collisions. Omitting the decoherence effects caused by {the
laser bandwidth and }collisions would overestimate the transition
probability by more than a factor of {two in the experimental
conditions. Moreover,} we also account for Doppler effects and provide
the matrix element, the saturation fluence, the elastic and inelastic
collision rates for the singlet and triplet states, and the resonance
linewidth. This calculation thus quantifies one of the key unknowns of
the HFS experiment, leading to a precise definition of the requirements
for the laser system and to an optimization of the hydrogen gas target
where \muμp
is formed and the laser spectroscopy will occur.