  We propose a generalization of the Tonks-Girardeau model that describes a
coherent gas of cold two-level Bosons which interact with two external fields
in a Ramsey interferometer. They also interact among themselves by contact
collisions with interchange of momentum and internal state. We study the
corresponding Ramsey fringes and the quantum projection noise which,
essentially unaffected by the interactions, remains that for ideal Bosons. The
dual system of this gas, an ideal gas of two-level Fermions coupled by the
interaction with the separated fields, produces the same fringes and noise
fluctuations. The cases of time-separated and spatially-separated fields are
studied. For spatially separated fields the fringes may be broadened slightly
by increasing the number of particles, but only for large particle numbers far
from present experiments with Tonks-Girardeau gases. The uncertainty in the
determination of the atomic transition frequency diminishes, essentially with
the inverse root of the particle number.
