Ramsey interferometry with a two-level Tonks-Girardeau gas

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.