  We describe a collective state atomic interferometer (COSAIN) with the signal
fringe as a function of phase-difference or rotation narrowed by $\sqrt{N}$
compared to a conventional interferometer - $N$ being the number of atoms -
without entanglement. This effect arises from the interferences among
collective states, and is a manifestation of interference at a Compton
frequency of ten nonillion Hz, or a de Broglie wavelength of ten attometer, for
$N=10^6$ and $v = 300 m/s$. The population of the collective state of interest
is detected by a null measurement scheme, in which an event corresponding to
detection of zero photons corresponds to the system being in that particular
collective state. The signal is detected by collecting fluorescence through
stimulated Raman scattering of Stokes photons, which are emitted predominantly
against the direction of the probe beam, for a high enough resonant optical
density. The sensitivity of the ideal COSAIN is found to be given by the
standard quantum limit. However, when detection efficiency and collection
efficiency are taken into account, the detection scheme of the COSAIN increases
the quantum efficiency of detection significantly in comparison to a typical
conventional Raman atomic interferometer employing fluorescence detection,
yielding a net improvement in stability by as much as a factor of $10$. We
discuss how the inhomogeneities arising from the non-uniformity in experimental
parameters affect the COSAIN signal. We also describe an alternate experimental
scheme to enhance resonant optical density in a COSAIN by using cross-linearly
polarized counter-propagating Raman beams.
