We present a protocol for an atomic interferometer that reaches the Heisenberg limit (HL), within a factor of $\ensuremath{\sim}\sqrt{2}$, via collective state detection and critical tuning of one-axis twist spin squeezing. It generates a Schr\"odinger cat state, as a superposition of two extremal collective states. When this Schr\"odinger-cat-state interferometer is used as a gyroscope, the interference occurs at an ultrahigh Compton frequency, corresponding to a mesoscopic single object with a mass of Nm, where N is the number of particles in the ensemble, and m is the mass of each particle. For 87Rb atoms, with N=106, for example, the interference would occur at a Compton frequency of $\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{31}$ Hz. Under this scheme, the signal is found to depend critically on the parity of N. We present two variants of the protocol. Under protocol A, the fringes are narrowed by a factor of N for one parity, while for the other parity the signal is zero. Under protocol B, the fringes are narrowed by a factor of N for one parity, and by a factor of N for the other parity. Both protocols can be modified in a manner that reverses the behavior of the signals for the two parities. Over repeated measurements under which the probability of being even or odd is equal, the averaged sensitivity is smaller than the HL by a factor of $\ensuremath{\sim}\sqrt{2}$ for both versions of the protocol. We describe an experimental scheme for realizing such an atomic interferometer, and discuss potential limitations due to experimental constraints imposed by the current state of the art, for both collective state detection and one-axis-twist squeezing. We show that when the Schr\"odinger-cat-state interferometer is configured as an accelerometer, the effective two-photon wave vector is enhanced by a factor of N, leading to the same degree of enhancement in sensitivity. We also show that such a mesoscopic single object can be used to increase the effective base frequency of an atomic clock by a factor of N, with a sensitivity that is equivalent to the HL, within a factor of $\ensuremath{\sim}\sqrt{2}$.
We describe a collective-state atomic interferometer (COSAIN) with the signal fringe as a function of phase difference or rotation narrowed by 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 $10\phantom{\rule{0.16em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.16em}{0ex}}{10}^{30}$ Hz, or a de Broglie wavelength of 4.5 femtometer, for N=106 and v=1 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 nonuniformity 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 counterpropagating Raman beams.