We present a protocol for an atomic interferometer that reaches the Heisenberg limit (HL), within a factor of $\sim \sqrt{2}$, via collective state detection and critical tuning of one-axis twist spin squeezing. It generates a Schrödinger cat state, as a superposition of two extremal collective states. When this Schrödinger-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 ${}^{87}\mathrm{Rb}$ atoms, with $N={10}^6$, for example, the interference would occur at a Compton frequency of $\sim 2\times {10}^{31}$ 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 $\sqrt{N}$ 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 $\sim \sqrt{2}$ 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 Schrödinger-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 $\sim \sqrt{2}$.

• Received 22 June 2018

DOI:https://doi.org/10.1103/PhysRevA.98.013636