  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 Schr\"odinger cat (SC) state, as a superposition of two extremal
collective states. When this SC 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 $^{87}$Rb
atoms, with $N=10^{6}$, for example, the intereference 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 show that when the SC 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 $\sim$
$\sqrt{2}$.
