  In a conventional atomic interferometer employing $N$ atoms, the phase
sensitivity is at the standard quantum limit: $1/\sqrt{N}$. Using
spin-squeezing, the sensitivity can be increased, either by lowering the
quantum noise or via phase amplification, or a combination thereof. Here, we
show how to increase the sensitivity, to the Heisenberg limit of $1/N$, while
increasing the quantum noise by $\sqrt{N}$, thereby suppressing by the same
factor the effect of excess noise. The protocol uses a Schr\"odinger Cat state
representing a superposition of two collective states of $N$ atoms, behaving as
a single entity with an $N$-fold increase in Compton frequency. The resulting
$N$-fold phase magnification is revealed by using atomic state detection
instead of collective state detection. We also show how to realize an atomic
clock based on such a Schr\"odinger Cat state, with an $N$-fold increase in the
effective transition frequency. We discuss potential experimental constraints
for implementing this scheme, using one axis twist squeezing employing the
cavity feedback scheme, and show that the effects of cavity decay and
spontaneous emission are highly suppressed. We find that the maximum
improvement in sensitivity can be close to the ideal limit, for as many as ten
million atoms.
