  We analyze methods to go beyond the standard quantum limit for a class of
atomic interferometers, where the quantity of interest is the difference of
phase shifts obtained by two independent atomic ensembles. An example is given
by an atomic Sagnac interferometer, where for two ensembles propagating in
opposite directions in the interferometer this phase difference encodes the
angular velocity of the experimental setup. We discuss methods of squeezing
separately or jointly observables of the two atomic ensembles, and compare in
detail advantages and drawbacks of such schemes. In particular we show that the
method of joint squeezing may improve the variance by up to a factor of 2. We
take into account fluctuations of the number of atoms in both the preparation
and the measurement stage, and obtain bounds on the difference of the numbers
of atoms in the two ensembles, as well as on the detection efficiency, which
have to be fulfilled in order to surpass the standard quantum limit. Under
realistic conditions, the performance of both schemes can be improved
significantly by reading out the phase difference via a quantum non-demolition
(QND) measurement. Finally, we discuss a scheme using macroscopically entangled
ensembles.
