  Planar squeezed states, i.e. quantum states which are squeezed in two
orthogonal spin components, have recently attracted attention due to their
applications in atomic interferometry and quantum information [Q. Y. He et al,
New J. Phys. 14, 093012 (2012)]. While canonical variables such as quadratures
of the radiation field can be squeezed in at most one component, simultaneous
squeezing in two orthogonal spin components can be achieved due to the angular
momentum commutation relations. We present a novel scheme for planar squeezing
via quantum non-demolition (QND) measurements in spin-1 systems. The QND
measurement is achieved via near-resonant paramagnetic Faraday rotation
probing, and the planar squeezing is obtained by sequential QND measurement of
two orthogonal spin components. We compute the achievable squeezing for a
variety of optical depths, initial conditions, and probing strategies. The
planar squeezed states generated in this way contain entanglement detectable by
spin-squeezing inequalities and give an advantage relative to non-squeezed
states for any precession phase angle, a benefit for single-shot and
high-bandwidth magnetometry.
