  We investigate the Kerr and magneto-optical effects for a probe laser field
with two orthogonally polarized components, propagating in a cold Rydberg
atomic gas with an inverted-Y-type level configuration via double
electromagnetically induced transparency (EIT). Through an approach beyond both
mean-field and ground-state approximations, we make detailed calculations on
third-order nonlinear optical susceptibilities and show that the system
possesses giant nonlocal selfand cross-Kerr nonlinearities contributed by
Rydberg-Rydberg interaction. The theoretical result of the cross-Kerr
nonlinearity obtained for 85Rb atomic gas is very close to the experimental one
reported recently. Moreover, we demonstrate that the probe laser field can
acquire a very large magnetooptical rotation via the double EIT, which may be
used to design atomic magnetometers with high precision. The results presented
here are promising not only for the development of nonlocal nonlinear
magneto-optics but also for applications in precision measurement and optical
information processing and transmission based on Rydberg atomic gases.
