We describe an apparatus for performing constant strain rate deformations of polymer glasses while simultaneously measuring the segmental mobility with an optical probe reorientation method. Poly(methyl methacrylate) glasses were deformed at T-g - 19 K, for local strain rates between 3.7 X 10(-5) and 1.2 X 10-(4) s(-1). In these experiments, the mobility initially increases in the preyield regime, by a factor of 40-160, as compared to the undeformed PMMA glass. The mobility then remains constant after yield, even as the stress is decreasing due to strain softening. This is consistent with the view that the sample is being pulled higher on the potential energy landscape in this regime. Higher strain rates lead to higher mobility in the postyield regime, and for the range of strain rates investigated, mobility and strain rate are linearly correlated. We observe that thermal history has no influence on mobility after yield and that deformation leads to a narrowing of the distribution of segmental relaxation times. These last three observations are consistent with previously reported constant stress experiments on PMMA glasses. The experimental features reported here are compared to computer simulations and theoretical models.