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We describe an ab initio and non-perturbative $R$-matrix with time-dependencetheory for ultrafast atomic processes in light fields of arbitrarypolarization. The theory is applicable to complex, multielectron atoms andatomic ions subject to ultrashort (particularly few-femtosecond and attosecond)laser pulses with any given ellipticity, and generalizes previoustime-dependent $R$-matrix techniques restricted to linearly polarized fields.We discuss both the fundamental equations, required to propagate themultielectron wavefunction in time, as well as the computational developmentsnecessary for their efficient numerical solution. To verify the accuracy of ourapproach, we investigate the two-photon ionization of He, irradiated by a pairof time-delayed, circularly polarized, femtosecond laser pulses, and comparephotoelectron momentum distributions, in the polarization plane, with thoseobtained from recent time-dependent close-coupling calculations. The predictivecapabilities of our approach are further demonstrated through a study ofsingle-photon detachment from F$^{-}$ in a circularly polarized, femtosecondlaser pulse, where the relative contribution of the co- and counter-rotating$2p$ electrons is quantified.

$R$-matrix with Time-dependence Theory for Ultrafast Atomic Processes in Arbitrary Light Fields