  We show how to realize a quantum interface between optical fields and the
polarized nuclear spins in a singly charged quantum dot, which is strongly
coupled to a high-finesse optical cavity. An effective direct coupling between
cavity and nuclear spins is obtained by adiabatically eliminating the (far
detuned) excitonic and electronic states. The requirements needed to map qubit
and continuous variable states of cavity or traveling-wave fields to the
collective nuclear spin are investigated: For cavity fields, we consider
adiabatic passage processes to transfer the states. It is seen that a
significant improvement in cavity lifetimes beyond present-day technology would
be required for a quantum interface. We then turn to a scheme which couples the
nuclei to the output field of the cavity and can tolerate significantly shorter
cavity lifetimes. We show that the lifetimes reported in the literature and the
recently achieved nuclear polarization of ~90% allow both high-fidelity
read-out and write-in of quantum information between the nuclear spins and the
output field. We discuss the performance of the scheme and provide a convenient
description of the dipolar dynamics of the nuclei for highly polarized spins,
demonstrating that this process does not affect the performance of our
protocol.
