  We present a study of even parity Rydberg exciton states in cuprous oxide
using time-resolved second harmonic generation (SHG). Excitonic states with
principal quantum number n = 5 - 12 were excited by nanosecond pulses around
1143 nm. Using time-resolved single-photon counting, the coherently generated
second harmonic was isolated both temporally and spectroscopically from
inelastic emission due to lower-lying free and bound excitonic states, which
included narrow resonances at 1.99 eV associated with an exceptional lifetime
of 641 $\pm$ 7 $\mu$s. The near transform-limited excitation bandwidth enabled
detailed measurements of the exciton lineshape and position, from which we
obtained values for the quantum defects of the S and D excitonic states
associated with the appropriate crystal symmetries. Odd parity P and F
excitonic states were also observed, in accordance with predicted
quadrupole-allowed two-photon excitation processes. We compared our
measurements to conventional one-photon spectroscopy in the same sample, and
find that the SHG spectrum is cut off at a lower principal quantum number (n =
12 vs n = 15). We attribute this effect to a combination of spatial
inhomogeneities and local heating, and discuss the prospects for observing
higher principal quantum number even parity states in future experiments.
