  The recent determination of a robust spin Hamiltonian for the
anti-ferromagnetic XY pyrochlore Er2Ti2O7 reveals a most convincing case of the
"order by quantum disorder" (ObQD) mechanism for ground state selection. This
mechanism relies on quantum fluctuations to remove an accidental symmetry of
the magnetic ground state, and selects a particular ordered spin structure
below T_N=1.2K. The removal of the continuous degeneracy results in an energy
gap in the spectrum of spin wave excitations, long wavelength pseudo-Goldstone
modes. We have measured the ObQD spin wave gap at a zone center in Er2Ti2O7,
using low incident energy neutrons and the time-of-flight inelastic scattering
method. We report a gap of Delta =0.053 +/- 0.006 meV, which is consistent with
upper bounds placed on it from heat capacity measurements and roughly
consistent with theoretical estimate of ~ 0.02 meV, further validating the spin
Hamiltonian that led to that prediction. The gap is observed to vary with
square of the order parameter, and goes to zero for T ~ T_N.
