Measurement of the positronium $1^3$${\mathit{S}}_1$-$2^3$${\mathit{S}}_1$ interval by continuous-wave two-photon excitation

Using continuous-wave excitation to eliminate the problems inherent with pulsed laser measurements of nonlinear transitions, we have measured the $1^3$ ${\mathit{S}}_{1\mathrm{-}}$ $2^3$ ${\mathit{S}}_1$ interval in positronium (Ps) to be 1 233 607 216.4±3.2 MHz. The quoted 2.6 ppb (parts per ${10}^9$) uncertainty is primarily due to the determination of the Ps resonance relative to the ${\mathrm{Te}}_2$ reference line, with a 1.5 ppb contribution from a recent calibration of the ${\mathrm{Te}}_2$ line relative to the hydrogen 1S-2S transition. The uncertainty corresponds to 3.5×${10}^{\mathrm{-}5}$ of the ${\mathrm{\alpha }}^2$ ${\mathit{R}}_{\mathrm{\infty }}$ QED contribution to the $1^3$ ${\mathit{S}}_{1\mathrm{-}}$ $2^3$ ${\mathit{S}}_1$ interval. Our measurement is sufficiently accurate to provide a test of the as yet uncalculated ${\mathrm{\alpha }}^4$ ${\mathit{R}}_{\mathrm{\infty }}$ QED corrections.