Observing the quantum regime of a free-electron laser
In plain words
If one photon's recoil is enough to knock an electron out of resonance, the laser behaves like a two-level quantum system with a much narrower spectrum; this regime has not been reached.
Precise statement
With $\rho$ the FEL Pierce parameter and $k_r$ the resonant wavenumber, the quantum FEL parameter is $\rho_{\mathrm{bar}} = \rho \gamma m c/(\hbar k_r)$; the quantum regime $\rho_{\mathrm{bar}} < 1$ also requires relative energy spread below $\hbar k_r/(\gamma m c)$. Determine whether any configuration (optical undulator driven by a laser, crystal or other short-period undulator) reaches $\rho_{\mathrm{bar}} < 1$ with gain length shorter than the interaction length, and observe the predicted single-line spectrum.
What would settle it
Measurement of a spectrum showing the narrowing and line structure predicted for $\rho_{\mathrm{bar}} < 1$.