Quantum Mechanical Treatment of Transit-Time Optical Stochastic Cooling of Muons

Ultra-fast stochastic cooling (i.e., on microsecond time-scales) would be desirable in certain applications, for example, in order to boost final luminosity in a muon collider or neutrino factory, where even with relativistic dilation, the short particle lifetimes severely limit the total time available to reduce beam phase space. But fast cooling requires very high-bandwidth amplifiers so as to limit the incoherent heating effects from neighboring particles. A method of transit-time optical stochastic cooling has been proposed which would employ high-gain, high-bandwidth, solid-state lasers to amplify the spontaneous radiation from the charged particle bunch in a strong-field magnetic wiggler. This amplified light is then fed back onto the same bunch inside a second wiggler, with appropriate phase delay to effect cooling. But before amplification, the usable signal from any one particle is quite small, on average much less than one photon for each pass through the wiggler, suggesting that the radiation should be treated quantum mechanically, and raising doubts as to whether this weak signal even contains sufficient phase information necessary for cooling, and whether it can be reliably amplified to provide the expected cooling on each pass. A careful examination of the dynamics, where the radiation and amplification processes are treated quantum mechanically, indicates that fast cooling is in principle possible, with cooling rates which essentially agree with classical calculations, provided that the effects of the unavoidable amplifier noise arising from quantum mechanical uncertainty are included. Thus, in this context quantum mechanical uncertainties do not present any insurmountable obstacles to optical cooling, nor do they lead to any significant differences than in the purely classical regime, but do establish a lower limit on cooling rates and achievable emittances.