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Quantum Effects in Compton Backscattering R. Bonifacio Outline 1. Compton Red-Shift 2. Novel Quantum Statistical Model 3. Classical and Quantum Limit 4. Implications for ELI-NP:Q.C.S. Introduction Conceptual design : Compton back-scattering High power laser + relativistic e-beam Backscattered field Laser “wiggler” lL lL lr 2 4 lr lL 2lw Compton backscattering shift lr lL 2lC lL 2lC Lab frame: (2) lr 2 2 (1 ) (1 ) lL lC l (1 ); l 2 1 lr l 4 lC 4lC Compton shift parameter: 2 l mc lL lL 1 lr l 2 4 3 3 2 Å l 0 . 5 m l 10 10 ELI-NP 10 L r Electron rest frame: (1) Spontaneous Emission : Quantum Model Spontaneous emission – Why? – random emission of photons leads to growth of energy spread spontaneous emission is a random process - after a distance z through wiggler, different electrons emit different numbers of photons - different number of recoils = momentum/energy spread Spontaneous Emission Models Previous models (Saldin, Madey) predict diffusive growth of energy spread due to spontaneous emission i.e. z 2 z Our model – can use Poisson process argument or physical argument Spontaneous Emission Models Physical argument – consider ensemble described by distribution function f ( p, z ) f ( p) will be increased by electrons with momentum p+hk emitting photons f ( p) will be decreased by electrons with momentum p emitting photons f Rf p k , z Rf p, z z Spontaneous Emission Models f Rf p k , z Rf p, z z Where R = spontaneous rate of emission of photons 4 a0 From Jackson R 3 lL 2 (a0 << 1) = fine structure constant = 1/137 Spontaneous Emission Models f p, z Rf p k Rf p z can be expressed in terms of the emitted number of photons, N, using the change of variable p p0 Nk so f N , z Rf N 1, z Rf N , z z which is the master equation for a Poisson process with solution N N e N Quantum Coherence f N , z N! 2 p k Rz where N N Rz so 2 p k 2 Rz Quantum Recoil Parameter We define classical and quantum regimes The parameter we use to identify the different regimes is the “quantum recoil parameter” ( P) k 1 : Classical regime 1 : Quantum effects Spontaneous Emission Models W ( p, Z ) 1 W p , Z W p,Z Z In scaled notation : where: Z Rz so: Note that when , p 1 p p k Z W ( p, Z ) d p 1 and p 2 Z 2 : W ( p, Z ) 1 W ( p, Z ) 1 2W ( p, Z ) 2 2 Z p 2 p Drift Diffusion (Saldin et al.) Spontaneous Emission Model – Momentum Evolution Classical limit : 5 Model produces diffusive growth of spread in classical limit – results of Saldin & Madey can also be recovered analytically Spontaneous Emission Models Quantum limit : 0 .1 1 Momentum distribution is a series of discrete lines separated by i.e. in unscaled units spacing is ħk. Spontaneous Emission Model – Momentum Evolution Quantum limit with large initial spread: 0.1, 0 10 Initial momentum spread > spacing between momentum levels Discreteness of momentum states is lost Condition for Discrete Lines 1 I ) Compton Shift negligible if 2 mc 1 ( ) 1 II ) Discrete lines if ( ) I ) and II ) compatible if ELI-NP: 10 lL 0.5m lr 10 3 ( ) 2 10 3 Å 10 2 Conclusions 1 1 ELI:NP 1 Continuous Classical drift-diffusion process Discrete Lines Centered on Poisson distribution if Nk ( ) 102 NOVEL QUANTUM EFFECT OBSERVABLE Quantum Compton Source Q.C.S. G. Robb and R. Bonifacio, EPL 94 34002 (2011) Coherent and Spontaneous Emission Coherent Spontaneous Spontaneous 2 a emission 0 12 parameter lLlr lC For sufficiently large , spontaneous emission kills FEL gain if 3 2 10