1. Quantum Effects in Compton Backscattering
R. Bonifacio

2. Outline
Compton Red-Shift
Novel Quantum Statistical Model
Classical and Quantum Limit
Implications for ELI-NP:Q.C.S.

3. Compton back-scattering
Introduction
Conceptual design :
Compton back-scattering
High power laser + relativistic e-beam
Backscattered
field lL lL
Laser “wiggler” lr lr

4. Compton backscattering shift
Electron rest frame: (1)
Lab frame: (2)
Compton shift parameter:
ELI-NP Å

5. Spontaneous Emission : Quantum Model
Spontaneous emission – random emission of photons leads to growth of energy spread
Why? – 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

6. Spontaneous Emission Models
Previous models (Saldin, Madey) predict diffusive growth of energy spread due to spontaneous emission i.e.
Our model – can use Poisson process argument or physical argument

7. Spontaneous Emission Models
Physical argument – consider ensemble described by distribution function
will be increased by electrons with momentum p+hk emitting photons
will be decreased by electrons with momentum p emitting photons

8. Spontaneous Emission Models
Where R = spontaneous rate of emission of photons
From Jackson
(a0 << 1)
a = fine structure constant = 1/137

9. Spontaneous Emission Models
can be expressed in terms of the emitted number of photons, N, using the change of variable so
which is the master equation for a Poisson process with solution
Quantum Coherence
where so

10. Quantum Recoil Parameter
We define classical and quantum regimes
The parameter we use to identify the different regimes is
the “quantum recoil parameter”
: Classical regime
: Quantum effects

11. Spontaneous Emission Models
In scaled notation :
where:
and ,
so:
Note that when :
(Saldin et al.)
Drift
Diffusion

12. Spontaneous Emission Model – Momentum Evolution
Classical limit :
Model produces diffusive growth of spread in classical limit
– results of Saldin & Madey can also be recovered analytically

13. Spontaneous Emission Models
Quantum limit :
Momentum distribution is a series of discrete lines separated by
i.e. in unscaled units spacing is ħk.

14. Spontaneous Emission Model – Momentum Evolution
Quantum limit with large initial spread:
Initial momentum spread > spacing between momentum levels
Discreteness of momentum states is lost

15. Condition for Discrete Lines
I ) Compton Shift negligible if
II ) Discrete lines if
I ) and II ) compatible if
ELI-NP: Å

16. Conclusions
Continuous Classical drift-diffusion process
Discrete Lines Centered on
Poisson distribution
ELI:NP if
NOVEL QUANTUM EFFECT OBSERVABLE
Quantum Compton Source Q.C.S.
G. Robb and R. Bonifacio, EPL (2011)

17. Coherent and Spontaneous Emission
parameter
For sufficiently large b,
spontaneous emission kills
FEL gain if