Download Towards Sub-Ångström Coherent Light Sources: The Quantum FEL

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

4lC
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  Nk
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  0.5m  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
Nk
 ( )
 102

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