Transcript Document

Quantum Effects in BECs and FELs
Nicola Piovella, Dipartimento di Fisica and INFN-Milano
Rodolfo Bonifacio, INFN-Milano
Luca Volpe (PhD student), Dipartimento di Fisica-Milano
Mary Cola (Post Doc), Dipartimento di Fisica-Milano
Gordon R. M. Robb, University of Strathclyde, Glasgow, Scotland.
work supported by INFN (QFEL project)
Outline
1. Introductory concepts
2. Classical FEL-CARL Model
3. Quantum FEL-CARL Model
4. Propagation Effects
5. Quantum SASE regime
Free Electron Laser (FEL)
w
r  2
2
Collective Atomic Recoli Laser (CARL)
Pump beam wp
R. Bonifacio et al,
Opt. Comm. 115, 505 (1995)
Probe beam w wp
Both FEL and CARL are examples of collective recoil lasing
Pump field
CARL
~p
Cold atoms
Backscattered field
(probe)
SN SNSN
FEL
Electron
beam
N S N SN S
“wiggler” magnet
(period w)
EM radiation
  w /2 << w
At first sight, CARL and FEL look very different…
Connection between CARL
and FEL can be seen
more easily by
transforming to a frame (L’)
moving with electrons
FEL
electrons
EM pump, ’w
(wiggler)
Backscattered
EM field
’  ’w
CARL
Pump
laser
Connection between FEL
and CARL is now clear
Cold atoms
Backscattered
field ~p
In FEL and CARL particles self-organize to form compact
bunches ~ which radiate coherently.
Collective Recoil Lasing = Optical gain + bunching
bunching factor b (0<|b|<1):
1
b
N
N
e
j 1
i j
Exponential growth of the emitted radiation:
Both FEL and CARL are described using the same
‘classical’ equations, but different independent
variables.
 j
FEL:
2
 ( Ae
i j
z
A A 1


z z1 N
2
 c.c.)
N
e
i j
  (k  kw ) z  wt ;
 iA
j 1
CARL:
  2kz ;
z  wrec  t
z1  wrec  ( z / c)
1/ 3
2k 2
PL n1/ 3
wrec 

2/3
m
a
z1 
z  v 0t
Lc
w
Lg  ;
4
z
z
Lg

Lc 
Lg ;
w
mc  0

  Bw 2 / 3n1/ 3
k
 | A| 
2
N photons
N
CARL-FEL instability animation
A
0
z1
Animation shows evolution of electron/atom positions
in the dynamic pendulum potential together with the
probe field intensity.
V (   )  2 | A | cos(   )
Linear Theory (classical)
Ae
iz
    2 1  0
runaway
solution
wp w
mc( 0   r )

( FEL) 
(CARL)
 k
wrec
See figure (a)
Maximum gain at =0
| A|  e
2
3z
Quantum model of FEL/CARL
We now describe electrons/atoms as QM wavepackets, rather than
classical particles.
Procedure :
Describe N particle system as a Q.M. ensemble
Write Schrodinger equation for macroscopic wavefunction
 ( , z )
Include propagation using a multiple-scaling approach
 ( , z , z1 )
Canonical Quantization
 
p


H
p
H
p    Ae  c.c.  



i
pz m c(   0 )


p


(
FEL
)


k
k




p2
H
 i Aei  c.c.
2
so
Quantization (with classical field A) :
p  pˆ  i
so


[ˆ, pˆ ]  i
H  Hˆ

 ˆ
i
 H
z


1  2
i
i


i

A
(
z
)
e
 c.c.
2
z
2  
dA

dz
2

d  ( , z ) e i  iA
2
0
R. Bonifacio, N. Piovella, G.R.M.Robb and M.Cola, Optics Comm, 252, 381 (2005)
Quantum FEL Propagation model
So far we have neglected slippage, so all sections of the e-beam
evolve identically (steady-state regime) if they are the same initially.
We have introduced propagation into the model, so
different parts of the electron beam can feel different fields :

i  2
i



[
A
(
z
,
z
)
e
 c.c.]
1
2
z
2  
A
A


z
z1
2

d  ( , z , z1 ) 2 e i  iA
0
Here  describes spatial evolution of  on scale of 
and z1 describes spatial evolution of A and  on scale of cooperation
length, Lc >> .
z1  ( z  vr t ) / Lc
where
Lc 

4
Quantum Dynamics
  0,2 
 ( , z , z1 ) 


cn ( z , z1 )ein
n  
ein
is momentum eigenstate corresponding to eigenvalue
Only discrete changes of momentum are possible :
pz
n=1
n=0
n=-1
n( k )
pz= n (k) , n=0,±1,..
k

cn
in 2

cn   Acn 1  A*cn 1
z
2


A A


cn cn*1  iA
z
z1 n  

| cn |2  pn probability to find a particle with p=n(ħk)
=10,  no propagation
 A

 0

 z1

1
steady-state evolution: 

10
(a)
-1
10
-3
|A|
classical limit
is recovered for
2
10
-5
10
-7
10
-9
  1
10
0
10
20
30
40
50
z
0.15
(b)
pn
many momentum states
occupied,
both with n>0 and n<0
0.10
0.05
0.00
-15
-10
-5
0
n
5
10
Quantum limit for
 1
 A




0
 z

1


Only TWO momentum states involved : n=0 and n= - 1
( , z )  c0 ( z )  c1 ( z )ei
n=0
n=-1
Dynamics are those of
a 2-level system coupled to
an optical field,described by
Maxwell-Bloch equations
  0.1
0.0
10
-0.2
8
2
|A|
-0.4
6
<p>
4
-0.6
2
-0.8
0
0
100
200
z
-1.0
0
100
200
z
Bunching and density grating
QUANTUM REGIME <1
CLASSICAL REGIME >>1
0.6
0.15
0.5
0.4
pn
pn
0.10
0.3
0.2
0.05
0.1
0.00
-20 -15 -10
-5
0
5
10
-5 -4 -3 -2 -1
15
10
2.0
| ( ) |2 8
| ( ) |21.5
6
N( )/N
N( )/N
1
2
3
4
5
n
n
4
1.0
0.5
2
0
0
0.0
0
1
2
3
 /2 
4
5
0
 ( )   cn ein
n
1
2
3
 /2 
4
5
A  e 
iz
Quantum Linear Theory
 2
1 
      2   1  0
4 

  1
Quantum regime for <1
1.0
0.8
Classical
limit
|Im|
0.6
(a)
(a) 2
(b) 2.
(c) 2
(d) 2
(e) 2
(f) 2
(b)
1
max at  
2
 m c( 0   r )  k / 2 ( FEL) 


 w w  w
(CARL) 
p
rec

(c)
0.4
(d)
(e)
(f)
0.2
0.0
-10
   2  1  0
width
-5
0
5

10
15
 
QUANTUM CARL HAS BEEN OBSERVED WITH BECs
IN SUPERRADIANT REGIME (MIT, LENS)
When the light escapes rapidly from the sample of length L,
we see a sequential Super-Radiant (SR) scattering,
with atoms recoiling by 2ħk, each time emitting a SR pulse
2

dcn
in
*

cn   Acn 1  A cn 1
dz
2


dA
*

cn cn 1  iA  KA
dz
n  

damping of radiation
c
K
L
SEQUENTIAL SUPERRADIANT SCATTERING
LASER
n=0
BEC
2k
n=-1
n=-2
| A |2  N 2 sec h2 ( gNz)
0
2k
0.002
2
|A|
<p>
-2
0.001
-4
0.000
0
250
500
z
0
250
500
z
Superradiant Rayleigh
Scattering in a BEC
(Ketterle, MIT 1991)
for K>>1 and
 K
Experimental evidence of quantum CARL at LENS
•
Production of an elongated 87Rb BEC in a magnetic trap
•
Laser pulse during first expansion of the condensate
•
Absorption imaging of the momentum components of the cloud
trap
BEC
g
laser beam w, k
Experimental
values:
 = 13 GHz
w = 750 mm
P = 13 mW
absorption imaging
p  2 k
L.Fallani et al, PRA 71 (2005) 033612
The experiment
Temporal evolution of the population in the first three atomic momentum
states during the application of the light pulse.
pump
light
n=0
(p=0)
n=-2
n=-1
(p=2ħk) (p=4ħk)
PROPAGATION EFFECTS IN FELs :
SUPERRADIANT INSTABILITY
Particles at the trailing edge of the beam never receive
radiation from particles behind them: they just radiate
in a SUPERRADIANT PULSE or SPIKE which propagates forward.
if Lb << Lc the SR pulse remains small (weak SR).
if Lb >> Lc the weak SR pulse gets amplified (strong SR) as it
propagates forward through beam with no saturation.
The SR pulse is a self-similar solution of the propagation equation.
SR in the classical model:
A A

 ei
z z1
Strong SR (Lb=30 Lc) from a coherent seed

z  vt 
 z1 

Lc 

R. Bonifacio, B.W. McNeil, and P.
Pierini PRA 40, 4467 (1989)
CLASSICAL SASE
Ingredients of Self Amplified Spontaneous Emission (SASE)
i) Start up from noise
ii) Propagation effects (slippage)
iii) SR instability

The electron bunch behaves as if each cooperation
length would radiate independently a SR spike
which is amplified propagating on the other electrons
without saturating. Spiky time structure and spectrum.
SASE is the basic method for producing coherent X-ray
radiation in a FEL
CLASSICAL SASE
Time profile with many random
spikes (approximately L/Lc)
Broad and noisy spectrum at
short wavelengths (X-FEL)
Example from DESY (Hamburg) for the SASE-FEL experiment
SASE : NUMERICAL SIMULATIONS
L  30Lc
CLASSICAL REGIME:
 5
QUANTUM REGIME:
  0.1
SASE: average momentum distribution
CLASSICAL REGIME:
 5
Classical behaviour :
both n<0 and n>0 occupied
QUANTUM REGIME:
  0.1
Quantum behaviour :
sequential SR decay, only n<0
  0.1 1/  10
  0.3 1/  3.3
  0.2 1/  5
  0.4 1/  2.5
Quantum SASE:
Spectral purification and multiple line spectrum
• In the quantum regime the gain bandwidth
3/ 2
decreases as 4 
line narrowing.
• Spectrum with multiple lines. When the width
of each line becomes larger or equal to the
line separation, continuous spectrum, i.e.,
classical limit. This happens when
4
3/ 2
1
  0.4
FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE
FOR AN X-RAY SOURCE (=1Ǻ)
CLASSICAL SASE
needs:
GeV Linac (Km)
Long undulator (100 m)
High cost (109 $)
yields:
Broad and chaotic spectrum
QUANTUM SASE
needs:
MeV Linac (m)
Laser undulator (~1mm)
lower cost (106 $)
yields:
quasi monocromatic spectrum
CONCLUSIONS
• Classical FEL/CARL model
- classical motion of electrons/atoms
- continuous momenta
• Quantum FEL/CARL model
- QM matter wave in a self consistent field
- discrete momentum state and line spectrum
• Quantum model with propagation
- new regime of SASE with quantum ”purification’’
- appearance of multiple narrow lines