Quantum Effects in BECs and FELs
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Transcript of Quantum Effects in BECs and FELs
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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)
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Outline
1. Introductory concepts
2. Classical FEL-CARL Model
3. Quantum FEL-CARL Model
4. Propagation Effects
5. Quantum SASE regime
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Free Electron Laser (FEL) 22 w
r
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Collective Atomic Recoli Laser (CARL)
Pump beam p
Probe beam p
R. Bonifacio et al, Opt. Comm. 115, 505 (1995)
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Both FEL and CARL are examples of collective recoil lasing
Cold atoms
Pump field
Backscattered field(probe)
CARL
FEL
“wiggler” magnet(period w)
Electron beam
EM radiation w /<< w N
S N
S N
S N
S N
S N
S
At first sight, CARL and FEL look very different…
~p
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electrons
EM pump, ’w
(wiggler)
BackscatteredEM field’ ’w
Connection between CARL and FEL can be seen
more easily by transforming to a frame (’)
moving with electrons
Cold atoms
Pumplaser
Backscatteredfield
Connection between FEL and CARL is now clear
FEL
CARL
~p
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Collective Recoil Lasing = Optical gain + bunching
In FEL and CARL particles self-organize to form compact bunches ~ which radiate coherently.
N
j
i jeN
b1
1 bunching factor b (0<|b|<1):
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Exponential growth of the emitted radiation:
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Both FEL and CARL are described using the same ‘classical’ equations, but different independent
variables.
AieNz
A
z
A
ccAez
N
j
i
ij
j
j
11
2
2
1
.).(
FEL:
;)( tzkk w
CARL:;2kz
N
NA photons2||
gL
zz
cL
tzz 0
1
v
;4w
gL ;gw
c LL
3/13/20 nBk
mcw
3/2
3/13/1
a
L nP
)/(1 czztz recrec
m
krec
22
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CARL-FEL instability animation
Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.
01
z
A
)cos(||2)( AV
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Linear Theory (classical)
2 1 0
Maximum gain at =0
runaway solution
See figure (a)
)()()( 0 CARLFEL
k
mc
rec
pr
zieA
zeA 32||
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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
Quantum model of FEL/CARL
Include propagation using a multiple-scaling approach
),,( 1zz
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Canonical Quantization
p
Hp
H
ccAep i ..
..2
2
ccAeip
H i
Quantization (with classical field A) :
ipp ˆ ip ]ˆ,ˆ[ HH ˆ
Hz
i ˆ
so
Aiezdzd
dA
ccezAiz
i
i
i
2
0
2
2
2
),(
..)(2
1
so
)()( 0 FEL
k
mc
k
pp z
R. Bonifacio, N. Piovella, G.R.M.Robb and M.Cola, Optics Comm, 252, 381 (2005)
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Quantum FEL Propagation model
Here describes spatial evolution of on scale of and describes spatial evolution of A and on scale of cooperation length, Lc >>
1z
1 ( ) /r cz z v t L
We have introduced propagation into the model, sodifferent parts of the electron beam can feel different fields :
So far we have neglected slippage, so all sections of the e-beamevolve identically (steady-state regime) if they are the same initially.
Aiezzdz
A
z
A
ccezzAi
z
i
i
2
0
21
1
12
2
),,(
.].),([2
where
4cL
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Quantum Dynamics
Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,..
pz kn=1n=0n=-1
is momentum eigenstate corresponding to eigenvalue ( )n kine
2| |n nc p
n
inn ezzczz ),(),,( 11
probability to find a particle with p=n(ħk)
2,0
Aiccz
A
z
A
cAAccin
z
c
nnn
nnnn
*1
1
1*
1
2
2
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classical limit is recovered for
many momentum states occupied,
both with n>0 and n<0
1
-15 -10 -5 0 5 100.00
0.05
0.10
0.15
(b)
n
p n
0 10 20 30 40 5010-9
10-7
10-5
10-3
10-1
101
=10, no propagation
(a)
z
|A|2
steady-state evolution:
01z
A
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Quantum limit for
iezczcz )()(),( 10
Only TWO momentum states involved : n=0 and n= - 1
n=0
n=-1
Dynamics are those of a 2-level system coupled to an optical field,described byMaxwell-Bloch equations
1
0 100 2000
2
4
6
8
10
z
|A|2
0 100 200
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
z
<p>
1.0
01z
A
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0 1 2 3 4 50
2
4
6
8
10
N(
)/N
/2
-20 -15 -10 -5 0 5 10 150.00
0.05
0.10
0.15
pn
n
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
N(
)/N
/2
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.1
0.2
0.3
0.4
0.5
0.6
pn
n
Bunching and density gratingCLASSICAL REGIME >>1 QUANTUM REGIME <1
2| ( ) | 2| ( ) |
( ) inn
n
c e
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zieA Quantum Linear Theory
014
12
2
-10 -5 0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
(a) (b) (c) (d) (e) (f)
(f)(e)
(d)
(c)
(b)(a)
|Im|
Classicallimit
Quantum regime for <1
max at
2
1
)(
)(2/)( 0
CARL
FELkmc
recp
r
width
012 1
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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
KA
L
cK
Aicczd
dA
cAAccin
zd
dc
nnn
nnnn
*1
1*
1
2
2
damping of radiation
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k2n=-2
n=0
n=-1
0 250 5000.000
0.001
0.002
z
|A|2
0 250 500
-4
-2
0
z
<p>
BECLASER
k2
)(sec|| 222 gNzhNA
SEQUENTIAL SUPERRADIANT SCATTERING
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Superradiant Rayleigh Scattering in a BEC(Ketterle, MIT 1991)
for K>>1 and
K
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• 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
Experimental values:
= 13 GHzw = 750 mP = 13 mW
laser beam kw,
BEC
absorption imaging
trap
g
Experimental evidence of quantum CARL at LENS
2p k
L.Fallani et al, PRA 71 (2005) 033612
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The experiment
pump light
n=0(p=0)
n=-1(p=2ħk)
n=-2(p=4ħk)
Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse.
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Particles at the trailing edge of the beam never receive radiation from particles behind them: they just radiatein 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.
PROPAGATION EFFECTS IN FELs : SUPERRADIANT INSTABILITY
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1
i
z
Ae
z
A
Strong SR (Lb=30 Lc) from a coherent seed
SR in the classical model:
c1 L
vtzz
R. Bonifacio, B.W. McNeil, and P. Pierini PRA 40, 4467 (1989)
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Ingredients of Self Amplified Spontaneous Emission (SASE)
i) Start up from noiseii) 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
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CLASSICAL SASE
Example from DESY (Hamburg) for the SASE-FEL experiment
Time profile with many randomspikes (approximately L/Lc)
Broad and noisy spectrum atshort wavelengths (X-FEL)
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SASE : NUMERICAL SIMULATIONS
cLL 30
CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0
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Classical behaviour : both n<0 and n>0 occupied
CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0
SASE: average momentum distribution
Quantum behaviour : sequential SR decay, only n<0
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0.1 1/ 10 0.2 1/ 5
0.3 1/ 3.3 0.4 1/ 2.5
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Quantum SASE:Spectral purification and multiple line spectrum
• In the quantum regime the gain bandwidth decreases as 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
3/ 24
3/ 24 1 0.4
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CLASSICAL SASEneeds:GeV Linac (Km)Long undulator (100 m)High cost (109 $)yields:Broad and chaotic spectrum
FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE FOR AN X-RAY SOURCE (=1Ǻ)
QUANTUM SASEneeds:MeV Linac (m)Laser undulator (~1m)lower cost (106 $)yields:quasi monocromatic spectrum
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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