introduction to classical and quantum high-gain fel theory
TRANSCRIPT
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Introduction to Classical and Quantum High-Gain FEL Theory
Rodolfo Bonifacio&
Gordon Robb
University of Strathclyde, Glasgow, Scotland.
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Outline
1.
Introductory concepts
2.Classical FEL Model
3.
Classical SASE
4.Quantum FEL Model
5.
Quantum SASE regime : Harmonics
6.
Coherent sub-Angstrom (γ-ray) source
7.
Experimental evidence of QFEL in a BEC
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1. Introduction
Magnetostatic
“wiggler”
field
Relativistic electron beam
EM radiationN
S N
S N
S N
S N
S
The Free Electron Laser (FEL) consists of a relativistic beam of electrons (v≈c)
moving through a spatially periodic magnetic field (wiggler).
Principal attraction of the FEL is tunability
:-
FELs
currently produce coherent light from microwaves
through visible to UV-
X-ray production via Self-
Amplified Spontaneous
Emission (SASE) (LCLS –
1.5Å)
(wavelength λw
)
λ ∝ λw
/γ2 << λw
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Exponential growth of the emitted radiation and bunching:
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Ingredients of a SASE-FEL :
• High-gain (single pass) (no mirrors)
• Propagation/slippage of radiation with respect to electrons
• Startup
from electron shot noise (no seed field)
Consequently, structure of talk is :
• Recap of high-gain FEL theory (classical & quantum)
• Propagation effects (slippage & superradiance)
• SASE (classical & quantum)
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(6) R. B., N. Piovella, G.R.M.Robb, and M.M.Cola, Europhysics Letters, 69, (2005) 55 and quant-ph/0407112
.(7)
R.B., N. Piovella, G.R.M. Robb & A. Schiavi, PRST-AB 9, 090701 (2006) (8)
R. B., N. Piovella, G.R.M.Robb, and M.M.Cola, Optics Commun. 252, 381 (2005)
Some references relevant to this talkHIGH-GAIN AND SASE FEL with “UNIVERSAL SCALING”
Classical Theory(1) R.B, C. Pellegrini and L. Narducci, Opt. Commun. 50, 373 (1984).(2) R.B, B.W. McNeil, and P. Pierini
PRA 40, 4467 (1989)(3) R.B, L. De Salvo, P.Pierini, N.Piovella,
C. Pellegrini, PRL 73, 70 (1994).(4, 5) R.B. et al,Physics
of High Gain FEL and Superradiance, La Rivista
del Nuovo
Cimento
vol. 13. n. 9 (1990) e vol. 15 n.11 (1992)
QUANTUM THEORY
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2. The High-Gain FELWe consider a relativistic electron beam moving in both a magnetostatic
wiggler field and an electromagnetic wave.
wigglerelectron beam
EM wave
( )..ˆ2
AA ww ccee zikw += −
Wiggler field (helical)
:
Radiation field :(circularly polarised plane wave)
where yixe ˆˆˆ +=
( )..ˆ),(2i-A )(
r cceetzA ctzik += −−
)2(w
wkλπ
=
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Energy of the electrons is 2γmcE =
Rate of electron energy change is dtdγmc
dtdE 2=
This must be equal to work done by EM wave on electrons i.e.
⊥⋅−= vEedtdγmc2
We want to know the beam-radiation energy exchange :
Problem : What is ?⊥vThe canonical momentum is a conserved quantity.
0constantAepΠ TT ==−=Ti.e.
Consequently : γmAe
vTT=
2.1 Classical Electron Dynamics (details in refs. 4,5)
TT mvp γ=
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( )t∂⋅∂
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂−−=
⋅−=
⊥⊥
⊥⊥
AA21
cme
γmAe
tA
mce
vEmc
edtdγ
22
2
2
2
γ
where rw AAA +=⊥
(wiggler + EM field)
Now rrrwww AAAA2AAAA ⋅+⋅+⋅=⋅ ⊥⊥
no time dependence
EM field << wiggler
so the only term of interest is ( )( )..),(AA2 ccetzAA tzkki
wrww −∝⋅ −+− ω
so ( )( )..2
)( cceaakdzd cktzkkiw w +−= −+
γγ
⎟⎠⎞
⎜⎝⎛ =
mceAa w
w(1)
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Whether electron gains or loses energy depends on the value ofthe phase variable
( ) tzkkw ωθ −+=
The EM wave (ω,k) and the wiggler “wave”
(0,kw
) interfere to produce a “ponderomotive
wave”
with a phase velocity
kkv
wph +
=ω
From the definition of θ,
it can be shown that :
r
rww kckk
γγγ −
=⎟⎟⎠
⎞⎜⎜⎝
⎛−+= 2
v1
dzdθ
z
( )( )..2
)( cceaakdzd cktzkkiw w +−= −+
γγ
( )kc=ω
where ⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
21 2
wwr
aλ
λγ is the resonant energy
(2)
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FEL resonance condition
( )2
2
21
γλλ w
wa+
= (magnetostatic
wiggler )
( )2
2
41
γλλ w
pumpa+
= (electromagnetic wiggler )
Example : for λ=1A, λw
=1cm, E~5GeV
Example : for λ=1A, λpump
=1μm, E=35MeV
1=waLet:
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2.2 Field Dynamics
The radiation field evolution is determined by Maxwell’s wave
equationJ
tA
cA r
r 02
2
22 1 μ−=
∂∂
−∇
The (transverse) current density is due to the motionof the (point-like) electrons in the wiggler magnet.
( )∑ −−=j
jj trrveJ )(δ
( )..ˆ),(2i-A )(
r cceetzA ctzikr += −−Radiation field :
(circularly polarised plane wave)
where γmAe
v w≈
Apply the SVEA : ),(),(),,(),( tzkAtzAz
tzAtzAt
<<∂∂
<<∂∂ ω
and average on scale of λr
to give
θ
γωω i
r
wp eaktzatcz
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛
∂∂
+∂∂
2
2
2),(1
2
0p
enem
ωε
=where(3) ( )∑
=
=N
jjN 1
....1....
(details in refs. 4,5)
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13
‘Classical’ universally scaled equations2
2
11
( . .)
1
j
j
ij
Ni
j
Ae c czA A ez z N
θ
θ
θ
−
=
∂= − +
∂∂ ∂
+ =∂ ∂ ∑
2| | Rad
Beam
PAP
ρ =
gLzz =
cLtzz 0
1v−
=
;4πρλw
gL =21
3312 2
W W
A Beam
aII
λργ πσ
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
j
Vθ
∂= −
∂
;4
rcL λ
πρ=
A is the normalised S.V.E. A. of FEL rad. –
self consistent
( )φθ += jAV sin2
j j rj
r
pz
θ γ γργ
∂ −= =
∂
Ref 1.
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We will now use these equations to investigate the high-gainregime.
We solve the equations with initial conditions
( ]πθ 2,0=j (uniform distribution of phases)
0=jp (cold, resonant beam)
1<<A (small input field)
and observe how the EM field and electrons evolve.
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Strong amplification of field is closely linked to phase bunching of electrons.
Bunched electrons mean that the emitted radiation is coherent.
For randomly spaced electrons : intensity ∝
NFor perfectly bunched electrons : intensity ~ N2
It can be shown that at saturation in classical case, intensity ∝
N4/3
As radiated intensity scales > N, this indicates collective behaviour
Exponential amplification in high-gain FEL is an example of a collective instability.
z>0
z=0
Ponderomotivepotential
∑=
−=N
j
i jeN
b1
1 θ
|b|<<1
|b|~1
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Collective Recoil Lasing = Optical gain + bunching
In FEL
and CARL
particles self-organize to form compact bunches ~λ
which radiate coherently.
∑=
−=N
j
i jeN
b1
1 θbunching factor b (0<|b|<1):
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FEL instability animation
Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.
01
=∂∂zA
)sin(||2)( ϕθϕθ +=+ AV
Steady State
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Classical high-gain FEL
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A fully Hamiltonian model of the classical FEL Bonifacio, Casagrande
& Casati, Optics Comm. 40
(1982)
(constant) C02
02 =+=+ pApA
Defining ϕiaeA = then pCa −=
Defining ϕθ += jjq then the FEL equations can be rewritten as
pC
qp
zddq
jj
−−=
sin
jj qpC
zddp
cos2 −−=
jpH
∂∂
=
jqH
∂∂
−=
where ∑⎥⎥⎦
⎤
⎢⎢⎣
⎡−+=
jj
j qpCp
H sin22
2
Equilibrium occurs when 2
3π=jq so 0=
zddp j
BUT 0>zd
dq jso 0<
zdpd
i.e. GAIN
Steady State
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zσp
z
|A|2
|b|z
The scaled radiation power |A|2, electron bunching |b| and the energy spread σp
for the classical high-gain FEL amplifier.
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Classical chaos in the FEL
If we calculate the distance, d (z), between different trajectories in the 2‐dimensional phase‐space ( ) ( )',';, jjjj qpqp
( ) ( ) ( )[ ]∑ −+−=j jj qqppzd 22 ''so where ( ) 10 <<= εd
In the exponential regime : ( ) ( )zzd αexp∝ 0; >α
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Linear Theory (classical) Ref(1)
Linear theory 0A i A iAδ− − =
( ) 2 1 0λ δ λ− + =
Maximum gain at δ=0
2 33
gtz
A e e∝ =
Quantum theory: different results(see later)
0 r
F r
γ γδρ γ
−=
runawaysolution
/(4 )g w Fλ πρ=⎡ ⎤⎣ ⎦
See figure (a)
zieA λ∝
λIm
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CLASSICAL REGIME, LONG PULSEL = 30LC , resonant (δ=0)
For long beams (L >> Lc
) Seeded Superradiant
Instability Ref(2):
Including propagation
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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 electronswithout 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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25
c
bs L
LNπ2
=
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2626
DRAWBACKS OF ‘CLASSICAL’
SASE
simulations from DESY
for the SASE experiment (λ
~ 1 A)
Time profile has many random spikes
Broad and noisy spectrum atshort wavelengths (x-ray FELs)
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27
what is QFEL?
QFEL is a novel macroscopic quantum coherent effect:
collective Compton backscattering
of a high- power laser wiggler by a low-energy electron
beam. The QFEL linewidth
can be four orders
of
magnitude smaller than that of the classical SASE FEL
27
Phys. Rev. ST Accel. Beams 9
(2006) 090701Nucl. Instr. And Meth. A 593 (2008) 69
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28
Why QUANTUM FEL theory?
In classical theory e-momentum recoil ΔP continuous variable
WRONG: if one electron emits n photons knP =Δ QUANTUM THEORY
kP
kmc )(σγρρ =⎟
⎠⎞
⎜⎝⎛=
QUANTUM FEL
parameter:21
3312 4
L W
A Beam
aII
λργ πσ
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
If 1>>ρ CLASSICAL LIMIT
1<<ρIf STRONG QUANTUM EFFECTS
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2929
why QFEL requires a LASER WIGGLER?why QFEL requires a LASER WIGGLER?
kkmc p
C
r
r
σλλργγρρ ===
r
2Ww
2)a1(
λ+λ
=γ
)a1(21
2WWr
C
+λλ
λ≤ρ⇒≤ρ andand
C
2W
3WrW
W 2)a1(
Lλ
+λλ≥
ρλ
≈
for a laser wigglerfor a laser wiggler 2/LW λ→λ
MAGNETIC WIGGLER:MAGNETIC WIGGLER:
λλW W ~~
1cm, E 1cm, E ~~1010
GeVGeV
ρρ
~~
1010--66
,,
LLW W ~~
1Km1Km
to lase atto lase at
λλrr
=0.1 Α:=0.1 Α:
LASER WIGGLERLASER WIGGLER
λλL L ~~
1 1 μμm, E m, E ~~100100
MeVMeV
ρρ
~~
1010--4 4 ,,
LLW W ~~
1 mm1 mm
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30
Conceptual design of a QFEL Conceptual design of a QFEL
( )202 1
4aL
r +=γλλ 1L mλ μ= If γ ≅
200 ( E ≅
100 MeV) ⇒ λr
≅
0.3 Å !
λr
λL
RTWPa L
Lλ)(4.20 = 210,1,100 0 =⇒=== amRmTWP LL μμλ
Compton back-scattering (COLLECTIVE)
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3131
Procedure : Describe N particle system as a Quantum Mechanical
ensemble
Write a Schrödinger-like equation for macroscopic wavefunction:Ψ
QUANTUM FEL MODEL
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32
1D QUANTUM FEL MODEL
{ }2
12
22
11 0
1 ( , ) . .2
| ( , , ) |
i
i
i i A z z e c cz
A A d z z ez z
θ
πθ
ρρ θ
θ θ −
∂Ψ ∂ Ψ= − − − Ψ
∂ ∂
∂ ∂+ = Ψ
∂ ∂ ∫
A : normalized FEL amplitude
( )
1
;8
42
Lg
g
z
c
rc
z
zz LL
z v tzL
L
z v t
λπρ
λπρ
πθλ
= =
−=
=
= −
R.Bonifacio, N.Piovella,
G.Robb, A. Schiavi, PRST-AB (2006)
( )..2
2
ccAeipH i −−= θρρ
[ ] ip =,θθ∂∂
−= ip
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{ }2
13/ 2 2
22
11 0
1 ( , ) . .2
| ( , , ) |
i
i
i i A z z e c cz
A A d z z ez z
θ
πθ
ρ θ
θ θ −
∂Ψ ∂ Ψ= − − − Ψ
∂ ∂
∂ ∂+ = Ψ
∂ ∂ ∫
Let ( )φin exp=Ψ and3/ 21v φ
θρ
∂=
∂
θθ ∂∂
−==∂∂
+∂∂
= TOTVvvzv
zdv Fd where ( )TOTV . .ii Ae ccθ=− −
2
3 2
1 1
2
nn θρ
⎡ ⎤∂+ ⎢ ⎥∂⎣ ⎦
( ) 0nvzn
=∂
∂+
∂∂
θ
∫ −=∂∂
+ θθ denz
d i
1
AzdA
See E. Madelung, Z. Phys 40, 322 (1927)
∞→ρClassical limit : no free parameters
MadelungMadelung
Quantum Fluid Description of QFEL*Quantum Fluid Description of QFEL**R. Bonifacio, N. Piovella, G. R. M. Robb, and A. Serbeto, Phys. Rev. A 79, 015801 (2009)
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WignerWigner
approach for 1D QUANTUM MODELapproach for 1D QUANTUM MODEL
Introducing the Wigner
function :
*1( , )2 2 2
iqp q qW p dq eθ θ θπ
⎛ ⎞ ⎛ ⎞= Ψ − Ψ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ρpp
2
2
( , ) ( )
( , ) ( )
dp W p
d W p p
θ θ
θ θ
= Ψ
= Ψ
∫∫
Using the equation for we obtain a finite-difference
equation for ),( zθΨ
),,( zpW θ
( ) 1 1,. . ,2 2
0iW Wp Ae c W pcz
W pθ ρ θ θρ ρθ
⎧ ⎫⎛ ⎞ ⎛ ⎞+ − −⎨ ⎬⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎩
∂ ∂+ − −
∂ ⎭=
∂
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for
ρ>>1:
The Wigner
equation becomes a Vlasov
equationdescribing the evolution of a classical
particle ensemble
The classical model is valid when Quantum
regime for
( ) 0.. =∂∂
−−∂∂
+∂
∂pWccAeWp
zW iθ
θ
(1 1, ),2 2
,W pWp
p W pρ θ θρ ρ
θ⎧ ⎫⎛ ⎞ ⎛ ⎞+ − −⎨ ⎬⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎩ ⎭
∂→
∂
1>>ρ1<ρ
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36
Quantum Dynamics
Only discrete
changes of momentum are possible : pz
= n ( k) , n=0,±1,..
pz
k
n=1
n=0
n=-1
is momentum eigenstate corresponding to eigenvalue ( )n kine θ
2| |n nc p=
kzecn
inn ==Ψ ∑
∞
−∞=
θθ ,
probability to find a particle with p=n(ħk)
( )πθ 2,0∈
( )
AicczA
zA
cAAccinzc
nnn
nnnn
δ
ρρ
+=∂∂
+∂∂
−−−=∂
∂
∑∞
−∞=−
+−
*1
1
1*
1
2
2
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37
classical limit is recovered for
many momentum states occupied,
both with n>0 and n<0
1>>ρ
-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
ρ=10, δ=0, no propagation
(a)
z
|A|2
steady-state evolution: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂ 0
1zA
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Quantum bunching
38
( )φθψ
ψ θ
++=
+=
cos21 102
10
cc
ecc i
φ: relative phase
Momentum wave interference
( )φθψ +=== 2210 cos2
21ccMaximum interference:
,zk=θ
Maximum bunching when 2-momentum eigenstates
are equally populated with fixed relative phase
121
20 =+ ccwhere
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39
0 1 2 3 4 50
2
4
6
8
10
N(θ
)/N
θ /2π
-20 -15 -10 -5 0 5 10 150.00
0.05
0.10
0.15
p n
n
0 1 2 3 4 50 .0
0 .5
1 .0
1 .5
2 .0
N(θ
)/N
θ /2 π
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
p n
n
Bunching and density grating
CLASSICAL REGIME ρ>>1 QUANTUM REGIME ρ<1
2| ( ) |ψ ϑ 2| ( ) |ψ ϑ
( ) inn
nc e ϑψ ϑ = ∑
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40
The physics of the Quantum FEL
,..)1,0( −=n
k)p(
kmc zσ
=γ
ρ=ρ
knk ( ) kn 1−
MomentumMomentum--energy levels:energy levels:(p(pzz
=n=nħħkk, , EEnn
∝∝ppzz
2 2 ∝∝nn22))
1>>ρCLASSICAL REGIME:CLASSICAL REGIME:many momentum level many momentum level
transitionstransitions→→
many spikesmany spikes
QUANTUM REGIME:QUANTUM REGIME:a single momentum level a single momentum level
transitiontransition→→
single spikesingle spike
1≤ρ
Frequencies equally spaced byFrequencies equally spaced by
⎟⎠⎞
⎜⎝⎛ −=
−= −
211
21 nEE nn
n ρρϖ
ρ1 with widthwith width 4 ρ
Increasing the lines overlap for Increasing the lines overlap for ρ 0.4ρ >
ρρ 23,
21 −−
= (harmonics)
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( )zieA λ∝Quantum Linear Theory
( ) 014
12
2 =+⎟⎟⎠
⎞⎜⎜⎝
⎛−Δ−
ρλλ
-10 -5 0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
(a) 1/2ρ=0(b) 1/2ρ=0.5(c) 1/2ρ=3(d) 1/2ρ=5(e) 1/2ρ=7(f) 1/2ρ=10
(f)(e)
(d)(c)
(b)(a)
|Imλ|
δ
Classicallimit
Quantum regime for ρ<1
max atρ21
=Δ
width ρ∝
( ) 012 =+Δ− λλ1>>ρ
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4242
0.2ρ =
0.4ρ =σ= 4 ρ
Continuous limit 4 1/ 0.4ρ ρ ρ→≥ ≥
discrete frequencies as in a cavity
max formax for ρ=Δ 2/1
)1n2(21
n −ρ
=ω
⎟⎟⎠
⎞⎜⎜⎝
⎛
ρω
ω−ω=ω
sp
sp
2
ω
ω
ω
( )zieA λ∝( ) 2
2
1 1 04
λ λρ
⎛ ⎞− Δ − + =⎜ ⎟⎝ ⎠
0.1ρ =1ρ
⎟⎟⎠
⎞⎜⎜⎝
⎛−=Δ ω
ρn
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4343Classical regime: both n<0 and n>0 occupied
CLASSICAL REGIME: 5=ρ QUANTUM REGIME: 1.0=ρ
momentum distribution for SASE
Quantum regime: sequential SR decay, only n<0
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4444/ 30cL L =
SASE Quantum purificationSASE Quantum purification
quantum regimequantum regime classical regimeclassical regime( )05.0=ρ ( )5=ρ
R.BonifacioR.Bonifacio, , N.PiovellaN.Piovella, , G.RobbG.Robb, NIMA(2005), NIMA(2005)
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4545
0.1 1/ 10ρ ρ= = 0.2 1/ 5 ρ ρ= =
0.3 1/ 3.3ρ ρ= = 0.4 1/ 2.5ρ ρ= =
,..]1,0n[2/)1n2(n −=ρ−=ω
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46-8 -7 -6 -5 -4 -3 -2
0,0
0,2
0,4
0,6
0,8
1,0
ρ≈ωωΔ 2
46
bLλ
≈ωωΔ
LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME
CLASSICAL ENVELOPE
QUANTUM SINGLE SPIKE
rλ
bL b
r
QFEL Lλ
≈⎟⎠⎞
⎜⎝⎛
ωωΔ
spikesc
b NL
Lquantumclassical
==π2
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47
QFEL requirements
Emittance:
)1(105)(
20
2/34
aEE
rL +⋅< −
λλρσ
20
23
32
300)(a
AILrλλρσ
=
Energy spread : ( )Arλ ( )mL μλ
( )mμσ
Condition to neglect diffraction :2
int4
LL
RZ Lπλ
= >
( ) ( )[ ] 31
pC03.0mradmm Qn ×≈ε (thermal)
Rosenzweig et al, NIM A 593, 39 (2008)
Not necessary with plasma guiding (D. Jaroszynski collaboration)
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48
Harmonics Production
,..)5,3,1h(h =ω
Distance between gain lines: ρ
=Δh
Gain bandwidth of each line:
h4 ρ
=σ
.
Possible frequencies
One photon recoil khLarger momentum level separation quantum effects easier
Extend Q.F. Model to harmonics [G Robb NIMA A 593, 87 (2008)]
Results (a0
>1)
Separated quantum lines if Δ<σ i.e.3/44.0 h≤ρ
4.01 ⇒=h 7.13⇒=h 4.35⇒=hPossible classical behaviour for fundamental BUT quantum for harmonics
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49
1=ρ
Fundamental3rd
harmonic5th
harmonic0.3A 0.1A 0.06Ae.g.
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Main limitations in classical regime :
πρλ
4w
gw LL =>> 43 10,10 −−≈ρ
ρσ<
EE)(
gL>*β
g
rn L
*
4β
πγλε ≤
1.
2.
3.
4.
Quantum FEL : as above with
2L
wλλ → ρρρ →
Quantum regime easier in the sub-A region and 1≈ρ
1<= ργρk
mc
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Parameters for QFEL
51
Q (pC) 1τ
(fs) 1.3 I (kA) 0.77 εn (mm mrad) 0.03E (MeV) 100σ
(μm) 0.5ΔE/E 4x10-4
λL
(μm) 1PL
(TW) 100 aw 2τ
(ps) 3.4R (μm) 12.6Lint (mm) 1
λr
(A) 0.3Pr
(MW) 30Δω/ω 7x 10-5
Nphot 6x 106
τ
(fs) 1
Electron beam Laser beam QFEL beam
Relaxed parameters with plasma channel (guiding) : Dino Jaroszynski
1=ρ
Note : 5th harmonic at 0.06 A
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CLASSICAL SASEneeds:GeV LinacLong undulator (100 m)yields:High PowerBroad and chaotic spectrum
FEL IN CLASSICAL\SASE
CAN GO TO λ=1.5Ǻ
(LCLS)
QUANTUM SASEneeds:100 MeV Linac Laser undulator (λ~1μm)yields:Lower powerVery narrow line spectrum
QUANTUM SASE WORKS BETTER FOR SUB-Ǻ
REGION
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Quantum FEL and Bose-Einstein Condensates (BEC)
It has been shown [8] that Collective Recoil Lasing (CARL)from a BEC driven by a pump laser and a Quantum FEL are described by the same theoretical model.
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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
λw
)
Electron beam
EM radiationλ ∝ λw
/γ2 << λwN
S N
S N
S N
S N
S N
S
At first sight, CARL and FEL look very different…
λ~λp
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electrons
EM pump, λ’w(wiggler)
BackscatteredEM fieldλ’
≈ λ’w
Connection between CARLand FEL
can be seen
more easily by transforming to a frame (Λ’)
moving with electrons
Cold atoms
Pumplaser
Backscatteredfield
Connection between FELand CARL
is now clear
FEL
CARL
λ~λp
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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:
Δ
= 13 GHzw = 750 mmP = 13 mW
laser beam kw,
BEC
absorption imaging
trap
g
Experimental Evidence of Quantum Dynamics Experimental Evidence of Quantum Dynamics The LENS ExperimentThe LENS Experiment
2p kΔ =R. B., F.S. Cataliotti, M.M. Cola, L. Fallani, C. Fort, N. Piovella, M. Inguscio,
Optics Comm. 233, 155(2004) and Phys. Rev. A 71, 033612 (2005)
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LENS experimentLENS experiment
pump light
p=0 p=-2hk p=-4hk
n=0 n=-1 n=-2
Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse.
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MIT experimentMIT experimentSuperradiant
Rayleigh
Scattering from a BECS. Inouye et al., Science 285, 571 (1999)
Back scattered intensity for different laser powers: 3.8 2.4 1.4 mW/cm2
Duration 550 μs
Number of recoiled particles for different laser intensity (25 & 45 mW/cm2). Total number of atoms 2·
107
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Superradiant
RayleighScattering
in a BEC(Ketterle, MIT 1991)
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Summarising:
A BEC driven by a laser field shows momentum quantisation and superradiant backscattering as in a QFEL, being described by the same system of equations.