Introduction to Classical and Quantum FEL Theory...Classical SASE FEL X-ray experiments (DESY,LCLS):...
Transcript of Introduction to Classical and Quantum FEL Theory...Classical SASE FEL X-ray experiments (DESY,LCLS):...
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R. Bonifacio University of Milano and INFN – LNF
Introduction to Classical and Quantum FEL Theory
Natal 2016
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OUTLINE
Classical SASE and spiking
Semi-classical FEL theory: quantum purification
Fully quantum theory
Conclusions
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SASE high-gain regime
( )22 1
2w
waλλγ
= +
wλ
w w wa Bλ=
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The FEL described using ‘classical’ universally scaled equations
2
2
11
( . .)
1
j
j
ij
Ni
j
Ae c czA A ez z N
θ
θ
θ
−
=
∂= − +
∂∂ ∂
+ =∂ ∂ ∑
jj zk=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
R.Bonifacio, C. Pellegrini and L. Narducci, Opt. Commun. 50, 373 (1984).
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Pendulum potential and self-bunching
11
1 jN
i
j
A A ez z N
θ−
=
∂ ∂+ =
∂ ∂ ∑
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SELF-BUNCHING • start up from noise • exponential growth of intensity and bunching • saturation (Prad ~ρ Pbeam) after several Lg
∑=
−=N
j
i jeN
b1
1 θ
b~0 b~0.8
wiggler length (several Lg)
bunching:
jj zk=θ
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c
bs L
LNπ2
=
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DRAWBACKS OF ‘CLASSICAL’ SASE
simulations from DESY for the SASE experiment (λ ~ 1 A)
Time profile has many random spikes
Broad and noisy spectrum at short wavelengths (x-ray FELs)
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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
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Conceptual design of a QFEL
( )202 1
4aL
r +=γλλ 1L mλ µ= If γ ≅ 200 ( E ≅ 100 MeV) ⇒ λr ≅ 0.1 Å !
λr
λL
RTWPa L
Lλ)(4.20 = 110,1,1 0 =⇒=== amRmTWP LL µµλ
Compton back-scattering (COLLECTIVE)
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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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1D QUANTUM FEL MODEL
kP
kmc
FEL
)(σγρρ =
=
1>>ρ
{ }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
∂ ∂ ∫
A : normalized FEL amplitude
: QUANTUM FEL parameter
( )
1
;4
42
wg
g
z
c
c
z
zz LLz v tz
L
L
z v t
λπρ
λπρ
πθλ
= =
−=
=
= −
the classical model is valid when
R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006)
G. Preparata from QFT, PRA (1988) 01
=∂∂zA
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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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{ }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 and 3/21v φ
θρ
∂≡
∂
θθ ∂∂
−==∂∂
+∂∂
= TOTVvvzv
zdv Fd where ( )TOTV . .ii Ae c cθ= − −
2
3 21 1
2
nn θρ
∂+ ∂
( ) 0nvzn
=∂
∂+
∂∂
θ
1
A Az
in e dz
θ θ−∂ ∂+ =
∂∂ ∫
See E. Madelung, Z. Phys 40, 322 (1927)
∞→ρClassical limit : See Dawson (1977) : no free parameters
Madelung Quantum Fluid Description of QFEL
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Quantum Dynamics
Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,..
pz
kn=1 n=0 n=-1
is momentum eigenstate corresponding to eigenvalue ( )n kine θ
2| |n nc p=
∑∞
−∞=
=n
inn ezzczz θθΨ ),(),,( 11
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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The physics of Quantum FEL
ρ
k)p(
kmc z
σ=
γρ=ρ
kn k ( ) kn 1−
Momentum-energy levels: (pz=nħk, En∝pz
2 ∝n2)
1>>ρCLASSICAL REGIME: many momentum level
transitions many spikes
QUANTUM REGIME: a single momentum level
transition single spike
1≤ρ
2 21 ( 1) 2 1 1, 3, 5,...n n nE E n n nω − = − ∝ − − ∝ − = − − −
In classical regime with universal scaling no dependence on
Equally spaced frequencies as in a cavity
,......)1,0( −=n
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0.1ρ =
0.2ρ =
0.4ρ =
( ) 22
1 1 04
λ λρ
− ∆ − + =
width 4 ρ
Continuous limit 4 1/ 0.4ρ ρ ρ→≥ ≥
discrete frequencies as in a cavity
1ρ
ω−
ρ=∆
2n
)1n2(21
n −ρ
=ω
ρω
ω−ω=ω
sp
sp
2
ω
ω
ω
( )zieA λ∝
ρ1
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18 18 Classical 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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19 19 / 30cL L =
SASE Quantum purification
quantum regime classical regime ( )05.0=ρ ( )5=ρ
R.Bonifacio, N.Piovella, G.Robb, NIMA(2005)
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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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21 21 -8 -7 -6 -5 -4 -3 -2
0,0
0,2
0,4
0,6
0,8
1,0
ρ≈ωω∆ 2
bLλ
≈ωω∆
LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME
CLASSICAL ENVELOPE (10-3 - 10-4)
QUANTUM SINGLE SPIKE (~10-7)
rλ
bL b
r
QFEL Lλ
≈
ωω∆
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why QFEL requires a LASER WIGGLER?
=λ
λλ
ργ=γ
ρ=ρmch
kmc
CC
r
r r
2Ww
2)a1(
λ+λ
=γ
)a1(21
2WWr
C
+λλ
λ≤ρ⇒≤ρ and
C
2W
3WrW
W 2)a1(
Lλ
+λλ≥
ρλ
≈
for a laser wiggler 2/LW λ→λ
MAGNETIC WIGGLER:
λW ~ 1cm, E ~10 GeV
ρ ~ 10-6 , LW ~ 1Km
to lase at λr=0.1 Α:
LASER WIGGLER
λL ~ 1 µm, E ~25 MeV
ρ ~ 10-4 , LInt ~ 1 mm
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Harmonics Production
,..)5,3,1h(h =ω
Distance between gain lines: ρ
=∆h
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 ∆<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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1=ρ
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a QFEL experiment •will generates a single spike almost monochromatic X-ray radiation (∆ω/ω~10-7). •Needs a laser wiggler •Reduces cost (~ 106 U$) for 0.1Å •Very compact apparatus (~ m)
Classical SASE FEL X-ray experiments (DESY,LCLS): • require very long Linac (~GeV, Km) and undulators (~100 m) •Generate cahotic radiation with broad and spiky spectrum (∆ω/ω~10-3). •Have very high cost (109 U$) and large size for 1Å
Classical versus quantum SASE
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Summary Classical regime : >> 1 Quantum regime: : discreteness of momentum exchange
relevant=> quantum effects. The system is prepared in a defined momentum state p0, making
transition to the lower state The system radiates a monocromatic train wave lambda, whose
length is Lb. Hence one has a single line with linewidth In the opposite case random transition from many momentum
states. Each transition gives a spike with different frequency. Total bandwidth:
QFEL has a linewidth 4 orders of magnitude smaller than the classical
The dimensions and cost are three order of magnitude smaller
310−≈ρ
ρ1≤ρ
710/ −≈br Lλ
For details see : Opt. Comm. 252, 381 (2005). (FEL and CARL)
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Quantum FEL and Bose-Einstein Condensates (BEC)
It has been shown that Collective Recoil Lasing (CARL) from a BEC driven by a pump laser and a Quantum FEL are described by the same theoretical model. [1] R. B., N. Piovella, G.R.M.Robb, and M.M.Cola, Optics Commun. 252, 381 (2005)
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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 GHz w = 750 mm P = 13 mW
Experimental Evidence of Quantum Dynamics The LENS Experiment
R. Bonifacio, 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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MIT experiment Superradiant Rayleigh Scattering from a BEC
S. 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 Rayleigh Scattering 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.
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To be published on EPL.
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Please note in: semiclassical theory the initial state with zero field and zero bunching is in equilibrium state. Here it is not because of spontaneous emission
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Conclusion
In semiclassical treatment the system behave as two level system where only transition to the first momentum recoil state. In quantum treatment in principal all recoil state can be achieved just properly detuning the system Further work need to be done.