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R. R. BonifacioBonifacioINFN-Milano and CBPF * – RJ – Brazil

N. N. PiovellaPiovella, M.M. Cola, L. VolpeDipartimento di Fisica & INFN-Milano

G.R.M. RobbUniversity of Strathclyde, Glasgow, UK

A. SchiaviUniversità di Roma “La Sapienza”

* Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, Brazil

The Quantum Free Electron LaserThe Quantum Free Electron Laser

(QFEL)(QFEL)

2

OUTLINEOUTLINE

Classical SASE and spikingClassical SASE and spiking

Quantum theory and QFEL parameterQuantum theory and QFEL parameter

Linear quantum theoryLinear quantum theory

Quantum purification and narrow Quantum purification and narrow linewidthlinewidth

Why a laser wiggler? Why a laser wiggler? EmittanceEmittance criteria, criteria,

possible experimental set up and parameterspossible experimental set up and parameters

ConclusionsConclusions

3

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 classicalSASE FEL

4

SASESASE highhigh--gaingain regimeregime

5

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:

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DRAWBACKS OF ‘CLASSICAL’ SASE

simulations from DESY for the SASE experiment

Time profile has manyrandom spikes (n= Lb/Lc)

Broad and noisy spectrum atshort wavelengths (x-ray FELs)

R.Bonifacio, L. De Salvo, P.Pierini, N.Piovella, C. Pellegrini, PRL (1994)

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Procedure : Describe N particle system as a Quantum Mechanical ensemble

Write a Schrödinger equation for macroscopic wavefunction:

or equivalently ..

the equation for the Wigner function (quantum distribution):

Ψ

QUANTUM FEL MODEL

Include slippage z1(i.e. propagation)

)z,z,p,(Wor

)z,z,(

1

1

θ

θΨW

8

1D QUANTUM FEL MODEL

⎟⎠⎞

⎜⎝⎛=

kmc

FELh

γρρ

1>>ρ

{ }

θπ

θ

θθ

θρ

i

i

ezzdzA

zA

ccezzAiz

i

−Ψ=∂∂

+∂∂

Ψ−−∂

Ψ∂−=

∂Ψ∂

∫ 21

2

01

12

2

2/3

|),,(|

..),(2

1

A : normalized FEL amplitude

: QUANTUM FEL parameter

gL

c

c

z1

g

LL

Ltvzz

Lzz

⎟⎟⎠

⎞⎜⎜⎝

⎛λλ

≈

−=

=

the classical model is valid when

R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006)

G. G. PreparataPreparata from QFT, PRA (1988)from QFT, PRA (1988)01

=∂∂zA

9

01)( 2 =+Δ− λλ

θθ ˆˆ ii eppe −− →ρρ

λλ21

21'01)'( max

2 =Δ−Δ=Δ=+Δ−

LINEAR THEORYLINEAR THEORY

R.BonifacioR.Bonifacio, , C.PellegriniC.Pellegrini, , L.NarducciL.Narducci, Opt. , Opt. CommunCommun. (1985). (1985)

Classical theoryClassical theory

Quantum theoryQuantum theory

Mistake of equation (2): incorrect quantizationMistake of equation (2): incorrect quantization

014

1)( 22 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−Δ−

ρλλ

( )peeppe iii ˆˆ21 ˆˆ θθθ −−− +→ WeylWeyl symmetrizationsymmetrization rulerule

Correct quantizationCorrect quantization

1)1)

Previous approachPrevious approach

2)2)

3)3)R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006)

C.B.SchroederC.B.Schroeder, , C.PellegriniC.Pellegrini, , P.ChenP.Chen, , PRE (2001)PRE (2001)

[ ] θθ ˆˆ ˆ, ii epe −− =

( )zieA λ∝

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( )zieA λ∝Quantum Linear Theory

( ) 0141

22 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛ρ

−λΔ−λ

Classicallimit

Quantum regime for ρ<1

Resonance: ρ=Δ

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width ρ∝-10 -5 0 5 10 15

0.0

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

11

0.1ρ =

0.2ρ =

0.4ρ =

( ) 22

1 1 04

λ λρ

⎛ ⎞− Δ − + =⎜ ⎟⎝ ⎠

width 4 ρ

Continuous limit 4 1/ 0.4ρ ρ ρ→≥ ≥

discrete frequencies as in a cavity

1ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛ω−

ρ=Δ

2n

maxmax forfor ρ=Δ 2/1

)1n2(21

n −ρ

=ω

⎟⎟⎠

⎞⎜⎜⎝

⎛

ρω

ω−ω=ω

sp

sp

2

ω

ω

ω

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The physics of Quantum FEL

ρ

k)p(

kmc z

hh

σ=

γρ=ρ

kn hkh ( ) kn h1−

MomentumMomentum--energyenergy levelslevels::((ppzz=n=nħħkk, , EEnn∝∝ppzz

2 2 ∝∝nn22))

1>>ρCLASSICAL REGIME:CLASSICAL REGIME:manymany momentummomentum levellevel

transitionstransitionsflfl manymany spikesspikes

QUANTUM REGIME:QUANTUM REGIME:a single a single momentummomentum levellevel

transitiontransitionflfl single single spikespike

1≤ρ

[ ] 12)1( 221 −∝−−∝−= − nnnEE nnnω

In In classicalclassical regime regime withwith universaluniversal scaling no scaling no dependencedependence onon

Equally spaced frequencies as in a cavityEqually spaced frequencies as in a cavity

,......)1,0( −=n

13Classical 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

14/ 30cL L =

SASE mode SASE mode operationoperationquantum regimequantum regime classicalclassical regimeregime( )05.0=ρ ( )5=ρ

R.BonifacioR.Bonifacio, , N.PiovellaN.Piovella, , G.RobbG.Robb, NIMA(2005), 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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CLASSICAL AND QUANTUM SASE

3102 −≈≈Δ ρωω

classical regime, ρ=5 quantum regime, ρ=0.1

710−≈≈Δ

bLλ

ωω

many spikes underthe gain curve≈ 2π(Lb/Lc) only one spike!

17-8 -7 -6 -5 -4 -3 -2

0,0

0,2

0,4

0,6

0,8

1,0

ρ≈ωωΔ 2

bLλ

≈ωωΔ

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

≈⎟⎠⎞

⎜⎝⎛

ωωΔ

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whywhy QFEL QFEL requiresrequires a LASER WIGGLER?a LASER WIGGLER?

⎟⎠⎞

⎜⎝⎛ =λ

λλ

ργ=γ

ρ=ρmch

kmc

CC

r

rh 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

≈

forfor a laser wigglera laser wiggler 2/LW λ→λ

MAGNETIC WIGGLER:MAGNETIC WIGGLER:

λλWW ~~ 1cm, E 1cm, E ~~3.5 3.5 GeVGeV

ρρ ~~ 1010--66 ,, LLW W ~~ 3Km3Km

toto laselase atat λλrr=1 Α:=1 Α:

LASER WIGGLERLASER WIGGLER

λλWW ~~ 1 1 μμm, E m, E ~~25 25 MeVMeV

ρρ ~~ 1010--4 4 ,, LLW W ~~ 2 mm2 mm

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typical parameters for QFELtypical parameters for QFEL

E [MeV] 21

I [A] 420

εn [mm mrad] 0.1-1

δγ/γ [%] 0.01

λr [A] 1.5Pr [MW] 3.5

Δω/ω 10-7

Brillance 1028

λL [μm] 1

PL [TW] 2

τ [ps] 40

w0 [μm] 30

Zr [mm] 3

Electron beamElectron beam Laser beamLaser beamQFEL beamQFEL beam

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EMITTANCE CRITERIA FOR A LASER WIGGLEREMITTANCE CRITERIA FOR A LASER WIGGLER

n

2b*

L

2

R ,R4Zε

γσ=β

λπ

=

2L

n R4⎟⎠⎞

⎜⎝⎛ σ

πγλ

≤ε*RZ β≤

forfor instanceinstance γγ=50, =50, λλLL=1 =1 μμm, m, IfIf RR~2~2σσ , , εεnn<1 mm <1 mm mradmrad !!!!2

n R4 ⎟

⎠⎞

⎜⎝⎛ σ

≤ε

)/2)(1(06.0

2

2gLwL

gain

radrn

LZaR

LZ

+≈

≤

σλ

πγλε2) 2) ResonanceResonance broadeningbroadening

due due toto offoff--axisaxis emissionemission smallersmallerthanthan the the naturalnatural linewidthlinewidth

1) 1) ee--beambeam containedcontained in the laser in the laser beambeam::

rrZ

λπσ 24

=

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independentindependent parametersparameters::

SCALING LAWS FOR A LASER WIGGLERSCALING LAWS FOR A LASER WIGGLER

ρμλλ ,a),m(),A( wLr&

( )2w

r

L a150 +λλ

≈γ

)a1(105

2wLr

4

+λλ

ρ⋅≈ρ −

)1()a1(a105)A(I 32

w5r

L2w

3

ρ+ρ+λλ⋅

≈

32wLr

2wL

1)a1(a3)TW(Pρ

ρ++λλ⋅≈

4/1

32w

5Lr

1)a1(4)m( ⎥⎦

⎤⎢⎣

⎡ρ

ρ++λλ≈μσ

32w

3Lrg

1)a1(1.0)mm(Lρ

ρ++λλ≈

2wLn a113.0)radmm( +λ⋅≈ε

32w

3Lr

1)a1(35.1)ps(ρ

ρ++λλ⋅≈τ

ee--beambeam::

laser laser beambeam::R. R. BonifacioBonifacio, , N.PiovellaN.Piovella, , M.ColaM.Cola, ,

L.VolpeL.Volpe, NIMA (2007), NIMA (2007)

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QFEL parameter ρ 0.2 5 0.2

Laser wave length λL (μm) 1 1 10

Radiation length λr (Å) 1.5 1.5 1.5

Wiggler parameter a0 0.3 0.8 0.3

FEL parameter ρ 7.5·10-5 1.5·10-3 2.4·10-5

Gain length Lg (mm) 1.2 0.03 37.5

Laser Rayleigh range ΖL (mm) 3 0.07 94

Laser radius R (μm) 15.4 2.4 274

Laser power PL (TW) 2 0.3 6

Laser duration τL (ps) 40 1 1200

Interaction length Lint (mm) 6 0.14 187

Ε−beam energy γ 42.6 52.3 135

E-beam radius σ (μm) 10.3 1.6 183

Peak current I (kA) 0.4 22 1.3

Emittance limit εn (mm mrad) 0.1 0.1 0.9

Gain band width Δγ/γ 3.4·10-5 1.5·10-3 1.1·10-5

FEL line width Δω/ω 2.1·10-7 1.5·10-3 6.6·10-7

Number of spikes Ns 1 278 1

FEL power (MW) 3.5 900 11

Photons’ number Nph 6·109 3·1010 6·109

Peak Brilliance B(*) 1028 1.6·1026 1.2·1023

σ5.1,52 == RLZ gL

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a QFEL experiment•will generates a single spike almost monochromatic X-ray radiation (Δω/ω~10-7).•Needs a laser wiggler•Reduces cost (~ 106 U$) •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 spikyspectrum (Δω/ω~10-3).•Have very high cost (109 U$) and large size

Classical versus quantum SASEClassical versus quantum SASE

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Classical regime : >> 1Classical regime : >> 1Quantum regime: : discreteness of momentum exchangeQuantum regime: : discreteness of momentum exchange

relevant=> quantum effects. relevant=> quantum effects. The system is prepared in a defined momentum state pThe system is prepared in a defined momentum state p00, making , making

transition to the lower statetransition to the lower stateThe system radiates a The system radiates a monocromaticmonocromatic train wave lambda, whose train wave lambda, whose

length is Llength is Lbb. Hence one has a single line with . Hence one has a single line with linewidthlinewidth

In the opposite case random transition from many momentum In the opposite case random transition from many momentum states. Each transition gives a spike with different frequency. states. Each transition gives a spike with different frequency. Total bandwidth: Total bandwidth:

QFEL has a QFEL has a linewidthlinewidth 4 orders of magnitude smaller than the 4 orders of magnitude smaller than the classicalclassical

The The dimensionsdimensions and and costcost are are threethree orderorder of of magnitudemagnitudesmallersmaller

310−≈ρ

ρ1≤ρ

SummarySummary

710/ −≈br Lλ