1 1

R. Bonifacio University of Milano and INFN – LNF

Introduction to Classical and Quantum FEL Theory

Natal 2016

2 2

OUTLINE

Classical SASE and spiking

Semi-classical FEL theory: quantum purification

Fully quantum theory

Conclusions

3 3

SASE high-gain regime

( )22 1

2w

waλλγ

= +

wλ

w w wa Bλ=

4

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).

5

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

7

c

bs L

LNπ2

=

8 8

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)

9 9

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)

11 11

Procedure : Describe N particle system as a Quantum Mechanical ensemble

Write a Schrödinger-like equation for macroscopic wavefunction: Ψ

QUANTUM FEL MODEL

12 12

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

14

{ }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

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

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

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∆

22 22

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)

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)

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)

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

Superradiant Rayleigh Scattering in a BEC (Ketterle, MIT 1991)

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.