Introduction to Classical and Quantum High-Gain FEL Theory

Introduction to Classical and Quantum High-Gain FEL Theory

Rodolfo Bonifacio&

Gordon Robb

University of Strathclyde, Glasgow, Scotland.

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

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

Exponential growth of the emitted radiation and bunching:

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)

(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

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

=

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=

( )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)

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)

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:

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)

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.

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.

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

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):

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

Classical high-gain FEL

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

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.

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

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

CLASSICAL REGIME, LONG PULSEL = 30LC , resonant (δ=0)

For long beams (L >> Lc

) Seeded Superradiant

Instability Ref(2):

Including propagation

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

25

c

bs L

LNπ2

=

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)

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

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

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

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)

3131

Procedure : Describe N particle system as a Quantum Mechanical

ensemble

Write a Schrödinger-like equation for macroscopic wavefunction:Ψ

QUANTUM FEL MODEL

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

{ }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)

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

⎧ ⎫⎛ ⎞ ⎛ ⎞+ − −⎨ ⎬⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎩

∂ ∂+ − −

∂ ⎭=

∂

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

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

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

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

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 = ∑

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)

( )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

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

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

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)

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

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

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)

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

49

1=ρ

Fundamental3rd

harmonic5th

harmonic0.3A 0.1A 0.06Ae.g.

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

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

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

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.

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

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

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)

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.

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

Superradiant

RayleighScattering

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.