Quantum Effects in BECs and FELs

Nicola Piovella, Dipartimento di Fisica and INFN-Milano

Rodolfo Bonifacio, INFN-Milano

Luca Volpe (PhD student), Dipartimento di Fisica-Milano

Mary Cola (Post Doc), Dipartimento di Fisica-Milano

Gordon R. M. Robb, University of Strathclyde, Glasgow, Scotland.

work supported by INFN (QFEL project)

Outline

1. Introductory concepts

2. Classical FEL-CARL Model

3. Quantum FEL-CARL Model

4. Propagation Effects

5. Quantum SASE regime

Free Electron Laser (FEL) 22 w

r

Collective Atomic Recoli Laser (CARL)

Pump beam p

Probe beam p

R. Bonifacio et al, Opt. Comm. 115, 505 (1995)

Both FEL and CARL are examples of collective recoil lasing

Cold atoms

Pump field

Backscattered field(probe)

CARL

FEL

“wiggler” magnet(period w)

Electron beam

EM radiation w /<< w N

S N

S N

S N

S N

S N

S

At first sight, CARL and FEL look very different…

~p

electrons

EM pump, ’w

(wiggler)

BackscatteredEM field’ ’w

Connection between CARL and FEL can be seen

more easily by transforming to a frame (’)

moving with electrons

Cold atoms

Pumplaser

Backscatteredfield

Connection between FEL and CARL is now clear

FEL

CARL

~p

Collective Recoil Lasing = Optical gain + bunching

In FEL and CARL particles self-organize to form compact bunches ~ which radiate coherently.

N

j

i jeN

b1

1 bunching factor b (0<|b|<1):

Exponential growth of the emitted radiation:

Both FEL and CARL are described using the same ‘classical’ equations, but different independent

variables.

AieNz

A

z

A

ccAez

N

j

i

ij

j

j

11

2

2

1

.).(

FEL:

;)( tzkk w

CARL:;2kz

N

NA photons2||

gL

zz

cL

tzz 0

1

v

;4w

gL ;gw

c LL

3/13/20 nBk

mcw

3/2

3/13/1

a

L nP

)/(1 czztz recrec

m

krec

22

CARL-FEL instability animation

Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.

01

z

A

)cos(||2)( AV

Linear Theory (classical)

2 1 0

Maximum gain at =0

runaway solution

See figure (a)

)()()( 0 CARLFEL

k

mc

rec

pr

zieA

zeA 32||

We now describe electrons/atoms as QM wavepackets, rather than classical particles.

Procedure :

Describe N particle system as a Q.M. ensemble

Write Schrodinger equation for macroscopic wavefunction

),( z

Quantum model of FEL/CARL

Include propagation using a multiple-scaling approach

),,( 1zz

Canonical Quantization

p

Hp

H

ccAep i ..

..2

2

ccAeip

H i

Quantization (with classical field A) :

ipp ˆ ip ]ˆ,ˆ[ HH ˆ

Hz

i ˆ

so

Aiezdzd

dA

ccezAiz

i

i

i

2

0

2

2

2

),(

..)(2

1

so

)()( 0 FEL

k

mc

k

pp z

R. Bonifacio, N. Piovella, G.R.M.Robb and M.Cola, Optics Comm, 252, 381 (2005)

Quantum FEL Propagation model

Here describes spatial evolution of on scale of and describes spatial evolution of A and on scale of cooperation length, Lc >>

1z

1 ( ) /r cz z v t L

We have introduced propagation into the model, sodifferent parts of the electron beam can feel different fields :

So far we have neglected slippage, so all sections of the e-beamevolve identically (steady-state regime) if they are the same initially.

Aiezzdz

A

z

A

ccezzAi

z

i

i

2

0

21

1

12

2

),,(

.].),([2

where

4cL

Quantum Dynamics

Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,..

pz kn=1n=0n=-1

is momentum eigenstate corresponding to eigenvalue ( )n kine

2| |n nc p

n

inn ezzczz ),(),,( 11

probability to find a particle with p=n(ħk)

2,0

Aiccz

A

z

A

cAAccin

z

c

nnn

nnnn

*1

1

1*

1

2

2

classical limit is recovered for

many momentum states occupied,

both with n>0 and n<0

1

-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

=10, no propagation

(a)

z

|A|2

steady-state evolution:

01z

A

Quantum limit for

iezczcz )()(),( 10

Only TWO momentum states involved : n=0 and n= - 1

n=0

n=-1

Dynamics are those of a 2-level system coupled to an optical field,described byMaxwell-Bloch equations

1

0 100 2000

2

4

6

8

10

z

|A|2

0 100 200

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

z

<p>

1.0

01z

A

0 1 2 3 4 50

2

4

6

8

10

N(

)/N

/2

-20 -15 -10 -5 0 5 10 150.00

0.05

0.10

0.15

pn

n

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

N(

)/N

/2

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

0.6

pn

n

Bunching and density gratingCLASSICAL REGIME >>1 QUANTUM REGIME <1

2| ( ) | 2| ( ) |

( ) inn

n

c e

zieA Quantum Linear Theory

014

12

2

-10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

(a) (b) (c) (d) (e) (f)

(f)(e)

(d)

(c)

(b)(a)

|Im|

Classicallimit

Quantum regime for <1

max at

2

1

)(

)(2/)( 0

CARL

FELkmc

recp

r

width

012 1

QUANTUM CARL HAS BEEN OBSERVED WITH BECs IN SUPERRADIANT REGIME (MIT, LENS)

When the light escapes rapidly from the sample of length L,we see a sequential Super-Radiant (SR) scattering, with atoms recoiling by 2ħk, each time emitting a SR pulse

KA

L

cK

Aicczd

dA

cAAccin

zd

dc

nnn

nnnn

*1

1*

1

2

2

damping of radiation

k2n=-2

n=0

n=-1

0 250 5000.000

0.001

0.002

z

|A|2

0 250 500

-4

-2

0

z

<p>

BECLASER

k2

)(sec|| 222 gNzhNA

SEQUENTIAL SUPERRADIANT SCATTERING

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

for K>>1 and

K

• 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 GHzw = 750 mP = 13 mW

laser beam kw,

BEC

absorption imaging

trap

g

Experimental evidence of quantum CARL at LENS

2p k

L.Fallani et al, PRA 71 (2005) 033612

The experiment

pump light

n=0(p=0)

n=-1(p=2ħk)

n=-2(p=4ħk)

Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse.

Particles at the trailing edge of the beam never receive radiation from particles behind them: they just radiatein a SUPERRADIANT PULSE or SPIKE which propagates forward.

if Lb << Lc the SR pulse remains small (weak SR).

if Lb >> Lc the weak SR pulse gets amplified (strong SR) as it propagates forward through beam with no saturation.

The SR pulse is a self-similar solution of the propagation equation.

PROPAGATION EFFECTS IN FELs : SUPERRADIANT INSTABILITY

1

i

z

Ae

z

A

Strong SR (Lb=30 Lc) from a coherent seed

SR in the classical model:

c1 L

vtzz

R. Bonifacio, B.W. McNeil, and P. Pierini PRA 40, 4467 (1989)

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 electrons without saturating. Spiky time structure and spectrum.

SASE is the basic method for producing coherent X-ray radiation in a FEL

CLASSICAL SASE

CLASSICAL SASE

Example from DESY (Hamburg) for the SASE-FEL experiment

Time profile with many randomspikes (approximately L/Lc)

Broad and noisy spectrum atshort wavelengths (X-FEL)

SASE : NUMERICAL SIMULATIONS

cLL 30

CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0

Classical behaviour : both n<0 and n>0 occupied

CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0

SASE: average momentum distribution

Quantum behaviour : sequential SR decay, only n<0

0.1 1/ 10 0.2 1/ 5

0.3 1/ 3.3 0.4 1/ 2.5

Quantum SASE:Spectral purification and multiple line spectrum

• In the quantum regime the gain bandwidth decreases as line narrowing.

• Spectrum with multiple lines. When the width of each line becomes larger or equal to the line separation, continuous spectrum, i.e., classical limit. This happens when

3/ 24

3/ 24 1 0.4

CLASSICAL SASEneeds:GeV Linac (Km)Long undulator (100 m)High cost (109 $)yields:Broad and chaotic spectrum

FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE FOR AN X-RAY SOURCE (=1Ǻ)

QUANTUM SASEneeds:MeV Linac (m)Laser undulator (~1m)lower cost (106 $)yields:quasi monocromatic spectrum

CONCLUSIONS

• Classical FEL/CARL model- classical motion of electrons/atoms- continuous momenta

• Quantum FEL/CARL model- QM matter wave in a self consistent field- discrete momentum state and line spectrum

• Quantum model with propagation- new regime of SASE with quantum ”purification’’ - appearance of multiple narrow lines