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					 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 (-ray) source

7. Experimental evidence of QFEL in a BEC
1. Introduction
         The Free Electron Laser (FEL) consists of
            a relativistic beam of electrons (v≈c)
 moving through a spatially periodic magnetic field (wiggler).

                     S    N    S    N      S


    Relativistic                               EM radiation
                    N S N S N
    electron                                   l  lw /2 << lw
    beam         Magnetostatic “wiggler” field
                         (wavelength lw)

  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)
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)
          Some references relevant to this talk
        HIGH-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


(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)
2. The High-Gain FEL
We consider a relativistic electron beam moving in both a
magnetostatic wiggler field and an electromagnetic wave.

                    EM wave


            electron
            beam                            wiggler
                                                                         2
   Wiggler field (helical) : Aw 
                                  A w ik w z
                                              ˆ
                                      e e  c.c.                  (k w 
                                                                          lw
                                                                             )
                                   2
                             
   Radiation field :
   (circularly polarised     Ar 
                                  -i
                                                                      
                                     A( z , t ) e ik ( z ct ) e  c.c.
                                                                ˆ
                                   2
    plane wave)

                     where        e  x  iy
                                  ˆ ˆ ˆ
2.1 Classical Electron Dynamics           (details in refs. 4,5)
We want to know the beam-radiation energy exchange :
Energy of the electrons is E  γmc 2
                                  dE     2 dγ
Rate of electron energy change is     mc
                                  dt       dt
This must be equal to work done by EM wave on electrons i.e.
                        dγ        
                     mc   2
                             e E  v
                        dt
                  
Problem : What is v  ?
 The canonical momentum is a conserved quantity.
                       
       i.e. ΠT  pT  e AT  constant  0
                                                pT  mvT
                               e AT
           Consequently : v T 
                                 γm
 dγ     e  
        2
            Ev
 dt    mc
                           
           e  A    e A                               
                                
         mc  2 
                    t   γm                where A  Aw  Ar
                                     
                              
           2                                          (wiggler + EM field)
         e      1  A  A
     
       m 2 c 2 2     t
                              
Now A  A  Aw  Aw  2Aw  Ar  Ar  Ar

                     no time               EM field << wiggler
                   dependence
so the only term of interest is
              
                                
            2Aw  Ar  Aw A( z, t ) ei kw k z t  c.c.      
         d
so
         dz
            
               k aw
               2 
                                                   
                    a e i ( k  k w ) z ckt  c.c.       (1)
                                                                      
                                                                       aw 
                                                                      
                                                                             eAw 
                                                                             mc 
                                                                                 
                 d
                 dz
                    
                       k aw
                       2 
                                                          
                            a ei ( k  k w ) z ckt  c.c.

Whether electron gains or loses energy depends on the value of
the phase variable
                         k w  k z  t                      kc
 The EM wave (,k) and the wiggler “wave” (0,kw) interfere to
 produce a “ponderomotive wave” with a phase velocity
                                            
                            v ph 
                                          kw  k
From the definition of , it can be shown that :

                dθ               c           r
                    k w  k 1 
                                      2k w
                                  vz           r                (2)
                dz                  
                    lw    1  aw 2   
            r                      
  where             l     2              is the resonant energy
                                     
FEL resonance condition

    l  lw
             1  a w
                         2
                                 (magnetostatic wiggler ) Let:   aw  1
                2   2


         Example : for l=1A, lw=1cm, E~5GeV




   l  l pump
                1  a  w
                             2
                                        (electromagnetic wiggler )
                 4      2



         Example : for l=1A, lpump=1mm, E=35MeV
2.2 Field Dynamics                                                  (details in refs. 4,5)
                                   
   Radiation field :
   (circularly polarised
                                   Ar 
                                        -i
                                                   
                                           Ar ( z , t ) e ik ( z ct ) e  c.c.
                                                                        ˆ                 
                                         2
    plane wave)
                                                                  
 The radiation field evolution is                            1  Ar       2
 determined by Maxwell’s wave                           Ar  2
                                                        2
                                                                       m0 J
                                                             c t 2
            equation
       The (transverse) current density is due to the motion
       of the (point-like) electrons in the wiggler magnet.
                                                     
           
           J   e  v j  r  rj (t ) where
                                                               e Aw
                                                                v
                                                                    γm
                    j
                                                   
Apply the SVEA :           A( z, t )  A( z, t ),    A( z, t )  kA( z, t )
                        t                          z
and average on scale of lr to give
                                                                                                 e 2 ne
                                                                         where  p              0 m
                        k  p          aw
                                   2
     1  
         a( z , t )   2               e  i              (3)                        N

                                                                                           .... 
                                                                                      1
  z c t             2 
                          
                                        r
                                       
                                                                               ....                  j
                                                                                      N   j 1
 ‘Classical’ universally scaled equations
                                                                     j r
 j
 2
                   i j               V                    pj     
                                                                        j

          ( Ae           c.c.)                              z      r
z   2
                                       j
                                                                     PRad
A A 1 N i j                                              | A| 2

      e                                                           PBeam
z z1 N j 1
                                                            V  2 A sin j   


     A is the normalised S.V.E. A. of FEL rad. – self consistent


                z  v 0t                      z
                                                  z
           z1 
                  Lc                             Lg
                                                    1                       2
      lw                   lr            1    I   lW aW 
                                                        3                       3
 Lg  ;            ; Lc                                   
     4                  4           2    I A   2 Beam                    Ref 1.
                                                                                         13
We will now use these equations to investigate the high-gain
regime.

We solve the equations with initial conditions
         j  0,2    (uniform distribution of phases)
           pj  0       (cold, resonant beam)
           A 1        (small input field)

and observe how the EM field and electrons evolve.
  Strong amplification of field is closely linked to phase bunching
                                  1 N  i
  of electrons.             b
                                N
                                      e
                                    j 1
                                            j




  Bunched electrons mean that the emitted radiation is coherent.
  For randomly spaced electrons            : intensity  N
  For perfectly bunched electrons          : intensity ~ N2
          |b|<<1    z=0
                                                       Ponderomotive
           |b|~1    z>0                                potential

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



    Collective Recoil Lasing = Optical gain + bunching

                                               N
                                         1
                                              e
                                                       i j
    bunching factor b (0<|b|<1):      b
                                         N     j 1
     FEL instability animation                               A
                                            Steady State         0
                                                             z1
Animation shows evolution of electron/atom positions
in the dynamic pendulum potential together with the
probe field intensity.          V (   )  2 | A | sin(   )
Classical high-gain FEL
                                                                          Bonifacio, Casagrande & Casati,
A fully Hamiltonian model of the classical FEL                            Optics Comm. 40 (1982)


                    A  p  A0  p0  C (constant)
                      2                    2
Steady State



    Defining   A  aei then a  C  p

    Defining q j   j   then the FEL equations can be rewritten as

                               dq j                 sin q         H
                                       pj                   
                               dz                   C p          p j
                            dp j                                   H
                                    2 C  p cos q j  
                            dz                                     q j
                                             p2                
                                      H     2 C  p sin q j 
                                               j
                       where
                                          j  2
                                                               
                                                                
                                                  3    dp j
                    Equilibrium occurs when q j     so      0
                                                   2    dz
                             dq j                      d p
                      BUT             0       so            0            i.e. GAIN
                               dz                       dz
|A|2




       z
 |b|       The scaled radiation
           power |A|2, electron
           bunching |b| and the
           energy spread σp for the
       z   classical high-gain FEL
 σp
           amplifier.



       z
Classical chaos in the FEL
 If we calculate the distance, d (z), between different trajectories
 in the 2-dimensional phase-space  p j , q j ;  p j ' , q j '


          so d z       p  p '  q  q '       where d 0    1
                                        2            2
                           j       j             j



 In the exponential regime :                   
                                       d z  exp  z     ;  0
           Linear Theory (classical) Ref(1)

                                                                     ilz
Linear theory        i A  iA  0
                   A                                Ae
                                                                        runaway
l   l 1  0
           2
                            0 r
                              F r
                                                       Iml                  solution




       See figure (a)               Maximum gain at =0
                                                                 z
                                                            3
                                                                g
                                    A e             e
                                         2      3t


                                     g  lw /(4 F ) 
                                                       
                                 Quantum theory: different results
                                                    (see later)
For long beams (L >> Lc) Seeded Superradiant Instability Ref(2):
                    Including propagation
             CLASSICAL REGIME, LONG PULSE
             L = 30LC , resonant (=0
                   CLASSICAL SASE

   Ingredients of Self Amplified Spontaneous Emission (SASE)

  i) Start up from noise
  ii) 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
      Lb
Ns 
     2 Lc




             25
DRAWBACKS OF ‘CLASSICAL’ SASE

 Time profile has many          Broad and noisy spectrum at
 random spikes                  short wavelengths (x-ray FELs)
simulations from DESY for the SASE experiment (λ ~ 1 A)




                                                          26 26
 Phys. Rev. ST Accel. Beams 9 (2006) 090701
  Nucl. Instr. And Meth. A 593 (2008) 69


                   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 27
                       Why QUANTUM FEL theory?




      In classical theory e-momentum recoil DP continuous variable


WRONG: if one electron emits n photons   DP  n  k       QUANTUM THEORY


  QUANTUM FEL parameter:
                                                            1                2
                                                       I   lL aW 
                                                                3                3
         mc       ( P)                    
                                                  1
                                                                      
                                           2    I A   4 Beam 
         k        k

       If   1 CLASSICAL LIMIT

       If   1 STRONG QUANTUM EFFECTS
                                                                                 28
   why QFEL requires a LASER WIGGLER?
      mc     l              p              l w (1  a 2 )
         r                                     W

      kr     lC             k                   2l r


                  2l C                         lW   l r l3W (1  a 2 )
 1  
                                  and     LW                     W
            l r l W (1  a 2 )
                           W                              2l C


for a laser wiggler               lW  lL / 2

                                    to lase at lr0.1 A:

MAGNETIC WIGGLER:                                LASER WIGGLER

   lW ~ 1cm, E ~10 GeV                          lL ~ 1 mm, E ~100 MeV

     ~ 10-6 , LW ~ 1Km                           ~ 10-4 , LW ~ 1 mm    29 29
                 Conceptual design of a QFEL
                    Compton back-scattering (COLLECTIVE)


                                                                         lr




                          lL



        lL
lr 
       4 2
            1  a02  lL  1m m      If   200 ( E  100 MeV)  lr  0.3 Å !

                     lL
a0  2.4 PL (TW )                  PL  100 TW , lL  1 mm , R  10 mm  a0  2
                      R

                                                                                  30
           QUANTUM FEL MODEL

                            Procedure :
Describe N particle system as a Quantum Mechanical ensemble




Write a Schrödinger-like equation for macroscopic wavefunction: 




                                                                31 31
                                 R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006)


       1D QUANTUM FEL MODEL

H
   p2
   2
            
       i Ae i  c.c.          , p   i       p  i
                                                              
                                                             



                                                                                  z              lL
                                                                          z            Lg 
        1            2                                                          ;

                                                                 
                                                                                 Lg             8
 i               i  A( z1 , z )ei  c.c.                                   z  vz t
   z    2   2                                                         z1 
                                                                                   Lc
                    2                                                          lr
 A A                                                                    Lc 
   
 z z1
                   
                    0
                      d |  ( , z1 , z ) |2 ei
                                                                           
                                                                               4
                                                                              2
                                                                                  z  vz t 
                                                                               l


                A        : normalized FEL amplitude
                                                                                               32
        Madelung Quantum Fluid Description of QFEL*
  *R. Bonifacio, N. Piovella, G. R. M. Robb, and A. Serbeto, Phys. Rev. A 79, 015801 (2009)


                        1 2
               i
                 z
                      3/ 2
                       2   2
                                                    
                                 i A( z1 , z )ei  c.c.                      
                                 2
               A A
                 
               z z1
                                0
                                   d |  ( , z1 , z ) |2 ei

                                                            1 
        Let   n exp i                  and v                              See E. Madelung,
                                                                          Z. Phys 40, 322 (1927)
                                                            3/ 2

    n  nv 
              0
    z   
dv v    v       VTOT
  
dz z
      v
         
            F 
                   
                                          where                        i
                                                        VTOT  i Ae  c.c.              1  1 2 n 
                                                                                         3
                                                                                         2 
                                                                                                   2 
                                                                                               n  


      dA  A
               n e i d
      d z z1
             Classical limit :                       no free parameters
    Wigner approach for 1D QUANTUM MODEL

Introducing the Wigner function :
                            1           *  q        q
               W ( , p)      dq e    2     2 
                                    iqp

                           2                         

        dp W ( , p)  ( )
                              2

                                                  p
                                               p
                                                   
                                2                 
                                                    
                       
         d W ( , p)  ( p)


 Using the equation for  ( , z )
 we obtain a finite-difference equation for   W ( , p, z )

W    W                                                         
           Ae  c.c.  W   , p 
               i                       1                        1
   p                                              W  , p         0
z                                 2                     2   
                      1            1   W ( , p)
for >>1:  W  , p    W  , p    
                     2           2     p



            W
             z
                p
                   W
                   
                          i
                             
                       Ae  c.c.
                                  W
                                  p
                                     0   
   The Wigner equation becomes a Vlasov equation
describing the evolution of a classical particle ensemble


           The classical model is valid when   1
                Quantum regime for   1
                                   Quantum Dynamics
                                                    
       0,2                                 cn ein ,   kz
                                                n  


       ein    is momentum eigenstate corresponding to eigenvalue     n ( k )
Only discrete changes of momentum are possible :        pz= n (k) , n=0,±1,..
                                     n=1
                              pz     n=0
                                     n=-1                           k
                    cn
                    z
                        
                           in 2
                           2
                                                
                                cn   Ac n 1  A*cn 1                  
                                
                    A A
                    z
                           
                         z1 n  
                                      
                                   cn cn 1  iA
                                       *



              | cn |2  pn     probability to find a particle with p=n(ħk)       36
                           A     
steady-state evolution:   
                           z  0 
                                                         =10, 0, no propagation
                             1              10
                                                  1




                                               -1
                                                          (a)
                                             10

          classical limit                      -3
                                             10
         is recovered for




                                       2
                                       |A|
                                               -5
                                             10

                                               -7
                                             10

             1                            10
                                               -9

                                                      0         10   20       30   40    50

                                                                          z
                                             0.15

                                                          (b)
         many momentum states
                occupied,                    0.10

          both with n>0 and n<0
                                       pn




                                             0.05


                                                                                        37
                                             0.00
             Quantum bunching

      c0  c1e i                k z,         c0  c1  1
                                                    2    2
                         where

      1  2 c0 c1 cos   
       2

                                            : relative phase
       Momentum wave interference


                                          2 cos2    
                                  1
                        c0  c1 
                                              2
Maximum interference:
                                   2


Maximum bunching when 2-momentum eigenstates are equally
populated with fixed relative phase


                                                                38
                                         Bunching and density grating
                              CLASSICAL REGIME >>1                                               QUANTUM REGIME <1

              0.15



              0.10                                                                        0.6

                                                                                          0.5
     pn




              0.05                                                                        0.4

                                                                                          0.3




                                                                          pn
              0.00                                                                        0.2
                 -20 -15 -10          -5       0      5       10   15
                                                                                          0.1
                                           n
                    10
                                                                                            -5 -4 -3 -2 -1     0      1   2   3   4   5
                                                                                                               n
|  ( ) |          2 8
                                                                        |  ( ) |    2

                                                                                          2.0
                     6
          N( )/N




                     4                                                                    1.5
                                                                            N( )/N



                     2                                                                    1.0

                     0
                          0     1    2            3       4        5                      0.5
                                          /2 

                                                                ( )   cn ein
                                                                        0.0
                                                                            0                     1     2             3       4       5 39
                                                                                      n                       /2 
 The physics of the Quantum FEL
                            mc (p z )
                            
                            k   k

Momentum-energy levels:                          n k
                                k
(pz=nħk, Enpz2 n2)                            n  1 k
      En  En 1 1    1                 1  3
 n              n   (n  0,1,..)    ,         (harmonics)
         2          2                 2 2

Frequencies equally spaced by 1  with width 4 
Increasing  the lines overlap for   0.4
CLASSICAL REGIME:   1             QUANTUM REGIME:   1
   many momentum level               a single momentum level
         transitions                     transition
→ many spikes                        → single spike
                                                               40
  Quantum Linear Theory                                          A  e il z



         2   1 
 l  D l  2   1  0
                                                                        l  D l2  1  0
            4 
                
                                                             1

                                                                 Quantum regime for <1
              Classical
                                                                                      1
              limit                                                      max at   D
                                                                                     2
        1.0
                                                  (a) 120
        0.8                                       (b) 120.
                                                  (c) 123
                                                  (d) 12

                                                                            
        0.6                                       (e) 12
                   (a)    (b)                     (f) 1210    width
|Iml|




                                (c)
        0.4
                                      (d)
                                            (e)
                                                   (f)
        0.2
           discrete frequencies as in a cavity         A  e  il z

                  2   1               n   
          l  D l  2   1  0   D   
                                            
                                                          
                                                           
                                                                 sp 
                                                                       
                     4                                  2 sp 
                                                                      
                      1
  0.1                                             max for D  1 / 2
                                                            1
                                                     n     (2n  1)
                                                           2
  0.2
                                                 
                                                           = 4 
  0.4
                                                 

    Continuous limit        4   1/     0.4
                                                                 42 42
          momentum distribution for SASE

CLASSICAL REGIME:    5       QUANTUM REGIME:       0.1




       Classical regime:             Quantum regime: 43 43
   both n<0 and n>0 occupied    sequential SR decay, only n<0
SASE Quantum purification        R.Bonifacio, N.Piovella, G.Robb, NIMA(2005)



 quantum regime     0.05    classical regime              5 




                                                                   44 44
 L / Lc  30
   0.1 1/  10   n  (2n  1) / 2  [n  0,1,..]     0.2 1/  5




  0.3 1/  3.3                                         0.4 1/  2.5




                                                                45 45
LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME
               lr
                                                       D     lr
                                                        
                   Lb
                                                        QFEL Lb


                                            D l   QUANTUM SINGLE SPIKE
                                              
 1,0                                         Lb
 0,8
                                               CLASSICAL ENVELOPE        D
 0,6                                                                         2
                                                                         
 0,4
                                             classical   Lb
 0,2                                                        N spikes
                                             quantum 2Lc
 0,0
    -8   -7   -6        -5   -4   -3   -2
                                                                          46 46
                             QFEL requirements
                      (E)                      3/ 2
                                                               lr  A lL m m
                                       4
Energy spread :               5 10
                       E                    lL lr (1  a )
                                                        2
                                                        0



                                     2 3
                       I ( A)  300 3 2 2                     mm
                                   lr lL a0

                                            4 R 2
Condition to neglect diffraction :     ZL          Lint
                                             lL
Not necessary with plasma guiding (D. Jaroszynski collaboration)

                   n mm mrad  0.03 QpC
                                                               1
                                                                3     (thermal)
Emittance:


 Rosenzweig et al, NIM A 593, 39 (2008)                                     47
                        Harmonics Production

Possible frequencies            h (h  1,3,5,..)
 One photon recoil        h k
      Larger momentum level separation                   quantum effects easier

             Extend Q.F. Model to harmonics           [G Robb NIMA A 593, 87 (2008)]

 Results (a0 >1)
                                        h
 Distance between gain lines:      D
                                        

 Gain bandwidth of each line:          4 
                                  
                                        h

                                                         0.4 h 4 / 3
                          .

      Separated quantum lines if         D   i.e.

              h  1 0.4 h  3  1.7            h  5  3.4
Possible classical behaviour for fundamental BUT quantum for harmonics
                                                                                       48
 1




                           3rd       5th
          Fundamental   harmonic   harmonic   49


   e.g.      0.3A         0.1A      0.06A
Main limitations in classical regime :

                1.                 lw
                     Lw  Lg             10 3 ,10 4
                                  4
                        ( E)
                2.              
                         E

                3.        *  Lg

                           lr  *
                4.    n 
                           4 Lg


Quantum FEL : as above with
                lL                            mc
         lw                                   1
                2                              k

   Quantum regime easier in the sub-A region and             1
                  Parameters for QFEL
                                          1
           Electron beam             Laser beam                   QFEL beam


  Q (pC)          1             λL (mm)     1            λr (A)       0.3
   (fs)          1.3           PL (TW)     100          Pr (MW)      30
  I (kA)          0.77          aw          2            D/         7x 10-5
  n (mm mrad) 0.03              (ps)      3.4          Nphot        6x 106
  E (MeV)         100           R (mm)      12.6          (fs)       1
   (mm)          0.5           Lint (mm)   1
  DE/E            4x10-4




                      Note : 5th harmonic at 0.06 A
                                                                                51
Relaxed parameters with plasma channel (guiding) : Dino Jaroszynski
   FEL IN CLASSICAL\SASE CAN GO TO l=1.5Ǻ (LCLS)
 QUANTUM SASE WORKS BETTER FOR SUB-Ǻ REGION

CLASSICAL SASE               QUANTUM SASE
needs:                       needs:
GeV Linac                    100 MeV Linac
Long undulator (100 m)       Laser undulator (l~1mm)
yields:                      yields:
High Power                   Lower power
Broad and chaotic spectrum   Very narrow line spectrum
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


                                              Pump field
    CARL
                                                 l~lp
                     Cold atoms   Backscattered field
                                       (probe)

                           SN SNSN

  FEL                                         EM radiation
             Electron      N S N SN S        l  lw /2 << lw
              beam        “wiggler” magnet
                             (period lw)

     At first sight, CARL and FEL look very different…
                                 FEL
Connection between CARL                           EM pump, l’w
   and FEL can be seen                              (wiggler)
       more easily by
transforming to a frame (L’)                Backscattered
   moving with electrons       electrons      EM field
                                               l’  l’w

                                CARL
                                                     Pump
 Connection between FEL                              laser
  and CARL is now clear
                                            Backscattered
                               Cold atoms       field l~lp
    Experimental Evidence of Quantum Dynamics
              The LENS Experiment
     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


                                                                 trap
                                                     BEC


                                                                              g

Experimental values:           laser beam w, k


D = 13 GHz
w = 750 mm
P = 13 mW                                                               absorption imaging

                                                                         Dp  2k
          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 experiment
Temporal evolution of the population in the first
three atomic momentum states during the application
of the light pulse.




pump
light




               n=0    n=-1    n=-2
               p=0    p=-2hk p=-4hk
                        MIT experiment
       Superradiant Rayleigh Scattering from a BEC
            S. Inouye et al., Science 285, 571 (1999)




Back scattered intensity for                  Number of recoiled particles for
different laser powers: 3.8 2.4               different laser intensity (25 &
1.4 mW/cm2 Duration 550 ms                    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.