Introduction to Classical and Quantum FEL Theory R. Bonifacio University of Milano and INFN LNF Natal 2016 1 1
OUTLINE Classical SASE and spiking Semi-classical FEL theory: quantum purification Fully quantum theory Conclusions 2 2
SASE high-gain regime λ w λw λ = 1+ 2 2γ ( a 2 w ) a = λ B w w w 3 3
The FEL described using classical universally scaled equations 2 θ j iθ j = ( Ae + c. c.) 2 z N A A 1 iθ j + = e z z N 1 j= 1 V = θ j ρ A = V P 2 Rad 2 A sin P = j Beam ( ) θ +φ A is the normalised S.V.E. A. of FEL rad. self consistent θ j = k z j λ L w g = ; 4πρ ; L c = λ r 4πρ z v 0 z1 = L c t 1 3 z = z L g 1 I λwa W ρ = 2γ I A 2πσ Beam 2 3 R.Bonifacio, C. Pellegrini and L. Narducci, Opt. Commun. 50, 373 (1984). 4
Pendulum potential and self-bunching A A + = e z z N 1 N 1 iθ j j= 1 5
SELF-BUNCHING start up from noise exponential growth of intensity and bunching saturation (P rad ~ρ P beam ) after several L g b~0 b~0.8 bunching: b = N 1 θ N j= 1 e i j wiggler length (several L g ) θ j = k z j 6
N s Lb = 2π L c 7
DRAWBACKS OF CLASSICAL SASE Time profile has many random spikes Broad and noisy spectrum at short wavelengths (x-ray FELs) simulations from DESY for the SASE experiment (λ ~ 1 A) 8 8
what is QFEL? QFEL is a novel macroscopic quantum coherent effect: collective Compton backscattering of a highpower 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 9 9
Conceptual design of a QFEL Compton back-scattering (COLLECTIVE) λ r λ L 2 ( a ) λl λ 1 r = + λ 2 0 L 1µ m 4 γ = If γ 200 ( E 100 MeV) λ r 0.1 Å! a = λl 0 2.4 PL ( TW ) R P = 1TW, = 1 m, R = 10 m a = 1 L λl µ µ 0 10
QUANTUM FEL MODEL Procedure : Describe N particle system as a Quantum Mechanical ensemble Write a Schrödinger-like equation for macroscopic wavefunction: Ψ 11 11
R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB (2006) 1D QUANTUM FEL MODEL 2 Ψ 1 Ψ { iθ i = i A( z } 3/2 2 1, z) e cc.. Ψ z 2ρ θ 2π A A + = Ψ(,, ) z z 1 0 2 iθ dθ θ z1 z e z z 1 L c z = ; Lg = L = g z vt z L c λ = 4πρ 2π θ = λ ( z vt) z λw 4πρ A : normalized FEL amplitude mcγ σ ( P) ρ = ρfel = k k A z 1 = 0 : QUANTUM FEL parameter the classical model is valid when ρ >>1 G. Preparata from QFT, PRA (1988) 12 12
steady-state evolution: A z 1 = 0 10 1 ρ=10, δ=0, no propagation 10-1 (a) classical limit is recovered for A 2 10-3 10-5 10-7 ρ >>1 10-9 0 10 20 30 40 50 z 0.15 (b) many momentum states occupied, both with n>0 and n<0 p n 0.10 0.05 0.00-15 -10-5 0 5 10 13 n
Madelung Quantum Fluid Description of QFEL dv dz n + z v z 2 Ψ 1 Ψ { iθ i = i A( z } 3/2 2 1, z) e cc.. Ψ z 2ρ θ 2π A A + = Ψ(,, ) z z 1 0 Let = n exp( iφ) ( nv) θ 2 iθ dθ θ z1 z e 1 Ψ and v 3/2 ρ = v V + v = F = θ θ = TOT 0 φ θ where ( iθ V ) TOT = i Ae c.. c See E. Madelung, Z. Phys 40, 322 (1927) 2 1 1 n + 3 2 2ρ n θ A z A + = z 1 ne iθ dθ Classical limit : ρ 14 See Dawson (1977) : no free parameters
Quantum Dynamics ( 0, ) θ 2π ( θ, z, z = 1) cn ( z, z1 n= Ψ ) e inθ in e θ is momentum eigenstate corresponding to eigenvalue n( k) Only discrete changes of momentum are possible : p z n=1 n=0 n=-1 p z = n ( k), n=0,±1,.. k c n 2 c z A + z = n p in = c 2ρ n A z 1 2 = n n= ρ c n c ( * Ac A c ) * n 1 n 1 + iδa n+ 1 probability to find a particle with p=n(ħk) 15
The physics of Quantum FEL mcγ ρ = ρ k Momentum-energy levels: (p z =nħk, E n p z 2 n 2 ) σ(pz ) = k k ωn = En En n n n = 2 2 1 ( 1) 2 1 1, 3, 5,... n k ( n 1) k ( n = 0, 1,...) Equally spaced frequencies as in a cavity CLASSICAL REGIME: ρ >>1 many momentum level transitions many spikes QUANTUM REGIME: a single momentum level transition single spike ρ 1 In classical regime with universal scaling no dependence on ρ 16 16
discrete frequencies as in a cavity ( λ ) λ 1 4ρ 1 0 2 + = 2 = n 2 ρ ω ( iλz A e ) ω ω ω = 2ρω sp sp ρ = 0.1 1 ρ 1 ρ ω ω n = 1 2ρ (2n 1) ρ = 0.2 ρ = 0.4 ω ω width 4 ρ Continuous limit 4 ρ 1/ ρ ρ 0.4 17 17
momentum distribution for SASE CLASSICAL REGIME: ρ = 5 QUANTUM REGIME: ρ = 0. 1 Classical regime: both n<0 and n>0 occupied Quantum regime: sequential SR decay, only n<0 18 18
SASE Quantum purification R.Bonifacio, N.Piovella, G.Robb, NIMA(2005) quantum regime ( = 0.05) ρ classical regime ( ρ = 5) L/ L c = 30 19 19
ω = (2n 1) / 2ρ ρ = 0.1 1/ ρ = 10 n ρ = 0.2 1/ ρ = 5 [n = 0, 1,..] ρ = 0.3 1/ ρ = 3.3 ρ = 0.4 1/ ρ = 2.5 20 20
LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME λ r L b ω ω QFEL λ L r b 1,0 0,8 ω ω λ L b QUANTUM SINGLE SPIKE (~10-7 ) 0,6 0,4 0,2 ω ω 2ρ CLASSICAL ENVELOPE (10-3 - 10-4 ) 0,0-8 -7-6 -5-4 -3-2 21 21
why QFEL requires a LASER WIGGLER? ρ = ρ mcγ k r = ργ λ λ r C λ C = h mc γ = λ w (1 + a 2λ r 2 W ) ρ 1 ρ λ r λ W 2λ C (1 + a 2 W ) and L W λ ρ W λ r λ 3 W (1 + a 2λ C 2 W ) for a laser wiggler λ / 2 W λ L to lase at λ r =0.1 Α: MAGNETIC WIGGLER: λ W ~ 1cm, E ~10 GeV LASER WIGGLER λ L ~ 1 µm, E ~25 MeV ρ ~ 10-6, L W ~ 1Km ρ ~ 10-4, L Int ~ 1 mm 22 22
Harmonics Production Possible frequencies One photon recoil h ω (h = h k Larger momentum level separation 1,3,5,..) quantum effects easier Extend Q.F. Model to harmonics [G Robb NIMA A 593, 87 (2008)] Results (a0 >1) Distance between gain lines: = h ρ Gain bandwidth of each line:. σ = Separated quantum lines if 4 ρ h σ < i.e. ρ 0.4h h =1 0.4 h = 3 1. 7 h = 5 3. 4 Possible classical behaviour for fundamental BUT quantum for harmonics 4/3 23
ρ =1 24
Classical versus quantum SASE 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 (10 9 U$) and large size for 1Å a QFEL experiment will generates a single spike almost monochromatic X- ray radiation ( ω/ω~10-7 ). Needs a laser wiggler Reduces cost (~ 10 6 U$) for 0.1Å Very compact apparatus (~ m) 25 25
ρ Summary Classical regime : >> 1 Quantum regime: ρ 1 : discreteness of momentum exchange relevant=> quantum effects. The system is prepared in a defined momentum state p 0, making transition to the lower state The system radiates a monocromatic train wave lambda, whose length is L b. Hence one has a single line with linewidth λ r / L b 10 In the opposite case random transition from many momentum states. Each transition gives a spike with different frequency. Total bandwidth: ρ 10 3 QFEL has a linewidth 4 orders of magnitude smaller than the classical The dimensions and cost are three order of magnitude smaller 7 For details see : Opt. Comm. 252, 381 (2005). (FEL and CARL) 26 26
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)
Experimental Evidence of Quantum Dynamics The LENS Experiment Production of an elongated 87 Rb 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 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/cm 2 Duration 550 µs Number of recoiled particles for different laser intensity (25 & 45 mw/cm 2 ). Total number of atoms 2 10 7
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.
To be published on EPL. 32
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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 36
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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. 38