Quantum Free Electron Laser From 1D to 3D

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1 Quantum Free Electron Laser From 1D to 3D Luca Volpe Dipartimento di Fisica and INFN Milano University of Milano (Italy) Tutore: Dott. Nicola Piovella Cotutore: Prof. Roberto Pozzoli Milan QFEL group: R.Bonifacio, N.Piovella, M.M. Cola, L. Volpe and R. Gaiba A.Schiavi Universitá La Sapienza, Roma G.R.M Robb University Stranthclyde, Glasgow Scottland Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.1/29

2 OUTLOOK Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

3 OUTLOOK REVIEW OF 1D QFEL MODEL Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

4 OUTLOOK REVIEW OF 1D QFEL MODEL EXTENSION FROM 1D 3D HOW TO Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

5 OUTLOOK REVIEW OF 1D QFEL MODEL EXTENSION FROM 1D 3D HOW TO DISCRETE WIGNER DISTRIBUTION Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

6 OUTLOOK REVIEW OF 1D QFEL MODEL EXTENSION FROM 1D 3D HOW TO DISCRETE WIGNER DISTRIBUTION NEW 3D MODEL FOR QFEL Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

7 OUTLOOK REVIEW OF 1D QFEL MODEL EXTENSION FROM 1D 3D HOW TO DISCRETE WIGNER DISTRIBUTION NEW 3D MODEL FOR QFEL WORKING EQUATIONS Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

8 OUTLOOK REVIEW OF 1D QFEL MODEL EXTENSION FROM 1D 3D HOW TO DISCRETE WIGNER DISTRIBUTION NEW 3D MODEL FOR QFEL WORKING EQUATIONS CONCLUSION Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.2/29

9 HISTORY OF 1D QFEL MODEL Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.3/29

10 HISTORY OF 1D QFEL MODEL First quantization with Collective Operators by Bonifacio and Casagrande (1984) Steady State Regime Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.3/29

11 HISTORY OF 1D QFEL MODEL First quantization with Collective Operators by Bonifacio and Casagrande (1984) Steady State Regime Path integral approach by Preparata PRA 38 (1988) Steady State Regime Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.3/29

12 HISTORY OF 1D QFEL MODEL First quantization with Collective Operators by Bonifacio and Casagrande (1984) Steady State Regime Path integral approach by Preparata PRA 38 (1988) Steady State Regime Second Quantization by Bonifacio, Piovella, Cola, PRA 67 (2003) Steady State Regime Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.3/29

13 HISTORY OF 1D QFEL MODEL First quantization with Collective Operators by Bonifacio and Casagrande (1984) Steady State Regime Path integral approach by Preparata PRA 38 (1988) Steady State Regime Second Quantization by Bonifacio, Piovella, Cola, PRA 67 (2003) Steady State Regime Propagation effects by Bonifacio, Piovella, Robb, Cola, Opt. Commun. 252 (2005) Multiple scaling approach Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.3/29

14 FEL PARAMETER λ r = 1 β γ r = β λ w λ w 1+a 2 w 2γ 2 λ l = 2λ w Resonant Wavelength λ w (1+a 2 w) 2λ Resonant energy p = mc(γ γ 0) k Momentum in k unit θ = (k + k w )z c(k k w )t Phase ρ = 1 γ r ( aw ω p 4k w c ) 2/3 Classical Fel parameter ρ = ρ mcγ r k = γ r ρ λ r λ c Quantum Fel parameter z = z/l g p = p/ ρ Rescaled Position and Momentum δ = γ 0 γ r ργ r L g = λ w /4πρ Detuning and Gain length Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.4/29

15 1D QFEL MODEL I Hamiltonian and particle equations H I (θ, p) = H I j (θ, p) = N Hj I (θ, p) [θ j, p k ] = iδ jk [a, a ] = 1 j=1 { p 2 j 2 ρ i ρ ( ) ae iθ j h.c. N e δa a } z θ j = p j ρ = p j H ρ ( ) z p j = ae iθ j + h.c. N e ρ N d z a = e iθ j + iδa N e j=1 = θj H Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.5/29

16 1D QFEL MODEL II N particle described by matter-wave field ˆψ(θ, z) ˆψ(θ, z) = n Z ĉ n ( z)e iθn, 2π 0 dθ ˆψ (θ) ˆψ(θ) = ˆN The electrons are treated like BOSONS!!! ] [ [ĉ n, ĉ n = δ nn ˆψ(θ), ˆψ (θ )] = δ(θ θ ) Ĥ II = 2π 0 dθ ˆψ (θ) H I (θ, i θ, a, a ) ˆψ(θ) N. Piovella, M.M. Cola and R. Bonifacio PRA 67. (2003) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.6/29

17 1D QFEL MODEL III 1 ρ ( i z ˆψ = 2 ρ 2 θ ˆψ i ae iθ c.c.) ˆψ N e ρ 2π d z a = dθ ˆψ (θ) e iθ ˆψ(θ) + iδa N e 0 Preparata Hypotesis (N e ) ˆψ N e ψ and â ρn e A introducing the time-dependence in field A( z) A( z, θ) i z ψ(θ, z) = 1 ( ) 2 ρ 2 θψ(θ, z) i ρ A( z, θ)e iθ c.c. ψ(θ, z) z A( z, θ) + 1 2ρ θa( z, θ) = ψ(θ, z) 2 e iθ + iδa( z, θ) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.7/29

18 MULTIPLE SCALING APPROACH Including propagation effect we introduce a slow scale: z 1 = ɛθ = z v rt ɛ = 2ρ L c = λ β r L c 4πρ 1 ɛ θ 1 ɛ θ + z 1 β r = v r = ck k + k w performing a perturbation expansion to the first order in ɛ we obtain: i z ψ(θ, z, z 1 ) = 1 2 ρ 2 θψ(θ, z, z 1 ) i ρ z A( z, z 1 ) + z1 A( z, z 1 ) = 2π 0 ( ) A( z, z 1 )e iθ c.c. ψ(θ, z, z 1 ) dθ ψ(θ, z, z 1 ) 2 e iθ + iδa( z, z 1 ) R. Bonifacio, N.Piovella, G.R.M. Robb, M.M. Cola, Opt. Commun. 252 (2005) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.8/29

19 1D WORKING EQUATIONS ψ(θ, z, z 1 ) = 1 2π n Z c n ( z, z 1 )e in(θ+δ z) 2π 0 dθ ψ(θ, z, z 1 ) 2 = I(z 1 ) z c n = i ( n 2 ) 2 ρ + nδ c n ρ {Ac n 1 A c n+1 } z A + z1 A = n Z c n c n 1 c n ( z, z 1 ) 2 = Probability to find an electron whit momentum p = n( k) at z and z 1 1 N e n Z c nc n 1 = Bunching operator Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.9/29

20 THE ENERGY SPREAD We must include the more physical consistent situation of an initial distribution for the electron energy. Infact each electron of the beam has different initial energy. δ = γ γ 0 ργ r δ i = γ γ 0 i ργ r c n ( z, z 1 ) c n (δ, z, z 1 ) n Z c nc n 1 R dδ G(δ) n Z c n(δ)c n 1 (δ) G(δ) is normalized distribution center around δ = 1/2 ρ Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.10/29

21 Self Aplified Spontaneus Emission SASE Ingredient of SASE: Starting from noise Propagation effects Superradiant instability (the electrons radiate as N 2 e ) Each cooperation length in the e-beam radiates a SR spike which is amplified when it propagates forward on the beam SASE in the quantum regime: In the quantum regime the FEL behaves like a two level system Electrons emit coherent photons as in a LASER in the quantum SASE mode the spectrum is intrinsically narrow (quantum purification) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.11/29

22 CLASSICAL AND QUANTUM REGIME ρ = ρmcγ r / k If ρ 1,since δγ γ ρ mc(δγ) k Classical behavior Many recoils implies many photons, hence classically, each electron emits many photons ρ A 2 N ph /N e 1 If ρ 1 mc(δγ) k Quantum behavior Each electron emits only a single photon therefore quantum FEL behaves like a two-level system ρ A 2 N ph /N e 1 Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.12/29

23 QUANTUM 1D LINEAR THEORY I Performing a linear analysis and looking for solutions proportional to e i(λ z+ ωz 1) we obtain the dispersion relation: (λ n ) ( ) λ ρ = 0 n = δ + Note that: ( ) For n = 0 and ω = 0 (λ δ) λ 2 4 ρ = 0 2 Quantum steady state dispersion relation And for ρ 1 (λ δ) λ = 0 Classical steady state dispersion relation n ρ ω, R. Bonifacio, N. Piovella, G.R.M. Robb and A. Schiavi PRST 9, (2006) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.13/29

24 QUANTUM 1D LINEAR THEORY II Im λ (a) (b) (c) (d) (e) (f) when ρ 1 δ = 1 2 ρ, width=4 ρ Peak of Iλ = ρ δ a : 0, b : 1/2, c : 3, d : 5, e : 7, f : 10 ρ a :, b : 1, c : 1/6, d : 1/10, e : 1/14, f : 1/20 δ Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.14/29

Each of this line is center in ω = (2n 1)/2 ρ with distance 1/ ρ and has a width of 4 ρ, this corrispond in

25 QUANTUM 1D LINEAR THEORY III The regions of the spectrum that corrisponding to gain Iλ > 0 appear like a series of discret line corresponding to different value of n. Each of this line is center in ω = (2n 1)/2 ρ with distance 1/ ρ and has a width of 4 ρ, this corrispond in the momentum space to shift of k/2 with a width 4 ρ 3/2 ( k) ρ = 0.1, 0.2, 0.4 Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.15/29

26 FROM 1D TO 3D, PROBLEM AND SOLU PROBLEM the tranverse motion is essentially classical. So we look for a model which describes the quantum behavior on the longitudinal dimension and at the same time the classical behavior on the tranverse dimension. extension via Shroedinger equation ˆψ(θ) ˆψ(θ, x, y)? NO because it describes a cold beam with a quantum emittance equal to compton vawelength λ c 2π SOLUTION! Write a Discrete Wigner function W n (θ, x t, p t ) and perform the classical limit only for the transverse motion in the limit: α λ c ɛ n 0 Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.16/29

27 1D DISCRETE WIGNER FUNCTION In longitudinal dimension θ is periodic then the momentum of the electron must be discrete Therefore we write the 1D Discrete Wigner function introduced for the first time in the (1994) by Bizarro. π/2 π/2 W n (θ) = 1 dθ e i2nθ θ + θ ˆϱ θ θ π = w n (θ) + sinc[(n n 1/2)π]w n +1/2(θ) w s (θ) = 1 2π π π n dθ e i2sθ θ + θ ˆϱ θ θ, s = k or s = k + 1/2 J.P. Bizarro, PRA (1994) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.17/29

28 1D WIGNER MODEL FOR QFEL Evolution equation ( z + s ρ θ ) w s ρ ( Ae iθ + c.c.) {ws+1/2 w s 1/2 } = 0 z A + z1 A = n π π dθ w n+1/2 (θ, z, z 1 )e iθ + iδa For ˆϱ = ψ ψ pure state, n Z W n(θ, z) = ψ(θ, z) 2 π π W n(θ, z) = c n (z) 2 Beam profile Density probability n Z π π W n(θ, z) = 1 Normalization Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.18/29

29 CLASSICAL LIMIT for ρ 1 p = s/ ρ becomes a continuos variable ρ ( ( ) w s+1/2 w s 1/2 ρ [W p + 1 ) ( W p 1 )] p W 2 ρ 2 ρ In the momentum space p ± 1 2 ρ become p ± k 2 MAXWELL-WIGNER MAXWELL-VLASOV ( ) ( z + p θ ) W ρ Ae iθ + c.c. p W = 0 ( z + z1 ) A = d p π π dθ W (θ, p, z, z 1 )e iθ + iδa Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.19/29

30 3D QFEL WIGNER MODEL HOW TO Write the 3d quantum Hamiltonian Define the statistic operator ˆϱ Define the 3D Discrete Wigner rappresentation of ˆϱ Write the 3D Wigner Evolution equations Perform the classical limit only on the transverse coordinate Or similary: w s (θ, z, z 1 ) w s (θ, z, z 1, x t, p t ) A( z, z 1 ) A( z, z 1, x t ), a w a w g l ( x t ) g l = spatial profile of laser Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.20/29

31 3D QFEL HAMILTONIAN ˆ H(θ, p, x t, p t ; z) = ˆp2 2 ρ + αb 2 ˆp2 t [ ξ + p 2ρ (1 g l 2 ) bx 2 α2 p 2 t ( gl Aeiθ c.c. + ρξ 2ρ 2 g l 2 + i ρ ] ), x t = x t /σ, p = mc(γ γ 0), p t = mcγ 0 k b = L g β, d x t dz, β = σ2, X = kɛ r, ξ = a2 w ɛ r 1 + a 2, α = w [θ, p] = [ x, p x] = i mcγ 0 ɛ r. Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.21/29

32 3D DISCRETE WIGNER FUNCTION W m (θ, x, p t ) = w m (θ, x, p t ) + n Z sinc[(m n µ/2)π]w n+µ/2 (θ, x t, p w s (θ, x, p t ) = 1 2π 3 +π π R 2 dθ e i2θ s ] From the evolution equation z ˆϱ = i [Ĥ, ˆϱ d x te i2 x t p t θ θ, x t + x t ˆϱ x t x t, θ + θ a = L g /Z r = b/2x, Z r = 4πσ/λ we obtain Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.22/29

33 QFEL MAXWELL-WIGNER EQUATION z w s + { s ρ + δ + ξ 2ρ (1 g l 2 ) bx } 2 p2 t θ w s ( ) [ws+1/2 ρ gl ] Aeiθ + c.c. w s 1/2 + iα ρ ( )] { [ xt gl w s+1/2 + w s 1/2 Aeiθ c.c. 2 [ + (1 + αx) b p t xt ξ 2ρX } self focusing ( xt g l 2) pt ] w s = 0, ( z + z1 ) A ia 2 x t A = g l m Z R 2 d 2 p t 2π 0 dθ w m+1/2 (θ, x t, p t )e iθ. Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.23/29

34 ANALYSIS OF TRANSVERSE TERM I θ = Affect the QFEL resonance { s ρ + δ + ξ 2ρ (1 g l 2 ) bx } 2 p2 t λ = λ l 4γ 2 0 [ 1 + a 2 w + γ 2 0 (θ2 x + θ 2 y) ] δ 0 = γ 0 γ r ργ 0 ( γ ργ 0 )1D 1 Longitudinal detuning ξ 2ρ (1 g l 2 ) ( ) γ ργ 0 a w 1 2ρ a 2 w 1+a 2 w Off-resonance due to laser profile variation ( ) bx 2 p2 t γ ργ 0 x,y 1 2ρ γ 2 0 θ 2 t 1+a 2 w Off-resonance due to beam divergence Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.24/29

35 ANALYSIS OF TRANSVERSE TERM II x t = b p t Tranverse motion of the electron (divergenze of beam) p t = ξ 2ρX ( xt g l 2) β = 1 2γ 0 x a 2 l (x) Ponderomotive force due to the transverse laser profile (defocusing) Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.25/29

36 QFEL 3D WORKING EQUATIONS I z w k s + ik + { s w k s = 1 2π k Z wk s e ikθ ρ + δ + ξ 2ρ (1 g l 2 ) bx } 2 p2 t θ ws k ( gl Awk 1 s+1/2 g l Awk 1 s 1/2 + g la w k+1 s+1/2 g la w k+1 s 1/2 ( xt g l 2) ] pt ρ [ b p t xt ξ 2ρX w k s = 0, ) ( z + z1 ) A ia 2 x t A = g l m Z R 2 d 2 p t w 1 m+1/2 (θ, x t, p t ). Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.26/29

37 QFEL 3D WORKING EQUATIONS II w 2k n k Z n + k ˆϱ n k w 2k+1 n+1/2 k Z n + k + 1 ˆϱ n k { w 0 n n ˆϱ k 1d c n 2 w 1 n+1/2 n + 1 ˆϱ n 1d c n+1 c n = c nc n 1 w0 0 = 0 ˆϱ 0 1d c 0 2 w 1 0 = 1 ˆϱ 1 1d c 1 2 two level n=0,-1 w 1 1/2 = 0 ˆϱ 1 1d c 0 c 1 Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.27/29

38 3D QFEL TWO LEVEL APPROXIMATION D = w0 0 w 1 0 Population difference B = w 1/2 1 Bunching (polarizzation) L g = L g / ρ Quntum Gain length ρ 0.4 n = 0, 1 g l = 1 static wigler ( z + b p t xt ) D + 2 (AB + c.c.) = 0 ( z + b p t xt i X2 p t 2 ) B = AD ( z + z 1 i 2 x t ) A = R 2 d 2 p t B A = A ρ z = z ρ = z/l g b = L g /β a = L g /Z r z 1 = z 1 ρ Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.28/29

39 CONCLUSION We propose a new 3D Quantum FEL Model based on a MAXWELL-WIGNER equations The Wigner approach let us mix the different behaviour between the longitudinal and tranverse dynamics The MAXWELL-WIGNER equations switch into the classical MAXWELL-VLASOV equations for ρ 1 Future Developments: Implementation 3D Maxwell-Wigner code and simulations Search of particular analytic solution Dottorato di ricerca in fisica XXI Ciclo, Seminario di fine II anno p.29/29

3D Wigner model for a quantum free electron laser with a laser wiggler

3D Wigner model for a quantum free electron laser with a laser wiggler M.M. Cola,, L. Volpe,, N. Piovella,, A. Schiavi 3, and R. Bonifacio,4 Dipartimento di Fisica, Università Degli Studi di Milano, via