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Quantum Effects in BECs and FELs

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

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Quantum Effects in BECs and FELs

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  1. 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)

  2. Outline • Introductory concepts • Classical FEL-CARL Model • Quantum FEL-CARL Model • Propagation Effects • Quantum SASE regime

  3. Free Electron Laser (FEL)

  4. Collective Atomic Recoli Laser (CARL) R. Bonifacio et al, Opt. Comm. 115, 505 (1995) Pump beam wp Probe beam wwp

  5. EM radiation llw /g2 << lw S S S N N N N N N S S S Both FEL and CARL are examples of collective recoil lasing Pump field CARL l~lp Cold atoms Backscattered field (probe) FEL Electron beam “wiggler” magnet (periodlw) At first sight, CARL and FEL look very different…

  6. FEL EM pump, l’w (wiggler) Connection between CARL and FEL can be seen more easily by transforming to a frame (L’) moving with electrons Backscattered EM field l’ »l’w electrons CARL Pump laser Connection between FEL and CARL is now clear Backscattered field Cold atoms l~lp

  7. In FEL and CARL particles self-organize to form compact bunches ~l which radiate coherently. Collective Recoil Lasing = Optical gain + bunching bunching factor b (0<|b|<1):

  8. Exponential growth of the emitted radiation:

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

  10. CARL-FEL instability animation Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.

  11. Linear Theory (classical) runaway solution Maximum gain at d=0 See figure (a)

  12. Quantum model of FEL/CARL 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 Include propagation using a multiple-scaling approach

  13. Canonical Quantization so Quantization (with classical field A) : so R. Bonifacio, N. Piovella, G.R.M.Robb and M.Cola, Optics Comm, 252, 381 (2005)

  14. Quantum FEL Propagation model So far we have neglected slippage, so all sections of the e-beam evolve identically (steady-state regime) if they are the same initially. We have introduced propagation into the model, so different parts of the electron beam can feel different fields : Here q describes spatial evolution of Y on scale of l and describes spatial evolution of A and Y on scale of cooperation length, Lc >> l. where

  15. Quantum Dynamics is momentum eigenstate corresponding to eigenvalue Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,.. n=1 pz n=0 n=-1 probability to find a particle with p=n(ħk)

  16. steady-state evolution: classical limit is recovered for many momentum states occupied, both with n>0 and n<0

  17. n=0 n=-1 Quantum limit for Only TWO momentum states involved : n=0 and n= - 1 Dynamics are those of a 2-level system coupled to an optical field,described by Maxwell-Bloch equations

  18. Bunching and density grating QUANTUM REGIMEr<1 CLASSICAL REGIMEr>>1

  19. Quantum Linear Theory Quantum regime for r<1 Classical limit max at width

  20. 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 damping of radiation

  21. n=0 n=-1 n=-2 SEQUENTIAL SUPERRADIANT SCATTERING LASER BEC

  22. Superradiant Rayleigh Scattering in a BEC (Ketterle, MIT 1991) for K>>1 and

  23. Experimental evidence of quantum CARL at LENS • 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: D = 13 GHz w = 750 mm P = 13 mW L.Fallani et al, PRA 71 (2005) 033612

  24. The experiment pump light Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse. n=-2 (p=4ħk) n=0 (p=0) n=-1 (p=2ħk)

  25. PROPAGATION EFFECTS IN FELs: SUPERRADIANT INSTABILITY Particles at the trailing edge of the beam never receive radiation from particles behind them: they just radiate in 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.

  26. SR in the classical model: Strong SR (Lb=30 Lc) from a coherent seed R. Bonifacio, B.W. McNeil, and P. Pierini PRA 40, 4467 (1989)

  27. CLASSICAL SASE • Ingredients of Self Amplified Spontaneous Emission (SASE) • Start up from noise • Propagation effects (slippage) • SR instability •  • The electron bunch behaves as if each cooperation • length would radiate independently a SRspike • 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

  28. CLASSICAL SASE Broad and noisy spectrum at short wavelengths (X-FEL) Time profile with many random spikes (approximately L/Lc) Example from DESY (Hamburg) for the SASE-FEL experiment

  29. QUANTUM REGIME: CLASSICAL REGIME: SASE : NUMERICAL SIMULATIONS

  30. QUANTUM REGIME: CLASSICAL REGIME: SASE: average momentum distribution Quantum behaviour : sequential SR decay, only n<0 Classical behaviour : both n<0 and n>0 occupied

  31. 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

  32. FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE FOR AN X-RAY SOURCE (l=1Ǻ) QUANTUM SASE needs: MeV Linac (m) Laser undulator (l~1mm) lower cost (106$) yields: quasi monocromatic spectrum CLASSICAL SASE needs: GeV Linac (Km) Long undulator (100 m) High cost (109$) yields: Broad and chaotic spectrum QFEL

  33. 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

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