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 1.Introductory concepts 2.Classical FEL-CARL Model 3.Quantum FEL-CARL Model 4.Propagation Effects 5.Quantum SASE regime

3. Free Electron Laser (FEL)

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

5. 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 SN SN SN SN SN S At first sight, CARL and FEL look very different… ~ p

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

7. Collective Recoil Lasing = Optical gain + bunching In FEL and CARL particles self-organize to form compact bunches ~ which radiate coherently. 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) Maximum gain at  =0 runaway solution See figure (a)

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

13. Canonical Quantization 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 Here  describes spatial evolution of  on scale of and describes spatial evolution of A and  on scale of cooperation length, L c >>  We have introduced propagation into the model, so different parts of the electron beam can feel different fields : So far we have neglected slippage, so all sections of the e-beam evolve identically (steady-state regime) if they are the same initially. where

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

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

17. Quantum limit for 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 by Maxwell-Bloch equations

18. Bunching and density grating CLASSICAL REGIME  >>1 QUANTUM REGIME  <1

19. Quantum Linear Theory Classical limit Quantum regime for  <1 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=-2 n=0 n=-1 BEC LASER SEQUENTIAL SUPERRADIANT SCATTERING

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

23. 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  m P = 13 mW Experimental evidence of quantum CARL at LENS L.Fallani et al, PRA 71 (2005) 033612

24. 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.

25. 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 L b << L c the SR pulse remains small (weak SR). if L b >> L c 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

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

27. 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 CLASSICAL SASE

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

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

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

32. 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

33. CLASSICAL SASE needs: GeV Linac (Km) Long undulator (100 m) High cost (10 9 $) yields: Broad and chaotic spectrum FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE FOR AN X-RAY SOURCE ( =1Ǻ) QUANTUM SASE needs: MeV Linac (m) Laser undulator ( ~1  m) lower cost (10 6 $) yields: quasi monocromatic spectrum

34. 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