Nina Rohringer and Robin Santra

1. Characterization of statistical properties of x-ray FEL radiation by means of few-photon processes Nina Rohringer and Robin Santra

2. Outline • Motivation • SASE FEL • Amplification starts from “Shot noise” • Theoretical tools • Classical and quantum mechanical field-correlation functions • Density matrix formalism • Quantum electrodynamics • Atomic physics • 1-photon absorption • Elastic scattering • 2-photon absorption • Characterization of FEL radiation – Feasibility study • Rate equations for Helium and Neon • Outlook

3. Single-shot measurements of SASE FEL Field intensities and phases of the 530 nm chaotic output of a SASE FEL at Low Energy Undulator Testline (APS) Random phases and amplitudes ! Statistical description necessary Yuelin Li et al. Phys. Rev. Lett. 91, 243602 (2003).

4. Electron bunch-duration Tb Gain bandwidth Theoretical methods to predict statistical properties of SASE FEL amplification starts from “Shot Noise” Gaussian random process: random arrival times of electrons at the entrance of the undulator Simulations in the non-saturated regime: Single-shot spectrum Average over shots E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The Physics of Free Electron Lasers, (Springer-Verlag, Berlin 2000). Krinsky, Gluckstern Phys. Rev. ST Accel. 6, 50701 (2003).

5. Questions we have to ask:(1) Which statistical information of the radiation field is necessary to interpret a given experiment ?(2) Which experiments would allow to determine those relevant statistical properties of the radiation field ?

6. Classicalfield-correlation functions 1st order time correlation function (Michelson interferometer) 1st order time-space correlation function Young’s double slit experiment 2nd order correlation function (Hanbury-Brown and Twiss experiment)

7. Quantum mechanical field-correlation functions - quantum mechanical concept of coherence -R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963). -P. Lambropoulos, C. Kikuchi, and R.K. Osborn, Coherence and two-photon absorption, Phys. Rev. 144, 1081 (1966). -G.S. Agarwal, Field-correlation effects in multiphoton absorption processes, Phys. Rev. A 1, 1445 (1970).

8. atom Multi-mode density-matrix in Fock representation Final state: Statistical description by density matrix formalism Initial state:

9. Perturbative Quantum Electrodynamics Approach H=Hatomic+HField+HI 1st and 2nd order perturbation theory in A and A2 terms to calculate transition matrix elements

10. Atomic part Field One-photon absorption Correlations of different angles of incident radiation Generalized cross correlation function of 1st order

11. Restriction to single propagation direction: Generalized cross correlation function of 1st order Average number of photons with frequency Spectral intensity distribution

12. Atomic part Field One-photon single ionization

13. Atomic part Field Elastic X-ray scattering Negligible if far from resonance

14. Two-photon absorption negligible ? Atomic part Field

15. Correlation Functions of coherent and chaotic single-mode radiation field Coherent-state representation of density matrix: (Glauber’s quasi-probability p-representation) Coherent field: (pure coherent state) Chaotic field:

16. Intensity distribution: 1-dimensional classical models of SASE FEL Predictions Relation of higher-order to 1st order correlation functions (Generalized Siegert Relations) E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The Physics of Free Electron Lasers, (Springer-Verlag, Berlin 2000). S. Krinsky and R.L. Gluckstern,Phys. Rev. ST Accel. 6, 50701 (2003). C. B. Schroeder, C. Pellegrini, and P. Chen, Phys. Rev. E 64, 56502 (2001). S. O. Rice, Bell Syst. Tech. J 24, 46 (1945).

17. In principle,only first order correlation function G1 is needed!ButExperimental verification needed.

18. Feasibility Study for Helium and NeonRate-equations Gaussian pulse envelope Helium: Neon: Auger-decay and valence-shell ionization included

19. Helium transition probabilities in dependence of intensity Rate equations for Gaussian-shaped pulse

20. 1mm 0.5 mm 0.25 mm He+ 1.7d7 3.9d6 6.9d5 He++ sequ. 3.2d3 2.8d3 1.7d3 He++ corr. 4.4d51.d5 1.8d4 Expected experimental event rates pulse-duration 100 fs energy 1.4 keV repetition rate 120 Hz photons/pulse 5. 1012 gas density 1014 cm-3 Helium not suitable,…

21. Neon transition probabilities in dependence of intensity

22. 2.5mm 2 mm 1 mm Ne2+ 3.8d9 5.2d8 5.4d2 Ne4+5.0d9 1.1d9 5.4d3 Ne6+ 4.8d9 1.9d9 1.1d5 Ne8+ 3.1d9 2.1d9 3.6d7 Ne9+ 1.2d9 1.5d9 1.3d7 Ne10+ 2.1d8 8.3d8 4.9d8 Expected experimental event rates pulse-duration 100 energy 1.4 keV repetition rate 120 Hz photons/pulse 5. 1012 gas density 1014 cm-3

23. Conclusions and Outlook • Density matrix approach for statistical treatment of radiation field • Perturbative quantum electrodynamics approach • For few photon processes: • Shot to shot characterization of radiation field not necessary • Necessary information: generalized correlation functions of the radiation field • Low order correlation functions could in principle be determined by means of single- and double ionization of well-studied atomic systems • Theoretical Challenges: • Accurate atomic matrix-elements for elementary processes needed • Inversion problem