1. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 The Angular Momentum of Gauge Fields: The Case of Twisted Photons Andrei Afanasev The George Washington University Washington, DC QCD Evolution Workshop Jefferson Lab, Newport News, VA May 9, 2012

2. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Objectives. Consider Abelian gauge fields (QED). Discuss the properties of of photon beams with a large angular momentum (>ħ) projection on the direction of propagation. Generation. Formalism. Absorption by atoms

3. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Introduction. Photons carry linear momentum p=ħk (k=wave vector). Photons carry both spin angular momentum (SAM) and orbital angular angular momentum (OAM) – may be separated in paraxial approximation. Circularly polarized plane-wave photons carry J z =±ħ along the propagation direction z (Beth’s experiment, 1936). Heitler, Quantum Theory of Radiation (1954): larger J z possible if the EM wave is constrained in the transverse plane (cylindrical waves). Spherical waves: expansion in terms of angular momentum eigenfunctions, position dependence of vector potential A μ (x) contains OAM information. Beams of light with azimuthal beam dependence exp(il ϕ ) (e.g, Laguerre-Gaussian modes) can carry large values of OAM (Allen et al, 1992). Review: Yao, Padgett, Advances in Optics and Photonics 3, 161–204 (2011) and references therein

4. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Orbital vs Spin Angular Momentum (from Yao’11 review)

5. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Twisted Photons. Quantization of light beams having azimuthal phase dependence exp(il ϕ ) lead to a concept of twisted photons G. Molina-Terriza, J.Torres, L. Torner, “Twisted Photons”, Nature Physics, May 2007. The typical transverse intensity pattern of a light beam with orbital angular momentum, (a) theory (b) experiment. The light beam exhibits a dark spot in the center, and a ring-like intensity profile. (c) Azimuthal dependence of beam phase results in a helical wavefront. (d) Orientation of the local momentum of the beam has a vortex pattern (hence another name, an optical vortex).

6. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Generation of Light Beams with Orbital Angular Momentum. A diffraction grating with fork dislocation centered on the beam axis, could convert the fundamental Gaussian mode from any laser into a helically phased mode [V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP. Lett. 52, 429–431 (1990).] Commonly accepted method for producing helically phased beams.. Spiral Phase Plates: Gaussian beam is passed through optical media, with azimuthal dependence in thickness

7. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Generation of Twisted Photons with Helical Undulators. E. Hemsing, A. Marinelli, and J. B. Rosenzweig, “Generating Optical Orbital Angular Momentum in a High-Gain Free-Electron Laser at the First Harmonic,” Phys. Rev. Lett. 106, 164803 (2011).. AA, Mikhailichenko, On Generation of Photons Carrying Orbital Angular Momentum in the Helical Undulator, E-print: arXiv 1109.1603. Considered properties of synchrotron radiation by charged particles passing through a helical undulator. Shown that all harmonics higher than the first one radiated in a helical undulator carry OAM. Large K-factors favor large values of OAM for generated radiation.

8. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Helical Undulators (cont). AA, Mikhailichenko, On Generation of Photons Carrying Orbital Angular Momentum in the Helical Undulator, E-print: arXiv 1109.1603

9. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Transfer of Angular Momentum. For optical wavelengths, transfer of Orbital AM differs from Spin AM.. Important: wavelength vs the target size. Wavelength >> target size: OAM transfer results in linear momentum of the target as a whole. Wavelength < target size: OAM results in target rotation

10. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Absorption of Twisted Photons. Translation or rotation?. Mechanism depends on the dimensions of target vs wavefront cross section S. Depending on a topological number and pitch angle, wavefront cross section may be from a few to a few hundred wavelengths. If the target size is of the order S or larger, an essential fraction of photon energy is transferred to rotation of the target.. For smaller targets, rotation is more likely is the target is next the center of the optical vortex

11. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Atomic Excitations with High-L Photons. Twisted photons may enhance (relatively) atomic transitions with large transfer of angular momentum (Picon et al, 2010; AA, Carlson, Mukherjee, arXiv:1304.0115).. An atom must be close (relative to wavelength) to the beam axis, but there is a dip in intensity there. Result: the probability to excite high-L atomic levels is suppressed. Reason: optical wavelength >> atomic size, atomic transitions are caused by long-wavelength photons because the bound electrons are non-relativistic. Situation will change in hadronic physics: need photons with wavelength < fm to excite a nucleon.

12. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Twisted Photon State. Use plane-wave expansion. Plane wave:. Twisted wave:

13. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Fields of the Twisted Wave. Vector potential Magnetic field. Poynting vector

14. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Transverse Beam Profile. OAM light beam is characterized with a special transverse profile (example from AA, Carlson, Mukherjee,). Intensity dip on the beam axis, with transverse size > wavelength

15. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Atomic Photoexcitation. Interaction Hamiltonian:. Matrix element of the transition:. b- an impact parameter w.r.t. the atomic center

16. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Matrix Element of Photoexcitation. Photo-excite a hydrogen atom from the ground to the state with a principal quantum number n f, OAM l f, and OAM projection m f. Incoming twisted photon is defined by AM projection m γ, energy ω and a pitch angle θ k (with κ=k perp ). Matrix element:. Atomic factors

17. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Calculation Results. Matrix elements as a function of an impact parameter b

18. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Helicity Asymmetry. Flip photon helicity Λ, keep the OAM projection the same =>. Results a different twisted photon state with m γ  m γ -2Λ. Photoabsorption cross sections are different for a given impact parameter => (parity-conserving) helicity asymmetry. The largest asymmetry is near the center of optical vortex. Asymmetry is zero after averaging over the impact parameter b. May be observed for small-size targets or near-field geometry

19. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Twisted Photons for Hadrons vs Atoms. For atomic transitions photon OAM is preferably transferred to internal degrees of freedom if the target is near the center of an optical vortex. Otherwise OAM results in linear momentum of the entire atom. For excitation of a baryon with a twisted photon, γ T N->N* OAM will be passed to internal degrees of freedom and will help to get insight into nucleon structure. Will need more helicity amplitudes to describe a baryon resonance excitation

20. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 How to Generate Twisted Photons in MeV-GeV?. Serbo et al proposal (2010): Compton backscattering. U.D. Jentschura V.G. Serbo Generation of High-Energy Photons with Large Orbital Angular Momentum by Compton Backscattering. Phys.Rev.Lett. 103 (2011) 013001, e-Print: arXiv:1008.4788; Compton Upconversion of Twisted Photons: Backscattering of Particles with Non- Planar Wave Functions. Eur.Phys.J. C71 (2011) 1571, arXiv:1101.1206.. Theoretically demonstrated that OAM properties of twisted photons are preserved in Compton backscattering.. If holds, it provides a new tool in nuclear, hadronic and high-energy physics, that are photon beams with pre-selected OAM along their direction of propagation.

21. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Summary. Twisted photons carry large orbital angular momenta along the axis of propagation. Can be applied at widely different scales, from dust particles, to nanoparticles, molecules, (Rydberg) atoms, and nuclei. Atomic photoexcitation considered here. Results in excitation of states with a range of quantum numbers, different from plane waves. Predicted parity-conserving helicity asymmetry in the central region of an optical vortex: flipping the helicity results in a different photon state. Accelerator-based light source are most efficient for generating twisted X-rays and gamma-rays

22. QCD Evolution Workshop, Jefferson Lab, May 9, 2013 Outlook. Is it possible to generate twisted photons with GeV energies to probe the structure of hadrons?. Are the cross sections or spin asymmetries of basic QED processes involving twisted photons affected by their additional angular momentum?. Can (inelastic) polarized electron scattering on a twisted photon provide an information on its angular momentum? (=polarization structure functions of a twisted photon, TMDs, GPDs, etc). Extension to non-Abelian fields=> “twisted gluons”