The Angular Momentum of Gauge Fields: The Case of Twisted Photons

1. 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. 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. 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 Jz=±ħalong the propagation direction z(Beth’s experiment, 1936) • Heitler, Quantum Theory of Radiation (1954): larger Jz 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 Aμ(x) contains OAM information • Beams of light with azimuthal beam dependence exp(ilϕ) (e.g, Laguerre-Gaussian modes) can carry large values of OAM (Allen et al, 1992). Review: Yao, Padgett, Advances in Optics and Photonics3, 161–204 (2011) and references therein

4. Orbital vs Spin Angular Momentum (from Yao’11 review)

5. Twisted Photons • Quantization of light beams having azimuthal phase dependence exp(ilϕ) 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. 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. 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. Helical Undulators (cont) • AA, Mikhailichenko, On Generation of Photons Carrying Orbital Angular Momentum in the Helical Undulator, E-print: arXiv 1109.1603

9. 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. Absorption of Twisted Photons • Translation or rotation? • Mechanism depends on the dimensions of target vswavefront 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. 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. Twisted Photon State • Use plane-wave expansion • Plane wave: • Twisted wave:

13. Fields of the Twisted Wave • Vector potential Magnetic field • Poynting vector

14. 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. Atomic Photoexcitation • Interaction Hamiltonian: • Matrix element of the transition: • b- an impact parameter w.r.t. the atomic center

16. Matrix Element of Photoexcitation • Photo-excite a hydrogen atom from the ground to the state with a principal quantum number nf, OAM lf, and OAM projection mf. Incoming twisted photon is defined by AM projection mγ, energy ω and a pitch angle θk(with κ=kperp) • Matrix element: • Atomic factors

17. Calculation Results • Matrix elements as a function of an impact parameter b

18. Helicity Asymmetry • Flip photon helicityΛ, keep the OAM projection the same => • Results a different twisted photon state with mγmγ-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. 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 • OtherwiseOAM results in linear momentum of the entire atom • For excitation of a baryon with a twisted photon, γTN->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. How to Generate Twisted Photons in MeV-GeV? • Serboet 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. • Theoreticallydemonstratedthat OAM properties of twistedphotonsarepreserved in Comptonbackscattering. • Ifholds, it provides a newtool in nuclear, hadronicandhigh-energyphysics, thatarephotonbeamswithpre-selected OAM alongtheirdirection of propagation.

21. 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. 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”