Angular Momenta, Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves Konstantin Y. Bliokh Advanced Science Institute, RIKEN, Wako-shi, Saitama, Japan In collaboration with: Y. Gorodetski, A. Niv, E. Hasman (Israel); M. Alonso (USA), E. Ostrovskaya (Australia), A. Aiello (Germany), M. R. Dennis (UK), F. Nori (Japan)

Basic concepts: Angular momenta, Berry phase, Spin- orbit interactions, Hall effects Theory of photon AM Application to Bessel beams Electron vortex beams Theory of electron AM

Angular momentum of light 2. Extrinsic orbital AM (trajectory) kckckckc rcrcrcrc 3. Intrinsic orbital AM (vortex) 1. Intrinsic spin AM (polarization)

Different mechanical action of the SAM and OAM on small particles: O’Neil et al., PRL 2002; Garces-Chavez et al., PRL 2003 spinning motion orbital motion Angular momentum: Spin and orbital

Geometric phase2D: AM-rotation coupling: Coriolis / angular-Doppler effect3D:

Φ C Geometric phase dynamics, S :  Berry connection, curvature (parallel transport) geometry, E :

Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 222  Spin-orbit Lagrangian and phase  SOI Lagrangian from Maxwell equations Bliokh, 2008  for Dirac equation Berry phase:

Liberman & Zeldovich, PRA 1992; Bliokh & Bliokh, PLA 2004, PRE 2004; Onoda, Murakami, Nagaosa, PRL 2004  ray equations (equations of motion) + 2222 Spin-Hall effect of light

Berry phase Spin-Hall effect Anisotropy Bliokh et al., Nature Photon Spin-Hall effect of light

Fedorov, 1955; Imbert, 1972; …… Onoda et al., PRL 2004; Bliokh & Bliokh, PRL 2006; Hosten & Kwiat, Science 2008 Imbert  Fedorov transverse shift Spin-Hall effect of light

geometry, phase: dynamics, AM: Spin-Hall effect of light

Remarkably, the beam shift and accuracy of measurements can be enormously increased using the method of quantum weak measurements: Angstrom accuracy! Spin-Hall effect of light

Orbit-orbit interaction and phase Φ C  OOI Lagrangian 2222  Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006; Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995  Berry phase

Orbital-Hall effect of light  vortex-depndent shifts and orbit-orbit interaction 01 2 –1–1–1–1 –2–2–2–2 Bliokh, PRL 2006 Fedoseyev, Opt. Commun. 2001; Dasgupta & Gupta, Opt. Commun. 2006

Basic concepts: Angular momenta, Berry phase, Spin- orbit interactions, Hall effects Theory of photon AM Application to Bessel beams Electron vortex beams Theory of electron AM

Angular momentum of light  total AM = OAM +SAM?  z -components and paraxial eigenmodes:  - polarization, l - vortex

“the separation of the total AM into orbital and spin parts has restricted physical meaning.... States with definite values of OAM and SAM do not satisfy the condition of transversality in the general case.” A. I. Akhiezer, V. B. Berestetskii, “Quantum Electrodynamics” (1965)  transversality constraint: 3D  2D Nonparaxial problem 1 though

“in the general nonparaxial case there is no separation into ℓ -dependent orbital and  -dependent spin part of AM” S. M. Barnett, L. Allen, Opt. Commun. (1994) Nonparaxial problem 2  nonparaxial vortex spin-to-orbital AM conversion?.. Y. Zhao et al., PRL (2007) though

Spin-dependent OAM and position signify the spin-orbit interaction (SOI) of light : (i) spin-to-orbital AM conversion, (ii) spin Hall effect.  spin-dependent position M.H.L. Pryce, PRSLA (1948), etc. Modified AM and position operators  projected SAM and OAM cf. S.J. van Enk, G. Nienhuis, JMO (1994) compatible with transversality:

Modified AM and position operators Modified operators exhibit non-canonical commutation relations: But they correspond to observable values and are transformed as vectors: cf. S.J. van Enk, G. Nienhuis (1994); I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)

Helicity representation 3D  2D reduction and diagonalization:  all operators become diagonal ! cf. I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)  Berry connection

Basic concepts: Angular momenta, Berry phase, Spin- orbit interactions, Hall effects Theory of photon AM Application to Bessel beams Electron vortex beams Theory of electron AM

Bessel beams in free space  mean values  field  Berry phase The spin-to-orbit AM conversion in nonparaxial fields originates from the Berry phase associated with the azimuthal distribution of partial waves.  SAM and OAM for Bessel beam

Quantization of caustics  quantized GO caustic: fine SOI splitting !  spin-dependent intensity

Plasmonic experiment Y. Gorodetski et al., PRL (2008)  circular plasmonic lens generates Bessel modes

Spin and orbital Hall effects  orbital and spin Hall effects of light  azimuthally truncated field (symmetry breaking along x ) B. Zel’dovich et al. (1994), K.Y. Bliokh et al. (2008)

Plasmonic experiment K. Y. Bliokh et al., PRL (2008) Plasmonic half-lens produces spin Hall effect:

Basic concepts: Angular momenta, Berry phase, Spin- orbit interactions, Hall effects Theory of photon AM Application to Bessel beams Electron vortex beams Theory of electron AM

Angular momentum of electron 2. Extrinsic orbital AM kckckckc rcrcrcrc 1. Spin AMkk ‘non-relativistic’ 3. Intrinsic orbital AM ? ‘relativistic’

Scalar electron vortex beams

Murakami, Nagaosa, Zhang, Science 2003  equations of motion Orbital-Hall effect for electrons Bliokh et al., PRL 2007

Scalar electron vortex beams

Basic concepts: Angular momenta, Berry phase, Spin- orbit interactions, Hall effects Theory of photon AM Application to Bessel beams Electron vortex beams Theory of electron AM

Bessel beams for Dirac electron  plane wave  polarization bi-spinor  Dirac equation: E > 0E > 0 E < 0E < 0 cross- terms cross- terms  spinor in the rest frame

Bessel beams for Dirac electron  spin-dependent intensity  SOI strength  spectrum Bliokh, Dennis, Nori, arXiv:

Angular momentum  Foldy-Wouthhuysen transformation  projector onto E>0 subspace E > 0E > 0 E < 0E < 0 cross- terms cross- terms  OAM and SAM for Bessel beams Bliokh, Dennis, Nori, arXiv:

Magnetic moment for Dirac electron  magnetic moment from current  operator !  final result slightly differs from Gosselin et al. (2008); Chuu, Chang, Niu, (2010)

Spin and orbital AM of light, geometric phase, spin- and orbital-Hall effects of light General theory for nonparaxial optical fields: modified SAM, OAM, and position operators. Bessel-beams example, spin-orbit splitting of caustics and Hall effects Electron vortex beams, AM of electron, orbital-Hall effect. Relativistic nonparaxial electron: Bessel beams from Dirac equation General theory of SAM, OAM, position, and magnetic moment of Dirac electron.

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