Angular Momenta , Geometric Phases, and Spin-Orbit Interactions of Light and Electron Waves

Angular Momenta, Geometric Phases,

and Spin-Orbit Interactions of Light and Electron Waves

Konstantin Y. BliokhAdvanced 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)

ext c c L r kkc

rc

3. Intrinsic orbital AM (vortex)

int c

c

dV

L r r k

κ

0 1

1 2

cS κ1. Intrinsic spin AM (polarization)

1 1 c c c/ kκ k

exp i

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 phase

,i ie E e E e e

2D:

x x

y y

x yi e e e

1 1

d ΩJ

AM-rotation coupling:Coriolis / angular-Doppler effect

3D: J Ω

Φ

zk

xk

E κykS κ

C

Geometric phase

, , : ii e e e κ e κ e e

2 1 c

2

oszd S

dynamics, S:

d k kAC

1 cos ,sink

eA 3k

kkF A Berry connection, curvature

(parallel transport)

geometry, E:

Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 1986

X

YZ

e

w

v

te

w

v

tew

v

t 2W20 0

Spin-orbit Lagrangian and phase

SOI Lagrangian from Maxwell equationsSOI k S ΩAL

Bliokh, 2008

SOI 2ˆ ˆ ˆ

4em

p σ pL A E2

ˆˆ 1 ,

2m

p E

S pA for Dirac equation

SOI d LBerry phase:

Liberman & Zeldovich, PRA 1992;Bliokh & Bliokh, PLA 2004, PRE 2004; Onoda, Murakami, Nagaosa, PRL 2004

c cln ,k n k r t k F ray equations (equations of motion)

X

YZ

D2D

+2D

D

Spin-Hall effect of light

ˆ ˆ,i j ijl lr r i F

c c c const J r k κ

Berry phase Spin-Hall effect Anisotropy

Bliokh et al., Nature Photon. 2008

Spin-Hall effect of light

Fedorov, 1955; Imbert, 1972; ……Onoda et al., PRL 2004;Bliokh & Bliokh, PRL 2006;Hosten & Kwiat, Science 2008

D r

ImbertFedorov transverse shift

Dr

Spin-Hall effect of light

ˆ / sin ,z yR

coty

coszS

1 cot

ykY k

c c :y yk k k egeometry, phase:

ext : J L S

dynamics, AM:sin coszJ k Y 0zJ

k

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

zk

xk

phase κyk

int L κ

C

OOI LagrangiannOOI i t k L ΩL A

X

YZ

w

v

te

w

v

tew

v

t 20 0

Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006;Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995

SOI d L Berry phase

Orbital-Hall effect of light

l vortex-depndent shifts and orbit-orbit interaction

X

YZ

012–1–2

Bliokh, PRL 2006Fedoseyev, Opt. Commun. 2001;Dasgupta & Gupta, Opt. Commun. 2006

c c c J r k κ

• 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 lightˆ ˆˆ ˆˆ ˆ J r p S L S

ˆ ,i kr ˆ ,p k ˆa aijijS i

total AM = OAM +SAM?

ˆˆ ,z z zijijL i S i

ix yi e

E e e

z-components and paraxial eigenmodes:

- polarization, l - vortex

1

10

1

1

( )E k

“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)

0 E k E κ transversality constraint: 3D 2D

Nonparaxial problem 1

ˆˆ , LE κ SE κ though ˆ JE κ

“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

i iii ee e

E e ee κ

nonparaxial vortex

spin-to-orbital AM conversion?..

Y. Zhao et al., PRL (2007)

, 1z zL S

zJ though

Spin-dependent OAM and position signify the spin-orbit interaction (SOI) of light: (i) spin-to-orbital AM conversion, (ii) spin Hall effect.

2/ˆ ˆˆ k k Sr r spin-dependent position M.H.L. Pryce, PRSLA (1948), etc.

Modified AM and position operators

ˆˆ ˆ ˆ ΔL L r k projected SAM and OAM cf. S.J. van Enk, G. Nienhuis, JMO (1994)

ˆ ˆ ˆˆ ˆ S S κ κ S κΔ ˆ ˆ Δ κ κ S

,ˆˆ ˆ J L S compatible with transversality:

Modified AM and position operators

3ˆ /ˆ ˆ, li j ijlr kr i k

Modified operators exhibit non-canonical commutation relations:

ˆ ,0ˆ,i jS S ˆ ˆ ,ˆˆ,i j i ljl lL L i L S

ˆ ,ˆˆ,i j lijlSL S i

ˆ ˆˆ ,i j ijl lJ O i O

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

ˆ : , , , ,x y zU κ e e e e e κ †ˆ ˆˆ ˆU UO O

00

0 0

3D 2D reductionand diagonalization:

ˆ ˆ ̂ L L kA all operators

become diagonal !

ˆ ˆ ̂ r r A

ˆ ̂S κ

ˆ diag ,11, 0

cf. I. Bialynicki-Birula, Z. Bialynicka-Birula (1987), B.-S. Skagerstam (1992), A. Berard, H. Mohrbach (2006)

kA - 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

ˆE E O O mean values 0 iE A e

field

( )E k

20 02 1 cos ~

C

d k A Berry phase

The spin-to-orbit AM conversion in nonparaxial fields originates from the Berry phase associated with the azimuthal distribution of partial waves.

zL 1 ,zS SAM and OAMfor Bessel beam

Quantization of caustics

k R quantized GO caustic: fine SOI splitting !

1 1

4

1

22

2 2 2 22b bI r a J r J r a J r

spin-dependent intensity 1 / 2)/ 2( , ab

Plasmonic experiment

Y. Gorodetski et al., PRL (2008)

2B 0 / 2 ,

1

- circular plasmonic lens generates Bessel modes

B

1 1

Spin and orbital Hall effects

Bk Y orbital and spin Hall effects of light

1

1

4 0 4

,

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)

0Plasmonic half-lens produces spin Hall effect:

~Y

1

1

• 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

ext c c L r k

kc

rc

zsS e1. Spin AM

k k

1/ 2s 1/ 2s ‘non-relativistic’

3. Intrinsic orbital AM ?

int L κ

0 1

1 2‘relativistic’

Scalar electron vortex beams

Scalar electron vortex beams

Scalar electron vortex beams

Murakami, Nagaosa, Zhang, Science 2003

, /e m k E r k k F equations of motion

Orbital-Hall effect for electrons

const J r k κ

Bliokh et al., PRL 2007

Scalar electron vortex beams

Scalar electron vortex beams

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

1i E tp W e

p rp plane wave

12 ˆ

E m wW

E E m w

σ κ

polarization bi-spinor

ˆˆ ˆti m α p Dirac equation: ... ... ... ...... ... ... ...

... ... ... ...

... ... ... ...

E > 0

E < 0cross-terms

cross-terms

w

spinor in the rest frame1 1

1 200 1,

2, zw S

Bessel beams for Dirac electron

2 221 / 2 / 2ssr J r J r D D

spin-dependent intensity

201 sinm

ED

SOI strength

1/ 2s 1/ 2s

4

1

0s s iW e p spectrum

( )E k

s

s

Bliokh, Dennis, Nori, arXiv:1105.0331

Angular momentum

ˆFWU p Foldy-Wouthhuysen

transformationˆP projector onto E>0 subspace

... ... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

E > 0

E < 0cross-terms

cross-terms

†ˆ ˆ ˆˆ ˆ :FW FWU UO OP

ˆ ˆ ˆ L L Δˆˆ i kr A

ˆ ,ˆˆ S S Δ

2

ˆˆ 12

mp E

p σA

ˆ ˆ Δ pA

OAM and SAM for Bessel beams , 1z zL S ss D D

Bliokh, Dennis, Nori, arXiv:1105.0331

Magnetic moment for Dirac electron

†ˆ ˆˆ ˆ ˆ ˆ ˆ 22FW FWeU UE

M M L SP

2e dV M r j † ˆ /E j α p

magnetic moment from currentˆ , M M ˆ ˆ ˆ

2e M r α

operator !

22e sE

DM

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.

THANK YOU!