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