Photon Wave Mechanics and Spin-Orbit Interaction in Single Photons

Michael RaymerCody Leary

Oregon Center for OpticsOregon Center for OpticsDepartment of PhysicsUniversity of Oregon

1: How the Photon is usually taught

2: Elementary Theory of the Wave Function of a Photon

3: “Advanced” Theory of the Wave Function of a Photon

OUTLINE

4: Spin-Orbit Interaction in a Single Photon

Maxwell-Boltzmann vs. Bose-Einstein

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How the photon is usually taught:

Light is made of EM waves. Light is made of corpuscles.

Maxwell-Boltzmann Bose-Einstein

Light is made of EM waves.Modes are distinguishable.M-B counting statistics applies.

Light is made of corpuscles.They are indistinguishable.B-E counting statistics applies.

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Planck spectrumprediction

EM fields as entities.Photons as state description.(Dirac)

inference

prediction

Photons as entities.Quantum field as “emergent.”

inference

Not a change of basis. A change of viewpoint

1. Photons are Bose particles with

2. Light is made of photons, but it also has wave properties, which are important when photons are flying through space, but not when they are detected.

Question:Must a photon be monochromatic?

E = hν

An atom initially in an excited

Question:If a photon can be in a fairly localized wave packet, what wave equation does this obey?

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in an excited state decays spontaneously.

begin with Einstein’s kinematic equation:

E = (m c2)2 + (c p)2

m=mass, p=momentum

PaulAl

electron: m>0, v<<c

Teaching Wave Mechanics for Particles - 1

(de Broglie) p = hk (Planck) E = hω(ignore polarization, spin, interactions)

p2photon: m=0, v=c

Dispersion relations --> Wave Equations in 1D

Electron Wave EquationQuickTime™ and a

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Erwin

ih∂∂ t

Ψ ≅ −h2

2m

∂2

∂x2Ψ

E ; mc2 + p2

2m+ ...

James Clerk

Photon Wave Equation

h2 ∂2

∂t 2Ψ(x,t) = h2 c2 ∂2

∂x2Ψ(x,t)

E2 = c2 p2

(h cancels out)

Teaching Wave Mechanics for Particles - 2

p = hk E = hω

ψ n (x,t) =2

bsin(knx)exp(−iω nt) kn = nπ / b, n = 1,2,3...

Photon E = cp∂2

∂t 2 Ψ(x,t) = c2 ∂2

∂x2 Ψ(x,t)

Electron

ih∂∂ t

Ψ ≅ −h2

2m∇ur 2

Ψ

n = 3

E = p2 / 2m

Particle in a Box

x = 0 bn = 1

ωn =En

h=

c(hkn )

h= n

cπb

n = 2

n = 3

(like a laser resonator)

ω n =En

h=

(hkn )2

2mh=

nπb

21

2mh

x = 0 bn = 1

n = 2

n = 3

(an electron “resonator”)

Probability: Born Interpretation

PhotonElectron

the probability density for finding the particle at position x at time t .Ψ(x,t)

2 =

Teaching Wave Mechanics for Particles - 3

Photon

Always relativistic: Problematic, but OK for eigenstates of energy (or states with small spread in energy)

Electron

• Nonrelativistic: OK

• Relativistic:Problematic -charge density not ≠mass density

Spin: Just tack it on

Electron (s = 1/2)

Spin is described by two new quantum numbers, s and m

Sr

= h s(s + 1) S

u$= hm

u$

Photon (s = 1)

Teaching Wave Mechanics for Particles - 4

Electron (s = 1/2)

Ψel = ψ+

1

2

(x,t)χ+

1

2

+ ψ−

1

2

(x,t)χ−

1

2

Sr

= h (1 / 2)(1 / 2+ 1)

S

u$= hm (any axis)

m = 1/2, -1/2 “spin projection”

Photon (s = 1)

Ψ ph = ψ +1(x,t)χ+1 + ψ −1(x,t)χ−1

Sr

= h (1)(1+ 1)

S

k$= hσ (propagation axis)

(no worse than what we do to an electron)

= 1, -1 (not 0) “helicity”σ

So, What is a Photon?

1. The name given to the n=1 states of the electromagnetic quantum field.

---- or ----2. A fundamental quantum particle, through which

the EM field emerges when many photons are the EM field emerges when many photons are present. (Like a nation emerging from an aggregate of many people.)

Analogous statements hold for electrons.

Many details have been swept under the rug…

ih∂∂ t

Ψ = (m c2)2 + c2(−ih∇ur

)2 Ψm≠Os=1/2v~c

electron

E = (m c2)2 + (c p)2 (Einstein)

(Planck)

ih∂∂ t

⇔ E(de Broglie) pur

⇔ −ih∇uru

Derivation of Quantum Wave Equations

v~cDirac Equation

ih∂∂ t

Ψ = cmβ Ψ − ihc (αur

g∇uru

)Ψ

v<<c 4 components

Schrödinger Equation

2 components

ih∂∂ t

Ψ(2) = −h2

2m∇ur 2

Ψ(2)

Require local number density

Ψuru

* Ψuru

=

ih∂∂ t

Ψ = (m c2)2 + c2(−ih∇ur

)2 Ψm≠Os=1/2v~c

electron

E = (m c2)2 + (c p)2 (Einstein)

(Planck)

ih∂∂ t

⇔ E(de Broglie) pur

⇔ −ih∇uru Can we

do the same for a single photon?

Derivation of Quantum Wave Equations

v~cDirac Equation

ih∂∂ t

Ψ = cmβ Ψ − ihc (αur

g∇uru

)Ψ

v<<c 4 components

Schrödinger Equation

2 components

ih∂∂ t

Ψ(2) = −h2

2m∇ur 2

Ψ(2)

Require local number density

Ψuru

* Ψuru

=

E = (m c2)2 + (c p)2

ih∂∂ t

Ψ = (m c2)2 + c2(−ih∇ur

)2 Ψ

m=Os=1v=c

∇uru

⋅ Ψuru

= 0

(Planck)

ih∂∂ t

⇔ E(de Broglie) pur

⇔ −ih∇uru

Parallel treatment for photon:

m≠Os=1/2v~c

electron

Derivation of Quantum Wave Equations

(Einstein)

Dirac Equation

ih∂∂ t

Ψ = cmβ Ψ − ihc (αur

g∇uru

)ΨPhoton Wv. Equation

ih∂∂ t

Ψuru

= hcσ ∇uru

×Ψuru

∇ ⋅ Ψ = 0

3 components for each helicity

Require local energy density

Ψuru

* Ψuru

=

( cancels) hσ = helicity (±1)

v~c

v<<c

Schrödinger Equation

4 components

2 components

ih∂∂ t

Ψ(2) = −h2

2m∇ur 2

Ψ(2)

Derivation of Photon Wave Equation

E = c p

uv⋅ puv

Eψ (p

uv,E) = c p

uv⋅ puv

ψ (puv

,E)

ψ (p

ur,E) = (ψ x ,ψ y ,ψ z )

OµOµψ T = i p

uv× (i p

uv× ψ T ) = (p

uv⋅ puv

)ψ T − puv

(puv

⋅ψ T ) = (puv

⋅ puv

)ψ T

ψ = ψ T + ψ L

puv

× ψ L = 0, puv

⋅ψ T = 0

Oµ B p

uv⋅ puv

= i puv

×?

momentum wave fnphoton, m=0, s=1, 3 components

i∂∂ t

ψuv

T (rv,t) = c ∇

uru×ψ

uvT (r

v,t)

Eψ T (p

uv,E) = c i p

uv×ψ T (p

uv,E)

ψuv

T (rv, t) ≡ dE d3p δ (E − c p

uv)exp(−iEt / h + i p

uv⋅ rv

/ h)∫∫ f (E)ψ T (puv

,E)

Photon Wave Equationf (E) = E

Require local energy density. -->

Ψuru

T* Ψ

uruT =

Arbitrary weight fn.f (E)

E = (m c2)2 + (c p)2

Photon Wv. Equation

ih∂∂ t

Ψuru

= hcσ ∇uru

×Ψuru

∇uru

⋅ Ψuru

= 0

3 components

Require Ψuru

* Ψuru

=

(Einstein)

∂∂ t

ψur

I = −c ∇uru

×ψur

R

∂∂ t

ψur

R = c ∇uru

×ψur

I

ψuv

(rv,t) = ψ

urR + iψ

urI

m=Os=1

Require local energy density

Ψ * Ψ =

Compare to Maxwell’s Equations in Free Space:

∂∂ t

Bur

= −c ∇uru

×Eur

∂∂ t

Eur

= c ∇uru

×Bur

Eur

= electric field

Bur

= magnetic field

For a single-photon field, the quantum wave function of the photon obeys the same wave equation as the complex electromagnetic field

σ = ±1

E + σiB

i∂

Ψuru

= σc ∇uru

×Ψuru

helicity (spin, polarization):

Maxwell, in 1862, discovered a fully relativistic, quantum mechanical theory of a single photon.

i∂ t

Ψ = σc ∇ ×Ψ

ih∂∂ t

Ψuru

= σhc ∇uru

×Ψuru

≡ Hµ Ψuru

MODES STATES

We can elevate the photon wave function to a quantum field, then the usual quantum field theory reappears. See the review:“Photon wave functions, wave-packet quantization of light, and coherence theory,”

Brian J. Smith and M. R., New J. Phys. 9, 414 (2007)

There are subtleties:• energy density, not particle number density. • Cannot localize a photon wave function to a point. • The scalar (inner) product has an unusual form.• There is NOT a Fourier-transform relation between momentum and position wave functions.

Ψuru

T* Ψ

uruT =

States (Modes) of Single Photons

Polarization Transverse Beam Shape

x

y

z

A photon has four degrees of freedom: momentum in x, y, and z; and spin (polarization).

HG10

HG01

LG01 HG’10

HG’01

LG10V

H

D

A

LC

RC

Laguerre-Gauss and Hermite-Gauss Spatial Modes

1. Spin Hall Effect for Electrons: opposite spin accumulation on opposing latteral surfaces of a current-carrying sample. Its origin is spin-orbit interaction.

Dyakonov and Perel (1971) Sov. Phys. JETP Lett. 13, 467

Hirsch (1999) PRL 83, 1834

2. Spin Hall Effect for Light: spin-dependent displacement perpendicular to the refractive index gradient for photons passing through an air-glass interface.

M. Onoda, S. Murakami, N. Nagaosa, PRL 93, 083901 (2004)

Observed: Hosten, Kwiat Science 319 (2008)

To what extent is there a photon-electron analogy?

Observed: Hosten, Kwiat Science 319 (2008)

Inhomogeneity in refractive index causes SOI.

Spin-Orbit Interaction (SOI) in Spherical Potentials

• ELECTRONIN AN INHOMOGENOUS SPHERICAL ELECTRIC POTENTIAL (ATOM)

H ' = −e2

2m2c2

1

r3 SgL

rr

× pur

= L = O AM

(atomic fine structure)

S = SAM

Coulomb potential

• PHOTONIN A DIELECTRIC SPHERE

(atomic fine structure)

Polarization-dependent mode-frequency shifts?

Spin-Orbit Interaction (SOI) in CylindricalPotentials

• ELECTRONIN AN CYLINDRCIAL WAVEGUIDE

Solve Dirac Equation for the traveling-wave states.

• PHOTONIN A CYLINDRICAL OPTICAL FIBER

Solve Maxwell’s Equations for the modes and send a single photon through.

Dirac Equation --> Schrodinger Equation with SOI

ih∂∂ t

Ψ = −h2

2m∇ur 2

Ψ +e

m2c2Sg∇

urV × p

ur( )Ψ −e

2m2c2Sg∇

urV × p

ur( )Ψ

Recall Coulomb potential:

force on magnetic moment relativisitic Thomas factor

Cylindr. potential: zr

ELECTRON in a CYLINDER STEP POTENTIAL

(C Leary, D Reeb, M Raymer, to appear NJP)

Recall Coulomb potential:

∇ur

V = −err

r3

Sg ∇ur

V × pur( )= − e 2

r 3 SgL

Cylindr. potential:

∇ur

V =ρur

ρ∂V

∂ρ

Sg ∇ur

V × pur( )=

1

ρ∂V

∂ρSg ρ

ur× p

ur( )=

1

ρ∂V

∂ρSg ρ

ur× p

urT( )+ p z term

=1

ρ∂V

∂ρS z L z + p z term

ρur

parallel or anti-parallel

rr

× pur

= L = O AMS = SAM

ELECTRON in a CYLINDER STEP POTENTIAL

ih∂∂ t

Ψ = −h2

2m∇ur 2

Ψ + H '; H ' =e

2m2c2

1

ρ∂V

∂ρSzLz

zr

For fixed propagation constant (z-momentum), perturbative Energy shift is:

where unperturbed states are

δE = Ψ H ' Ψ

Ψσ =+1=

1

0

J

mlκ r( )eiml φe

i β z −ω t( )

∂V

∂ρ= V0δ (ρ − a)

ρur

Then

0 l

Ψσ =−1=

0

1

J

mlκ r( )eiml φe

i β z −ω t( )

δE = Ψσ

†H 'Ψσ ∝ (ml σ ) Jmlκa( )( )2

∫

ml

σ ml

σ ml

σ ml

σ

parallel AM antiparallel AM

ELECTRON in a CYLINDER STEP POTENTIAL

ih∂∂ t

Ψ = −h2

2m∇ur 2

Ψ + H '; H ' =e

2m2c2

1

ρ∂V

∂ρSzLz

For fixed propagation constant (z-momentum), perturbative Energy shift is:

where unperturbed states are

δE = Ψ H ' Ψ

Ψσ =+1=

1

0

J

mlκ r( )eiml φe

i β z −ω t( )

Then

0 l

Ψσ =−1=

0

1

J

mlκ r( )eiml φe

i β z −ω t( )

δE = Ψσ

†H 'Ψσ ∝ (ml σ ) Jmlκa( )( )2

∫

E

β

• For a given energy, a parallel-AM electron state has a smaller z-propagation constant than that of an anti-parallel state. • For a given z-propagation constant, a parallel-AM electron state has a larger energy than that of an anti-parallel state. • Non-perturbative solution of Dirac equation gives same result.

par

anti-par

e i m l φ + β1 z + e − i m l φ + β 2 z a cos m l φ + σβ

1− β

2

2z

+ = σ = −1

ELECTRON STATE ROTATION in a CYLINDER WAVEGUIDE

Superposition of degenerate positive-helicity states with opposite OAM:

ml = ± 2

+ = σ ⋅ ml = −2

σ ⋅ ml = +2

+ = σ ⋅ ml = −2

σ ⋅ ml = +2

σ = −1

σ = +1

z

z

Superposition of two degenerate positive-helicity states with opposite OAM:

ml= +2,−2

ELECTRON STATE ROTATION in a CYLINDER WAVEGUIDE

σ = +1 σ = −1

• Kapany and Burke first predicted polarization-dependent spatial mode rotation of optical modes in fiber. (1972)

• Did not explain in terms of SOI.

z

SPATIAL MODES ROTATION FOR LIGHT?

coren

claddingn

z

• Zel’dovich, Liberman (1990; PRA 46, 5199, 1992) first predicted optical SOI:– Treated a many-mode fiber with a parabolic index profile.– Predicted spatial mode rotation, due to SOI. – Observed rotation of speckle pattern, but not of single modes.

step-index 200 um

Dooghin et al PRA 1992

• Complementary to Rytov-Berry rotation of polarization by topological phase.• See also works by A.V. Volyar.

Maxwell’s Equations in an Inhomogeneous Medium, interpreted as the Quantum Wave Equation for a single photon

D = εE, H = B / µ, ∇ ⋅ D = 0, ∇ ⋅ B = 0

∂D

∂t= ∇ × H ,

∂B

∂t= −∇ × E

ih∂∂ t

Ψuru

= hc ∇uru

×Ψuru

+ hc ∇uru

N ×Ψuru

∇uru

+ ∇uru

N( )⋅ Ψuru

= 0

--> Photon Wave Equation:

I. Bialynicki-Birula (Prog.Opt.1996)

(6 components)

• Maxwell Wave Equation:

Perturbation Theory for Optical SOI in Step-Index Fiber(C Leary)

coren

z

a

∇2E + ω 2ε(ρ)E + ∇ ∇ lnε(ρ)gE[ ]= 0

E = ET + EL( )ei (β z−ω t ), EL << ET

⇒ Hµ 0ET + Hµ 'ET = β 2ET

Hµ 0 = ∇T2 + ω 2ε(ρ)( )

ρφ

• Unperturbed eigenmodeshave well defined components of spin and orbital angular momentum along z axis.

coren

claddingn

ml

σ

E(ρ,φ,z) = eσ J

mlκ ρ( )eiml φe

i βz−ωt( )

H 0 = ∇T + ω ε(ρ)( )Hµ 'ET = ∇T ∇T lnε(ρ)gET[ ]

where circular polarization vector

eσ =1

0

or

0

1

Perturbed Modes in Circular-Pol Basis States:

Perturbation Theory for Optical SOI in Step-Index Fiber(C Leary)

coren

z

a

E+1,ml=

1

0

J

mlκ ρ( )eiml φe

i [β+δβ+1] z−ωt( )

E =0 J κ ρ( )eiml φe

i [β+δβ−1] z−ωt( )

Eσ ,ml

ρφ

coren

claddingn E−1,ml

=1

Jml

κ ρ( )e l e −1( )

δβσ ∝ Eσ ,mlHµ ' Eσ ,ml

∝ Eσ ,mlσµ3 Lµz

1

ρ∂ε∂ρ

Eσ ,ml

∝ (σ ml ) Jmlκ a( )( )2

∂ε∂ρ

∝ δ (ρ − a)

H 'electron ∝ S$z Lµz1

ρ∂V

∂ρsame as for

electron!

Nonperturbative Solutions for Optical SOI in Step-Index Fiber

propagation constant is different when SAM and OAM are parallel or antiparallel (for fixed )

β

ω β

paranti-parω

z

σ = +1

RCP

z

ml = +2 ml = +2

σ = −1

LCPparallel anti-parallel

e+ i2φe

i βz −ω t( )

Photon spin angular momentum (SAM) and orbital angular momentum (OAM) can carry quantum

information.

If photon SAM and OAM interact, then quantum gate interactions can perhaps be based on such interactions.

Single-Photon Spin-Controlled Hadamard Gate

e i m l φ + β +1z + e − i m l φ + β −1z ∝ cos m l φ + σβ +1

− β −1

2z

+ = = +

σ ⋅ = − σ ⋅ m = +1 0 1+ −45o

σ = −1

ml = 1

σ ⋅ ml = −1 σ ⋅ ml = +1 0 1+

Flipping the photon spin (circular polarization) flips the direction of rotation of the superposition spatial mode.

0

+ = = −

σ ⋅ ml = −1 σ ⋅ ml = +1 0 1− +45o

0

σ = +1

+ =σ ⋅ m = −2 σ ⋅ m = +2

= +

0 1 −22.5o

σ = −1

Single-Photon Spin-controlled Hadamard gate ml = 2

e i m l φ + β +1z + e − i m l φ + β −1z ∝ cos m l φ + σβ +1

− β −1

2z

+ = σ ⋅ ml = −2

σ ⋅ ml = +2

= −

0 1−22.5o

Flipping the photon spin (circular polarization) flips the direction of rotation of the superposition spatial mode.

0

σ ⋅ ml = −2 σ ⋅ ml = +2 0 1+

−22.50

σ = +1

SAM-OAM Entangling by Hadamard gate

+ σ = −1 σ = +1

ml = 2

+ σ = −1 σ = +1

+

0 1−

±22.5o

00 1+

0

+

Summary: Spin-Orbit Interaction in Cylindrical Waveguides

• Phase-velocity splitting proportional to . • Parallelor anti-parallelSAM and OAM give rise to differentpropagation constants, for fixed frequency.• Depends on total AM, .• SOI-split states (modes) have a longitudinally varying relative phase difference, which creates rotation of

σ ml

j lm m σ= +

relative phase difference, which creates rotation of superposition states (modes).• Can be used to implement a single-photon spin-controlled spatial rotation, for entangling spin and spatial modes.• Electron-photon analogy strengthens the photon-as-particle viewpoint.

Poincare Sphere for Polarization

H

RC

H V

+i =RC

H V

-i =LC

H

V

D A

LC

H V

+ =

H V

- =

D

A

Rotation=Hadamard

Poincare Sphere for L=1 Modes

HG10

LG01

HG01

HG’10 HG’01

LG10

+ =HG10 HG01 HG’10

- =HG10 HG01 HG’01

Rotation=Hadamard

Van Enk; Galvez..

What are the proper Scalar Product and Normalization? Bialynicki-Birula (1996)+refs.

• should be bilinear • should be Lorentz invariant

Ψuru

j Ψuru

m( )≡Ψuru

m*(rr')

rr

− rr'

2 d 3r '∫

Ψuru

j (rr)d3r = δ j ,m∫

Ψuru

Ψuru( )≡

Ψuru*

(rr')

r r d3r '∫

Ψuru

(rr)d3r = 1∫

No local particle density

Invariant,Non-local

Norm:

Ψ Ψ( )≡Ψ (r ')

rr

− rr'

2 d3r '∫

Ψ(r)d3r = 1∫

density

(deal with it)Norm:

The mean Energy of the photon is:

Ψuru

Hµ Ψuru( )=

Ψuru*

(rr')

rr

− rr'

2 d 3r '∫

hc∇ × Ψ

uru(rr)d 3r =∫ d3r Ψ

uru(rr)* Ψ

uru(rr)∫ = H

Is a local energy density.

Not invariant (OK)is the probability amplitude for

localizing Energy, not particle position. Ψuru

(rr)

Ψuru

(rr)

2

Quantum Field Theory: Dirac used Monochromatic Modes

Eur (+ )

(rr,t) = d 3k∫ a$(k

r,σ ) k

σ∑ ε

rσ exp(ik

r⋅ rr

− iω t)

a$(kr,σ ), a$

†(kr

',σ ')

= δ (kr,kr

')δσ ,σ '

Bosonic operators:

Quantum Field Theory using Temporal-Spatial (Wave-Packet) Modes

(pur

= hkr)

Non-Monochromatic modes (wave packets):

Unitary transformation:

Eur (+ )

(rr,t) = b$j

j∑ v

rj (r

r,t) , b$j , b$m

†

=δ j ,m

b$j = d 3k R j* (k

r,σ ) aµ(k

r,σ )∫

σ∑

vr

j (rr,t) = d3k R j (k

r,σ ) k∫

σ∑ ε

rσ exp(ik

r⋅ rr

− iω t)

U. M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041 (1966)

Quantum Field Theory using Temporal-Spatial (Wave-Packet) Modes

The T-G wave-packet modes are orthogonal under the same scalar product as are the photon wave functions

Ψuru

j Ψuru

m( )≡Ψuru

m*(rr')

rr

− rr'

2 d3r '∫

Ψuru

j (rr)d 3r = δ j ,m∫

L. Invariant,

Photon Wave Functions:

Non-Monochromatic wave packet modes:

vr

j (rr,t) = d3k Rj (k

r,λ) k∫

λ∑ u

rkr,λ (r

r,t)

L. Invariant, Non-local

vr

j vr

m( )≡vr

m*(rr')

rr

− rr'

2 d 3r '∫

vr

j (rr)d3r = δ j ,m∫

If we quantize the one-photon wave function, we obtain standard Dirac Quantum Field Theory

Bosonic operators

energy e-states:

i∂∂ t

ψuv

j (rv,t) = c ∇

uru×ψ

uvj (r

v,t)

Ψµ (rv,t) = b$j

j∑ ψ

urj (r

r,t) , b$j , b$m

†

=δ j,m

ψur

m

*(rr)ψur

j (rr)d3r = hω j δ j ,m∫

Ψµ (rv,t) = k j b$j

j∑

ψur

j (rr,t)

k j

ψur

j * ( rr,t)

k j

ψur

m (rr,t)

km

d3r = δ j ,m∫

Ψµ (rv,t) = k j b$j

j∑ ϕ

urj (r

r,t)

Dirac form