Entangled structures in classical and quantum optics

Antonio Zelaquett Khoury

Preface

• Quantum information: Encoding and processing information in

physical systems governed by Quantum Mechanics.

• Basic quantum information unit: qubits.

• Operations: Unitary (quantum gates) + measurements → algorithms.

• Informational advantages: Superpositions and entanglement.

• Examples: Quantum cryptography (BB84) and teleportation

• Platforms: photons, trapped ions, superconducting circuits, etc…

• Drawback: Decoherence

• Our “daily bread”: Optical vortices and photonic DoFs.

0 1 +

Outline

Lecture 1:

Optical vortices as entangled structures in classical optics

Lecture 2:

Quantum-like simulations and the role of quantum inequalities

Lecture 3:

Quantization of the electromagnetic field

Lecture 4:

Vector beam quantization and the unified framework

Outline

Lecture 1:

Optical vortices as entangled structures in classical optics

Paraxial Equation

(Paraxial Equation)

022 =

+⊥

zik

0

12

2

2

2 =

−

t

E

cE

tienruE −= ˆ)(

022 =+ ukuikzezyxru ),,()( =

zk

z

2

2

Fundamental Gaussian Mode

Beam width Wavefront radius Rayleigh range

( )2 2 2 2

arctan( / )

0 2

2 1exp exp

( ) ( ) 2 ( )Ri z zx y x y

ik ew z w z R z

− + += −

r

General Paraxial Modes

(Paraxial Equation)

Hermite-Gauss (HG)

Rectangular

Laguerre-Gauss (LG)

Cylindrical

022 =

+⊥

zik

HGnm(x,y) LGpl(r,f)

Hermite-Gaussian Modes

Gouy phase N=m+n

N=0

N=1

N=2

( )( ) ( 1)arctan /N Rz N z z = +

( ) ( )* 2

mn m n mm nnHG HG d = r r r

( ) ( ) ( )*

,

mn mn

m n

HG HG = − r r r r

Complete

Orthonormal

( )2 2 2 2

( )

2

2 2exp exp

( ) ( ) ( ) ( ) 2 ( )Ni zmn

mn m n

A x y x y x yHG H H ik e

w z w z w z w z R z

− + +

= −

r

Laguerre-Gaussian Modes

Gouy phase N=2p+|l|

N=0

N=1

N=2

( )( ) ( 1)arctan /N Rz N z z = +

( ) ( )* 2

pl p l pp llLG LG d = r r r

( ) ( ) ( )*

,

pl pl

p l

LG LG = − r r r r

Complete

Orthonormal

( )

| |2 2 2

( )| |

2 2

2 2exp exp

( ) ( ) ( ) ( ) 2 ( )N

l

pl i zl il

pl p

ALG L ik e e

w z w z w z w z R z

fr r r r −

= −

r

Orbital angular momentum

l=1

Twisted wavefrontkz+lf=0kz=0

Linear momentum

Angular momentum

3

0 d= P E(r)×B(r) r

( ) 3

0 0 0( d= −J r ) r r × E(r)×B(r) r

S O= +J J JParaxial

propagationSPIN + ORBITAL

pol wavefront

Intensity and phase of LG modes

Intensidade

Intensity

Phase (theo)

Phase (exp)

( )2 2/

0, 1 , i r wLG r r e eff −

( )2 22 2 2 /

0, 1 , r wLG r r ef −

BS

Holographic production of LG and HG beams

N.R. Heckenberg et al, Opt. Lett. 17, 221 (1992)

Confocal lens

l = -1

l = 1

l = 0

l = 0

l = 0

l = 0

Algebraic structure of paraxial wave functions

Hermite-Gauss (HG) Laguerre-Gauss (LG)

LG-HG Unitary transformation

E. Abramochkin and V. Volostnikov,

Opt. Commun. 83, 123 (1991)

← SU(3) →

← SU(2) →

← SU(1) →

( )

( )

+

−

r

r

( )

( )

H

V

r

r

Astigmatic mode transformations

( )

( )

2 2 2

2 2 2

( )/

( )/

x y w

H

x y w

V

x e

y e

− +

− +

r

r

2

H Vi

=HG-LG

HG-HG 452

H V

=

Mode Converter

cylindrical lenses at 45o

(2

1)+ i+(

2

1)

MC eingenvectors

452

H V

=

Poincaré representation

,ˆ ˆ ˆcos sin

2 2

i

H Ve

= +e e e

Poincaré sphere for

polarization modes

Poincaré sphere for

first order modes

, cos sin2 2

i

H Ve

= +

Quantum computation unit: QUBIT

, cos 0 sin 12 2

ie f = +

Bloch sphere

Spin-Orbit Entanglement

Spin-Orbit Modes

Tensor product in QM AB A B =

AB A B Entanglement

2

1001

2

1100

=

=

Bell states

Tensor product in CO ( ) ˆsep = Ψ r ε

( ) ˆent Ψ r ε

( ) ( )

( ) ( )

1ˆ ˆ

2

1ˆ ˆ

2

H H V V

H V V H

=

=

Ψ r e r e

Φ r e r e

)( onpolarizatispatial

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆH H H V V H V V = + + +Ψ r e r e r e r e

10

2

−=

C

C

11100100 +++=

Bell modes

concurrence

−= 2C

Polarization vortices

+ −− +

( ) ( )1

ˆ ˆ2

H H V V = Ψ r e r e ( ) ( )1

ˆ ˆ2

H V V H = Φ r e r e

+

S-Plate

Spin-orbit coupling in liquid crystals

L. Marrucci, C. Manzo, and D. Paparo,

Phys. Rev. Lett. 96, 163905 (2006)

State tomography: Polarimetry

Spin and orbital Stokes parameters

Spin Orbital

H

V

AD

L

R

H

V

AD

L

R

( )

( )

( )

1

2

3

/

/

/

H V TOT

D A TOT

R L TOT

S I I I

S I I I

S I I I

= −

= −

= −

( )

( )

( )

1

2

3

/

/

/

H V TOT

D A TOT

R L TOT

O I I I

O I I I

O I I I

= −

= −

= −

Polarization Stokes Parameters

𝑆1 =𝐼𝐻 − 𝐼𝑉𝐼𝑇𝑂𝑇

𝑆2 =𝐼𝐷 − 𝐼𝐴𝐼𝑇𝑂𝑇

𝑆3 =𝐼𝐿 − 𝐼𝑅𝐼𝑇𝑂𝑇

𝐻/𝑉

𝐷/𝐴

𝐿/𝑅

+/−

WP

PBS

𝑆12 + 𝑆2

2+𝑆32 ≤ 1

H

V

AD

L

R

2*ˆ

j jI = e E

Polarization projection

Parity mode selector

Spatial-pol parity

HH VV

HV VH

WP

Mirror reflection

ˆ ˆ

ˆ ˆ

H H

V V

→−

→ +

e e

e e

H H

V V

→ −

→ +

Spatial parity

Orbital Stokes Parameters

𝑂1 =𝐼𝐻 − 𝐼𝑉𝐼𝑇𝑂𝑇

𝑂2 =𝐼𝐷 − 𝐼𝐴𝐼𝑇𝑂𝑇

𝑂3 =𝐼𝐿 − 𝐼𝑅𝐼𝑇𝑂𝑇

𝐻/𝑉

𝐷/𝐴

𝐿/𝑅

𝑂12 + 𝑂2

2+𝑂32 ≤ 1

WP

+/−

MC

H

V

AD

L

R

2* 2( ) ( )j jI E d= r r r Spatial mode

projection

State tomography in QM

( )

( )*

1 / 2 / 2

/ 2 1 / 2

, , 1 ,

aa ab

ba bb

aa bb aa bb ba ab

z x iy

x iy z

r rr

r r

r r r r r r

+ − = =

+ −

+ = =

Mutually unbiased bases

General state

, , , , ,z z x x y y

+ − + − + −

2 2

1( )

2

z z z z

x y

j jj j k

i

j k

+ − + − = =

= =

( )

( ) ( )( )

ab ba x

ab ba y

aa bb z

x Tr

y i Tr

z Tr

r r r

r r r

r r r

= + =

= − =

= − =

Tomographic measurements

sin cos

sin sin

cos

x

y

z

f

f

=

=

=

Stokes

vector beam polarimetry

Polarization Stokes parameters for vector beams

𝑆12 + 𝑆2

2+𝑆32 = 1

𝑆12 + 𝑆2

2+𝑆32 < 1

Fully polarized

𝑆12 + 𝑆2

2+𝑆32 = 0

Partially polarized

Fully unpolarized

𝑆12 + 𝑆2

2+𝑆32 =?Vector beams

𝑆12 + 𝑆2

2+𝑆32 = 0 !

22* 2 * * 2ˆ ˆ( ) ( ) ( )j j j k

k

I d d= = e r r e r r r

Orbital Stokes parameters for vector beams

2* * 2ˆ ( ) ( )j k j

k

I d= e r r r

WP

+/−

MC

𝑂12 + 𝑂2

2+𝑂32 = 0 !

Quantum mechanical counterpart

( )

( )

, ; ,, ; ,

, ; ,,

AB AB i j k l A B A Bi j k l

A B AB AB AB i j k kB B A Ak i j k

i k j l

Tr k k i j

r r

r r r r

=

= = =

1A

0A

1B

0B

Partial D.M.

Total D.M.

00 11

22A BAB

r r+

= = =1 Locally

Incoherent

Globally

Coherent

Spatial partial trace

𝑆12 + 𝑆2

2+𝑆32 = 0 !

22* 2 * * 2ˆ ˆ( ) ( ) ( )j j j k

k

I d d= = e r r e r r r

Spatial partial trace

Orbital Stokes parameters for vector beams

2* * 2ˆ ( ) ( )j k j

k

I d= e r r r

WP

+/−

MC

𝑂12 + 𝑂2

2+𝑂32 = 0 !

Polarization partial trace

Local measurement

B B

1A

0A

1B

0B

Remote projection

Local

measurement

00 11

2

0 0 1 1

AB A B

A B B

+ =

= +

General case ( )AB B AB B BATrr r →

Spatial and polarization filtering

𝑆12(𝒓) + 𝑆2

2(𝒓)+𝑆32 𝒓 > 0

1,2,3( ) ( )jI Sr r

WP

+/−

MC

𝑂12(𝒓) + 𝑂2

2(𝒓)+𝑂32 𝒓 > 0

Rotation invariance

2-qubit invariance

00 11 , ,

2 2

+

+ + = =

Vector beam invariance

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ

2 2

H H V V +

+ += =

r e r e r e r eΨ

ˆ ˆ ˆcos sin

ˆ ˆ ˆsin cos

H V

H V

= +

= − +

e e e

e e e

cos 0 sin 1

sin 0 cos 1

= +

= − +

( ) ( ) ( )

( ) ( ) ( )

cos sin

sin cos

H V

H V

= +

= − +

r r r

r r r

Polarization filtering of a vector beam

polarizer

Creating OAM through polarization operations

2-qubit invariance

, ,00 11

2 2

i + −

+

++ = =

Vector beam invariance

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ

2 2

H H V Vi + −

+

+ + = =

r e r e r e r eΨ

ˆ ˆ ˆcos sin

ˆ ˆ ˆsin cos

H V

H V

= +

= − +

e e e

e e e

cos 0 sin 1

sin 0 cos 1

i

i

+

−

= +

= − +

( ) ( ) ( )

( ) ( ) ( )

cos sin

sin cos

H V

H V

i

i

+

−

= +

= − +

r r r

r r r

45 OAM =

Polarization filtering of a vector beam

l/4 polarizer

+Ψ

+Ψ

Conclusions

• Fundamentals of optical OAM and transverse modes.

• Spin-orbit structural non separability in classical laser beams .

• Analogies with principles of quantum measurement theory.