Shashi Prabhakar, S. Gangi Reddy, A. Aadhi, Ashok Kumar, Chithrabhanu P.,

G. K. Samanta and R. P. Singh

Physical Research Laboratory,Ahmedabad. 380 009.

Feb 27, 2014IPQI 2014

Orbital angular momentum of light: Applications in quantum information

1 R. P. Singh

Whirlpools

Tornadoes

Outline of the talk

• How light acquires orbital angular momentum (OAM)• Experimental techniques to produce light with OAM

• Spontaneous Parametric Down-Conversion (SPDC)– Why– What– How

• Experiments and results• Hyper and hybrid entanglement• Applications – recent experiments• Future plan• Conclusion

3 R. P. Singh

Poynting showed classically for a beam of circularly polarized light

1

EnergyMomentumAngular

WJ z

Spin Angular Momentum

4 R. P. Singh

Angular momentum, Polarized: per photon

BethPhys. Rev. 50, 115, 1936

Can a light beam possess orbital angular momentum?

What would it mean?

L = r x p

Does each photon in the beam have the same orbital angular momentum?

Is the orbital angular momentum an integral number of ?

5 R. P. Singh

Orbital Angular Momentum

For a field amplitude distribution where

ilzruzru exp ,, 0

zz l

WJ

Energy

MomentumAngular

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. WoerdmanPhys. Rev. 45, 8185, 1992

6 R. P. Singh

Orbital Angular Momentum contd…

7 R. P. Singh

Difference in SAM and OAM

Intensity and phase plot of a beam carrying OAM

Helical Wavefront

Each photon carries anOrbital Angular Momentumof lħ, l order of vortex, can be any integer

2π 4π

0

2

Topological charge8 R. P. Singh

Optical Vortex

6π

Optical vortices are generated as natural structures when light passes through a rough surface or due to phase modification while propagating through a medium.

Controlled generation

1. Computer generated hologram (CGH)

2. Spiral phase plate

3. Astigmatic mode converter

4. Liquid crystal (Spatial light modulator)

9 R. P. Singh

Generation of Vortices in light

He-Ne Laser

10 R. P. Singh

Generation using CGH

He-Ne Laser B1

M1 M2

B2

ACGH L

Screen

CCD

11 R. P. Singh

Finding vortex order with Interferometry

The number of rings present in the Fourier transform of intensity

The number dark lobes present at the focus of a tilted lens

Opt. Lett. 36, 4398-4400 (2011)  Phys. Lett. A 377, 1154-1156 (2013) 

m=1 m=2

m=2 m=3

Finding order, other than Interferometry

12 R. P. Singh

Entanglement

While generation of entangled particles

• Total energy is conserved• Total (spin/orbital/linear) momentum is conserved• Annihilation happens• Generated simultaneously from the source• Preserve non-classical correlation with propagation

13 R. P. Singh

Entanglement contd…

Variables that can be chosen for entanglement• Polarization• Spin• Orbital angular momentum• Position and momentum

1. Among these, polarization is the one which can be easily handled and manipulated in the lab using λ/2, λ/4 plates and polarizing beam-splitters.

2. The most common method to generate entangled photons in lab is Spontaneous parametric down conversion (SPDC).

14 R. P. Singh

Spontaneous parametric down conversion

Energy Conservation

p: Pump beams: Signal beam (High ω)i: Idler beam (Low ω)

Phase-matching condition

ωi

ω pωs

Phy. Rev. A 31, 2409 (1985)

ii k,pp k,

ss k,

isp isp kkk

iksk

pk

15 R. P. Singh

Phase matching (Birefringence)

birefringence Δn = ne – no

16 R. P. Singh

Incident light

e-ray(polarized)

o-ray(polarized)

Optics axis

Type-I SPDC

λ2λ

BBO crystal

2λ

|H>

|V>

|H>

• e o + o type interaction• Produces single cone• The two output photons (signal and idler) generated will be non-

collinear

Collimated pump Strongly focused pump

Phy. Rev. A 83, 033837 (2011)

17 R. P. Singh

Type-II SPDC

λ2λ

BBO crystal

2λ

|V>

|V>

|H>

• e o + e type interaction• Produces double cone• The two output photons (signal and idler) generated can be both

non-collinear and collinear

Phy. Rev. A 68, 013804 (2003)

18 R. P. Singh

e-ray

o-ray

pump

e-ray

o-ray

Specification of components usedBBO Crystal• Size: 8×4×5 mm3

• θ = 26˚ (cut for 532 nm)• Cut for type-1 SPDC• Optical transparency: ~190–

3300 nm• ne = 1.5534, no = 1.6776

Diode Laser• Wavelength: 405 nm• Output Power: 50 mW

Interference filter• Wavelength range 810±5

nm19 R. P. Singh

20 R. P. Singh

First OAM entanglement experiment

Mair et al., Nature, 2001

10 0,1 2, 1 1,2 3, 21 0 0 1 2 1 1 2 3 2 ....C C C C C

12

Polarization entanglement :

Mair et al., Nature 2001

21 R. P. Singh

First OAM entanglement experiment contd…

Fig. 1 Left panel: Schematic sketch of the setup.

R Fickler et al. Science 2012;338:640-643

22 R. P. Singh

Quantum Entanglement of High Angular Momenta

Robert Fickler, Radek Lapkiewicz, William N. Plick, Mario Krenn, Christoph Schaeff, Sven Ramelow, Anton Zeilinger, Science 338, 640-643 (2012).

R Fickler et al. Science 2012;338:640-643

Published by AAAS

23 R. P. Singh

Quantum Entanglement of High Angular Momenta contd

Measured coincidence counts as a function of the angle of one mask and different angles of the other mask.

Related works at PRL

• Spatial distribution of down-converted photons by• Gaussian pump beam• Optical vortex pump beam• Bell inequality violation for light with OAM• OAM qubit generation

24 R. P. Singh

Generating correlated photon pairs

25 R. P. Singh

Generating correlated photons

Generating correlated photons

Blue Laser

405 nm & 50 mW

Lensf = 5 cm

BBOcrystal

IF

EMCCD

λ/2plate

Angle(λ/2) = 45˚ and 0˚ Background subtracted

IF: Interference filter 810±5 nmEMCCD: Electron Multiplying

CCD

26 R. P. Singh

Observing SPDC at varying pump intensity

3mW 5mW 8mW

Width of the SPDC ring is independent of the intensity of the light beam.

50 100 150

Width of the SPDC ring is independent of number of accumulations taken by EMCCD camera.

27 R. P. Singh

SPDC with Gaussian pump beam

1.0 mm

1.0 mm

28 R. P. Singh

SPDC with Gaussian pump beam (theory)

1.0 mm

1.0 mm

29 R. P. Singh

SPDC with gaussian pump beam

300 400 500 600 700

300

400

500

600 rin

g ( m

)

pump ( m)

Numerical Experimental

30 R. P. Singh

SPDC with optical vortex beam

BBOcrystal

IF λ/2plate

EMCCDCamera

Lens

Collimating Lens Combination

M

M

SLM

A

A

λ=405 nm, P=50 mW

Blue Laser

A

Lens

31 R. P. Singh

S. Prabhakar et al., Optics Communications

SPDC with optical vortex pump beam

1.0 mm

1.0 mm

Order of vortex m=1 m=3 m=5

32 R. P. Singh

SPDC with optical vortex pump beam

33 R. P. Singh

0 1 2 3 4 5

600

800

1000

1200

1400

1600

1800

2000

2200

F

WH

M (m

)

Order (m)

Numerical Experimental

Orbital angular momentum conservation: mp = ms + mi

Our approach:

34 R. P. Singh

Multi-photon, multi- dimensional entanglement can be achieved using OPO

R. P. Singh 35

Classical Entanglement

2,,,, baEbaEbaEbaEBThe Bell-CHSH inequality

For continuous variables, Wigner Distribution Function can be used instead of E(a, b)

2

,2;,2,2;,1 ,1;,2,1;,1

2221

1211

YXYX

YXYX

PYPXWPYPXWPYPXWPYPXW

B

Here, (X, PX) and (Y, PY) are conjugate pairs of dimensionless quadratures

P. Chowdhury et al. Phys. Rev. A 88, 013803 (2013).

Violation of Bell’s inequality for light beams with OAM

36

Classical Bell’s Violation for Optical Vortex beams

Wigner Distribution Function (WDF) can be defined as

exp,,,,,, 212121,,

dRdRpRpRiRRyxppyxW yxmnyxmn

2/,2/2/,2/,,,

as defined and (TPCF)function correationpoint -Two is where

21,*

21,21,

,

RyRxERyRxERRyx mnmnmn

mn

In other words, WDF is the Fourier Transform of TPCF. Experimentally, TPCF can be determined by using Shearing-Sagnac Interferometry.n (azimuthal) and m (radial) are the two indices in the electric field for LG beams with OAM.

R. P. Singh

Violation of Bell’s inequality contd…

R. P. Singh 37

Experimental setup for determining TPCFViolation of Bell’s inequality Experiment

R. P. Singh 38

Variation of non-locality with order of vortex (n)

Magnitude of violation of Bell inequality increases with the increase in the order of vortex

Violation of Bell inequality contd…

39

ResultsOrder (n) Theoretical (|Bmax|) Experimental (|Bmax|)0 2 2.01350 ± 0.012691 2.17 2.18460 ± 0.059332 2.24 2.26326 ± 0.08063

Violation of Bell’s inequality contd…

R. P. Singh

m=0, n=1, X1 = 0; PX1 = 0; X2 = X; PX2 = 0; Y1 =0; PY1 = 0; Y2 = 0; PY2 = PY ,

xPY

All the OAM Qubits on the Poincare sphere can be realized by projecting the non separable state of polarization and OAM into different polarization basis.

Non separable polarization – OAM state 22 VHThis state can be generated from Q-plate or modified Sagnac interferometer with vortex lens.

Polarization Poincare sphere OAM Poincare sphere

R. P. Singh

Generation of OAM qubits

40

OAM qubit

OV lens λ/2

PBS State Preparation

λ/2 (α)λ/4 (β)

PBSProjective measurements in polarization basis

2l

Horizontal polarization will acquire OAM of +2 Vertical polarization will get OAM of -2.

HWP (λ/2(α)) and QWP (λ/2(β)) with PBS will project the state in to different polarization basis.

Each combination of HWP and QWP will generate corresponding points on the Poincare sphere of OAM.

Generation of non separable state

H

V2l

22 VH

R. P. Singh 41

α=0 ̊ α= 22.5 ̊ α=45 ̊ α=67.5 ̊ α=90 ̊ α=112.5 ̊ α=135 ̊ α=157.5 ̊ α=45 ̊ β=0 ̊ β = 0 ̊ β =0 ̊ β =0 ̊ β =0 ̊ β =0 ̊ β =0 ̊ β=0 ̊ β =90 ̊

Experimental results

Conclusion and future outlook• Optical Vortices and orbital angular momentum of

light• Spontaneous Parametric Down-conversion can be

used to generate entangled photons in different degrees of freedom

• Spatial distribution of SPDC ring with higher order optical vortices

• Proposal to generate multi-photon, multi- dimensional entanglement

• Bell inequality violation for light beams with OAM• OAM qubit generation with non separable OAM-

polarization state • Using hybrid entanglement for quantum teleportation

and quantum key distribution

43 R. P. Singh

Thank you!

44 R. P. Singh

OAM entanglement

Future plan

l = -2 -1 +1 +2

The rotation in phase provides orbital angular momentum of lћ to the photons.

Rotation of phase front as the beam propagates

45 R. P. Singh

Generating correlated photon pairs

Blue Laser

405 nm & 50 mW

Lensf = 5 cm

BBOcrystal

IF

EMCCD

λ/2plate

IF: Interference filter 810±5 nmEMCCD: Electron Multiplying

CCD

46 R. P. Singh

SPDC with gaussian pump beam

λ=405 nm, P=50 mWBBO

crystalIFλ/2

plate

A

EMCCDCamera

Blue Laser

47 R. P. Singh

Generating optical vortices

Computer generated holography technique for the generation of optical vortices.

2 ,21 lxkModT xblazed

48 R. P. Singh