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

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R. P. Singh

Orbital angular momentum of light:

Applications in quantum information

- ShashiPrabhakar, S. Gangi Reddy, A. Aadhi, Ashok Kumar, Chithrabhanu P.,
- G. K. Samanta and R. P. Singh
Physical Research Laboratory,

Ahmedabad. 380 009.

Feb 27, 2014

IPQI 2014

Whirlpools

Tornadoes

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

R. P. Singh

3

Spin Angular Momentum

Poyntingshowed classically for a beam of

circularly polarized light

Angular momentum

per photon

Polarized:

Beth

Phys. Rev. 50, 115, 1936

R. P. Singh

4

Orbital Angular Momentum

Can a light beam possess orbital angular momentum?

What would it mean?

L = rxp

Does each photon in the beam have

the same orbital angular momentum?

Is the orbital angular momentum an integral number of

?

R. P. Singh

5

Orbital Angular Momentum contd…

For a field amplitude distribution where

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. Woerdman

Phys. Rev. 45, 8185, 1992

R. P. Singh

6

Difference in SAM and OAM

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7

Optical Vortex

Intensity and phase plot of a beam carrying OAM

Helical Wavefront

2π

4π

6π

Each photon carries an

Orbital Angular Momentum

oflħ, l order of vortex, can

be any integer

Topological charge

R. P. Singh

8

Generation of Vortices in light

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

- Computer generated hologram (CGH)
- Spiral phase plate
- Astigmatic mode converter
- Liquid crystal (Spatial light modulator)

R. P. Singh

9

Generation using CGH

He-Ne Laser

R. P. Singh

10

M1

M2

CCD

CGH

A

L

B1

B2

He-Ne Laser

Screen

Finding vortex order with Interferometry

R. P. Singh

11

Finding order, other than Interferometry

- The number of rings present in the Fourier transform of intensity

m=1 m=2

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

m=2 m=3

Opt. Lett. 36, 4398-4400 (2011)

Phys. Lett. A 377, 1154-1156 (2013)

R. P. Singh

12

R. P. Singh

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

R. P. Singh

Variables that can be chosen for entanglement

- Polarization
- Spin
- Orbital angular momentum
- Position and momentum

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

R. P. Singh

p:Pump beam

s:Signal beam (High ω)

i:Idler beam (Low ω)

Phase-matching condition

Energy Conservation

Phy. Rev. A 31, 2409 (1985)

R. P. Singh

Optics axis

e-ray

(polarized)

Incident light

o-ray

(polarized)

birefringence Δn = ne – no

R. P. Singh

|H>

2λ

λ

|V>

|H>

2λ

BBO crystal

Collimated pump Strongly focused pump

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

Phy. Rev. A 83, 033837 (2011)

R. P. Singh

|V>

2λ

λ

|V>

|H>

2λ

BBO crystal

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

e-ray

e-ray

pump

o-ray

o-ray

Phy. Rev. A 68, 013804 (2003)

R. P. Singh

BBO 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 nm

First OAM entanglement experiment

Mair et al., Nature, 2001

Polarization entanglement :

R. P. Singh

20

First OAM entanglement experiment contd…

Mair et al., Nature 2001

R. P. Singh

21

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. P. Singh

22

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.

R. P. Singh

23

R. P. Singh

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

R. P. Singh

R. P. Singh

Lens

f = 5 cm

λ/2

plate

Blue Laser

EMCCD

405 nm & 50 mW

BBO

crystal

IF

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

IF: Interference filter 810±5 nm

EMCCD: Electron Multiplying CCD

Generating correlated photons

R. P. Singh

Width of the SPDC ring is

independent of the intensity

of the light beam.

3mW 5mW8mW

Width of the SPDC ring is

independent of number of accumulations taken by EMCCD camera.

50 100 150

R. P. Singh

1.0 mm

1.0 mm

R. P. Singh

1.0 mm

1.0 mm

R. P. Singh

R. P. Singh

S. Prabhakar et al., Optics Communications

R. P. Singh

1.0 mm

1.0 mm

Order of vortexm=1 m=3 m=5

R. P. Singh

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

Our approach:

Orbital angular momentum conservation: mp = ms + mi

R. P. Singh

34

R. P. Singh

Violation of Bell’s inequality for light beams with OAM

The Bell-CHSH inequality

For continuous variables, Wigner Distribution Function can be used instead of E(a, 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 contd…

Wigner Distribution Function (WDF) can be defined as

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

R. P. Singh

Violation of Bell’s inequality Experiment

R. P. Singh

Violation of Bell inequality contd…

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

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 ,

PY

x

R. P. Singh

Generation of OAM qubits

Polarization Poincare sphere OAM Poincare sphere

All the OAM Qubitson 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

This state can be generated from Q-plate or modified Sagnacinterferometer with vortex lens.

R. P. Singh

Projective measurements in polarization basis

PBS

λ/2 (α)

λ/4 (β)

PBS

State Preparation

λ/2

OV lens

Generation of non separable state

OAM qubit

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.

H

V

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

R. P. Singh

- 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

R. P. Singh

R. P. Singh

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

l = -2-1+1+2

Rotation of phase front as the beam propagates

Future plan

R. P. Singh

Lens

f = 5 cm

λ/2

plate

Blue Laser

EMCCD

405 nm & 50 mW

BBO

crystal

IF

IF: Interference filter 810±5 nm

EMCCD: Electron Multiplying CCD

R. P. Singh

R. P. Singh

Computer generated holography technique for the generation of optical vortices.