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Orbital angular momentum of light: Applications in quantum information. 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, 2014 IPQI 2014. Whirlpools. Tornadoes. - PowerPoint PPT Presentation
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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