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
R. P. Singh
1
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
R. P. Singh
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Spin Angular Momentum
Poynting showed classically for a beam of
circularly polarized light
J z Angular Momentum
1


W
Energy

Angular momentum
  ,   Polarized:   per photon
Beth
Phys. Rev. 50, 115, 1936
R. P. Singh


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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
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Orbital Angular Momentum contd…
For a field amplitude distribution where
ur , z   u0 r , z  exp  il 
J z Angular Momentum
l  z


W
Energy

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. Woerdman
Phys. Rev. 45, 8185, 1992
R. P. Singh
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Difference in SAM and OAM
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OpticalIntensity
Vortex and phase plot of a beam carrying OAM
2
0
2π
4π
6π
Helical Wavefront
Each photon carries an
Orbital Angular Momentum
of lħ, l order of vortex, can
be any integer
Topological charge
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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
1. Computer generated hologram (CGH)
2. Spiral phase plate
3. Astigmatic mode converter
4. Liquid crystal (Spatial light modulator)
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Generation using CGH
He-Ne Laser
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Finding vortex order with Interferometry
M1
M2
CGH
He-Ne Laser
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B1
A
CCD
L
B2
Screen
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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
R. P. Singh
m=3
Opt. Lett. 36, 4398-4400 (2011)
Phys. Lett. A 377, 1154-1156 (2013)
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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
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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 beamsplitters.
2. The most common method to generate entangled photons in lab is
Spontaneous parametric down conversion (SPDC).
R. P. Singh
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Spontaneous parametric down conversion
s , k s
p,k p
i , k i
Energy Conservation
 p  s  i
ωp
ωs
ωi
p: Pump beam
s: Signal beam (High ω)
i: Idler beam (Low ω)
Phase-matching condition
k p  k s  k i
ki
ks
kp
Phy. Rev. A 31, 2409 (1985)
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Phase matching (Birefringence)
Optics axis
e-ray
(polarized)
Incident light
o-ray
(polarized)
birefringence Δn = ne – no
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Type-I SPDC
• e  o + o type interaction
• Produces single cone
• The two output photons (signal and idler) generated will be non-collinear
|H>
2λ
λ
|V>
2λ
|H>
BBO crystal
Collimated pump
Strongly focused pump
Phy. Rev. A 83, 033837 (2011)
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Type-II SPDC
• e  o + e type interaction
• Produces double cone
• The two output photons (signal and idler) generated can be both noncollinear and collinear
e-ray
e-ray
|V>
2λ
λ
pump
|V>
2λ
|H>
BBO crystal
o-ray
o-ray
Phy. Rev. A 68, 013804 (2003)
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Specification of components used
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
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First OAM entanglement experiment
Mair et al., Nature, 2001
  C10 1 0  C0,1 0 1  C2,1 2 1  C1,2 1 2  C3,2 3 2  ....
Polarization entanglement :
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

1
   
2

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First OAM entanglement experiment contd…
Mair et al., Nature 2001
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Quantum Entanglement of High Angular Momenta
Fig. 1 Left panel: Schematic sketch of the setup.
R Fickler
et al. Science
2012;338:640-643
Robert
Fickler,
Radek
Lapkiewicz, William N. Plick, Mario Krenn, Christoph
Schaeff, Sven Ramelow, Anton Zeilinger, Science 338, 640-643 (2012).
R. P. Singh
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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
R Fickler et al. Science 2012;338:640-643
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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
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Generating correlated photon pairs
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Generating correlated photons
λ/2
plate
Lens
f = 5 cm
EMCCD
Blue Laser
405 nm & 50 mW
BBO
crystal
Angle(λ/2) = 45˚ and
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IF
0˚
IF: Interference filter 810±5 nm
EMCCD: Electron Multiplying CCD
Background subtracted
Generating correlated photons
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Observing SPDC at varying pump intensity
Width of the SPDC ring is
independent of the intensity
of the light beam.
3mW
5mW
8mW
Width of the SPDC ring is
independent of number of
accumulations taken by
EMCCD camera.
50
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100
150
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SPDC with Gaussian pump beam
1.0 mm
1.0 mm
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SPDC with Gaussian pump beam (theory)
1.0 mm
1.0 mm
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SPDC with gaussian pump beam
600
ring ( m)
500
400
300
300
Numerical
Experimental
400
500
600
700
pump ( m)
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SPDC with optical vortex beam
SLM
A
A
Blue Laser
λ=405 nm, P=50 mW
M
A
Collimating Lens Combination
M
EMCCD IF Lens
Camera
BBO
crystal
λ/2 Lens
plate
S. Prabhakar et al., Optics Communications
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SPDC with optical vortex pump beam
1.0 mm
1.0 mm
Order of vortex
R. P. Singh
m=1
m=3
m=5
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SPDC with optical vortex pump beam
2200
2000
FWHM (m)
1800
1600
1400
1200
1000
Numerical
Experimental
800
600
0
1
2
3
4
5
Order (m)
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Multi-photon, multi- dimensional entanglement can be
achieved using OPO
Our approach:
Orbital angular momentum conservation: mp = ms + mi
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Classical
Violation Entanglement
of Bell’s inequality for light beams with
OAM
The Bell-CHSH inequality
B  Ea, b  Ea, b  Ea, b  Ea, b  2
For continuous variables, Wigner Distribution Function can be used
instead of E(a, b)
B
W  X 1, PX 1; Y1, PY 1   W  X 2, PX 2 ; Y1, PY 1 
 W  X 1, PX 1; Y 2, PY 2   W  X 2, PX 2 ; Y 2, PY 2 
2
Here, (X, PX) and (Y, PY) are conjugate pairs of dimensionless
quadratures
P. Chowdhury et al. Phys. Rev. A 88,
013803 (2013).
R. P. Singh
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Classical Bell’s
Violation
for Optical
Vortex beams
Violation
of Bell’s
inequality
contd…
Wigner Distribution Function (WDF) can be defined as
Wn ,m x, y, p x , p y  
 
   x, y, R , R exp iR p
n,m
1
2
1
x

 R2 p y  dR1 dR2
  
where  n,m is Two - point correation function (TPCF) and defined as
 n,m x, y, R1 , R2   En,m x  R1 / 2, y  R2 / 2E *n,m x  R1 / 2, y  R2 / 2
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.
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Experimental
setup inequality
for determining
TPCF
Violation of Bell’s
Experiment
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Variation
of Bell
non-locality
with
order of vortex (n)
Violation of
inequality
contd…
Magnitude of violation of Bell inequality increases with the increase
in the order of vortex
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Results of Bell’s inequality contd…
Violation
Order (n)
Theoretical (|Bmax|)
Experimental (|Bmax|)
0
2
2.01350 ± 0.01269
1
2.17
2.18460 ± 0.05933
2
2.24
2.26326 ± 0.08063
m=0, n=1, X1 = 0; PX1 = 0; X2 = X; PX2 = 0;
Y1 =0; PY1 = 0; Y2 = 0; PY2 = PY ,
PY
x
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Generation of OAM qubits
Polarization Poincare sphere
OAM Poincare sphere
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
  H 2  V 2
This state can be generated from Q-plate or modified Sagnac interferometer
with vortex lens.
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Generation of non separable state
OAM qubit
Projective measurements in
polarization basis
PBS
λ/2 (α)
λ/4 (β)
Horizontal polarization will acquire
OAM of +2 Vertical polarization will
get OAM of -2.
  H 2  V 2
PBS
State Preparation
H
λ/2
V
l  2
OV lens
l  2
R. P. Singh
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.
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Experimental results
α=0 ̊
β=0 ̊
α= 22.5 ̊ α=45 ̊ α=67.5 ̊ α=90 ̊ α=112.5 ̊ α=135 ̊ α=157.5 ̊ α=45 ̊
β=0̊
β =0 ̊ β =0 ̊
β =0 ̊ β =0 ̊
β =0 ̊
β=0 ̊
β =90 ̊
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
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Thank you!
R. P. Singh
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OAM entanglement
The rotation in phase provides orbital angular momentum of lћ to the
photons.
l=
-2
-1
+1
+2
Rotation of phase front as the beam propagates
R. P. Singh
Future plan
45
Generating correlated photon pairs
λ/2
plate
Lens
f = 5 cm
EMCCD
Blue Laser
405 nm & 50 mW
BBO
crystal
R. P. Singh
IF
IF: Interference filter 810±5 nm
EMCCD: Electron Multiplying CCD
46
SPDC with gaussian pump beam
Blue Laser
λ=405 nm, P=50 mW
A
λ/2
BBO IF EMCCD
plate crystal
Camera
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Generating optical vortices
Tblazed 
1
Mod k x x  l , 2 
2
Computer generated holography technique for the generation of optical vortices.
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