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Quantum Key Distribution with Full Laguerre-Gauss Encoding
Presented at the OSA Imaging and Applied Optics Congress, Orlando, FL, USA June 26, 2018.
Robert W. Boyd
Department of Physics andMax-Planck Centre for Extreme and Quantum Photonics
University of Ottawa
The Institute of Optics and Department of Physics and Astronomy
University of Rochester
Department of Physics and AstronomyUniversity of Glasgow
The visuals of this talk are available at boydnlo.ca/presentations/
1. QKD with Twisted Light2. Why We Need to Encode in Azimuth and Radius3. New Developments in Mode Sorters4. Advances in Free-Space QKD
Prospectus
1. QKD with Twisted Light
Our Research: BB84 in a High-Dimensional State Space
• Instead of using the two-dimensional state space of polarization, we usea (potentially) infinite dimensional state space of the orbital angular mo-mentum (OAM) modes of the photon.
• One motivation is to send more than one bit of information per photon.
• Another motivation is to increase the security of the protocol.
What Are the Orbital Angular Momentum (OAM) States of Light?
t Light can carry spin angular momentum (SAM) by means of its circularpolarization.
t Light can also carry orbital angular momentum (OAM) by means of thephase winding of the optical wavefront.
t A well-known example are the Laguerre-Gauss modes. These modescontain a phase factor of exp(ilφ ) and carry angular momentum of h̄k perphoton. (Here φ is the azimuthal coordinate.)
l =0 l = +1 l = +2
Phase-front structure of some OAM states
See, for instance, A.M. Yao and M.J. Padgett, Advances in Photonics 3, 161 (2011).
Laguerre-Gauss Modes
The paraxial approximation to the Helmholtz equation (∇2 + k2)E(k) = 0 givesthe paraxial wave equation which is written in the cartesian coordinate system as
!
∂2
∂x2+
∂2
∂y2+ 2ik
∂
∂z
"
E(x, y, z) = 0. (1)
The paraxial wave equation is satisfied by the Laguerre-Gaussian modes, a familyof orthogonal modes that have a well defined orbital angular momentum. The fieldamplitude LGl
p(ρ, φ, z) of a normalized Laguerre-Gaussian modes is given by
LGlp(ρ, φ, z) =
#
2p!
π(|l| + p)!
1
w(z)
$√2ρ
w(z)
%|l|
Llp
&
2ρ2
w2(z)
'
× exp
&
−ρ2
w2(z)
'
exp
&
−ik2ρ2z
2(z2 + z2
R)
'
exp
&
i(2p + |l| + 1)tan−1
!
z
zR
"'
e−ilφ, (2)
where k is the wave-vector magnitude of the field, zR the Rayleigh range, w(z) theradius of the beam at z, l is the azimuthal quantum number, and p is the radialquantum number. Ll
p is the associated Laguerre polynomial.
1
Laguerre- Gauss
Spiral phase plate ( )
LG
How to create a beam carrying orbital angular momentum?
Pass beam through a spiral phase plate
Use a spatial light modulator acting as a computer generated hologram(more versatile)
Exact solution to simultaneous intensity and phase masking with a single phase-only hologram, E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, Optics Letters 38, 3546 (2013).
Laguerre-Gaussian Basis
0 1 2 12 13 14 25 26 27
0 1 2 12 13 14 25 26 27
“Angular” Basis (mutually unbiased with respect to LG)
High Capacity QKD Protocol
. . . . . . . .
. . . . . . . .
We are developing a free-space quantum key distribution system that can carry many bits per photon (think about it!).
We encode either in the Laguerre-Gauss modes or in their linear superpositions (or in other transverse modes).
We are developing means to mitigate the influence of atmospheric turbulence
Spatially-Based QKD System
Source Weak Coherent Light Heralded Single Photon
Protocol Modified BB84 as discussed
Challenges 1. State Preparation 2. State Detection 3. Turbulence
Mode Sorting
A mode sorter
Sorting OAM using Phase Unwrapping Optically implement the transformation
*Berkhout et al. PRL 105, 153601 (2010). O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
Position of spot determines OAM
Experimental Results (CCD images in output plane)
- Can also sort angular position states.
- Limited by the overlap of neighboring states.
OAM
ANG
-3
-3
-2
-2
-1
-1
3
3
2
2
1
1
0
0
Our Laboratory Setup
We use a seven-dimensionalstate space.
Mirhosseini et al., New Journal of Physics 17, 033033 (2015).
Laboratory Results - OAM-Based QKD%R
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2. Why We Need to Encode in Azimuth and Radius
Why We Need to Encode in Azimuth and Radius
• Large telescopes are expensive; we want to make full use of our resources
This? Or this?
• In concept we want to encode in communication modes, but for large Fresnel numbers they approximate Laguerre-Gauss modes.
• Laguerre-Gauss modes are described by two mode indices, one (l) for azimuthal variation and one (p) for radial variation.
• We can roughly square the size of Hilbert space by encoding in both l and p.
• Miles Padgett showed earlier how to sort in l . Only recently have people shown how to sort in p.
3. New Developments in Mode Sorters
A genetic algorithm determines the pattern on the control SLM
SLM1
SLM2
LG (l=0) p=0 p=1 p=2 p=3Use Gerchberg-Saxton phase-retrieval algorithm
numerical experimentCrosstalk Matrices
Phase Screens
Laboratory Setup
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Results for sorting both azimuthal and radial dependencet��8F�VTF�B�QPMBSJ[BUJPO�MFOT�UP�DSFBUF�B�DPNNPO�QBUI FRFT (ractional Fourier transform) module
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4. Advances in Free-Space QKD
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0
1
0
1
H
V
H
V
H
V
H
V
DA DA DA DA
Conjugate basis0 , 2p � 0 ,2p
1 , 2p - 1 ,2p
, DLp , ALp , DRp , ARp
Sim.
Exp.
Sim. Sim. Sim.
Exp. Exp. Exp.
Sim. Sim. Sim. Sim.
Exp. Exp. Exp. Exp.
Main basis
• Encode in the transverse variation of amplitude, phase, and polarization• Transfer 2.15 bits per detected photon
• Perform QKD in an 8-dimensional state space.• Crosstalk matrix
• Secure image transmission
• New quantum-state sorters for full Laguerre-Gauss determination allow enhanced performance in free-space quantum communications.
• This technology is useful more generally in advanced imaging by allowing a complex optical field to be decomposed into a complete set of orthogonal modes.
Summary