quantum key distribution with full laguerre-gauss encoding · 2018-06-26 · ta well-known example...

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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 and Max-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 Astronomy University of Glasgow The visuals of this talk are available at boydnlo.ca/presentations/

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Page 1: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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/

Page 2: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 3: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

1. QKD with Twisted Light

Page 4: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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.

Page 5: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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).

Page 6: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 7: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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).

Page 8: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 9: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 10: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

Mode Sorting

A mode sorter

Page 11: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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.

Page 12: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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).

Page 13: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

Laboratory Results - OAM-Based QKD%R

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Page 14: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

2. Why We Need to Encode in Azimuth and Radius

Page 15: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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.

Page 16: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

3. New Developments in Mode Sorters

Page 17: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

A genetic algorithm determines the pattern on the control SLM

Robert Boyd
Robert Boyd
fidelity = 97% efficiency = 0.2%�
Page 18: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 19: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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Page 20: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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Page 21: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

4. Advances in Free-Space QKD

Page 22: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and
Page 23: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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Page 24: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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Page 25: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

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

Page 26: Quantum Key Distribution with Full Laguerre-Gauss Encoding · 2018-06-26 · tA well-known example are the Laguerre-Gauss modes. These modes contain a phase factor of exp(ilφ) and

• 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