+49(0)3641·947985Alexander Szameit alexander.szameit@uni-jena.de +49(0)3641·947991

Integrated optical circuitsfor classical and quantum light

Part 2: Integrated quantum optics

Alexander Szameit

Outline

• Introduction

• Implementation of integrated wave plates

• Realisation of high-order single-photon W-states

• Integrated discrete fractional Fourier transforms

• Summary

Rapid growth of global transmitted data

Computer power for data processing reaches limit

Different platforms for quantum computation

ions qubits in super conductorsspins in solids

Photons

Photons as qubits

NATURE | VOL 409 | 4 JANUARY 2001

=

Problem: settings are usually large and complex

Source: Vienna University

How would a photonic quantum computer look like?

Source: www.wikipedia.de

Size of our photonic quantum computer chips

Our (current) setting

The integrated beam splitter

coupling constant C

C

Jones, J. Opt. Soc. Am. 155, 261 (1965).

out in

out in

a aT iRb biR T

bulk optics waveguides

Generalized beam splitter:R. Heilmann et al., Appl. Phys. Lett. 105, 061111 (2014).

Integrated quantum-photonics devices

Outline

• Introduction

• Implementation of integrated wave plates

• Realisation of high-order single-photon W-states

• Integrated discrete fractional Fourier transforms

• Summary

Herstellung mittels ultrakurzer LaserpulseFabrication using ultrashort laser pulses

Polarization dependency

index raise and mode profiledepend on polarisation

polarization dependent coupling

additional degree of freedom

Polarization dependent coupling

Polarizing beam splitters

polarization dependent transmission in a directional coupler

PPBS

coupling constant C

C

General phase gate

ie

G0

01phase

general phase gate• PBSs and geometrical length shift

polarization dependent transmission in a directional coupler

PPBS

Arbitrary wave plate: fundamentals

polarized light Jones formalism

wave plate:• fast axis orientation α• phase shifts φo, φe

retardation Δφ = φe − φo

out

out

inx x

J iny y

E EM

E E

o e o e

o e o e

2 2

waveplate

2 2

cos sin sin cos

sin cos sin cos

i i i i

J i i i i

e e e eM

e e e e

o

2 2

2 2

cos sin 1 sin cos

1 sin cos sin cos

i i

i

i i

e ee

e e

Particular wave plates

polarized light Jones formalism

wave plate:• fast axis orientation α• phase shifts φo, φe

retardation Δφ = φe − φo• HWP: Δφ = π

2 2waveplate

2 2

cos sin 2sin cos2sin cos sin cosJM

cos2 sin 22sin cos2

out

out

inx x

J iny y

E EM

E E

HWP 22.5 1 111 12JM

• Hadamard: α = 22.5°

HWP 45 0 11 0JM

• Pauli-X: α = 45°

Birefringence in waveguides

Fernandes et al., Opt. Express 20, 24103 (2012).

Tunable birefringence in laser-written waveguides

fast axis

cross section index ellipsoidFernandes et al., Opt. Express 20, 24103 (2012).

Tunable birefringence in laser-written waveguides

fast axis

Heilmann et al., Scientific Reports 4, 4118 (2014)

cross section index ellipsoid

Classical light measurements

Heilmann et al., Scientific Reports 4, 4118 (2014)

Hadamard

Hadamard

Pauli-X

Pauli-X=45°

=22.5°

Quantum light measurements

,X 0.992(7)H V F ,

Had 0.999(5)H V F

Outline

• Introduction

• Implementation of integrated wave plates

• Realisation of high-order single-photon W-states

• Integrated discrete fractional Fourier transforms

• Summary

Multipartite entangled W-states

• robust against loss [Dür et al., Phys. Rev. A 62, 062314 (2000)]• secure communication [Yuan et al., Int. J. Quantum Inform. 9, 607 (2011)]

• teleportation [Shi & Tomita Phys. Lett. A 296, 161 (2002); Joo et al., New J. Phys. 5, 136 (2005)]

• quantum cloning machines [Bruß et al., Phys. Rev. A 57, 2368 (1998)]

• genuine random number generation

coherent superposition of eigenstates

Generation of even-order W-states

W-states via waveguide arrays

C

C

C

C

prob.

Perez-Leija et al., PRA 87, 013842 (2013).

Ĉ

C

C

Ĉ

Engineered coupling coefficients

Generation of odd-order W-states

Experimental results

WR in optics

WR in ultra cold atoms

Gräfe et al., Nature Photon. 8, 791 (2014).

Coherence test

Gräfe et al., Nature Photon. 8, 791 (2014).

Inspired by Lougovski et al., New J. Phys. 11, 063029 (2009)

Entanglement verification due to fidelity criterion:

– number of modes

Entanglement verification

Quantum Random Number Generation using W-states

in practice: output 1 to 8 numbers 0 to 7

photon time steps

range of QRNG

1 0 … 7

2 0 … 63

3 0 … 511

…

…

N 0 … 8N‐1 for M output channels:  0 … MN‐1

eigenstates have equal probability amplitude

no post‐processing required (e.g. Hash‐function)

statistical tests by NIST ✓

generation of QRNG on demand & on chip limitation in speed only by single-photon source & detector efficiency

Outline

• Introduction

• Implementation of integrated wave plates

• Realisation of high-order single-photon W-states

• Integrated discrete fractional Fourier transforms

• Summary

The Fourier transform: Useful everywhere

Optics Electrodynamics Quantum Mechanics

Image and Signal Processing Statististics & Finance Theory Economics

The fractional Fourier transform

0

2⁄

V. Namias, J.Inst.Maths. Applics 25,241 (1980).

1 tan e e e d

Applications of the discrete Fourier transform

Quantum wavefield reconstruction

Phase estimation Encryption theory

Joint frequency-time analysisBeam synthesis and shaping

FrFT

Differential equations

H. A. Ozaktas, The fractional Fourier transform and its applications, Wiley  (2003).

Fractional Fourier transform

Namias, J. Inst. Appl. Math. 25, 241 (1980)

fractional Fourier transform

harmonic oscillatorJx‐operator

Fourier operator:

The fractional Fourier transform

Jx operator fractional Fourier transform

Discrete fractional Fourier transform

discrete fractional Fourier transform

Jx‐operator

Atakishiyev & Wolf, J. Opt. Soc. am. A 14, 1467 (1997)

harmonic oscillator

Transferring the Hamiltonian to photonics

Perez‐Leija et al., Phys. Rev. A 87, 012309 (2013). Perez‐Leija et al., Phys. Rev. A 87, 022303 (2013).

Jx photonic lattices

1 n‐2 n‐1 nJn‐2

Nn+1 n+2Jn‐1 Jn Jn+1

parabolic coupling distribution

1 N

coup

ling J n

n

with

The discrete fractional Fourier transform on chip

Classical experiments

Weimann et al., Nature Commun. 7, 11027 (2015). Arrays with 21 elements and L= 7.5 cm

Classical experiments

Z

21 waveguides7.5 cm long

input:top‐hat function

output:sinc‐like function

Z

+ phase ramp!

Jx lattice & eigenstates

eigenfunctionof the Jx lattice

‐ element of the unitary propagation operator:

transition probability amplitude from site     to   

Weimann et al., Nature Commun. 7, 11027 (2015).

2-photon quantum interferometry

N00N

separable

Brobov et al., Phys. Rev. A 89, 043814 (2014).

Photon correlations in our FT device

1 2 3 N‐2 N‐1 N

Weimann et al., Nature Commun. 7, 11027 (2015).Fourier Suppression Law:Tichy et al., Phys. Rev Lett. 113, 020502 (2014).

Conclusions

• fs-laser written waveguides for integrated quantum circuits

• realization of integrated PPBS

• arbitrary wave plates in optical integrated devices using induced birefringence

• generation of W-states with 16 modes

• discrete fractional Fourier transform