Integrated optical circuits for classical and quantum ... · 10/16/2015 · • Integrated discrete...
Transcript of Integrated optical circuits for classical and quantum ... · 10/16/2015 · • Integrated discrete...
+49(0)3641·947985Alexander Szameit [email protected] +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
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).
Outline
• Introduction
• Implementation of integrated wave plates
• Realisation of high-order single-photon W-states
• Integrated discrete fractional Fourier transforms
• Summary
Polarization dependency
index raise and mode profiledepend on polarisation
polarization dependent coupling
additional degree of freedom
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°
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°
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
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
Experimental results
WR in optics
WR in ultra cold atoms
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
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:
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
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).
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).