Towards two-dimensional quantum walks in multicore fibre

Towards two-dimensional quantum walks in multicore fibre Peter J Mosley, Itandehui Gris-Sanchez, Robert J Francis-Jones, James Stone, Jonathan Knight, and Tim Birks Centre for Photonics and Photonic Materials, University of Bath UK pjm36@bath.ac.uk 12th September 2013 QuAMP Swansea

Overview Random walks - classical vs quantum Quantum walks with photons Quantum walks in coupled waveguides Multi-dimensional quantum walks Quantum walks in multicore fibre

Random walks Galton board (quincunx) Individual balls follow classical random walk Repeated iterations form normal distribution Sir Francis Galton (1822-1911)

Random walks Galton board (quincunx) Individual balls follow classical random walk Repeated iterations form normal distribution Feed in ball Repeat (N steps) Coin flip Move left or right Repeat (large n) Sir Francis Galton (1822-1911) Read position

Quantum walks Replace ball with quantum object (photon) Control over initial quantum state Path of quantum walker can interfere at nodes Can have coin to control quantum state after each step Feed in quantum state Repeat (N steps) Coin flip Move left and right Repeat (large n) Read position

Features of quantum walks Probability distributions after 10-step discrete time walks... (cl) / p N (qu) / N 10 5 0 5 10 Position Definite paths, no interference 10 5 0 5 10 Position Superposition, path interference Reasons to care about quantum walks: Quantum computation can be recast as quantum walk algorithms Simulation of quantum systems BosonSampling as test of speedup in limited form of quantum computation

Quantum walks with photons Do, JOSA B 22, 499 (2005)

Quantum walks with photons Do, JOSA B 22, 499 (2005) Broome, PRL 104, 152602 (2010)

Quantum walks with photons Do, JOSA B 22, 499 (2005) Broome, PRL 104, 152602 (2010) Schreiber, PRL 104, 050502 (2010)

Quantum walks with photons Do, JOSA B 22, 499 (2005) Broome, PRL 104, 152602 (2010) Schreiber, PRL 104, 050502 (2010) Peruzzo, Science 329, 1500 (2010)

Quantum walks with photons in waveguides Waveguides close enough to couple via mode overlap Photons undergo continuous time quantum walk 0 1 0 A 1 da dz = ima A 2 A = B M = B @ A 3 C A @. 1 g 12 g 13 g 21 2 g 23 g 31 g 32 3.... 1 C A Mode index 10 5 0 5 1 0.5 qu N 10 0 0.5 1 Propagation length 0 0 0.5 1 Propagation length But one-photon input yields same output intensity patterns as bright light

Two-photon quantum walks Two simultaneous walks Two photons input in two modes (i,j) Correlation between output modes (k,l) Distinguishable photons (di) k,l = A i,k A j,l 2 + A j,k A i,l 2 Indistinguishable photons (in) k,l = A i,k A j,l + A j,k A i,l 2 Mode k Mode l Mode k Mode l Peruzzo, Science 329, 1500 (2010) Bromberg, PRL 102, 253904 (2009) No photon-photon interference q 1 3 (di) k,k (di) l,l (di) k,l < 0 Hong-Ou-Mandel bunching Violation of inequality demonstrates quantum behaviour

Extending dimensionality Higher-dimensional walks allow simulations of more complex quantum systems (more than one particle) more speed-up for some algorithms (e.g. search) Multiple walkers increase dimensionality but cost time (sources 1% efficient) Instead (or as well!) dimensionality can be increased using geometry of walk Schreiber, Science 336, 55 (2012)

Increase dimensionality with multicore fibre Coupled waveguides in square array allow walk in two independent dimensions

Increase dimensionality with multicore fibre Coupled waveguides in square array allow walk in two independent dimensions da dz = ima M = 0 B @ 1 g 12 g 13 g 21 2 g 23 g 31 g 32 3.... 1 C A

Variations in fibre cores Inevitable core-to-core variations will reduce inequality violation Differences in wavenumber (beta) most problematic x 10 5 Mean inequality violation 8 6 4 2 0 0 1 2 x 10 4 Find core-to-core variation in fibres fabricated in Bath Start with triangular lattice fibre (less difficult to fabricate!)

Quantum walk on triangular lattice Fabrication 37-core fibre Ge-doped silica cores Pure silica cladding 8 microns

Characterizing variations - experiment 37-core fibre, 6.9 mm length Output intensity recorded for input in every individual core Fibre laser 1064nm 5ps, 50nJ Grating compressor 1064nm 250fs, 25nJ Supercontinuum 37-core fibre Interference filter CCD Interference filter allowed selection of wavelength between 530-780nm Corresponds to huge change in coupling length Wavelength of 650nm selected for 8 micron pitch fibre 37 images make up one data set

Experimental data Coupling into centre

Experimental data Coupling into centre Coupling into edge

Characterizing variations - analysis Assume coupling strength the same for all nearest neighbours Monte-Carlo reconstruction of core-to-core variation in propagation constants Minimise residual R = 1 mx mx P (ex) i,k P (calc) i,k 2m Gives normalised measure of fraction of light in wrong cores per input state Example of differences in reconstructed propagation constants: i=1 k=1 0.4 0.2 0 0.2 mean = 14, 000 mm 1 0.4 5 10 15 20 25 30 35 Core index

Implications of core variations Reconstructed betas have typical variation of 2 10-5 Simulations show inequality can be violated in fabricated fibre q 1 (in) (in) (in) k,k l,l k,l 3 Perfect array Our fibre Output mode index l 35 30 25 20 15 10 x 10 3 2 1.5 1 0.5 Output mode index l 35 30 25 20 15 10 x 10 3 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 5 5 0.2 10 20 30 Output mode index k 10 20 30 Output mode index k 0

Conclusion 2-d quantum walks increase dimensionality without requiring more photons Multicore fibre provides a scalable platform for performing such walks Core-to-core variations appear sufficiently low to allow clear observation of quantum behaviour Future work: propagate single photons and photon pairs in triangular lattice fibre fabricate multicore fibre with square array for 2 independent dimensions

Bonus slide 1 - fabrication Capillaries Stack Cane Fibre