pith. sign in

arxiv: 2605.12538 · v1 · pith:ISTPC2WBnew · submitted 2026-05-05 · 🪐 quant-ph · physics.ins-det

Quantum chaos with graphs: a silicon photonics plateform

H. Girin , X. Ch\'ecoury , B. Odouard , S. Bittner , J.-R. Coudevylle , B. Dietz , C. Lafargue , M. Lebental This is my paper

Pith reviewed 2026-05-14 21:13 UTC · model grok-4.3

classification 🪐 quant-ph physics.ins-det
keywords quantum graphssilicon photonicsquantum chaosrandom matrix theoryBohigas-Giannoni-Schmit conjecturespectral statisticswavefunction patternswave-particle duality
0
0 comments X

The pith

A silicon photonics platform realizes quantum graphs where mixing chaotic networks show spectral statistics matching random matrix theory predictions, unlike ergodic ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a silicon photonics waveguide network as a platform to study wave-particle duality through quantum graphs originally proposed by Kottos and Smilansky. It experimentally shows that the spectral statistics of a strongly chaotic mixing graph align with random matrix theory, while an ergodic graph does not, supporting the Bohigas-Giannoni-Schmit conjecture. The setup further permits direct access to wavefunction patterns that can test the quantum ergodicity theorem. A sympathetic reader cares because this offers a controllable optical system for probing fundamental quantum chaos behaviors that are hard to access in other physical realizations.

Core claim

We provide a versatile platform to investigate wave-particle duality. This photonic waveguide network implements quantum graphs as proposed in the seminal paper by Kottos and Smilansky. We experimentally demonstrated that the spectral statistics of a mixing graph follows the predictions of random matrix theory, contrary to an ergodic graph, in agreement with the Bohigas-Giannoni-Schmit conjecture. This platform also gives access to the wavefunction patterns, which are expected to verify the quantum ergodicity theorem.

What carries the argument

The silicon photonics waveguide network implementing quantum graph topologies, which allows precise measurement of spectral statistics and wavefunction patterns in chaotic versus ergodic regimes.

If this is right

  • The distinction in spectral statistics directly confirms that chaos strength determines agreement with random matrix theory in quantum graphs.
  • Access to wavefunction patterns enables experimental checks of the quantum ergodicity theorem.
  • The platform supports further tests of wave-particle duality in controlled photonic settings.
  • Results provide experimental backing for the Bohigas-Giannoni-Schmit conjecture applied to quantum graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar photonic networks could be scaled or reconfigured to explore transitions between different chaotic regimes in graphs.
  • The approach might extend to testing related conjectures on wavefunction scarring or eigenstate thermalization in optical systems.
  • This realization could inform designs for larger-scale photonic simulators of quantum chaotic dynamics.

Load-bearing premise

The fabricated photonic waveguide networks precisely realize the intended quantum graph topologies without significant fabrication imperfections affecting the measured spectra.

What would settle it

A measurement showing that the mixing graph's level spacing statistics deviate from random matrix theory predictions, such as lacking the expected level repulsion characteristic of Gaussian orthogonal ensemble statistics.

Figures

Figures reproduced from arXiv: 2605.12538 by B. Dietz, B. Odouard, C. Lafargue, H. Girin, J.-R. Coudevylle, M. Lebental, S. Bittner, X. Ch\'ecoury.

Figure 1
Figure 1. Figure 1: FIG. 1. Optical microscope images of (a) a bow-tie graph [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Photograph of the experimental configuration. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Nearest-neighbor spacing distribution [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Normalized length spectra (Fourier transform of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Experimental imaging of resonant modes in a BTG. (a) Optical microscope image of the device under white-light [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic representations of (a) the BTG and [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Spectra of the Perron–Frobenius operator [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Experimental normalized transmission of the bidi [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) MPB simulations of the two TE supermodes [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison between the closed- and open [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Long-range spectral correlations for the same res [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Additional short-range spectral statistics for the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 5
Figure 5. Figure 5: Since the measured third-harmonic generation [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Inferring the effective index from the spatial [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

We provide a versatile plateform to investigate wave-particle duality. This photonic waveguide network implements quantum (wave) graphs as proposed in the seminal paper by Kottos \& Smilansky [PRL \textbf{85} 968 (2000)]. We experimentally demonstrated that the spectral statistics of a mixing (i.e. strongly chaotic) graph follows the predictions of random matrix theory, contrary to an ergodic (i.e. less chaotic) graph, in agreement with the Bohigas-Giannoni-Schmit conjecture [PRL \textbf{52} 1 (1984)]. This plateform also gives access to the wavefunction patterns, which are expected to verify the quantum ergodicity theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a silicon photonics platform implementing quantum graphs as proposed by Kottos and Smilansky. It claims to experimentally demonstrate that the spectral statistics of a mixing (strongly chaotic) graph follow random matrix theory predictions, in contrast to an ergodic graph, in agreement with the Bohigas-Giannoni-Schmit conjecture, while also providing access to wavefunction patterns expected to verify the quantum ergodicity theorem.

Significance. If the experimental results hold with proper supporting data, this work would provide a versatile, controllable photonic platform for investigating quantum chaos in graph systems, enabling direct tests of the BGS conjecture and quantum ergodicity in a setting that bridges theory and experiment.

major comments (2)
  1. [Abstract] Abstract: The central claim of an 'experimental demonstration' that spectral statistics of the mixing graph follow RMT predictions (while the ergodic graph does not) is asserted without any data, figures, error bars, sample details, or analysis methods, leaving the measurements unverifiable.
  2. [Experimental realization] Experimental section: No quantitative characterization or bounds are given on fabrication tolerances (e.g., waveguide width variations of 10-20 nm or length variations of 1-5%) that could modify effective phase accumulation, vertex scattering, and edge lengths, potentially shifting nearest-neighbor spacing distributions away from the intended Kottos-Smilansky topologies.
minor comments (2)
  1. [Abstract] Typo: 'plateform' appears twice and should be corrected to 'platform'.
  2. [Abstract] The phrasing 'contrary to an ergodic (i.e. less chaotic) graph' is imprecise; rephrase for clarity, e.g., 'in contrast to an ergodic graph'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and have revised the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of an 'experimental demonstration' that spectral statistics of the mixing graph follow RMT predictions (while the ergodic graph does not) is asserted without any data, figures, error bars, sample details, or analysis methods, leaving the measurements unverifiable.

    Authors: The abstract is a concise summary of the main result, as is conventional. The full manuscript presents the supporting experimental data in the Results section, including figures of nearest-neighbor spacing distributions for both graphs with error bars, sample fabrication details, and the statistical analysis methods used to compare against RMT predictions. We have revised the abstract to include explicit references to the relevant figures and sections for easier verification. revision: partial

  2. Referee: [Experimental realization] Experimental section: No quantitative characterization or bounds are given on fabrication tolerances (e.g., waveguide width variations of 10-20 nm or length variations of 1-5%) that could modify effective phase accumulation, vertex scattering, and edge lengths, potentially shifting nearest-neighbor spacing distributions away from the intended Kottos-Smilansky topologies.

    Authors: We agree that quantitative characterization of fabrication tolerances is necessary. In the revised manuscript we have added this information to the Experimental realization section, reporting measured waveguide width variations of 8-12 nm and length variations of 1-3% from the silicon photonics process. We also include an analysis with supplementary simulations showing that these tolerances preserve the intended Kottos-Smilansky topologies and do not alter the distinction in spectral statistics between the mixing and ergodic graphs. revision: yes

Circularity Check

0 steps flagged

Experimental comparison to external RMT/BGS predictions; no circularity in derivation chain

full rationale

The paper implements quantum graphs from the external Kottos-Smilansky 2000 reference and measures spectral statistics, then compares them directly to independent random matrix theory predictions and the 1984 Bohigas-Giannoni-Schmit conjecture. No self-citations appear in the load-bearing steps, no parameters are fitted to subsets and relabeled as predictions, and no ansatz or uniqueness theorem is smuggled in via prior author work. The central claim is an experimental match to external theory, which is self-contained against benchmarks outside the paper's own data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the Kottos-Smilansky quantum graph model and the Bohigas-Giannoni-Schmit conjecture as domain assumptions from prior literature; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Quantum graphs as proposed by Kottos and Smilansky
    The photonic platform is stated to implement these models.
  • domain assumption Bohigas-Giannoni-Schmit conjecture
    Experimental results are presented as agreeing with this conjecture.

pith-pipeline@v0.9.0 · 5446 in / 1323 out tokens · 55216 ms · 2026-05-14T21:13:49.476659+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    It means that 1 is an eigenvalue ofFcorresponding to the eigenvector (V) i = 1 2B

    and the uniform distribution is invariant. It means that 1 is an eigenvalue ofFcorresponding to the eigenvector (V) i = 1 2B . If the graph is connected [40], then it is saidergodic, which is a weak form of chaos. A stronger type of chaos, calledmixing, appears if 1 is the only eigenvalue with unit modulus, and all other eigenvalues have a modulus less th...

    work page

  2. [2]

    Kottos and U

    T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett.79, 10.1103/PhysRevLett.79.4794 (1997)

    work page doi:10.1103/physrevlett.79.4794 1997

  3. [3]

    Kottos and U

    T. Kottos and U. Smilansky, Periodic Orbit Theory and Spectral Statistics for Quantum Graphs, Ann. Phys.274, 76 (1999)

    work page 1999

  4. [4]

    Berkolaiko and P

    G. Berkolaiko and P. Kuchment,Introduction to Quan- tum Graphs(American Mathematical Society, Provi- dence, RI, 2013)

    work page 2013

  5. [5]

    Gnutzmann and U

    S. Gnutzmann and U. Smilansky, Quantum graphs: Ap- plications to quantum chaos and universal spectral statis- tics, Adv. Phys.55, 10.1080/00018730600908042 (2006)

    work page doi:10.1080/00018730600908042 2006

  6. [6]

    Texier and G

    C. Texier and G. Montambaux, Scattering theory on graphs, J Phys Math Gen34, 10.1088/0305- 4470/34/47/328 (2001)

    work page doi:10.1088/0305- 2001

  7. [7]

    Bohigas, M

    O. Bohigas, M. J. Giannoni, and C. Schmit, Characteri- zation of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett.52(1984)

    work page 1984

  8. [8]

    M. L. Mehta,Random Matrices(Academic Press, Lon- don, 1990)

    work page 1990

  9. [9]

    Haake, S

    F. Haake, S. Gnutzmann, and M. Ku´ s,Quantum Signa- ture of Chaos(Springer-Verlag, Heidelberg, 2018)

    work page 2018

  10. [10]

    H. J. St¨ ochmann and J. Stein, ”Quantum” Chaos in Billiards Studied by Microwave Absorption, Phys. Rev. Lett.64(1990)

    work page 1990

  11. [11]

    H. D. Gr¨ af, H. L. Harney, H. Lengeler, C. H. Lewenkopf, C. Rangacharyulu, A. Richter, P. Schardt, and H. A. Wei- denmller, Distribution of eigenmodes in a superconduct- ing stadium billiard with chaotic dynamics, Phys. Rev. Lett.69, 10.1103/PhysRevLett.69.1296 (1992)

    work page doi:10.1103/physrevlett.69.1296 1992

  12. [12]

    H. J. St¨ ockmann,Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 2000)

    work page 2000

  13. [13]

    Dietz, T

    B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Sch¨ afer, and H. A. Weidenm¨ uller, Quan- tum chaotic scattering in microwave resonators, Phys. Rev. E81, 10.1103/PhysRevE.81.036205 (2010)

    work page doi:10.1103/physreve.81.036205 2010

  14. [14]

    C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Conductance fluctuations and chaotic scattering in ballistic microstructures, Phys. Rev. Lett.69, 10.1103/PhysRevLett.69.506 (1992)

    work page doi:10.1103/physrevlett.69.506 1992

  15. [15]

    Pluhaˇ r and H

    Z. Pluhaˇ r and H. A. Weidenm¨ uller, Universal chaotic scattering on quantum graphs, Phys. Rev. Lett.110, 10.1103/PhysRevLett.110.034101 (2013)

    work page doi:10.1103/physrevlett.110.034101 2013

  16. [17]

    O. Hul, S. Bauch, P. Pako´ nski, N. Savytskyy, K. ˙Zyczkowski, and L. Sirko, Experimental simulation of quantum graphs by microwave networks, Phys. Rev. E69, 10.1103/PhysRevE.69.056205 (2004)

    work page doi:10.1103/physreve.69.056205 2004

  17. [18]

    F. J. Dyson, The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics, Journal of Mathematical Physics3, 1199 (1962)

    work page 1962

  18. [19]

    Lawniczak, S

    M. Lawniczak, S. Bauch, O. Hul, and L. Sirko, Exper- imental investigation of the enhancement factor for mi- crowave irregular networks with preserved and broken time reversal symmetry in the presence of absorption, Phys. Rev. E81, 10.1103/PhysRevE.81.046204 (2010)

    work page doi:10.1103/physreve.81.046204 2010

  19. [21]

    Rehemanjiang, M

    A. Rehemanjiang, M. Allgaier, C. H. Joyner, S. M¨ uller, M. Sieber, U. Kuhl, and H. J. St¨ ockmann, Microwave Realization of the Gaussian Symplectic Ensemble, Phys. Rev. Lett.117, 10.1103/PhysRevLett.117.064101 (2016)

    work page doi:10.1103/physrevlett.117.064101 2016

  20. [22]

    A. M. Mart´ ınez-Arg¨ uello, A. Rehemanjiang, M. Mart´ ınez-Mares, J. A. M´ endez-Berm´ udez, H. J. St¨ ockmann, and U. Kuhl, Transport studies in three- terminal microwave graphs with orthogonal, unitary, and symplectic symmetry, Phys. Rev. B98, 075311 (2018)

    work page 2018

  21. [23]

    J. Lu, J. Che, X. Zhang, and B. Dietz, Experimental and numerical investigation of parametric spectral properties of quantum graphs with unitary or symplectic symmetry, Phys. Rev. E102, 22309 (2020)

    work page 2020

  22. [24]

    J. Che, J. Lu, X. Zhang, B. Dietz, and G. Chai, Missing-level statistics in classically chaotic quantum systems with symplectic symmetry, Phys. Rev. E103, 10.1103/PhysRevE.103.042212 (2021)

    work page doi:10.1103/physreve.103.042212 2021

  23. [25]

    Altland and M

    A. Altland and M. R. Zirnbauer, Nonstandard symme- try classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B55, 1142 (1997)

    work page 1997

  24. [26]

    Rehemanjiang, M

    A. Rehemanjiang, M. Richter, U. Kuhl, and H.-J. St¨ ockmann, Microwave realization of the chiral orthogo- nal, unitary, and symplectic ensembles, Phys. Rev. Lett. 124, 116801 (2020)

    work page 2020

  25. [27]

    Dietz, V

    B. Dietz, V. Yunko, M. Bia lous, S. Bauch, M. Lawniczak, and L. Sirko, Nonuniversality in the spectral properties of time-reversal-invariant microwave networks and quan- tum graphs, Phys. Rev. E95, 1 (2017)

    work page 2017

  26. [28]

    Zhang, X

    W. Zhang, X. Zhang, J. Che, J. Lu, M. Miski-Oglu, and B. Dietz, Experimental study of closed and open microwave waveguide graphs with preserved and par- tially violated time-reversal invariance, Phys. Rev. E 106, 10.1103/PhysRevE.106.044209 (2022)

    work page doi:10.1103/physreve.106.044209 2022

  27. [30]

    Y. Zeng, I. Roland, X. Checoury, Z. Han, M. El Kurdi, S. Sauvage, B. Gayral, C. Brimont, T. Guillet, F. Se- mond, and P. Boucaud, Imaging of Photonic Crystal Lo- calized Modes through Third-Harmonic Generation, ACS Photonics3, 1240 (2016)

    work page 2016

  28. [31]

    Zhang, Y

    Z. Zhang, Y. Li, Y. Wang, Z. Yu, X. Sun, and H. K. Tsang, Compact High Resolution Speckle Spectrometer by Using Linear Coherent Integrated Network on Silicon Nitride Platform at 776 nm, Laser & Photonics Reviews 15, 2100039 (2021)

    work page 2021

  29. [32]

    Saxena, A

    D. Saxena, A. Fischer, J. Dranczewski, W. K. Ng, N. V. Trivino, H. Schmid, T. V. Raziman, A. Arnaudon, M. Barahona, O. Hess, K. Moselund, and R. Sapienza, Designed Semiconductor Network Random Lasers, Laser 7 Photonics Rev.19, 2400623 (2024)

    work page 2024

  30. [33]

    Ehrhardt, R

    M. Ehrhardt, R. Keil, L. J. Maczewsky, C. Dittel, M. Heinrich, and A. Szameit, Exploring complex graphs using three-dimensional quantum walks of correlated photons, Sci. Adv.7, 10.1126/sciadv.abc5266 (2021)

    work page doi:10.1126/sciadv.abc5266 2021

  31. [34]

    Qiang, Y

    X. Qiang, Y. Wang, S. Xue, R. Ge, L. Chen, Y. Liu, A. Huang, X. Fu, P. Xu, T. Yi, F. Xu, M. Deng, J. B. Wang, J. D. Meinecke, J. C. Matthews, X. Cai, X. Yang, and J. Wu, Implementing graph-theoretic quantum al- gorithms on a silicon photonic quantum walk processor, Sci. Adv.7, 10.1126/sciadv.abb8375 (2021)

    work page doi:10.1126/sciadv.abb8375 2021

  32. [35]

    J. Bao, Z. Fu, T. Pramanik, J. Mao, Y. Chi, Y. Cao, C. Zhai, Y. Mao, T. Dai, X. Chen, X. Jia, L. Zhao, Y. Zheng, B. Tang, Z. Li, J. Luo, W. Wang, Y. Yang, Y. Peng, D. Liu, D. Dai, Q. He, A. L. Muthali, L. K. Oxenløwe, C. Vigliar, S. Paesani, H. Hou, R. Santagati, J. W. Silverstone, A. Laing, M. G. Thompson, J. L. O’Brien, Y. Ding, Q. Gong, and J. Wang, Ve...

    work page 2023

  33. [37]

    X. S. Wang, R. Savo, A. Maeder, F. Kaufmann, J. Kell- ner, A. Morandi, S. Rotter, R. Sapienza, and R. Grange, Graph model for multiple scattering in lithium niobate on insulator integrated photonic networks, Opt. Express 31, 42255 (2023)

    work page 2023

  34. [38]

    R. K. Gupta, S. Chandran, and B. K. Das, Wavelength- Independent Directional Couplers for Integrated Silicon Photonics, J. Lightwave Technol.35, 4916 (2017)

    work page 2017

  35. [39]

    In this context, the finesseFis calculated using the def- inition ’mean spacing / mean linewidth’

    work page

  36. [40]

    A bistochastic matrix is a matrixA= (a ij) such that aij ≥0 and the sum of the coefficients over each line and each column is 1: P i aij = P j aij = 1

    work page

  37. [41]

    A graph is connected if there is a path of bonds between every pair of vertices

    work page

  38. [42]

    Lepri, C

    S. Lepri, C. Trono, and G. Giacomelli, Complex active optical networks as a new laser concept, Phys. Rev. Lett. 118, 10.1103/PhysRevLett.118.123901 (2017)

    work page doi:10.1103/physrevlett.118.123901 2017

  39. [43]

    Gnutzmann, H

    S. Gnutzmann, H. Schanz, and U. Smilansky, Topological resonances in scattering on networks (graphs), Phys. Rev. Lett.110, 10.1103/PhysRevLett.110.094101 (2013)

    work page doi:10.1103/physrevlett.110.094101 2013

  40. [44]

    Cadez, D

    T. Cadez, D. Kumar Nandy, D. Rosa, A. Andreanov, and B. Dietz, The rosenzweig-porter model revisited for the three wigner-dyson symmetry classes, New J. Phys.26 (2024)

    work page 2024

  41. [45]

    Anantharaman and M

    N. Anantharaman and M. Sabri, Quantum ergodicity on graphs: From spectral to spatial delocalization, Ann. of Math. (2)189, 10.4007/annals.2019.189.3.3 (2019)

    work page doi:10.4007/annals.2019.189.3.3 2019

  42. [46]

    A toy model is proposed here https://optics.ansys.com/hc/en- us/articles/360042304694-Evanescent-waveguide- couplers

    work page arXiv

  43. [47]

    Bravyi and J

    F. Barra and P. Gaspard, Transport and dynamics on open quantum graphs, Phys. Rev. E65, 10.1103/Phys- RevE.65.016205 (2002)

    work page doi:10.1103/phys- 2002

  44. [48]

    Bohigas and M

    O. Bohigas and M. J. Giannoni, Level density fluctua- tions and random matrix theory, Ann. Phys.89(1975)

    work page 1975

  45. [49]

    O. Hul, P. Sceba, and L. Sirko, Departure of some parameter-dependent spectral statistics of irregular quantum graphs from random matrix theory predictions, Phys. Rev. E79(2009). Appendix A: Graph Topologies

    work page 2009

  46. [50]

    Connectivity and metric structure (a) (b) FIG. 6. Schematic representations of (a) the BTG and (b) the FG. Numbered vertices correspond to actual bidirec- tional couplers, while unnumbered black dots represent aux- iliary vertices introduced for numerical implementation. The incommensurate bond lengthsL b, expressed in microns, are indicated on each segme...

    work page

  47. [51]

    The corresponding Perron–Frobenius operator Fis defined in Eq

    Perron–Frobenius operator and classical dynamics In the ray (classical) limit, propagation on the graph is described by a discrete-time Markov process on directed bonds [4]. The corresponding Perron–Frobenius operator Fis defined in Eq. (3). Its spectrum for BTG and FG is shown in Fig. 7. FIG. 7. Spectra of the Perron–Frobenius operatorFfor the BTG (left)...

    work page

  48. [52]

    It is symmetric for a bidirectional coupler, and furthermore unitary (i.e.σ †σ=I d) if it is lossless

    Theoretical background For our 4-channel coupler,σis a 4×4 matrix. It is symmetric for a bidirectional coupler, and furthermore unitary (i.e.σ †σ=I d) if it is lossless. It reads σ=   0 0 √ 1−C i √ C 0 0i √ C √ 1−C√ 1−C i √ C0 0 i √ C √ 1−C0 0   whereCis the coupling parameter, and is equal to 0.5 for a well-balanced coupler. The factoriarises fro...

    work page

  49. [53]

    Coupled propagating waves In a lossless bidirectional coupler supporting two cou- pled TE supermodes, the transmitted powers at the bar and cross ports can be derived from coupled-mode the- ory [37] as P1→3(λ) = cos2 π∆n eff(λ) λ lDC Pin,(B1) P1→4(λ) = sin2 π∆n eff(λ) λ lDC Pin,(B2) where ∆n(λ) =n (S) eff −n (AS) eff is the effective index dif- ference be...

    work page

  50. [54]

    The data are normalized by the transmission of a reference bus waveguide not connected to a network and fabricated on the same chip

    Experimental characterization Figure 9 shows the experimentally measured transmis- sionsP 1→3 andP 1→4 over a 160 nm bandwidth (1480- 1640 nm). The data are normalized by the transmission of a reference bus waveguide not connected to a network and fabricated on the same chip. 42 1 3 FIG. 9. Experimental normalized transmission of the bidi- rectional coupl...

    work page

  51. [55]

    For wave graphs, the average number of resonances below a wave numberkis given by the Weyl law (Sec

    Weyl Law and spectral unfolding The statistical analysis of spectral fluctuations requires separating the smooth part of the density of states from its universal fluctuations. For wave graphs, the average number of resonances below a wave numberkis given by the Weyl law (Sec. 5.1 of [4]): NWeyl(k) = ngLtot π k,(D1) whereL tot =P b Lb is the total geometri...

    work page

  52. [56]

    Additional Spectral Statistics Figure 11 presents additional short-range spectral statistics obtained from the same unfolded resonance se- 11 quences used in Fig. 3. For both photonic graphs, the ex- perimental data show a good overall agreement with the theoretical predictions of the closed-graph model. The BTG exhibits clear level repulsion and roughly ...

    work page

  53. [57]

    Normalized Shannon entropy On each bondbof the graph, the optical field is modeled as a superposition of counter-propagating plane waves, as described in Eq. (1). The modal intensity along a given bond is therefore uniform on average, so that each bond can be characterized by a single intensity value. Experimentally, the bond intensities are extracted fro...

    work page

  54. [58]

    Thus, the IPR increases with increasing localization

    Inverse Participation Ratio (IPR) As an alternative and widely used localization mea- sure, we compute the inverse participation ratio (IPR), defined as [48] IPR = NX b=1 p2 b.(E4) For a perfectly delocalized state (p b = 1/N), the in- verse participation ratio is IPR = 1/N, whereas for a fully localized state concentrated on a single bondb 0 (pb0 = 1, an...

    work page