Geodesic
Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The analytic specifications of photonic lattices with fractional revival (FR) and perfect state transfer (PST) are reviewed. The approach to their design which is based on orthogonal polynomials is highlighted. A compendium of analytic models with PST is offered. New results on their FR properties are included. The nearest-neighbour approximation is adopted in most of the review; one analytic example with next-to-nearest neighbour interactions is also presented.
Keywords: perfect state transfer; fractional revival; quantum information. 
@article{SIGMA_2017_13_a73,
     author = {\'Eric-Olivier Boss\'e and Luc Vinet},
     title = {Coherent {Transport} in {Photonic} {Lattices:} {A} {Survey} of {Recent} {Analytic} {Results}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a73/}
}
TY  - JOUR
AU  - Éric-Olivier Bossé
AU  - Luc Vinet
TI  - Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a73/
LA  - en
ID  - SIGMA_2017_13_a73
ER  - 
%0 Journal Article
%A Éric-Olivier Bossé
%A Luc Vinet
%T Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a73/
%G en
%F SIGMA_2017_13_a73
Éric-Olivier Bossé; Luc Vinet. Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a73/

[1] Albanese C., Christandl M., Datta N., Ekert A., “Mirror inversion of quantum states in linear registers”, Phys. Rev. Lett., 93 (2004), 230502, 4 pp., arXiv: quant-ph/0405029 | DOI | MR

[2] Banchi L., Compagno E., Bose S., “Perfect wave-packet splitting and reconstruction in a one-dimensional lattice”, Phys. Rev. A, 91 (2015), 052323, 8 pp., arXiv: 1502.03061 | DOI

[3] Bellec M., Nikolopoulos G. M., Tzortzakis S., “Faithful communication Hamiltonian in photonic lattices”, Opt. Lett., 37 (2012), 4504–4506, arXiv: 1210.1009 | DOI | MR

[4] Bose S., “Quantum communication through an unmodulated spin chain”, Phys. Rev. Lett., 91 (2003), 207901, 4 pp., arXiv: quant-ph/0212041 | DOI

[5] Bose S., “Quantum communication through spin chain dynamics: an introductory overview”, Contemp. Phys., 48 (2007), 13–30, arXiv: 0802.1224 | DOI

[6] Chakrabarti R., Van der Jeugt J., “Quantum communication through a spin chain with interaction determined by a Jacobi matrix”, J. Phys. A: Math. Theor., 43 (2010), 085302, 20 pp., arXiv: 0912.0837 | DOI | MR | Zbl

[7] Chapman R. J., Santandrea M., Huang Z., Corrielli G., Crespi A., Yung M.-H., Osellame R., Peruzzo A., “Experimental perfect state transfer of an entangled photonic qubit”, Nature Comm., 7 (2016), 11339, 6 pp., arXiv: 1603.00089 | DOI

[8] Chen B., Song Z., Sun C. P., “Fractional revivals of the quantum state in a tight-binding chain”, Phys. Rev. A, 75 (2007), 012113, 9 pp., arXiv: quant-ph/0603033 | DOI

[9] Chihara T. S., An introduction to orthogonal polynomials, Mathematics and its Applications, 13, Gordon and Breach Science Publishers, New York–London–Paris, 1978 | MR | Zbl

[10] Christandl M., Datta N., Ekert A., Landahl A. J., “Perfect state transfer in quantum spin networks”, Phys. Rev. Lett., 92 (2004), 187902, 4 pp., arXiv: quant-ph/0309131 | DOI | MR

[11] Christandl M., Vinet L., Zhedanov A., Analytic next-to-nearest neighbour XX models with perfect state transfer and fractional revival, arXiv: 1607.02639

[12] Christodoulides D. N., Lederer F., Silberberg Y., “Discretizing light behaviour in linear and nonlinear waveguide lattices”, Nature, 424 (2003), 817–823 | DOI | MR

[13] Corrielli G., Crespi A., Della Valle G., Longhi S., Osellame R., “Fractional Bloch oscillations in photonic lattices”, Nature Comm., 4 (2013), 11339, 6 pp., arXiv: 1303.1958 | DOI

[14] Coutinho G. M., State transfer on graphs, Ph. D. Thesis, University of Waterloo, 2014

[15] Dai L., Feng Y. P., Kwek L.-C., “Engineering quantum cloning through maximal entanglement between boundary qubits in an open spin chain”, J. Phys. A: Math. Theor., 43 (2010), 035302, 14 pp. | DOI | MR | Zbl

[16] De Chiara G., Fazio R., Macchiavello C., Montangero S., Palma G. M., “Quantum cloning in spin networks”, Phys. Rev. A, 70 (2004), 062308, 4 pp., arXiv: quant-ph/0402071 | DOI

[17] Efremidis N. K., Christodoulides D. N., “Revivals in engineered waveguide arrays”, Opt. Comm., 246 (2005), 345–356 | DOI

[18] Genest V. X., Miki H., Vinet L., Zhedanov A., “Spin lattices, state transfer, and bivariate Krawtchouk polynomials”, Can. J. Phys., 93 (2015), 979–984, arXiv: 1410.4703 | DOI

[19] Genest V. X., Tsujimoto S., Vinet L., Zhedanov A., “Persymmetric Jacobi matrices, isospectral deformations and orthogonal polynomials”, J. Math. Anal. Appl., 450 (2017), 915–928, arXiv: 1605.00708 | DOI | MR | Zbl

[20] Genest V. X., Vinet L., Zhedanov A., “Exact fractional revival in spin chains”, Modern Phys. Lett. B, 30 (2016), 1650315, 7 pp., arXiv: 1506.08434 | DOI | MR

[21] Genest V. X., Vinet L., Zhedanov A., “Quantum spin chains with fractional revival”, Ann. Physics, 371 (2016), 348–367, arXiv: 1507.05919 | DOI | MR

[22] Gladwell G. M. L., Inverse problems in vibration, Solid Mechanics and its Applications, 119, 2nd ed., Kluwer Academic Publishers, Dordrecht, 2005 | DOI | MR

[23] Godsil C., “State transfer on graphs”, Discrete Math., 312 (2012), 129–147, arXiv: 1102.4898 | DOI | MR | Zbl

[24] Huang W.-P., “Coupled-mode theory for optical waveguides: an overview”, J. Opt. Soc. Amer. A, 11 (1994), 963–983 | DOI

[25] Iwanow R., May-Arrioja D. A., Christodoulides D. N., Stegeman G. I., Min Y., Sohler W., “Discrete Talbot effect in waveguide arrays”, Phys. Rev. Lett., 95 (2005), 053902, 4 pp. | DOI

[26] Jafarov E. I., Van der Jeugt J., “Quantum state transfer in spin chains with $q$-deformed interaction terms”, J. Phys. A: Math. Theor., 43 (2010), 405301, 18 pp., arXiv: 1005.2912 | DOI | MR | Zbl

[27] Karbach P., Stolze J., “Spin chains as perfect quantum state mirrors”, Phys. Rev. A, 72 (2005), 030301, 4 pp., arXiv: quant-ph/0501007 | DOI | MR

[28] Kay A., “Perfect state transfer: beyond nearest-neighbor couplings”, Phys. Rev. A, 73 (2006), 032306, 7 pp., arXiv: quant-ph/0509065 | DOI | MR

[29] Kay A., “Perfect, efficient, state transfer and its application as a constructive tool”, Int. J. Quantum Inform., 8 (2010), 641–676, arXiv: 0903.4274 | DOI | Zbl

[30] Kay A., “Generating quantum states through spin chain dynamics”, New J. Phys., 19 (2017), 043019, 15 pp., arXiv: 1609.01398 | DOI

[31] Kay A., “Tailoring spin chain dynamics for fractional revivals”, Quantum, 1 (2017), 24, 11 pp., arXiv: 1609.01397 | DOI | Zbl

[32] Koekoek R., Lesky P. A., Swarttouw R. F., “Hypergeometric orthogonal polynomials and their $q$-analogues”, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[33] Lederer F., Stegeman G. I., Christodoulides D. N., Assanto G., Segev M., Silberberg Y., “Discrete solitons in optics”, Phys. Rep., 463 (2008), 1–126 | DOI

[34] Lemay J.-M., Vinet L., Zhedanov A., “An analytic spin chain model with fractional revival”, J. Phys. A: Math. Theor., 49 (2016), 335302, 15 pp., arXiv: 1509.08965 | DOI | MR | Zbl

[35] Lemay J.-M., Vinet L., Zhedanov A., “The para-Racah polynomials”, J. Math. Anal. Appl., 438 (2016), 565–577, arXiv: 1511.05215 | DOI | MR | Zbl

[36] Longhi S., “Quantum-optical analogies using photonic structures”, Laser Photonics Rev., 3 (2009), 243–261 | DOI

[37] Miki H., Tsujimoto S., Vinet L., Zhedanov A., “Quantum-state transfer in a two-dimensional regular spin lattice of triangular shape”, Phys. Rev. A, 85 (2012), 062306, 4 pp., arXiv: 1203.2128 | DOI

[38] Nikolopoulos G. M., Jex I. (eds.), Quantum state transfer and network engineering, Quantum Science and Technology, Springer, Heidelberg, 2014 | DOI | MR | Zbl

[39] Perez-Leija A., Keil R., Kay A., Moya-Cessa H., Nolte S., Kwek L.-C., Rodríguez-Lara B. M., Szameit A., Christodoulides D. N., “Coherent quantum transport in photonic lattices”, Phys. Rev. A, 87 (2013), 012309, 8 pp., arXiv: 1207.6080 | DOI

[40] Peschel U., Pertsch T., Lederer F., “Optical {B}loch oscillations in waveguide arrays”, Opt. Lett., 23 (1998), 1701–1703 | DOI

[41] Petrovic J., “Multiport waveguide couplers with periodic energy exchange”, Opt. Lett., 40 (2015), 139–142 | DOI

[42] Post S., “Quantum perfect state transfer in a 2D lattice”, Acta Appl. Math., 135 (2015), 209–224 | DOI | MR | Zbl

[43] Robinett R. W., “Quantum wave packet revivals”, Phys. Rep., 392 (2004), 1–119, arXiv: quant-ph/0401031 | DOI | MR

[44] Rodríguez-Lara B. M., “Exact dynamics of finite Glauber–Fock photonic lattices”, Phys. Rev. A, 84 (2011), 053845, 6 pp., arXiv: 1108.3004 | DOI

[45] Rodríguez-Lara B. M., Soto-Eguibar F., Christodoulides D. N., “Quantum optics as a tool for photonic lattice design”, Phys. Scr., 90 (2015), 068014, 7 pp. | DOI

[46] Shi T., Li Y., Song Z., Sun C.-P., “Quantum-state transfer via the ferromagnetic chain in a spatially modulated field”, Phys. Rev. A, 71 (2005), 032309, 5 pp., arXiv: quant-ph/0408152 | DOI

[47] Stoilova N. I., Van der Jeugt J., “An exactly solvable spin chain related to Hahn polynomials”, SIGMA, 7 (2011), 033, 13 pp., arXiv: 1101.4469 | DOI | MR | Zbl

[48] Szameit A., Pertsch T., Nolte S., Tünnermann A., Peschel U., Lederer F., “Optical Bloch oscillations in general waveguide lattices”, J. Opt. Soc. Amer. B, 24 (2007), 2632–2639 | DOI

[49] Vinet L., Zhedanov A., “Dual-1 Hahn polynomials and perfect state transfer”, J. Phys. Conf. Ser., 343 (2012), 012125, 10 pp., arXiv: 1110.6477 | DOI

[50] Vinet L., Zhedanov A., “How to construct spin chains with perfect state transfer”, Phys. Rev. A, 85 (2012), 012323, 7 pp., arXiv: 1110.6474 | DOI

[51] Vinet L., Zhedanov A., “Para-Krawtchouk polynomials on a bi-lattice and a quantum spin chain with perfect state transfer”, J. Phys. A: Math. Theor., 45 (2012), 265304, 11 pp., arXiv: 1110.6475 | DOI | MR | Zbl

[52] Yung M.-H., “Quantum speed limit for perfect state transfer in one dimension”, Phys. Rev. A, 74 (2006), 030303, 4 pp., arXiv: quant-ph/0603179 | DOI