- Open Access
Topological zero-dimensional photonic modes in chiral coupled-cavity arrays
© Yannopapas; licensee Springer. 2015
- Received: 27 May 2014
- Accepted: 14 January 2015
- Published: 1 March 2015
We show theoretically that a properly chosen one-dimensional array of coupled photonic resonators (cavities) may possess localized zero-dimensional topological modes bearing resemblance with the corresponding edge modes/Majorana states in semiconductor nanowires atop a superconducting substrate. These modes constitute a manifold of degenerate states and are robust to the geometrical characteristics of the array, to the dielectric properties of the cavities and of the host medium. Such arrays can be realized in the laboratory as chains of microwave cavities within a metallic wire-network or as lattices of sinusoidally curved dielectric waveguides in the optical regime.
- topological states
The topological properties of the quantum states define a new paradigm in the description and classification of condensed matter. Namely, atomic crystals possessing a topologically nontrivial electronic band structure constitute a new class of materials whose salient properties are robust to phase transitions which modify the symmetry order of the atomic solid. Typical examples of such topological solids are the integer/fractional quantum Hall (I/FQHE) systems and the topological insulators. Well-known examples of topological properties are the existence of chiral edge states in QHE systems and the presence of gapless surface states in topological insulators which are both insensitive to the presence of randomness/disorder.
Over the years there has been a continuous transfer of ideas and phenomena from atomic solids to their man-made electromagnetic (EM) counterparts, i.e., photonic crystals and metamaterials. Among several phenomena in traditional condensed matter physics which have found their analogues in artificial electromagnetic systems, the topological properties of matter have been the focus of intensive research in the past years motivated by two main scopes: the investigation of new states of photons in the context of quantum simulation and the design of disorder-immune integrated photonic devices. Examples of topological electronic systems which have been simulated in photonics are 2D lattices of gyromagnetic cylinders simulating the integer QHE [1–12], metamaterials simulating topological insulators [13–18], metamaterial-based microwave networks simulating the fractional QHE , and many others.
The main application in topological condensed matter comes from the presence of electron states at the boundaries of finite solids, i.e., chiral edge states in 2D systems (e.g., quantum spin Hall effect) and gapless surface states in 3D systems (topological insulators), which are robust to disorder. In finite one-dimensional (1D) systems with non-trivial topology, zero-dimensional localized states may appear in model Hamiltonians such as Kitaev’s model . In particular, Kitaev predicted that when a semiconductor nanowire is placed atop a superconducting substrate, Majorana-type fermionic states emerge at the edges of the nanowire; this is a phenomenon with important application in topologically fault-tolerant quantum computing [21–23]. Unfortunately, a proof-of-principle experiment for these zero-dimensional edge states requires a highly sophisticated laboratory setup. Unlike electron systems, topological zero-dimensional states can more easily simulated with photons either in the microwave regime with coupled-cavity arrays  or in the optical regime with arrays of metallic nanoparticles [25, 26]. Namely, the EM Green’s tensor describing the coupling among the electric dipoles either in EM cavities  or nanoparticles  as well as the derived photonic frequency bands are in analogy with Bogoliubov-de Gennes equations of Kitaev’s model of Majorana edge states in semiconductor nanowires. Besides classical electrodynamics, quantum simulators for Majorana states have been proposed in the context of cold atoms  and trapped ions .
In the present work we show that an array of coupled cavities immersed in a chiral host medium may support a manifold of zero-dimensional EM modes similarly to the fractional QHE, with obvious application in fault-tolerant quantum computing which is a non-local, decoherence-free type of quantum computing [21–23]. We note that we are working in a regime of high EM-radiation intensity in which case the classical theory of light (Maxwell’s equations) is applicable. However, for low intensities where the quantum nature of light (photons) is employed, additional phenomena based on quantum correlations may arise which can possibly enrich the range of applications of the presented system such as, e.g., in multipartite entanglement . The paper is structured as follows. Section 2 presents a tight-binding coupled-dipole formalism which, for suitable EM designs, transforms a classical electrodynamic problem described by Maxwell’s equations to an eigenvalue problem equivalent to an electronic tight-binding Hamiltonian of ordinary atomic solids. Section 3 applies the formalism of Section 2 to a coupled-cavity array design which exhibits nontrivial topological frequency bands. Section 4 discusses the emergence of zero-dimensional edges modes in a coupled-cavity array. Section 5 assesses possible experimental realizations of topological coupled-cavity arrays and Section 6 concludes the paper.
We note that the above zero-dimensional modes are present with (\(\eta \neq 0\)) or without the presence of chirality (\(\eta = 0\)) in the host medium. In the non-chiral case, however, these modes may be fragile since they have a infinitesimal frequency difference from the EM modes spreading over the entire chain (and correspond to the Dirac-cone frequency bands). It is therefore mandatory to generate a frequency band gap at \(\Omega=0\) to isolate the degenerate manifold of states from the rest of states. And of course, the band gap should be also topologically nontrivial in order to preserve the degenerate zero-dimensional states. This is achieved by introducing chirality in our system (by assuming that \(\eta \neq 0\) in Eqs. (8) and (10)), which is the optical analogue of spin-orbit coupling in electron systems . Due to the non-trivial topological nature of these zero-dimensional EM modes, they are immune to the presence of disorder and/or possible fabrication imperfections [24, 25] facilitating this way their experimental observation. We note that, similarly to the Majorana edge states in Kitaev’s model, the observed emergence of zero-dimensional modes in the middle of the frequency gap of Figure 2 signifies the non-trivial topology of the corresponding frequency bands of Figure 2 .
In order to realize the proposed coupled-cavity arrays in the laboratory, it is important to create a negative phase (argument) of the hopping strength. As stated in Section 2, a negative phase can be achieved in a medium supporting backward-propagating waves, i.e. waves where the phase velocity is opposite to the group velocity. Such media are the so-called negative refractive-index metamaterials or left-handed metamaterials which are artificial dielectrics supporting backward-propagating waves. Therefore, in order to achieve a negative phase coupling, i.e., \(\exp ( - i \phi)\), between two NN cavities, the latter must be physically connected via a metamaterial element. Such can be a waveguide loaded with a negative-index metamaterial or a left-handed transmission line .
We have presented a one-dimensional photonic system, namely an array of coupled cavities in a chiral medium, which is topologically non-trivial and exhibits a manifold of degenerate zero-dimensional modes which are reminiscent of the Majorana states in semiconducting nanowires in touch with a superconducting substrate. This exotic photonic simulator can be realized and probed experimentally system with already fabricated structures such as arrays of sinusoidally curved dielectric waveguides or with cavities coupled with metamaterial elements or transmission lines, embedded in metallic wire networks.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
- Haldane FDM, Raghu S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys Rev Lett. 2008;100:013904. ADSView ArticleGoogle Scholar
- Wang Z, Chong YD, Joannopoulos JD, Soljačić M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys Rev Lett. 2008;100:013905. ADSView ArticleGoogle Scholar
- Wang Z, Chong YD, Joannopoulos JD, Soljačić M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature (London). 2009;461:772. ADSView ArticleGoogle Scholar
- Yu Z, Veronis G, Wang Z, Fan S. One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal. Phys Rev Lett. 2008;100:023902. ADSView ArticleGoogle Scholar
- Takeda H, John S. Compact optical one-way waveguide isolators for photonic-band-gap microchips. Phys Rev A. 2008;78:023804. ADSView ArticleGoogle Scholar
- Han D, Lai Y, Zi J, Zhang ZQ, Chan CT. Dirac spectra and edge states in honeycomb plasmonic lattices. Phys Rev Lett. 2009;102:123904. ADSView ArticleGoogle Scholar
- Ao X, Lin Z, Chan CT. One-way edge mode in a magneto-optical honeycomb photonic crystal. Phys Rev B. 2009;80:033105. ADSView ArticleGoogle Scholar
- Onoda M, Ochiai T. Designing spinning Bloch states in 2D photonic crystals for stirring nanoparticles. Phys Rev Lett. 2009;103:033903. ADSView ArticleGoogle Scholar
- Ochiai T, Onoda M. Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states. Phys Rev B. 2009;80:155103. ADSView ArticleGoogle Scholar
- Shen R, Shao LB, Wang B, Xing DY. Single Dirac cone with a flat band touching on line-centered-square optical lattices. Phys Rev B. 2010;81:041410. ADSView ArticleGoogle Scholar
- Poo Y, Wu RX, Lin Z, Yang Y, Chan CT. Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys Rev Lett. 2011;106:093903. ADSView ArticleGoogle Scholar
- Yannopapas V. Photonic analog of a spin-polarized system with Rashba spin-orbit coupling. Phys Rev B. 2011;83:113101. ADSView ArticleGoogle Scholar
- Yannopapas V. Gapless surface states in a lattice of coupled cavities: a photonic analog of topological crystalline insulators. Phys Rev B. 2011;84:195126. ADSView ArticleGoogle Scholar
- Chen WJ, Hang ZH, Dong JW, Xiao X, Wang HZ, Chan CT. Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking. Phys Rev Lett. 2011;107:023901. ADSView ArticleGoogle Scholar
- Zhong W, Zhang X. Dirac-cone photonic surface states in three-dimensional photonic crystal slab. Opt Express. 2011;19:13738. ADSView ArticleGoogle Scholar
- Khanikaev AB, Hossein Mousavi S, Tse WK, Kargarian M, MacDonald AH, Shvets G. Photonic topological insulators. Nat Mater. 2012;12:233. ADSView ArticleGoogle Scholar
- Liang GQ, Chong YD. Optical resonator analog of a two-dimensional topological insulator. Phys Rev Lett. 2013;110:203904. ADSView ArticleGoogle Scholar
- Rechtsman MC, Zeuner JM, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A. Photonic Floquet topological insulators. Nature (London). 2013;496:196. ADSView ArticleGoogle Scholar
- Yannopapas V. Topological photonic bands in two-dimensional networks of metamaterial elements. New J Phys. 2012;14:113017. MathSciNetView ArticleGoogle Scholar
- Kitaev AY. Unpaired Majorana fermions in quantum wires. Phys Usp. 2001;44:131. ADSView ArticleGoogle Scholar
- Kitaev AY. Fault-tolerant quantum computation by anyons. Ann Phys. 2003;303:2. ADSMathSciNetView ArticleMATHGoogle Scholar
- Alicea J, Oreg Y, von Refael G, Oppen F, Fisher MPA. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nat Phys. 2011;7:412. View ArticleGoogle Scholar
- Pachos JK. Introduction to topological quantum computation. Cambridge: Cambridge University Press; 2012. View ArticleMATHGoogle Scholar
- Yannopapas V. Dirac points, topological edge modes and nonreciprocal transmission in one-dimensional metamaterial-based coupled-cavity arrays. Int J Mod Phys B. 2014;28:1441006. ADSMathSciNetView ArticleGoogle Scholar
- Poddubny A, Miroshnichenko A, Slobozhanyuk A, Kivshar Y. Topological Majorana states in zigzag chains of plasmonic nanoparticles. ACS Photonics. 2014;1:101. View ArticleGoogle Scholar
- Ling CW, Xiao M, Chan CT, Yu SF, Fung KH. Topological edge plasmon modes between diatomic chains of nanoparticles. arXiv:1401.7520.
- Bardyn CE, Baranov MA, Rico E, İmamoǧlu ZP, Diehl S. Majorana modes in driven-dissipative atomic superfluids with a zero Chern number. Phys Rev Lett. 2012;109:130402. ADSView ArticleGoogle Scholar
- Noh C, Rodríguez-Lara BM, Angelakis DG. Proposal for realization of the Majorana equation in a tabletop experiment. Phys Rev A. 2013;87:040102(R). ADSView ArticleGoogle Scholar
- Liew TCH, Savona V. Multimode entanglement in coupled cavity arrays. New J Phys. 2013;15:025015. MathSciNetView ArticleGoogle Scholar
- Purcell EM, Pennypacker CR. Scattering and absorption of light by non-spherical dielectric grains. Astrophys J. 1973;186:705. ADSView ArticleGoogle Scholar
- Dmitriev V. Space-time reversal symmetry properties of eletromagnetic Green’s tensors for complex and bianisotropic media. Prog Electromagn Res. 2004;48:145. View ArticleGoogle Scholar
- Iyer AK, Eleftheriades GV. Negative refraction metamaterials: fundamental principles and applications. New York: Wiley; 2005. Google Scholar
- Caloz C, Itoh T. Electromagnetic metamaterials: transmission line theory and microwave applications. New Jersey: Wiley-IEEE Press; 2006. Google Scholar
- Sakoda K. Universality of mode symmetries in creating photonic Dirac cones. J Opt Soc Am B. 2012;29:2770. ADSView ArticleGoogle Scholar
- Sakoda K. Proof of the universality of mode symmetries in creating photonic Dirac cones. Opt Express. 2012;20:25181. ADSView ArticleGoogle Scholar
- Laughlin RB. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys Rev Lett. 1983;50:1395. ADSView ArticleGoogle Scholar
- Cho J, Angelakis DG, Bose S. Fractional quantum Hall state in coupled cavities. Phys Rev Lett. 2008;101:246809. ADSView ArticleGoogle Scholar
- Stefanou N, Modinos A, Yannopapas V. Optical transparency of mesoporous metals. Solid State Commun. 2001;118:69. ADSView ArticleGoogle Scholar
- Pendry JB, Holden AJ, Stewart WJ, Youngs I. Extremely low frequency plasmons in metallic mesostructures. Phys Rev Lett. 1996;76:4773. ADSView ArticleGoogle Scholar
- Wu C, Li H, Wei Z, Yu X, Chan CT. Theory and experimental realization of negative refraction in a metallic helix array. Phys Rev Lett. 2010;105:247401. ADSView ArticleGoogle Scholar
- Wu C, Li H, Yu X, Li F, Chen C, Chan CT. Metallic helix array as a broadband wave plate. Phys Rev Lett. 2011;107:177401. ADSView ArticleGoogle Scholar
- Efremidis NK, Zhang P, Chen Z, Christodoulides DN, Rüter CE, Kip D. Wave propagation in waveguide arrays with alternating positive and negative couplings. Phys Rev A. 2010;81:053817. ADSView ArticleGoogle Scholar
- Zeuner JM, Efremidis NK, Keil R, Dreisow F, Christodoulides DN, Tünnermann A, Nolte S, Szameit A. Optical analogues for massless Dirac particles and conical diffraction in one dimension. Phys Rev Lett. 2012;109:023602. ADSView ArticleGoogle Scholar
- Longhi S, Marangoni M, Lobino M, Ramponi R, Laporta P, Cianci E, Foglietti V. Observation of dynamic localization in periodically curved waveguide arrays. Phys Rev Lett. 2006;96:243901. ADSView ArticleGoogle Scholar