Topological Photonics with Heavy-Photon Bands

Topological Photonics with Heavy-Photon Bands Vassilios Yannopapas Dept. of Physics, National Technical University of Athens (NTUA) Quantum simulations and many-body physics with light, 4-11/6/2016, Hania, Crete

Outline Electrons in atomic solids Integer Quantum Hall effect in 2D electron gas and graphene Rashba-type spin-orbit coupling and spintronics/ Anomalous quantum Hall effect Fractional Quantum Hall effect in a Haldane lattice (possibly in 2D layers of LiV 2 O 4, MgTi 2 O 4, etc.) Topological insulators Kitaev s model, Majorana fermions: semiconducting nanowire atop a superconductor Dirac equation for massless relativistic particles Photons in artificial dielectrics Photonic quantum Hall effect in magnetoelectric photonic crystals Chiral, Faraday-active metamaterials of plasmonic spheres Microwave photons in superconducting QED systems described by Jaynes- Cummings-Hubbard model. 3D lattice of weakly coupled cavities Topological 0D states in 1D coupledcavity chain with metamaterial couplings Negative-zero-positive index metamaterials

Photonic Analog of Integer Quantum Hall Effect (IQHE)

Topological insulators vs Band insulators nm 0 genus g=0 2 xy nmne / h nm 1 genus g=1 Chern number: Berry flux Berry curvature MZ Hasan and CL Kane, RMP 82, 3045 (2010) Bloch wave function of electrons

Photonic graphene: 2D photonic crystal with magnetoelectric materials Photonic graphene: honeycomb lattice of magnetoelectric rods FDM Haldane and S Raghu, PRL 100, 013904 (2008) Time-reversal TR symmetry breaking leads to a photonic topological insulator 0 0 ˆ 0 0 0 0 i 0 ˆ i 0 0 0

Chiral edge states in ribbons of photonic graphene S21: forward direction S12: backward direction Y Poo et al, PRL 106, 093903 (2011)

Photonic graphene without a Dirac point: 2D photonic crystal with magnetoelectric materials in a plasmonic host Photonic graphene: square lattice of magnetoelectric rods in a lossless plasmonic host. Band structure of slab and surface guides modes of a 5-layer sample of the crystal. 0 0 ˆ 0 0 0 0 i 0 ˆ i 0 0 0 Time-reversal TR symmetry breaking leads to non-reciprocal bands: one-way slab and surface modes. VY, J Opt, submitted

Anomalous quantum Hall effect: spin-polarized electron gas with Rashba spin-orbit coupling Magnetic field B n= 2 2D spin-polarized electron gas with spin-orbit coupling (SOC) described by the time-reversal symmetry breaking Hamiltonian: D Xiao, MC Chang, Q Miu, RMP 82, 1959 (2010) n=-2 Topological phase of matter Kinetic energy Spin-orbit coupling (SOC) Exchange field (Ferromagnetic electron gas) Thin Fe films

and its photonic analog Quantum system 2D spin-polarized electron gas with SO coupling: Supports longitudinal excitation at : ( ) 0 L L Photonic system Chiral medium with longitudinal excitations (e.g.,plasmons) By solving Maxwell s equations it turns out: Dispersion relation D i E k ˆ E Spatial dispersion due to chirality ( ) 0 i ˆ 0 ( ) 0 i 0 ( ) Faraday activity which explicitly breaks time-reversal symmetry 2D electron gas Berry curvature Chern number Topological photonic modes VY, PRB 83, 113101 (2011)

Realization of the photonic analog: chiral metamaterial Chiral lattice of plasmonic spheres Electrodynamic solution (LKKR) Effective-medium approximation 2 p 2 ( ) 1 Time-reversal symmetry breaking with Faraday rotation (iη) VY, PRB 83, 113101 (2011)

Further analogy: plasmonics and spintronics Chiral lattice of plasmonic spheres 0.57 ω/ω p 0.56 k k k L R 2 p 2 ( ) 1 Polar semiconductor with Rashba SOC 0.55-1.0-0.5 0.0 0.5 1.0 4k z d/π BiTeI BiTeI k k k BiTeI K Ishizaka et al., Nat Mater 10, 521 (2011) S Datta, B. Das, APL 56, 665 (1990)

Photonic Analog of the Fractional Quantum Hall Effect (FQHE)

Bosons/ Spinless fermions in flat bands with nontrivial topology: a framework for FQHE Flat bands with non-zero Chern number (non-trivial topology) may substitute Landau levels in the QHE. Bulk electron band structure K Sun et al., PRL 106, 236803 (2011) Slab electron band structure

Strongly interacting bosons in flat bands with non-trivial topology: occurrence of FQHE Spectrum gaps for the ½-FQHE YF Wang et al., PRL 107, 146803 (2011) Phase diagram: intensity width of the spectrum gaps for the ½- FQHE

Photons in flat bands with non-trivial topology: a framework for photonic FQHE Atomic Hamiltonian Superconducting QED Hamiltonian: Jaynes-Cummings-Hubbard model VY, New J. Physics 14, 113017 (2012). JC model: 2-level atom in single-mode cavity

Photonic FQHE in superconducting-circuit QED network Cooper-pair box: 2-level excitation in a transmission-line resonator + photon blockade VY, New J. Physics 14, 113017 (2012). GV Eleftheriades, IEEE MWCL 13, 51 (2003)

Photonic Analog of Topological Insulators

Topological insulators: topological phases without TRS breaking 2005: Kane & Mele showed that time-reversal symmetry (TRS) breaking is not prerequisite for topological phases of matter For spin ½ electrons, the T symmetry is expressed as antiunitary operator Θ: For a Hamiltonian H which preserves TRS we have: exp( i S / ) K S y : spin operator y K : complex conjugation 1 H( k) ( k) The eigenvalues of a Hamiltonian H preserving TRS are at least double degenerate (Kramer s theorem) In the absence of SOC it has trivial application since spin up and down states are double degenerate However, in the presence of SOC it has dramatic consequences since spin up and down states are no longer degenerate Non-protected topological surface states Protected topological surface states MZ Hasan and CL Kane, RMP 82, 3045 (2010)

Topological insulators: topological phases without TRS breaking Weak TI (Surface states easily removed by disorder) Strong TI (Surface states robust to disorder) MZ Hasan and CL Kane, RMP 82, 3045 (2010) Weak TI Strong TI 4-band tight-binding model of s states on a diamond lattice with SOC L Fu, CL Kane & EJ Mele, PRL 98, 106803 (2007)

Topological crystalline insulators There exists other topological classes preserving different symmetries such as Topological superconductors (particle-hole symmetry) Topological magnetic insulators (magnetic translational symmetry) Topological crystalline insulators (point-group symmetry) For topological crystalline insulators SOC is not necessary since the Hamiltonian is invariant under operations of the point-group symmetry. E.g., for a crystal with C 4 symmetry, we have: Tetragonal crystal of atoms with p x and p y orbitals L Fu, PRL 106, 10682 (2011)

Photonic analog of topological crystalline insulators We consider a tetragonal crystal of weakly coupled uniaxial dielectric cavities embedded within a lossless plasmonic metal Uniaxial response : Each cavity is simulated by two dipoles d x and d y By employing the discerete dipole approximation in the context of tight-binding approach we end up with the following eigenvalue problem which provides the frequency band structure for a 3D crystal as well as for finite slabs of. VY, Phys. Rev. B 84, 195126 (2011). FT of the EM Green s tensor Bloch eigenfunction of the polarization field

Band gap and gapless surface states Frequency band structure of a 3D tetragonal lattice of weakly coupled cavities in plasma: Frequency band gap Frequency band structure of slab ABAB ABB from 80 bilayers Quadratic degeneracy of surface states Topological states of the EM field? Yes! The Z 2 topological invariant is ν 0 = 1 (nonzero) VY, Phys. Rev. B 84, 195126 (2011).

Metamaterial design for a photonic topological crystalline insulator First of all we need a lossless plasma! 2 p Metals should be the obvious choice. However, when metals are described as plasma, i.e., ( ) 1 which is the case in the optical regime, they are extremely lossy. ( i ) In the infrared regime and below (e.g., microwave regime) they are almost losses but not a plasma! Artificial plasma! 3D network of metallic wires of diameter d~10μm operating in the GHz regime Lossless plasma since metals in the GHz regime are perfect conductors and hence suffer small losses. 2 p 2 Therefore, ( ) 1, where ω p lies in the GHz regime. JB Pendry et al., PRL 76, 4773 (1996)

Metamaterial design for a photonic topological crystalline insulator Unit cell Uniaxial cavity Artificial plasma Guiding elements for orientationdependent intralayer coupling between the cavities

1D lattices of cavities: photonic simulators for Kitaev s model - Majorana-like states

Photons in a 1D chain of coupled cavities with metamaterial-based couplings FT of the EM Green s tensor Bloch eigenfunction of the polarization field Dispersion relation of photons analogous to the Bogoliubov-de Gennes dispersion relation of Kitaev s model. Non-trivial values of Zak s phase: VY, Int. J. Mod. Phys. B 28, 1441006 (2014).

E (arb. units) Ω P P Localized edge states in a 1D chain 2 0.0012 Ω=-0.707 Ω= 0.707 0.3 1 0.0008 0.2 0-1 0.0004 0.1-2 (a) -3-2 -1 0 1 2 3 ka 0D Edge states 0.0000 (b) 20 80 100 0.0 Cavity index Signature of non-trivial topology: robustness to disorder 100 10 Edge states δφ/φ=0 δφ/φ=0.25 δφ/φ=0.5 1 0.1-3 -2-1 0 1 2 3 Ω VY, Int. J. Mod. Phys. B 28, 1441006 (2014).

Physical realization Coupled-cavities connected with alternating NRI and PRI waveguides in a plasmonic host. 1D array of sinusoidally coupled waveguides VY, EPJ Quant. Tech. 2, 6 (2015).

Dirac physics in metamaterials

Photon Energy (ev) Dirac point in the dispersion relation of an optical metamaterial Orthorhombic lattice of close-packed gold nanoclusters Dispersion lines of the metamaterial 3.5 3.0 2.5 =1 =3 =4 =5 =8 Dielectric with permittivity ε 2.0 1.5 Re(n eff )>0 Single gold particle of 8nm radius Average cluster radius: 43nm 1.0-0.030-0.015 0.000 0.015 0.030 k z (nm -1 ) Dirac singularity: dispersion relations for a massless relativistic particle Re(n eff )<0 VY and AG Vanakaras, PRB 84, 045128 (2011); ibid, PRB 84, 085119 (2011).

Transmittance Absorbance Reflectance k z (nm -1 ) Simulation of light transmission around the Dirac point Transmitted wave 0.04 Real dispersion lines + T,R,A for a finite slab 0.02 0.00-0.02-0.04 (a) Incident wave 0.6 0.3 (b) Reflected wave 0.9 0.6 0.3 (c) 1 plane 2 planes 4 planes 8 planes 10-5 10-10 10-15 10-20 (d) 1.0 1.5 2.0 2.5 3.0 3.5 Photon Energy (ev) VY and AG Vanakaras, PRB 84, 045128 (2011); ibid, PRB 84, 085119 (2011).

Photon Energy (ev) Dirac equation for massless particles Metamaterial response described by the Dirac equation 0 ivd x 1 1 ( D) ivd x 0 2 2 3.5 3.0 Dispersion lines Re Im Im (approx.) v v iv D D D 2.5 Metamaterial response calculated by the electrodynamic solution 2.0 1.5 1.0-0.030-0.015 0.000 0.015 0.030 k z (nm -1 ) Offset due to impedance mismatch between air and metamaterial which is not taken into account by the Dirac model. VY and AG Vanakaras, PRB 84, 045128 (2011); ibid, PRB 84, 085119 (2011).

Conclusions Photonic simulators for the IQHE, FQHE, topological insulators and Majorana-like edge states. Realization in the microwave regime via coupled cavity arrays, transmission lines, supeconducting QED systems, dielectric waveguides, etc. Photonic tight-binding models for coupled cavities (framework of the EM Green s tensor dyadic) reproduce the tight-binding Hamiltonians for topological atomic matter. Photons, unlike electrons interact very weakly with each other (with pros and cons)

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