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

LiV2O4, MgTi2O4, 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 coupled-

cavity chain with metamaterial couplings

•Negative-zero-positive index

metamaterials

Photonic Analog of Integer Quantum Hall Effect (IQHE)

Topological insulators vs Band insulators

MZ Hasan and CL Kane, RMP 82, 3045 (2010)

genus g=0

genus g=1

Chern number:

Berry flux

Berry curvature

Bloch wave function of

electrons

2 /xy mn Ne h

0mn

1mn

Photonic graphene: 2D photonic crystal with magnetoelectric materials

Photonic graphene: honeycomb lattice

of magnetoelectric rods

0 0

ˆ 0 0

0 0

FDM Haldane and S Raghu, PRL 100, 013904 (2008)

Time-reversal TR symmetry breaking

leads to a photonic topological insulator

0

ˆ 0

0 0

i

i

Chiral edge states in ribbons of photonic graphene

Y Poo et al, PRL 106, 093903 (2011)

S21: forward direction

S12: backward direction

Band structure of slab and surface guides

modes of a 5-layer sample of the crystal.

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.

0 0

ˆ 0 0

0 0

VY, J Opt, submitted

0

ˆ 0

0 0

i

i

Time-reversal TR symmetry breaking leads to

non-reciprocal bands: one-way slab and surface

modes.

Anomalous quantum Hall effect: spin-polarized electron gas with Rashba spin-orbit coupling

2D spin-polarized electron gas with spin-orbit coupling

(SOC) described by the time-reversal symmetry

breaking Hamiltonian:

Magnetic field B

Kinetic

energy

Spin-orbit

coupling

(SOC)

Exchange field

(Ferromagnetic

electron gas)

D Xiao, MC Chang, Q Miu, RMP 82, 1959 (2010)

n= 2

n=-2

•Topological phase of

matter

•Thin Fe films

… and its photonic analog

Spatial dispersion

due to chirality

VY, PRB 83, 113101 (2011)

Quantum system

2D spin-polarized electron gas with SO coupling:

Photonic system

Chiral medium with longitudinal excitations (e.g.,plasmons) ˆi D E k E

( ) 0

ˆ 0 ( ) 0

0 ( )

i

i

Faraday activity which explicitly

breaks time-reversal symmetry

By solving Maxwell’s equations it turns out:

Supports longitudinal

excitation at : ( ) 0L L

Dispersion relation

Berry curvature

Chern number

Topological photonic modes

2D electron gas

Realization of the photonic analog: chiral metamaterial

Chiral lattice of plasmonic spheres

2

2( ) 1

p

Effective-medium approximation

Electrodynamic solution (LKKR)

Time-reversal symmetry breaking

with Faraday rotation (iη)

VY, PRB 83, 113101 (2011)

Further analogy: plasmonics and spintronics

Chiral lattice of plasmonic spheres

2

2( ) 1

p

Polar semiconductor with Rashba SOC

BiTeI

K Ishizaka et al., Nat Mater 10, 521 (2011)

0.55

0.56

0.57

-1.0 -0.5 0.0 0.5 1.0

4kzd/π

ω/ω

p

L Rk k k

kk k

BiTeI

BiTeI

S Datta, B. Das, APL 56, 665 (1990)

Photonic Analog of the Fractional Quantum Hall Effect (FQHE)

Bulk electron band structure

Slab electron band structure

Bosons/ Spinless fermions in flat bands with non-trivial topology: a framework for FQHE

K Sun et al., PRL 106, 236803 (2011)

Flat bands with non-zero Chern number (non-trivial topology) may substitute

Landau levels in the QHE.

Strongly interacting bosons in flat bands with non-trivial topology: occurrence of FQHE

YF Wang et al., PRL 107, 146803 (2011)

Spectrum gaps for the ½-FQHE 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

VY, New J. Physics 14, 113017 (2012).

Atomic Hamiltonian

Superconducting QED Hamiltonian:

Jaynes-Cummings-Hubbard model

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

GV Eleftheriades, IEEE MWCL 13, 51 (2003)VY, New J. Physics 14, 113017 (2012).

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:

exp( / )

: spin operator

: complex conjugation

y

y

i S K

S

K

For a Hamiltonian H which preserves TRS we have: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

Protected

topological

surface states

Non-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)

MZ Hasan and CL Kane, RMP 82, 3045 (2010)

Strong TI

(Surface states robust to disorder)

L Fu, CL Kane & EJ Mele, PRL 98, 106803 (2007)

Weak TI

Strong TI

4-band tight-binding model of s states on a

diamond lattice with SOC

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 C4 symmetry, we have:

Tetragonal crystal of atoms with px and py 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 dx and dy

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.

FT of the EM Green’s tensor Bloch eigenfunction of the polarization field

VY, Phys. Rev. B 84, 195126 (2011).

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 Z2 topological invariant is ν0 = 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!

Metals should be the obvious choice. However, when metals are described as plasma, i.e.,

which is the case in the optical regime, they are extremely lossy.

In the infrared regime and below (e.g., microwave regime) they are almost losses but not a plasma!

2

( ) 1( )

p

i

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.

•Therefore, , where ωp lies in the GHz regime.

2

2( ) 1

p

JB Pendry et al., PRL 76, 4773 (1996)

Metamaterial design for a ‘photonic’ topological crystalline insulator

Uniaxial cavity

Guiding elements

for orientation-

dependent intra-

layer coupling

between the cavities

Artificial plasma

Unit cell

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

VY, Int. J. Mod. Phys. B 28, 1441006 (2014).

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:

Localized edge states in a 1D chain

VY, Int. J. Mod. Phys. B 28, 1441006 (2014).

0D Edge states

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

Ω

ka

(a)

0.0000

0.0004

0.0008

0.0012

20 80 1000.0

0.1

0.2

0.3

Ω=-0.707

(b)

|P|

Ω= 0.707

|P|

Cavity index

-3 -2 -1 0 1 2 3

0.1

1

10

100

Edge states

|E

| (ar

b.

un

its)

Ω

δφ/φ=0

δφ/φ=0.25

δφ/φ=0.5Signature of non-trivial topology: robustness to disorder

Physical realization

VY, EPJ Quant. Tech. 2, 6 (2015).

Coupled-cavities connected with alternating NRI and PRI waveguides in a plasmonic host.

1D array of sinusoidally coupled waveguides

Dirac physics in metamaterials

Dirac point in the dispersion relation of an optical metamaterial

Dispersion lines of the metamaterial

1.0

1.5

2.0

2.5

3.0

3.5

-0.030 -0.015 0.000 0.015 0.030

kz (nm

-1)

Photo

n E

ner

gy (

eV)

=1

=3

=4

=5

=8

Orthorhombic lattice of close-packed gold nanoclusters

Dirac singularity: dispersion relations for a massless

relativistic particle

Re(neff )<0

Re(neff )>0

Average cluster radius: 43nm

Single gold particle of 8nm radius

Dielectric with

permittivity ε

VY and AG Vanakaras, PRB 84, 045128 (2011); ibid, PRB 84, 085119 (2011).

Simulation of light transmission around the Dirac point

-0.04

-0.02

0.00

0.02

0.04

0.3

0.6

0.3

0.6

0.9

1.0 1.5 2.0 2.5 3.0 3.510

-20

10-15

10-10

10-5

(d)

(c)

(b)

k z (

nm

-1)

(a)

Ref

lect

ance

Abso

rban

ce

Tra

nsm

itta

nce

Photon Energy (eV)

1 plane

2 planes

4 planes

8 planes

Real dispersion lines + T,R,A for a finite slabTransmitted wave

Reflected wave

Incident

wave

VY and AG Vanakaras, PRB 84, 045128 (2011); ibid, PRB 84, 085119 (2011).

Dirac equation for massless particles

1.0

1.5

2.0

2.5

3.0

3.5

-0.030 -0.015 0.000 0.015 0.030

kz (nm

-1)

Ph

oto

n E

ner

gy

(eV

)

Re

Im

Im (approx.)

Dispersion lines

Metamaterial response

calculated by the

electrodynamic solution

1 1

2 2

0( )

0

D x

D

D x

iv

iv

Metamaterial response described by the Dirac

equation

D D Dv v iv

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-likeedge 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’stensor dyadic) reproduce the tight-binding Hamiltonians for topological atomicmatter.

•Photons, unlike electrons interact very weakly with each other (with pros and cons)

Thank you for your attention