Photonic Topological InsulatorsY. Plotnik1, J.M. Zeuner2, M.C. Rechtsman1, Y. Lumer1, S. Nolte2, M. Segev1, A. Szameit21Department of Physics, Technion – Israel Institute of Technology, Haifa, Israel2Institute of Applied Physics, Friedrich-Schiller-Universität, Jena, Germany

Outline-What are Topological Insulators?-Topological protection of photons?-How can we get unidirectional edge states in photonics? Floquet! -Description of our experimental system: photonic lattices-First observation of topological insulators-This is also the first observation of optical unidirectional edge states in optics! -Future directions

What are Topological insulators?

Valance band

Conduction band

Ef

Regular insulatorSpin Orbit Interaction:Topological Insulator

Scattering protectedEdge states

Kane and Mele, PRL (2005)

Magnetic field:Quantum Hall Effect

Unidirectional edge state

Von Klitzing et al. PRL (1980)

Main characteristics:•Edge conductance only•Immune to scattering/defects:

• No back-scattering• No scattering into the bulk

Only for Topological insulators:•No need for external fields

Motivation: No back scattering

No back scattering → Robust Photon transport!

Topological?

( ). .

i ik k k k

B z

u u dk Berry curvature dsp pg= Ñ =ò òòrr

Ef Ef

Background: photonic topological protectionby magnetic field

Raghu, Haldane PRL (2008)

Wang et. al., PRL (2008)

Unidirectional edge state:

Wang et. al. Nature (2009)

For optical frequencies, magnetic response is weak

von Klitzing et. al., PRL (1980) Kane and Mele, PRL (2005) We need a type of Kane-Mele transition,but how, without Kramers’ degeneracy ?

We need a solution without a magnetic field

(1) Hafezi, Demler, Lukin, Taylor, Nature Phys. (2011): aperiodic coupled resonator system (2) Umucalilar and Carusotto, PRA (2011): using polarization as spin in PCs(3) Fang, Yu, Fan, Nature Photon. (2012): electrical modulation of refractive index in PCs(4) Khanikev et. al. Nature Mat. (2012): birefringent metamaterials

Quantum hall No magnetic field Topological Insulator

Enter Floquet Topological Insulators

Lindner, Refael, Galitski, Nature Phys. (2011).

We can explicitly break TR by modulating! New Floquet eigenvalue equation:

Gu, Fertig, Arovas, Auerbach, PRL (2011).

Kitagawa, Berg, Rudner, Demler, PRB (2010).

+

( ) ( )H t H t T= +

ß

Experimental system: photonic lattices

Peleg et. al., PRL (2007)

Paraxial Schrödinger equation:

Array of coupled waveguides

·

· 0t

t

E E B

B B B

re

me

¶¶

¶¶

Ñ Ñ

Ñ ´ =Ñ

= ´ =-

=( )0i k x teE wy -=+ + ParaxialapproximationField envelopeMaxwell

=

Helical rotation induces a gauge field

' cos

' sin

'

x x R z

y y R z

z z

= + W

= + W

=

Paraxial Schrödingerequation

Coordinate Transformation

+

( )( ) ( ) 2 20 0

0 0

2 ,12 2

k n x y k Rz k ni i zAy y y yD W¶ = Ñ+ - -

( ) ( )0 sin ,cosz k R z zA = W W W

( ) ( )· †

,

nmi zn m

n m

z te rAH y y=å

Tight Binding Model (Peierls substitution)

Graphene opens a Floquet gap for helical waveguides

kx

ky

Band gap

kxa

Top edge

Edge states

Bottom edge

kxa

Experimental results: rectangular arraysMicroscope image

- No scattering from the corner- Armchair edge confinement

“Time”-domain simulations

Experimental results: group velocity vs. helix radius, R

R = 0µm(b) R = 2µm(c) R = 4µm(d) R = 6µm(e)

R = 8µm(f) R =10µm(g) R = 12µm(h) R = 14µm(i) R = 16µm(j)

(a)

R =10µmR =0R,

b c d e f g h i j

R = 0µm

Experimental results: triangular arrays with defects

missing waveguide R = 8 µmz = 10cm

Interactions: focusing nonlinearity gives solitons

kxky

Band gap

Y. Lumer et. al., (in preparation)

- Disorder: Topological Anderson insulator?- Topological cloak?

- What effect do interactions have on edge states?- many modes on-site.

Conclusion and Future work

- Non-scattering in optoelectronics

- First Optical Topological Insulator- First robust one way optical edge states (without any magnetic field!)Future Work:

Acknowledgments

Discussions: Daniel Podolsky

Challenge of scaling down: Faraday effect is weakFaraday effect Largest Verdet constant(e.g. in TGG) is ~100

Optical wavelengths are the keyto all nanophotonics applications

The effect is too weak.We need another way!

·

rad

T m

Theoretical proposals(1) Two copies of the QHE

Hafezi, Demler, Lukin, Taylor, Nature Phys. (2005).

(2) Modulation to break TR

Other theoretical papers in different systems:(3) Koch, Houck, Le Hur, Girvin, PRA (2010): cavity QED system(4) Umucalilar and Carusotto, PRA (2011): using spin as polarization in PCs(5) Khanikev et. al. Nature Mat. (2012): birefringent metamaterials

Fang, Yu, Fan, Nature Photon. (2012).