photonic topological insulators y. plotnik 1, j.m. zeuner 2, m.c. rechtsman 1, y. lumer 1, s. nolte...
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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).