Photonic Topological Insulators
Y. Plotnik1, J.M. Zeuner2, M.C. Rechtsman1, Y. Lumer1, S. Nolte2, M. Segev1, A. Szameit2
1Department
of Physics, Technion – Israel Institute of Technology, Haifa, Israel
2Institute 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?
Regular insulator
Magnetic field:
Quantum Hall Effect
Spin Orbit Interaction:
Topological Insulator
Unidirectional
edge state
Scattering protected
Edge states
Conduction
band
Ef
Valance
band
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
Von Klitzing et al. PRL (1980)
Kane and Mele, PRL (2005)
Motivation: No back scattering
No back scattering → Robust Photon transport!
Topological?
g=
i
p
ò
r
r
uk Ñ k uk dk =
i
p
òò (Berry curvature)ds
k
B. z .
Ef
Ef
Background: photonic topological protection
by magnetic field
For optical frequencies,
magnetic response is weak
Raghu, Haldane PRL (2008)
Unidirectional edge state:
Wang et. al. Nature (2009)
Wang et. al., PRL (2008)
We need a solution without a magnetic field
Quantum hall
No magnetic field Topological Insulator
Kane and Mele, PRL (2005)
von Klitzing et. al., PRL (1980)
We need a type of Kane-Mele transition,
but how, without Kramers’ degeneracy ?
(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
Enter Floquet Topological Insulators
We can explicitly break TR by modulating!
H (t )= H (t + T )
New Floquet eigenvalue equation:
+
ß
Kitagawa, Berg, Rudner, Demler, PRB (2010).
Lindner, Refael, Galitski, Nature Phys. (2011).
Gu, Fertig, Arovas, Auerbach,
PRL (2011).
Experimental system: photonic lattices
Array of coupled waveguides
Peleg et. al., PRL (2007)
Ñ·E =
r
e
Ñ´ E = -
¶
¶t
B
+ E = y ei(k0 x- wt ) + Paraxial
Ñ·B = 0 Ñ ´ B = me B
approximation
Maxwell
Field envelope
¶
¶t
=
Paraxial Schrödinger equation:
Helical rotation induces a gauge field
Paraxial
Schrödinger
equation
Coordinate
Transformation
+
Tight Binding Model (Peierls substitution)
x ' = x + R cos Wz
y ' = y + R sin Wz
H (z ) =
z'= z
å
n,m
i¶ z y =
1
2 k0
2
(iÑ + A (z ))
y-
k0D n(x , y )
n0
A(z)= k0 RW(sin Wz,cos Wz)
y-
k0 R2W2
2
y
te
iA(z )·rnm
y n†y m
Graphene opens a Floquet gap for helical waveguides
Band gap
kx
ky
Edge states
Top
edge
kxa
Bottom
edge
kxa
Experimental results: rectangular arrays
Microscope image
- No scattering from the corner
- Armchair edge confinement
“Time”-domain simulations
Experimental results: group velocity vs. helix radius, R
(a)
(f)
R = 8µm
(b)
RR==0µm
0µm
(c)
R = 2µm
(d)
R = 4µm (e)
R = 6µm
(g)
R =10µm (h)
R = 12µm
(i)
R = 14µm (j)
R = 16µm
b c d e f g h i
R =0
R =10µm
R,
j
Experimental results: triangular arrays with defects
missing waveguide
R = 8 µm
z = 10cm
Interactions: focusing nonlinearity gives solitons
Band gap
kx
Y. Lumer et. al., (in preparation)
ky
Conclusion and Future work
- First Optical Topological Insulator
- First robust one way optical edge states (without any magnetic field!)
Future Work:
- Non-scattering in optoelectronics
- Topological cloak?
- Disorder: Topological Anderson insulator?
- What effect do interactions have on edge states?
- many modes on-site.
Acknowledgments
Discussions: Daniel Podolsky
Challenge of scaling down:
Faraday effect is weak
Faraday effect
Largest Verdet constant
rad
(e.g. in TGG) is ~100 T ·m
Optical wavelengths are the key
to all nanophotonics applications
The effect is too weak.
We need another way!
Theoretical proposals
(1) Two copies of the QHE
Hafezi, Demler, Lukin, Taylor,
Nature Phys. (2005).
(2) Modulation to break TR
Fang, Yu, Fan, Nature Photon.
(2012).
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