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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