Topological Phases of Matter with Ultracold Atoms and Photons - - PowerPoint PPT Presentation
Topological Phases of Matter with Ultracold Atoms and Photons Hannah Price Currently : INO-CNR BEC Center & University of Trento, Italy From October : University of Birmingham, UK Advanced School and Workshop on Quantum Science and Quantum
Hannah Price
Currently: INO-CNR BEC Center & University of Trento, Italy From October: University of Birmingham, UK
Advanced School and Workshop on Quantum Science and Quantum Technologies, ICTP Trieste, September 2017
Lectures 1 & 2 Introduction to Topological Phases of Matter Lecture 3 Topological Phases of Matter with Ultracold Atoms Lecture 4 Topological Phases of Matter with Photons
Lu, Joannopoulos, & Soljačić, Nature Photonics 8, 821 (2014) Lu, Joannopoulos, & Soljačić, Nature Physics 12, 626 (2016) Carusotto & Ciuti, RMP 85, 299 (2013) Review of quantum fluids of light: Reviews of topological photonics: …and ours coming later this year…
Tuneable effective interactions (mediated by nonlinearities of the medium) Designer structures Always bosons Access field in momentum space (far-field imaging) Access field in real space (near-field imaging) (Fabrication imperfections) Photon pumping and losses
Detector System System Detector
Carusotto & Ciuti, RMP 85, 299 (2013)
Remember that we will be talking about:
β β
Remember that we will be talking about:
Lectures 1 & 2: Quadratic Hamiltonians ˆ
H = Ψ†HΨ
But more generally, we can talk about topology if a system obeys a linear equation: (where ω is the normal mode frequency),
H =
Properties of waves, not of quantum mechanics
Topology in classical photonics, phononics, mechanics…
e.g. Introduction to topological classical mechanics:
Huber et al., Nature Physics 12, 621–623 (2016)
E = ∂B ∂t , · D = 0 H = ∂D ∂t · B = 0 D = εE
B = µH
Assuming linear, isotropic, loss-free medium, Maxwell’s equations without source:
i
H
µ(r) E H
µ(r)−1
H
H
Topological photonics started with seminal theoretical works of Haldane and Raghu:
Haldane and Raghu, PRL 100, 013904 (2008) Raghu and Haldane, PRA 78, 033834 (2008)
Concept: Try to engineer photonic energy bands with non-trivial Chern number
Time- reversal Particle- hole Chiral How to make this topological? permittivity permeability
magneto-optic material in presence of magnetic field Applications in optical isolators
Figure from: http://www.fiber-optic-components.com/tag/optical-isolator Figure from: https://en.wikipedia.org/wiki/ Faraday_effect#/media/File:Faraday-effect.svg
Faraday effect
a b
y x z Antenna A Antenna B CES waveguide Metal wall Scatterer of variable length l
Gyromagnetic 2D photonic crystal
Wang et al., Nature 461, 772–775 (2009).
a b
A A
Frequency (GHz)
0.5 Wavevector (2π/ a)
b
1 2 3 4 5 6 1
1
Chern bands Many related experiments since… see e.g. Lu et al, Nature
Phys., 12, 626 (2016)
b 4cm
Waveguide
[NB photonic crystals also used for first observation of Weyl points!} Lu, et al., Science 349, 622 (2015)
Experimental realisation of Haldane-Raghu idea by MIT group:
rods [lattice optimised numerically]
ferrite rods —> gyromagnetic
Magneto-optical effect works well for breaking TRS at microwave frequencies but (i) this effect is weak at optical frequencies and (ii) having real magnetic fields is not good for many applications (e.g. on-chip devices)…. so we need other tricks!
c.f. Lecture 3!
U(T) = T exp
T dtH(t)
System modulated periodically in time
T = 2π/ω
For lots more about Floquet theory, see e.g:
H = H0 + V (t) V (t + T) = V (t)
periodic driving static
Concept: Design driving to engineer an artificial magnetic field in the effective Hamiltonian
U(T) = exp (−iTHeff)
Stroboscopic evolution captured by time-independent effective Hamiltonian: and can be in different topological classes
Heff H0 ω
Typically assume high-frequency driving ( all other frequencies) and then calculate effective Hamiltonian perturbatively, e.g. at lowest order:
Heff = 1 T Z T H(t)dt
Propagation of light in source-free non-magnetic material
r ⇥ r ⇥ E = ε ⇣ω c ⌘2 E,
In the “paraxial” approximation, for light propagating along z
E(x, y, z) = ψ(x, y, z) exp [ik0z] x
k0 kx,y
i∂zψ = 1 2k0 r2
⊥ψ k0∆n
n1 ψ.
slowly varying envelope refractive index deviation background refractive index
ε = ε1 + ∆ε(x, y, z)
n = √ε Shaking in time
Rechtsman, et al., Nature 496, 196 (2013)
β β
b
a b c d
Pumping on edge : chiral edge state spatially-varying refractive index along direction of propagation, z Propagating waveguides also recently used to realise anomalous Floquet topological states
Maczewsky et al., Nat. Comm. 8 (2017). Mukherjee et al., Nat. Comm. 8 (2017).
Like Schrodinger but with roles of t and z reversed!
Resonator Lattice Lattice sites are resonators, e.g.
Coupled together e.g. by waveguides
Often can be well-described by tight- binding Hamiltonians
ωA
microwave RLC resonators
resonators
(d) Combining superlattices and resonant driving
General concept same as: superlattice + resonant modulation (cold atoms) H = vA
a†
i ai + vB
b†
j bj
+
Vcos(Vt + fij)(a†
i bj + b† j ai)
(
a ωA ωB ϕ −2ϕ −3ϕ 3ϕ −2ϕ −ϕ 2ϕ −4ϕ −3ϕ 4ϕ 4ϕ 2ϕ −ϕ ϕ −4ϕ 3ϕ
H =
V 2 (e−ifijc†
i cj + eifijc† j ci)
(
In rotating wave approx, in rotating frame:
Dynamical modulation for resonator lattices Proposed model:
modulated hoppings
Fang, et al., Nat. Photon. 6, 782 (2012)
1 2 3 4
1 2 3 4
First proposed by: Boada et al., PRL, 108, 133001 (2012), Celi et al., PRL, 112, 043001 (2014)
Concept:
j j +1 BS –1 –1 +1 +1 BS4
j
BS4
j +1
BS1
j
BS3
j
BS3
j +1
SLM2
j +1
SLM2
j
BS2
j
BS1
j +1
BS2
j +1
SLM1
j +1
SLM1
j
Optical cavities — orbital angular momentum
Luo et al., Nature Comm. 6, 7704 (2015)
Optomechanics — photons & phonons
Schmidt et al., Optica 2, 635 (2015)
As yet no experimental realisation for photons, but interesting proposals…
10 µm
angular momentum frequency comb
ωw = ωw0 + ∆ω (|w| − w0) + . . .
∆ω = 2πc/neffR
free spectral range (FSR)
χ(3) χ(2)
Couple modes by modulating the refractive index with frequency
phase modulators e.g. through material nonlinearities
∆ω
Ring resonator lattice — different modes of ring resonators
Figure from: Hafezi et al, Nat.
Ozawa et al, PRA 93, 043827 (2016) Yuan et al. Opt. Lett. 41, 741 (2016) Ozawa et al., PRL 118, 013601 (2017)
4D topology? Applications in
isolators?
T 2 = +1
Remember from Lecture 2, for bosons there is no Kramer’s theorem as Problem: Two counter-propagating bosonic edge states (e.g. like in a topological insulator) will generally couple and backscatter —> not topologically-robust! Work-around solution: Design the system to suppress the inter-mode coupling Caveat: The following photonics set-ups are inspired by Class AII systems but they are not truly topological
Figure adapted from
314, 5806, 1692 (2006)
For fermions, two counter-propagating states on the same edge can be topologically- protected Time- reversal Particle- hole Chiral
Ingredients for a photonic “topological insulator”: 1) A “pseudo” spin-1/2?
e¬ e¬ x z y Hx Hx Ez Ez Hz Hy Hy
+ ¬
Ex Ey Hz Ex Ey TE TM TM TE <=> <=> ψ+ ψ¬
a
Usually, TE and TM modes have different wave-vectors, but in special metamaterials (where ε = μ), the wave-vectors are identical. Then:
Proposal: Khanikaev, et al. , Nature Materials 12, 233 (2013)
2) A “spin-orbit” coupling?
i
H
χ(r) χ(r)† µ(r) E H
|ψ¬| k||
χxy ≠ 0 χxy = 0
3.0 3.5 4.0 4.5 4.5 5.5 0.36 0.38 0.40 0.42 0.44 q||a0
a b
¬3.0 ¬3.5 ¬4.0 ¬5.0 ¬5.5 ¬4.5 K' K ψ+ ψ+ ψ¬ ψ¬ a0/2πc ω
|ψ+|
a
k||
χxy ≠ 0 χxy = 0
Bianisotropy
Experimental setup
z y x
Geometries of meta-atoms
6.2 mm 1 mm 27.6 mm 4.4 mm 19 mm 18.7 mm 6.2 mm 4.6 mm 6.2 mm 1 mm 26.8mm 5.4 mm 3 mm 6.2 mm Newtwork analyzer E5071C
Edge between PTI and POI
PTI and low index waveguide Edge without defect Edge with defect POI and low index waveguide Edge between PTI and POI
Experimental realisation (Hong Kong): Chen, et al., Nature Comm. 5, 5782 (2014) Microwave regime
2πα 2πα L1 + η L2 L2 L1 J
a c b
Port 3 Port 1 Port 4 Port 2 Link resonator Site resonator Probing waveguide 4 1 2 3 10 µm 30 µm x12 x34 R κin κex
Today, 67, 68–69 (2014)
Maryland group: Hafezi, et al., Nat. Photonics 7, 1001 (2013). Due to displacement of link resonators, photons travel a different distance and hence acquire a different phase from (1) to (2) than from (2) to (1) —> Analogous to the Peierls phase By making this displacement vary over the lattice, can engineer an artificial magnetic field
Φ
Quantum spin Hall system: 2 copies of Harper- Hofstadter model
Φ
For modes circulating clockwise in the site resonators (pseudo-spin-up) For modes circulating anti-clockwise in the site resonators (pseudo-spin-down)
Maryland group: Hafezi, et al., Nat. Photonics 7, 1001 (2013). Due to displacement of link resonators, photons travel a different distance and hence acquire a different phase from (1) to (2) than from (2) to (1) —> Analogous to the Peierls phase By making this displacement vary over the lattice, can engineer an artificial magnetic field
Φ
For modes circulating clockwise in the site resonators (pseudo-spin-up) Injecting light in pseudo- spin-up channel, see chiral edge states
Φ
For modes circulating anti-clockwise in the site resonators (pseudo-spin-down) but not protected — backscattering is just weak Quantum spin Hall system: 2 copies of Harper- Hofstadter model
Landau levels of photons
La
after a roundtrip —> Analogous to rotation of a gas (see Lecture 3) Chicago group: Schine, et al., Nature 534, 671 (2016).
l = 0 l = 3 l = 6 l = 9 l = 12 l = 15 l = 18 l = 21 l = 24 Mode –3 ΔL (μm)
32
(Residual harmonic trapping)
Coriolis force in rotating frame —> Lorentz force
Photons traversing the cavity in one direction experience opposite “magnetic field” to the opposite direction: quantum spin Hall
Particularly promising set-up for observing strongly-correlated states (when combined with atoms & Rydberg-EIT)
Zero mode DOS 10 20 30
In photonic crystals, metamaterials, exciton-polaritons, microwave resonator arrays, etc…
Dielectric resonator Selected zero mode Absorptive material Microwave antenna α - configuration β - configuration Interface A B t1 t2 d1 d2 n = ... ... –1 –2 –3 1 2 3 ... ...
e.g. Nice: Poli, et al., Nature Comm. 6, 6710 (2015) Spacing between resonators controls hopping e.g. Marcoussis: St-Jean, et al., arXiv:1704.07310
(g)
QWs DBR DBR
E-1578 (meV)
τ τ
τ τ exciton- polariton micropillars
Linewidth (μeV) 4 E - 1580 (meV) Position (μm)
P-bands
(c) (d) P = 0.1Pth P = 1.5Pth 10 2
Gap Gap
S-band D-bands
20 10 20 1
Lasing in a topological edge state!
Time- reversal Particle- hole Chiral
Remember that we exchange z and t So pumping in time is now pumping in space Spacing between waveguides controls hopping
Kraus et al, PRL, 109, 106402 (2012) [NB before the cold atom experiments]
1D Pump —> Dynamical 2D QH Effect (First Chern Number)
b
2D Pump —> Dynamical 4D QH Effect (Second Chern Number)
Zilberberg et al, arXiv:1705.08361
In cold atom experiments, probed bulk response, here probed edge states —> very complementary!
A nonlinear cavity under parametric driving can have a Hamiltonian of the form:
Hcavity = iχ(2) β∗ˆ a2 − βˆ a†2
Aligning cavities to form a lattice, the momentum-space Hamiltonian takes Hlattice = 1 2
Ψ†
k
ˆ Ψ−k A(k) B(k) B(−k)∗ A(−k)t ˆ Ψk ˆ Ψ†
−k
This reminds us of BdG in topological superconductors…. … but actually very different physics, e.g. no limit to occupancy of a state means there can be instabilities. Need new topological classification for bosons!
i.e. can inject two photons at a time
Peano, et al., Nature Comm. 7, 10779 (2016) & PRX, 6, 041026, (2016)
Can exploit instabilities to make non-reciprocal travelling-wave parametric amplifiers?
Also see Bardyn, et al., PRL 109, 253606 (2012) for proposal linked to Kitaev chain with parametric driving & strong interactions
Most straightforward: pump at the edge of the system to see topological edge states
Hafezi, et al., Nat. Photonics 7, 1001 (2013).
a b c d
Rechtsman, et al., Nature 496, 196 (2013)
a b
A A
Wang et al., Nature 461, 772–775 (2009).
See e.g. topological robustness to disorder
Loss
x 2πν1F ABZγ
BZ volume 1st Chern number external force pump
σxy = −e2 h X
n∈occupied
νn
in solid state:
measurements of currents and voltages Hall bar electrons fill bands up to Fermi level
With photons:
Proposal: Ozawa & Carusotto, PRL, 112, 133902, (2014)
∆Eband < γ < ∆Egap
{ { { {
e.g. combine synthetic gauge fields & photon blockade?
See e.g. Kapit, et al PRX (2014) See e.g. review of Carusotto et al, RMP (2013)
Lectures 1 & 2 Introduction to Topological Phases of Matter Lecture 3 Topological Phases of Matter with Ultracold Atoms Lecture 4 Topological Phases of Matter with Photons