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 Technologies, ICTP Trieste, September 2017

Overview 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 Reviews of topological photonics: Review of quantum fluids of light: Lu, Joannopoulos, & Solja č i ć , Nature Physics 12 , 626 (2016) Carusotto & Ciuti, RMP 85 , 299 (2013) Lu, Joannopoulos, & Solja č i ć , Nature Photonics 8 , 821 (2014) …and ours coming later this year…

Photonics Carusotto & Ciuti, RMP 85 , 299 (2013) Always bosons Tuneable effective interactions Designer structures (mediated by nonlinearities of the medium) (Fabrication imperfections) Access field in momentum Access field in real space Photon pumping and losses space (far-field imaging) (near-field imaging) Detector Detector System System

Topological Photonics Remember that we will be talking about: 1. Different parts of the EM spectrum —> very different physical systems β β

Topological Photonics Remember that we will be talking about: 2. Mostly about classical effects Properties of waves, not of quantum mechanics H = Ψ † H Ψ Lectures 1 & 2: Quadratic Hamiltonians ˆ classification of topological properties of this matrix H � � = � � � But more generally, we can talk about topology if a system obeys a linear equation: (where ω is the normal mode frequency), Topology in classical photonics, phononics, mechanics… e.g. Introduction to topological classical mechanics: Huber et al., Nature Physics 12, 621–623 (2016)

Lecture 4 •How can we engineer topology for photons? • Quantum Hall systems • Quantum spin Hall systems • SSH Model & Topological Pumps • Topological superconductors? •How can we probe topology with photons? •Future perspectives

Lecture 4 •How can we engineer topology for photons? • Quantum Hall systems • Quantum spin Hall systems • SSH Model & Topological Pumps • Topological superconductors? •How can we probe topology with photons? •Future perspectives

Photonic quantum Hall system 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) Maxwell’s equations without source: Assuming linear, isotropic, loss-free medium, � � E = � ∂ B � � H = ∂ D B = µ H ∂ t , D = ε E ∂ t permittivity permeability � · D = 0 � · B = 0 Find normal modes How to make this topological? � � � � � � � � � � � � � � � � ε ( r ) − 1 0 ε ( r ) 0 0 0 �� E E �� E E = ω = ω i i µ ( r ) − 1 0 0 µ ( r ) 0 0 ��� H H ��� H H Concept: Try to engineer photonic energy bands with non-trivial Chern number • Electrons in a lattice —> photons in a periodic structure • Magnetic field —> time-reversal symmetry breaking ( magneto-optical materials ) Particle- Time- Chiral hole reversal

Magneto-optic material magneto-optic material in Applications in optical isolators presence of magnetic field Faraday effect Figure from: https://en.wikipedia.org/wiki/ Figure from: http://www.fiber-optic-components.com/tag/optical-isolator Faraday_effect#/media/File:Faraday-effect.svg

Magnetic Photonic Crystals a Experimental realisation of Haldane-Raghu idea by Scatterer of Antenna B variable length l MIT group: Wang et al., Nature 461, 772–775 (2009). CES waveguide Antenna A • Microwaves propagate through a lattice of ferrite Metal wall rods [lattice optimised numerically] • TRS broken by strong magnetic field, coupling to z ferrite rods —> gyromagnetic Gyromagnetic 2D b y x photonic crystal A b a Many related experiments since… see e.g. Lu et al, Nature 4cm Phys., 12, 626 (2016) Waveguide Chern bands b A 0.5 b a ) Wavevector (2 π / [NB photonic crystals also 0 used for first observation of Weyl points!} 0 1 -2 1 Lu, et al., Science 349 , 622 (2015) -0.5 0 1 2 3 4 5 6 Frequency (GHz)

The end of the story? 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! • Floquet engineering c.f. Lecture 3! • Synthetic dimensions • ….

Floquet engineering periodic V ( t + T ) = V ( t ) static Very(!) brief intro to Floquet theory: driving T = 2 π / ω System modulated periodically in time H = H 0 + V ( t ) � T � � U ( T ) = T exp dtH ( t ) − i 0 Stroboscopic evolution captured by time-independent effective Hamiltonian: and can be in di ff erent topological classes H 0 H e ff U ( T ) = exp ( − iTH e ff ) Typically assume high-frequency driving ( all other frequencies) and then calculate ω � Z T effective Hamiltonian perturbatively, e.g. at lowest order: H e ff = 1 H ( t )d t T 0 Concept: Design driving to engineer an artificial magnetic field in the effective Hamiltonian For lots more about Floquet theory, see e.g: M. Bukov et al. Advances in Physics, 64, 139, (2015) N. Goldman et al., arXiv:1507.07805

Shaking: propagating waveguides ⌘ 2 ⇣ ω r ⇥ r ⇥ E = ε E , Propagation of light in source-free non-magnetic material c In the “paraxial” approximation, E ( x, y, z ) = ψ ( x, y, z ) exp [ ik 0 z ] x k 0 � k x,y for light propagating along z slowly varying envelope ⊥ ψ � k 0 ∆ n i ∂ z ψ = � 1 refractive index deviation ε = ε 1 + ∆ ε ( x, y, z ) r 2 ψ . n = √ ε 2 k 0 n 1 background refractive index b Like Schrodinger but with roles of t and z reversed! spatially-varying refractive index Shaking in along direction of propagation, z time Rechtsman, et al., Nature 496, 196 (2013) Pumping on edge : chiral edge state c d a b 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). 
 β β

Dynamical modulation: resonator lattices Lattice sites are resonators, e.g. Coupled together e.g. by waveguides Resonator or auxiliary resonators Lattice ω A Often can be well-described by tight- binding Hamiltonians optical/near-IR ring microwave RLC resonators resonators (d) Combining superlattices and resonant driving Dynamical modulation for resonator lattices General concept same as: superlattice + resonant modulation (cold atoms) ω A ω B Fang, et al., Nat. Photon. 6, 782 (2012) Proposed model: a � a † � b † H = v A i a i + v B j b j − 2 ϕ 3 ϕ − 4 ϕ ϕ i j modulated � V cos ( V t + f ij )( a † i b j + b † − ϕ − 3 ϕ 4 ϕ 2 ϕ j a i ) + ( hoppings k ij l − 2 ϕ 3 ϕ − 4 ϕ ϕ In rotating wave approx, in rotating frame: V � 2 ( e − i f ij c † i c j + e i f ij c † H = j c i ) ( − ϕ 2 ϕ − 3 ϕ 4 ϕ k ij l

Synthetic Dimensions in Photonics Concept: 1. Identify a set of states and reinterpret as sites in a synthetic dimension First proposed by: 0 1 2 3 4 Boada et al., PRL, 108, 133001 (2012), Celi et al., PRL, 112, 043001 (2014) 2. Couple these modes to simulate a tight-binding “hopping” J e − i φ J e i φ w 0 1 2 3 4 3. Combine with real spatial dimensions or more synthetic dimensions as desired As yet no experimental realisation for photons, but interesting proposals… Optomechanics — photons & phonons Optical cavities — orbital angular momentum Schmidt et al., Optica 2 , 635 (2015) Luo et al., Nature Comm. 6, 7704 (2015) j j +1 j j +1 BS 1 BS 1 BS j +1 j +1 –1 –1 SLM 2 SLM 1 j j SLM 1 SLM 2 j j j +1 j +1 BS 4 BS 2 BS 2 BS 4 +1 +1 j +1 j BS 3 BS 3

Synthetic Dimensions in Photonics Ozawa et al, PRA 93 , 043827 (2016) Ring resonator lattice — different modes of ring resonators Yuan et al. Opt. Lett. 41 , 741 (2016) Ozawa et al., PRL 118 , 013601 (2017) frequency ω w = ω w 0 + ∆ ω ( | w | − w 0 ) + . . . comb free spectral ∆ ω = 2 π c/n e ff R angular range (FSR) momentum 10 µ m Couple modes by modulating the ∆ ω refractive index with frequency Figure from: Hafezi et al, Nat. Photon. 7, 1001, (2013) e.g. through material or electro-optic χ (3) χ (2) nonlinearities phase modulators 4D topology? Applications in on-chip optical isolators? { { { {

Lecture 4 •How can we engineer topology for photons? • Quantum Hall systems • Quantum spin Hall systems • SSH Model & Topological Pumps • Topological superconductors? •How can we probe topology with photons? •Future perspectives