Anomalous Floquet topological systems with ultracold atoms Monika Aidelsburger Ludwig-Maximilians Universität München Munich Center for Quantum Science & Technology GEFÖRDERT VOM für Bildung und Forschung
Quantum simulation with ultracold atoms in optical lattices: • Atoms confined in periodic potentials → \/\/\/\/\ λ/2 Bose-Einstein Condensate: I. Bloch et al. Rev. Mod. Phys. 80, 885 (2008); C. Gross and I. Bloch, Science 357 (2017)
Quantum simulation with ultracold atoms in optical lattices: • Atoms confined in periodic potentialsババババλ/2 • Quantum simulation of Hubbard models • Access to local observables using quantum gas microscopes
I. Bloch et al. Rev. Mod. Phys. 80, 885 (2008); C. Gross and I. Bloch, Science 357 (2017)
Anti-ferromagnetic correlations in the Fermi-Hubbard model: A. Maruzenko, ... M. Greiner, Nature (2017) Thermalization of isolated quantum-many body systems: J. Y. Choi, ..., I. Bloch, Science 352, 1547 (2016)
Integer & fractional quantum Hall insulators Magnetic Field (T) K. Klitzing, Rev. Mod. Phys. (1986) Stormer et al., Rev. Mod. Phys. (1999) Topological insulators in 2D & 3D M. König et al., Science (2007) A. Roth et al., Science (2009) Weyl semimetals L. Lu et al., Science (2015) S.-Y. Xu et al., Science (2015)
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Review: X.-L. Qi & S.-C. Zhang, Rev. Mod. Phys. (2011) How to engineer topological phases?
• Time-periodic driven Hamiltonian Â(t) = Â(t + T) • Stroboscopic time evolution governed by effective Floquet Hamiltonian  Ĥ F
Engineer H f with topological properties! N. Goldman et al. PRX (2014); M. Bukov et al. Adv. in Phys. (2015); A. Eckardt, Rev. Mod. Phys. (2017)

Hofstadter model
Harper, Proc. Phys. Soc., Sect.A 68, 874 (1955) Azbel, Zh. Eksp. Teor. Fiz. 46, 929 (1964) Hofstadter, PRB 14, 2239 (1976) 2π Φ Зπ/2 Φ T
Review: N. Cooper et al. Rev. Mod. Phys. 91, 015005 (2019) Haldane model Haldane, PRL 61, 2015 (1988)
MA et al., PRL (2013); H. Miyake et al., PRL (2013) E. M. Tai et al., Nature (2017) G. Jotzu et al., Nature (2014) ;Tarnowski et al., Nat. Comm. (2019) |u μ (q)) : periodic Bloch function µ: band index Chern number: Ωµ = i ( {∂qx ∪µ |∂qy ∪µ ⟩ — {∂qy ∪µ |∂qx ∪µ ⟩ Berry curvature: M.  Atala, et al., Nat. Phys. (2013); L. Duca et al., Science (2015) G. Jotzu et al., Nature (2014); M. A. et al., Nature Phys. (2015) N. Fläschner, Science (2016); T. Li, Science (2016) T ARNOWSKI ET AL., N AT. C OMM. (2019); L. Asteria et al., Nat. Phys. (2019); B. Rem et al., Nat. Phys. (2019); .....
WEITENBERG/SENGSTOCK Semiclassical dynamics •  Bloch oscillations captured by band velocity µ: band index •  Anomalous transverse velocity proportional to Berry curv. Ω μ Karplus & Luttinger, Phys. Rev. (1954); D. Xiao et al., Rev. Mod. Phys. (2010) A. Dauphin & N. Goldman, PRL (2013); H. Price & N. Cooper, PRA (2012)
Semiclassical dynamics •  Bloch oscillations captured by band velocity µ: band index Anomalous velocity • Anomalous transverse velocity proportional to Berry curv. Ω μ Prepare condensate at momentum q and probe Berry curvature locally! Karplus & Luttinger, Phys. Rev. (1954); D. Xiao et al., Rev. Mod. Phys. (2010) A. Dauphin & N. Goldman, PRL (2013); H. Price & N. Cooper, PRA (2012)
Contribution from band velocity vanishes ∂Eμ/∂kx,yd²k = 0 Quasimomentum k Transverse center-of-mass motion ( th band): M. A. et al., Nature Phys. (2015) See also: G. Jotzu et al., Nature (2014)
M. A. et al., Nature Phys. (2015) See also: G. Jotzu et al., Nature (2014) Can we go beyond emulating static topological systems? Floquet engineering: • High-frequency limit: → known bulk-edge correspondence • Chern number uniquely determines # net edge modes in the gap Beyond high-frequency limit: T. Kitagawa et al., PRB 82, 235114 (2010) M. Rudner et al., PRX 3, 031005 (2013) • Edge modes, but zero Chern numbers! • Robust edge transport  ←→ localized bulk!
→ two degenerate flat bands at ε=0, with trivial topology! • Topology not determined by properties of Heff ! • Full time-dependence is important • Non-trivial winding of quasi- energy spectrum T. Kitagawa et al., Phys. Rev. B 82, 235114 (2010), M. Rudner et al., PRX 3, 031005 (2013)
C ００００００００ ００００００００ ００００００００ 0 0 0 0 0 0 0 0 0 0 0 0 9--O O O O O ００００００００ ０００００ Q-φ 0 ００００ ０ � ０ 0 0 0 0 0 0 0 ００００００００ O  O  O  O  O  O - 0.8 - 0.4 → Robust edge transport Cold atoms: • Reveal bulk and edge-state physics ● Study many-body phases S. Mukherjee et al., Nat. Commun. 8, 13918 (2017), L. J. Maczewsky et al., Nat. Commun. 8, 13756 (2017)
How to characterize the topological properties? Winding number ebunts
How to characterize the topological properties? Winding number ebunts T. Kitagawa et al., Phys. Rev. B 82, 235114 (2010); al., PRL 122, 253601 (2019) Experimental realization
Continuous modulation scheme: Modulate NN tunnelings betweenｽJ and J T. Kitagawa et al., PRB 82, 235114 (2010)
K. Wintersperger, ...., M.A., Nature Phys. 16, 1058-1063 (2020)
K. Wintersperger, ...., M.A., Nature Phys. 16, 1058-1063 (2020)
Hold time
Г-point: Modulation amplitude  m
• We always measure the smaller energy gap:  min(g°, g™) • In the high-frequency limit we probe go K. Wintersperger, ...., M.A., Nature Phys. 16, 1058-1063 (2020) N. Ünal et al., PRL 122, 253601 (2019) • We always measure the smaller energy gap:  min(g°, g™) • In the high-frequency limit we probe go • Transition, when g 0 = g T = hw/2 N. Ünal et al., PRL 122, 253601 (2019) K. Wintersperger, ...., M.A., Nature Phys. 16, 1058-1063 (2020)
Г-point: Modulation amplitude  m Determine bulk geometric properties
Band-touching singularities
Band-touching singularities • Consider 3D-parameter space: • Topological charge describes change in Winding number Topological charge Qs = ±1 F. Nathan et al, New J. Phys. 17, 125014 (2015); N. Ünal et al., PRL 122, 253601 (2019)
Measure Berry flux associated with Qs Measure Berry-flux locally shortly before and after the gap-closing point!
Anomalous transverse velocity proportional to Berry curv. Ωμ F. Nathan et al, New J. Phys. 17, 125014 (2015); N. Ünal et al., PRL 122, 253601 (2019)
Г-point: Complete set of topological invariants! K. Wintersperger, ...., M.A., Nature Phys. 16, 1058 (2020) Can we observe topological edge modes?
Mancini, ..., Fallani, Science 349 (2015) Stuhl, ..., Spielman, Science 349 (2015) See also: Chalopin, ..., Nascimbène, Nature Physics (2020)
Realizing a sharp edge: DMD + Width of the edge: 2-3 lattice sites!
Lattice depth: 6Er; λ=745nm; ωｧ= 2π × 17Hz; ω ε = 2π × 250Hz
f=7kHz, m=0.25
Bulk K. WINTERSPERGER, ...., MA, Nature Phys. 16, 1058-1063 (2020) Edge

Outlook:
• Interplay of topology & disorder • Realization of many-body top. phases Bulk expansion in the modulated lattice:  ∨band Haldane: Anomalous: C. A. BRACAMONTES ET AL. Phys. Rev. Lett. 128 (2022)

Experiment
Christoph Braun, Karen Wintersperger, Alexander Hesse Raphaël Saint-Jalm Immanuel Bloch Theory anomalous Floquet top. systems
• Nathan Goldman (ULB Brussels) • Marco Di Liberto (U of Padova) • André Eckardt (TU Berlin) K, Wintersperger, ..., M.A., Nature Phys. 16, 1058-1063 (2020); Ch. Braun et al. (in preparation)