Topological physics with a BEC: geometric pumping and edge states Hsin-I Lu with Max Schemmer, Benjamin K. Stuhl, Lauren M. Aycock, Dina Genkina, and Ian B. Spielman NTNU Condensed Matter Seminar 30 September 2015

2 Topological physics in condensed matter systems Hasan et al., Rev. Mod. Phys. 82, 3045 (2010) Topological insulators Nadj-Perge et al., Science 346, 602 (2014) Alicea et al., Nature Phys. 7, 413 (2011) 1D Topological superconductor Quantum Hall effect Hasan et al., Rev. Mod. Phys. 82, 3045 (2010) B

3 Probing topological features in cold atomic systems Aidelsburger et al., Nature Phys. 11, 162 (2015) Chern number of Hofstadter bands Berry phase, Berry curvature Wilson lines in hexagonal lattice Duca et al., Science 347, 288 (2015) Li et al., arXiv: (2015) Fläschner et al., arXiv: (2015) Topological band structure of Haldane model Jotzu et al., Nature 515, 237 (2014) Atala et al., Nature Phys. 9, 795 (2013) Zak phase of 1D bipartite lattice

4 Thouless, Phys. Rev. B 27, 6083 (1983) Quantized charge transport for topological band insulators Thouless “topological” charge pump

5 Thouless, Phys. Rev. B 27, 6083 (1983) Quantized charge transport for topological band insulators Thouless pump in cold atom insulators Lohse et al., arXiv: (2015) Nakajima et al., arXiv: (2015) Very recently

6 This talk: geometric pump with BEC Thouless pump This exp: geometric pump Band topology Quantized displacementLess than one site/cycle Lu et al., arXiv: (2015) Modulation of magnetization (polarization) Key difference Sensitive to Displacement

7 Origin of displacement Per-cycle displacement Filled bands: Displacement operator!

8 Geometric origin of displacement Lowest two bands during pump

9 Adiabatic potential during pumping One regime: non-moving lattice

10 Adiabatic potential during pumping The other regime-moving lattice

Experimental geometry 11

12 Effective magnetic field Effective Hamiltonian of double Raman+RF

Magnetic lattice 13

Adiabatic potential: bipartite magnetic lattice 14

Zak phase of the bipartite magnetic lattice 15

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Implementing magnetic lattice 17 TOF imaging spin resolved momentum distribution

Adiabatically loaded ground state 18

Geometric pump-magnetization evolution 19 Trajectory

Geometric pump: real space simulation 20

Geometric pump: position measurement 21 Imaging in-situ position before & after pumping

Geometric pump: displacement per cycle 22 Sample trajectories

23 Classical, geometric, and topological pump

24 Directly visualizing edge states of an elongated 2D lattice in the quantum Hall regime Photo: Emily Edwards Stuhl et al., Science 349, 1514 (2015) Mancini et al., Science 349, 1510 (2015)

25 How to simulate effective magnetic field? Aharonov-Bohm phases! j m Atoms in lattice:

26 Raman coupling Phase from Raman coupling

27 Raman coupling Phase in 2D elongated lattice D lattice

28 Raman coupling Phase in 2D elongated lattice D lattice

29 A 2-D hybrid lattice: zero flux Momentum distribution along e x Population on lattice site m -2 Lattice site m Fractional population -22

30 Distributions of top and bottom edge states decay exponentially away from edge

31 Dynamics: skipping orbits Effective tilted box potential Release from edge and observe skipping orbits m position

32 Dynamics: skipping orbits Effective tilted box potential Release from edge and observe skipping orbits m position

33 Outlook Hard wall boundary → end states with spin character Filled bands → quantized charge transport Periodic boundary condition → Hofstadter’s butterfly Bloch oscillation → Laughlin’s pump Soliton in syn. dim→ chiral edge soliton Topological magnetic lattice Synthetic dimension Lu et al., arXiv: (2015) Stuhl et al., Science 349, 1514 (2015) ; Mancini et al., Science 349, 1510 (2015)

34 The team Funding thanks to NIST, NSF, AFOSR, and ARO

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