Experimental reconstruction of the Berry curvature in a topological Bloch band Christof Weitenberg Workshop Geometry and Quantum Dynamics Natal 29.10.2015 arxiv:1509.05763 (2015)
Topological Insulators Topology of the bulk leads to chiral edge states Experimental access is mostly limited to the edge states Solid State Model systems Quantum Hall effect, Von Klitzing, PRL (1980). Helical waveguides Rechtsman, Nature 496, 196 (2013). Silicon photonics Hafezi,Nat. Photon. 7, 1001 (2013). Unidirectional backscattering in Polariton system Wang, Nature 461, 772 (2009). Topological RF circuit Ningyuan, PRX 5, 021031 (2015). Mechanical topological insulator Süsstrunk, Science 349, 47 (2015).
Wavefunction microscope To see the bulk topology, we need a wavefunction microscope! Experiments with cold atoms might provide new insight Our wavefunction microscope See also work with superconducting qubits Roushan, Nature (2014)
Topological bands are a hot topic in cold atom research! Theory Experiments/Lattice Engineering of topological bands: Jaksch, Zoller, New J. Phys. 5, 56 (2003). Kitagawa et al. PRB 82, 235114 (2010). Dalibard et al. Rev. Mod. Phys. 83, 1523 (2011). Cooper, PRL 106, 175301 (2011). Rudner et al. PRX 3, 031005 (2013). Goldman, Dalibard, PRX 4, 031027 (2014). Baur et al. PRA 89, 051605(R) (2014). Bukov et al. Adv. Phys. 64, 139 (2015). Detection of topology: Alba et al. PRL 107, 235301 (2011). Price, Cooper, PRA 85, 033620 (2012). Goldman et al. PRL 108, 255303 (2012). Dauphin, Goldman, PRL 110, 135302 (2013). Wang et al. PRL 110, 166802 (2013). Goldman et al. PNAS 110, 6736 (2013). Price, Cooper, PRL 111, 220407 (2013). Hauke et al. PRL 113, 045303 (2014). Non-Abelian Gauge fields: Osterloh et al. PRL 95, 010403 (2005). Nayak et al. Rev. Mod. Phys. 80, 1083 (2008). Goldman et al. PRL 103, 035301 (2009). Hauke et al. PRL 109, 145301 (2012). Topology and Interactions: Raghu et al. PRL 100, 156401 (2008). Rachel, Le Hur, PRB 82, 075106 (2010). Neupert et al. PRL 106, 236804 (2011). Cooper, Dalibard, PRL 110, 185301 (2013). Bergholtz et al. Intern. J. Mod. Phys. B 27, 1330017 (2013). Grushin et al. PRL 112, 156801 (2014). Soltan-Panahi et al., Nat. Phys (2011). Soltan-Panahi et al., Nat. Phys (2012). Jo et al., PRL 108, 045305 (2012). Struck et al. PRL 108, 225304 (2012). Cheuk et al. PRL 109, 095302 (2012). Struck et al. Nature Phys. 9, 738 (2013). Parker et al. Nature Phys. 9, 769 (2013). Atala et al. Nature Phys. 9, 795 (2013). Aidelsburger et al. PRL 111, 185301 (2013). Miyake et al. PRL 111, 185302 (2013). Jotzu et al. Nature 515, 237 (2014). Atala et al. Nature Phys. 10, 588 (2014). Aidelsburger et al. Nature Phys. 11, 162 (2015). Kennedy et al. Nature Phys. (2015). Stuhl et al., Science (2015). Mancini et al., Science (2015). Jotzu et al. PRL 115, 073002 (2015). Duca et al. Science 347, 288 (2015). Li et al. arxiv:1509.02185 (2015). Taie et al. arxiv:1506.00587 (2015). Nakajima et al. arxiv:1507.02223 (2015). Lohse et al. arxiv:1507.02225 (2015). Lu et al. arxiv:1508.04480 (2015).
Experiments with topological bands (I) Hofstadter model Haldane model Cyclotron orbits in Hofstadter Model Aidelsburger, PRL (2013) Bloch group Chern number of Hofstadter bands Aidelsburger, Nature Phys. (2015) Bloch group Condensation in Hofstadter Model Kennedy, Nature Phys. (2015) Ketterle group Chern number in Haldane Model Jotzu, Nature (2014) Esslinger group Chiral edge states Chiral edge states Stuhl, Science (2015) Spielman group Chiral edge states Mancini, Science (2015) Inguscio group Meissner effect Atala, Nature Phys. (2014) Bloch group
Experiments with topological bands (II) Magnetism via lattice driving 1D Gauge potentials Ising XY spin-models Struck, Nature Phys. (2013) Sengstock group Ferromagnetic domains Parker, Nature Phys. (2013) Chin group 1D Gauge potential Struck, PRL (2012) Sengstock group Spin-dependent driving Jotzu, PRL (2015) Esslinger group Spin-orbit coupled lattice Zak and Berry phase Spin-orbit coupled lattice Cheuk, PRL (2012) Zwierlein group Zak phase Atala, Nature Phys. (2013) Bloch group Aharonov-Bohm interferometer Duca, Science (2015) Bloch group Wilson lines Li, arxiv (2015) Bloch group
,Berry-ology * But how are all the different properties related to one-another? Berry connection: NOT gauge invariant Berry curvature: Berry phase Chern Number Would be nice to see this Berry curvature * Fuchs et al., EPJB 77, 351 (2010)
Calculated Berry Curvature for different systems Boron nitride (tight-binding model) Fuchs et al., Euro Phys. J B 77, 351 (2010) Ferromagnetic bcc Fe Yao et al., Phys. Rev. Lett. 92, 037204 (2004) Strained graphene Guinea et al., Nature Phys. 6, 30 (2009) Monolayer MoS 2 Feng et al., Phys. Rev. B 86, 165108 (2012)
Map of the full Berry curvature Fläschner et al., arxiv:1509.05763 (2015) related work: Li et al. arxiv:1509.02185 (2015)
How do we do it? Tunable hexagonal lattice for fermionic 40 K See: Soltan-Panahi et al.,nat.phys 7, 434 (2011) Baur et al., PRA 89, 051605(R) (2014). Offset between A and B Boron-nitride Massive Dirac points A B Well separated flat s-bands Tomography Berry curvature flattens out
Floquet engineering of dressed bands Quasi-energy Circular shaking Breaks time-reversal symmetry k-dependent coupling Berry curvature engineering Dirac point at K annihilated Dirac point at Γ created Three-fold symmetry
Eigenstates in Bloch sphere representation Two-band Hamiltonian allows for a Bloch sphere representation States of flat bands lie at the north- and south-pole For each k, the ground state is given by: k is given by:
Eigenstate reconstruction We follow the proposal by P. Hauke, M. Lewenstein, A.Eckardt, PRL 113, 045303 (2014): Related proposal Alba et al., PRL 107, 235301 (2011)
Reconstruction of full Hamiltonian The oscillations become visible in k-space after time-of-flight Momentum space density Fitting the oscillations gives and for each momentum (pixel)
Berry curvature Now we can reconstruct the Berry curvature using:
Amplitude + Phase = Berry curvature N. Fläschner, B. Rem, M. Tarnowski, D. Vogel, D. Lühmann, K. Sengstock., C. Weitenberg, arxiv:1509.05763 (2015)
Engineering of Berry Curvature Berry curvature (1/ b ²) Increasing amplitude 200 nm 100 nm
Conclusion Full state tomography Full measurement of Berry curvature Allows us to determine Chern number Engineering of Berry curvature Annihilation and creation of Dirac points Localization
Outlook What next? A B We want to further explore topological bands Study interacting Fermions, Bosons, or mixtures in these bands How can we prepare a Floquet topological Insulator? What can we learn from quenches into the nontrivial regime? We still have the spin degree of freedom: study high-spin systems or engineer additional spin-orbit-coupling Explore other interesting geometries using the tunable lattice
The BFM-team Barcelona Dresden Hamburg In collaboration with Ludwig Mathey Klaus Sengstock Christof Weitenberg Benno Rem Dirk-Sören Lühmann Matthias Tarnowski Nick Fläschner Dominik Vogel André Eckardt Maciej Lewenstein