Mapping the Berry Curvature of Optical Lattices

Transcription

1 Mapping the Berry Curvature of Optical Lattices Nigel Cooper Cavendish Laboratory, University of Cambridge Quantum Simulations with Ultracold Atoms ICTP, Trieste, 16 July 2012 Hannah Price & NRC, PRA 85, (2012). NRC, PRL 106, (2011) NRC & Jean Dalibard, EPL 95, (2011)

2 Outline Topology of 2D Bands in 2D Experimental Considerations Optical Flux Lattices, C 0

3 Topological Invariants Gaussian curvature κ = 1 R 1 R 2 negative, zero and positive κ 1 κ da = (2 2g) Gauss-Bonnet Theorem 2π closed surface genus g = 0, 1, 2,... for sphere, torus, 2-hole torus... Topological invariant: g cannot change under smooth deformations

4 Topological Features of 2D Bands Chern number C = 1 d 2 k Ω k 2π BZ [Thouless, Kohmoto, Nightingale & den Nijs (1982)] Berry curvature Ω k = i k u k u ẑ Crystal momentum k, Bloch state u k Topological invariant: C cannot change under smooth variations of the band C can be non-zero if time-reversal symmetry is broken

5 Topological Bands: Physical Consequences Integer quantum Hall effect [Thouless, Kohmoto, Nightingale & den Nijs (1982)] σ xy = C e2 h Gapless chiral edge state Bragg spectroscopy i.e. j x = C F y h [Goldman, Beugnon & Gerbier, arxiv: ] Expansion imaging [Zhao et al., PRA 84, (2011); Alba et al., PRL 107, (2011)] In simple cases one can reconstruct u k Bloch oscillations [Hannah Price & NRC, PRA 85, (2012)] Measure Ω k

6 (1D) in 2D Experimental Considerations Wavepacket centered on momentum k and position x k = F ẋ = 1 dε k dk Oscillations in ẋ (and x) with period T B = 2 k L F

7 (1D) in 2D Experimental Considerations Accelerated 1D lattice [Ben Dahan, Peik, Reichel, Castin & Salomon, PRL 76, 4508 (1996)] ˆv = 1 m ˆp from expansion images

8 in 2D in 2D Experimental Considerations Modified by the geometry of the Bloch wave functions u k [Chang & Niu, PRL 75, 1348 (1995)] k = F ṙ = 1 ε k k ( k ẑ)ω k Berry curvature Ω k = i k u k u ẑ Crystal momentum k, Bloch state u k The physical properties of a band depend on both ε k and Ω k

9 in 2D in 2D Experimental Considerations k = F ṙ = 1 ε k k ( k ẑ)ω k Complicated trajectories even for Ω k = 0 e.g. ε k = 2J [cos k x a + cos k y a] ṙ = 2Ja (sin k xa, sin k y a) = 2Ja ( sin F xta, sin F ) y ta y / a Lissajous figures when F not along 0 a high-symmetry direction x / a 50 25

10 Time-Reversal Protocol in 2D Experimental Considerations [Hannah Price & NRC, PRA 85, (2012)] k = F ṙ = 1 ε k k ( k ẑ)ω k Measure v k (+F) and v k ( F) k y v k (+F) + v k ( F) = 2 ε k k v k (+F) v k ( F) = 2 (F ẑ)ω k k x

11 Size of the effect Topology of 2D Bands in 2D Experimental Considerations Bloch period T B = h af Group velocity v g εa h x g = v g T B ε F F Berry curvature velocity v Ω = Ω k F a2 v Ω = x Ω Fa v g x g ε a 6 F " -? a x Ω a For Fa ε the effects of Berry curvature are of the same scale as conventional Bloch oscillations.

12 Outline Topology of 2D Bands Topology of 2D Bands in 2D Experimental Considerations Optical Flux Lattices, C 0

13 Honeycomb Lattice B A a 2 c X X a R 1 1 R3 yr 2 Topology of 2D Bands Y K Γ K 2. K K 1 ε k = ± V k T.. D. V k = J[e ik R1 + e ik R2 + e ik R3 ] Dirac points at K, K [Tarruell, Greif, Uehlinger, Jotzu & Esslinger, Nature 483, 302 (2012)] d a 2 A B a 1 x 2 nd Honeycomb 1D chains Square 1 VX [ER ] Chequerboard H.c. 1D c. 0 0 Square 8 V X [E R ] V Y = 2 E R 1 st B.Z. 2q B E G W q y Dirac points q x E q x q y VX,X,Y = [7,0.5,2] ER

14 Asymmetric Honeycomb Lattice B A a 2 a 1 R 1 R 2 R3 K Γ K 2 K K 1 Asymmetric, V A = V B = W. ε k = ± W 2 + V k 2 Fully gapped Ω k 0

15 in Asymmetric Honeycomb Lattice K 4 F = ( 0, 2 3) J a W = 1 2 J 1 Γ 2 K y / a 2 F x / a 2 v x / v R 2 0 +F -F v y / v R 2 0 Both +F and -F k y a k y a

16 Outline Topology of 2D Bands Topology of 2D Bands in 2D Experimental Considerations Optical Flux Lattices, C 0

17 Optically Induced Gauge Fields [J. Dalibard, F. Gerbier, G. Juzeliūnas & P. Öhberg, RMP 83, 1523 (2011)] [Y.-J. Lin, R.L. Compton, K. Jiménez-García, J.V. Porto & I.B. Spielman, Nature 462, 628 (2009)]

18 Optical Flux Lattices Topology of 2D Bands [NRC, PRL 106, (2011); NRC & Jean Dalibard, EPL 95, (2011)] Coherent coupling of internal states of the atom Ĥ = p2 2M Î + ˆV (r) Simple laser configurations, shallow lattices, V E R Landau levels: Narrow bands with C = 1 (n φ 10 9 cm 2 FQH states at high particle densities)

19 Two-Photon Dressed States J e = 1/2 [NRC & Jean Dalibard, EPL 95, (2011)] g g + Light at two frequencies: ω L with Rabi freqs. κ m (m = 0, ±1) ω L + δ with Rabi freq. E in σ ˆV = κ2 tot 3 Î + ( κ 2 κ + 2 ) Eκ 0 3 Eκ 0 κ + 2 κ 2

20 [NRC & Jean Dalibard, EPL 95, (2011)] [cf. Soltan-Panah et al., Nat. Phys. (2011)] Bloch vector ψ 0 (r) ˆ σ ψ 0 (r) wraps the sphere once within the unit cell Berry phase 2π (2-level system) N φ = 1 flux quantum

21 Bandstructure Topology of 2D Bands J g = 1/2 (e.g. 6 Li, 171 Yb, 199 Hg) V = 2E R, θ = π/4, ɛ = 1.3 DoS (arb.) x 1/ E/E R Narrow lowest energy band, with Chern number C = 1 Optical flux lattice analogue of the lowest Landau level

22 Bloch Oscilllations Topology of 2D Bands [V = 1.8E R, θ = 0.3, ɛ = 0.4] [Hannah Price & NRC, PRA 85, (2012)] ε k Ω k y / a 0-1 F x / a F = (0, 0.1E R /a) Net transverse drift: nonzero mean Berry curvature

23 Summary Topology of 2D Bands Two dimensional bands can have a topological character, encoded in the Berry curvature Ω k. The Berry curvature modifies the Bloch oscillations of an atomic wave packet. A time-reversal protocol cleanly extracts the effects of the Berry curvature. For non-zero Chern number, there is a net drift of the wave packet transverse to the force. This drift is the lattice analogue of the edge state of the IQH effect.