Topological Bandstructures for Ultracold Atoms

Transcription

1 Topological Bandstructures for Ultracold Atoms Nigel Cooper Cavendish Laboratory, University of Cambridge New quantum states of matter in and out of equilibrium GGI, Florence, 12 April 2012 NRC, PRL 106, (2011) Benjamin Béri & NRC, PRL 107, (2011) NRC & Jean Dalibard, EPL 95, (2011)

2 Motivation: fractional quantum Hall regime Rotating BECs n φ = 2MΩ h [K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)] FQH states of bosons for n 2D n φ < 6 [NRC, Wilkin & Gunn, PRL (2001)] [Laughlin, composite fermion, Moore-Read and Read-Rezayi] Ω 2π 100Hz n φ < cm 2

3 Optically Induced Gauge Fields [Y.-J. Lin, R.L. Compton, K. Jiménez-García, J.V. Porto and I.B. Spielman, Nature 462, 628 (2009)]

4 [NRC, PRL 106, (2011); NRC & Jean Dalibard, EPL 95, (2011)] Ĥ = p2 2M Î + ˆV (r) Landau levels: Narrow bands with unit Chern number n φ 10 9 cm 2 FQH states at high particle densities Distinct from previous tight-binding proposals [Jaksch & Zoller (2003); Mueller (2004); Sørensen, Demler & Lukin (2005); Gerbier & Dalibard (2010)] Generalizes to Z 2 topological invariant [Benjamin Béri & NRC, PRL 107, (2011)] Nearly free electron approach to topological bands

5 Outline Optically Induced Gauge Fields

6 Optically Induced Gauge Fields [J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, RMP 83, 1523 (2011)] Ĥ = p2 2M Î + ˆV (r) ˆV (r): optical coupling of N internal states 3P 0 e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom [F. Gerbier & J. Dalibard, NJP 12, (2010)] Ω R 1S 0 ω

7 3P 0 e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom [F. Gerbier & J. Dalibard, NJP 12, (2010)] Ω R ω ( ˆV = ( ( ΩR e iωt + Ω R e iωt) ) 1 ( 0 2 Ω R e iωt + Ω R e iωt) ω ( Ω R + Ω R e 2iωt) ( ΩR + Ω R e 2iωt) 2 ( ΩR (r) ) ) 1S 0 RWA ω, Ω R ˆV 2 Ω R (r)

8 [J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, RMP 83, 1523 (2011)] Ĥ = p2 2M Î + ˆV (r) ˆV (r) local spectrum E n (r) and dressed states n r ψ(r) = n ψ n (r) n r Adiabatic motion H n ψ n = n r Ĥψ n n r H n = (p qa)2 2M + V n (r) qa = i n r n r

9 Maximum flux density: Back of the envelope Vector potential qa = i 0 r 0 r qa < Cloud of radius R λ N φ n φ d 2 r = q h n φ N φ πr 2 < h λ A ds = q h 1 Rλ cm 2 A dr < 1 λ (2πR) [R 10µm λ 0.5µm]

10 Maximum flux density: Carefully this time! Optical wavelength λ qa < A can have singularities if the optical fields have vortices. h λ e.g. Ω R (r) (x + iy) Vanishing net flux. Can be removed by a gauge transformation. [cf. Dirac strings ]

11 Gauge-independent approach (two-level system) Bloch vector n(r) = 0 r ˆ σ 0 r r n φ = 1 8π ɛ ijkɛ µν n i µ n j ν n k Region A n n φ < Solid Angle Ω 1 λ 2 area A n φ d 2 r = Ω 4π The number of flux quanta in region A is the number of times the Bloch vector wraps over the sphere.

12 Optical flux lattices [NRC, Phys. Rev. Lett. 106, (2011)] Spatially periodic light fields which cause the Bloch vector to wrap the sphere a nonzero integer number, N φ, times in each unit cell. n φ = N φ A cell 1 λ cm 2 vectors (n x, n y ) contours n z N φ = 2 a contours n φ. (a) a (b)

13 Optical Flux Lattice: One-Photon Implementation Ĥ = p2 2M Î + V ˆM(r) ˆM = M(r) ˆ σ 3P 0 e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom [Gerbier & Dalibard, New Journal of Physics 12, (2010)] Ω R ω M x, M y : Rabi coupling, ω ω 0 M z : standing waves at anti-magic frequency, ω am ( VM = 2 V Ω(r) am(r) 2 Ω (r) V am(r) ) 1S 0

14 Square Lattice ( sin(κx) sin(κy) cos(κx) i cos(κy) ˆM sq = cos(κx) + i cos(κy) sin(κx) sin(κy) where κ 2π/a. ) vectors (n x, n y ) contours n z N φ = 2 a contours n φ (a) a (b)

15 Triangular lattice ( ˆM tri = cos[r (κ 1 + κ 2 )] cos(r κ 1 ) i cos(r κ 2 ) cos(r κ 1 ) + i cos(r κ 2 ) cos[r (κ 1 + κ 2 )] ) κ κ 1 + κ 2 2 (a) (b) θ κ 1 θ 2π/3 vectors: (n x, n y ) contours: n z N φ = 2 a a

16 Bandstructure (Triangular Lattice) Ĥ = p2 2M Î + V [c 1ˆσ x + c 2ˆσ y + c 12ˆσ z ] Tight-binding limit V > E R 2 κ 2 2M c i cos(κ i r), c 12 cos[(κ 1 + κ 2 ) r] 2Π Π E tb Π 2Π k ya k a x Lowest energy band has narrow width and Chern number of 1.

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

18 Triangular lattice with N φ = 1 per unit cell.

19 Bandstructure, J g = 1/2 DoS (arb.) x 1/10 V = 2E R, θ = π/4, ɛ = E/E R Narrow lowest energy band, with Chern number of 1 Can also be applied to bosons J g = 1 (e.g. 87 Rb)

20 Experimental Consequences Non-interacting fermions (IQHE) Filled band has chiral edge state: Precession of collective modes Bloch oscillations [Hannah Price & NRC, PRA 85, (2012)] Interacting fermions/bosons Strongly correlated phases if interactions large compared to bandwidth: likely candidates for FQHE states. Incompressible states (density plateaus) Chiral edge modes

21 Outline Optically Induced Gauge Fields

22 Topological Insulators [Hasan & Kane, RMP 82, 3045 (2010); Qi & Zhang, RMP 83, 1057 (2011)] TI: Band insulator with gapless surface states. IQHE: 2D, broken time reversal symmetry (TRS) Chern number number of chiral edge states Z 2 TI: fermions (S = 1 2, 3 2,...) with TRS (Kramers deg.) Band insulators are: trivial; or non-trivial (metallic surface) 2D: counterpropagating edge channels of opposite spin; Spin up Spin down 3D: relativistic (Dirac) 2D surface state.

23 Ĥ = p2 2M ÎN + V ˆM(r) [Benjamin Béri & NRC, PRL 107, (2011)] Time-reversal ˆθ = i ˆσ y ˆK TRS: ˆM = ˆθ 1 ˆM ˆθ N = 4 ( (A + B)Î ˆM = 2 CÎ 2 i ˆ σ D ) CÎ 2 + i ˆ σ D (A B)Î 2 = AÎ 4 + B ˆΣ 3 + C ˆΣ 1 + D ˆΣ 2ˆ σ [A, B, C, D = (D x, D y, D z ) real] Dressed states are Kramers doublets non-abelian gauge field.

24 171 Yb has nuclear spin I = 1/2 3P 0 V ˆM = ( ( 2 + V am) Î2 i ˆ σ Ed ) r i ˆ σ E ( d r 2 + V ) am Î2 1S 0 I = 1/2 +1/2 z TRS preserved if all components of E have a common phase.

25 Two Dimensions d r E = V (δ, cos(r κ1 ), cos(r κ 2 )) κ 1 = (1, 0, 0)κ κ 2 = (cos θ, sin θ, 0)κ κ 2 E z θ Ey κ 1 E x κ 3 =z 2 + V am(r) = V cos[r (κ 1 + κ 2 )] For Yb, θ 2π/3

26 Û = 2 1/2 (Î 4 i ˆΣ 3ˆσ 2 ) ˆM = Û ˆMÛ = c 1 ˆΣ 1 + c 2 ˆΣ 2ˆσ 3 + c 12 ˆΣ 3 + δ ˆΣ 2ˆσ 1. c i cos(κ i r), c 12 cos[(κ 1 + κ 2 ) r] (i) Decoupled spins, δ = 0 ˆM = c 1 ˆΣ 1 ± c 2 ˆΣ 2 + c 12 ˆΣ 3 OFLs of opposite flux for spin σ 3 = ±1. σ 3 = ±1 bands are degenerate, but with opposite Chern numbers. quantum spin Hall system [e.g. Levin & Stern, PRL (2009)]

27 (ii) Spin-orbit coupling, δ E/E R _ -0.4 (1,1) _ + + (0,1) (0,0) (1,0) (1,1) _ Γ nm = 1 2 (nκ 1 + mκ 2 ) Inversion symmetry [Fu & Kane, PRB (2007)] n,m=0,1 α filled ξ (α) nm = 1

28 Three Dimensions This nearly-free electron viewpoint leads to a general method to construct Z 2 non-trivial bands in 3D. [Benjamin Béri & NRC, PRL 107, (2011)] e.g. δ δ 0 cos(κ 3 r) c 12 c 12 + δ 0 (µ + c 13 + c 23 ) -0.6 E/E R V = 0.9E R δ 0 = 1, µ = k 1

29 Summary Simple forms of optical dressing lead to optical flux lattices : periodic magnetic flux with high mean density, n φ 1/λ 2. The low energy bands are analogous to the lowest Landau level of a charged particle in a uniform magnetic field. The approach can be generalized to generate Z 2 nontrivial bandstructures in 2D and 3D. Ultracold atomic gases can readily be used to explore strong correlation phenomena in topological bands.