Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators 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 , 175301 (2011) Benjamin B´ eri & NRC, PRL 107 , 145301 (2011) NRC & Jean Dalibard, EPL 95 , 66004 (2011) Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Motivation: fractional quantum Hall regime Rotating BECs n φ = 2 M Ω h [K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84 , 806 (2000)] FQH states of bosons for n 2D < ∼ 6 [NRC, Wilkin & Gunn, PRL (2001)] n φ [Laughlin, composite fermion, Moore-Read and Read-Rezayi] Ω ≃ 2 π × 100Hz ⇒ n φ < ∼ 2 × 10 7 cm − 2 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Optically Induced Gauge Fields [Y.-J. Lin, R.L. Compton, K. Jim´ enez-Garc´ ıa, J.V. Porto and I.B. Spielman, Nature 462 , 628 (2009)] Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators “Optical Flux Lattices” [NRC, PRL 106 , 175301 (2011); NRC & Jean Dalibard, EPL 95 , 66004 (2011)] H = p 2 ˆ ˆ I + ˆ V ( r ) 2 M • 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 B´ eri & NRC, PRL 107 , 145301 (2011)] • “Nearly free electron” approach to topological bands Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Outline Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Optically Induced Gauge Fields unas, P. ¨ [J. Dalibard, F. Gerbier, G. Juzeli¯ Ohberg, RMP 83 , 1523 (2011)] H = p 2 ˆ ˆ I + ˆ V ( r ) 2 M 3 P 0 ˆ V ( r ): optical coupling of N internal states ∆ Ω R ω e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom [F. Gerbier & J. Dalibard, NJP 12 , 033007 (2010)] 1 S 0 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators 3 P 0 ∆ Ω e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom R ω [F. Gerbier & J. Dalibard, NJP 12 , 033007 (2010)] 1 S 0 Ω R e i ω t + Ω ∗ 1 R e − i ω t � � � � 0 ˆ 2 V = � R e − i ω t + Ω R e i ω t � 1 � Ω ∗ ω 0 2 − ∆ 1 � � R e − 2 i ω t � � Ω R + Ω ∗ → 2 2 � 1 � R + Ω R e 2 i ω t � ∆ Ω ∗ 2 2 � � V → � − ∆ Ω R ( r ) ˆ RWA ω ≫ ∆ , Ω R Ω ∗ R ( r ) ∆ 2 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators unas, P. ¨ [J. Dalibard, F. Gerbier, G. Juzeli¯ Ohberg, RMP 83 , 1523 (2011)] H = p 2 ˆ ˆ I + ˆ V ( r ) 2 M ˆ V ( r ) ⇒ local spectrum E n ( r ) and dressed states | n r � � | ψ ( r ) � = ψ n ( r ) | n r � n Adiabatic motion H n ψ n = � n r | ˆ H ψ n | n r � H n = ( p − q A ) 2 + V n ( r ) q A = i � � n r | ∇ n r � 2 M Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Maximum flux density: Back of the envelope h Vector potential q A = i � � 0 r | ∇ 0 r �⇒| q A | < ∼ λ Cloud of radius R ≫ λ 1 � n φ d 2 r = q � ∇ × A · d S = q � A · d r < N φ ≡ λ (2 π R ) ∼ h h N φ 1 R λ ≃ 2 × 10 7 cm − 2 < ⇒ ¯ ≡ n φ ∼ [ R ≃ 10 µ m λ ≃ 0 . 5 µ m] π R 2 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Maximum flux density: Carefully this time! h Optical wavelength λ ⇒| q A | < ∼ λ A can have singularities – if the optical fields have vortices. e.g. Ω R ( r ) ∼ ( x + iy ) Vanishing net flux. Can be removed by a gauge transformation. [cf. “Dirac strings”] Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Gauge-independent approach (two-level system) n ( r ) = � 0 r | ˆ Bloch vector � � σ | 0 r � n φ = 1 1 | n φ | < 8 πǫ ijk ǫ µν n i ∂ µ n j ∂ ν n k ∼ λ 2 Solid Angle Ω � n φ d 2 r = Ω n Region A 4 π area A r The number of flux quanta in region A is the number of times the Bloch vector wraps over the sphere. Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Optical flux lattices [NRC, Phys. Rev. Lett. 106 , 175301 (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 ∼ 1 λ 2 ≃ 10 9 cm − 2 ¯ A cell vectors ( n x , n y ) a contours n φ contours n z N φ = 2 . (a) a (b) Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Optical Flux Lattice: One-Photon Implementation H = p 2 M ( r ) · ˆ ˆ ˆ I + V ˆ M = � ˆ M ( r ) σ � 3 P 0 ∆ 2 M e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom Ω R ω [Gerbier & Dalibard, New Journal of Physics 12 , 033007 (2010)] 1 S 0 M x , M y : Rabi coupling, ω ≃ ω 0 M z : standing waves at “anti-magic” frequency, ω am � � � Ω( r ) − � ∆ 2 − V am ( r ) 2 V M = � Ω ∗ ( r ) � ∆ 2 + V am ( r ) 2 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Square Lattice � � sin( κ x ) sin( κ y ) cos( κ x ) − i cos( κ y ) ˆ M sq = cos( κ x ) + i cos( κ y ) − sin( κ x ) sin( κ y ) where κ ≡ 2 π/ a . vectors ( n x , n y ) a contours n φ contours n z N φ = 2 (a) a (b) Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Triangular lattice � � cos[ r · ( κ 1 + κ 2 )] cos( r · κ 1 ) − i cos( r · κ 2 ) ˆ M tri = cos( r · κ 1 ) + i cos( r · κ 2 ) − cos[ r · ( κ 1 + κ 2 )] (a) (b) κ + κ 2 κ 2 1 κ 1 θ a θ ≃ 2 π/ 3 vectors: ( n x , n y ) contours: n z a N φ = 2 Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
Optically Induced Gauge Fields Optical Flux Lattices Z 2 Topological Insulators Bandstructure (Triangular Lattice) H = p 2 ˆ ˆ I + V [ c 1 ˆ σ x + c 2 ˆ σ y + c 12 ˆ σ z ] 2 M c i ≡ cos( κ i · r ) , c 12 ≡ cos[( κ 1 + κ 2 ) · r ] ∼ E R ≡ � 2 κ 2 V > Tight-binding limit 2 M E tb 3 2 1 2 Π k y a � 2 Π � 1 �Π Π � 2 k a � 3 x Lowest energy band has narrow width and Chern number of 1. Nigel Cooper Cavendish Laboratory, University of Cambridge Topological Bandstructures for Ultracold Atoms New quantum states of matter in and out of equilibrium GGI,
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