for Cold Atom Gases Nigel Cooper Cavendish Laboratory, University of Cambridge Artificial Magnetism for Cold Atom Gases Collège de France, 11 June 2014 Jean Dalibard (Collège de France) Roderich Moessner (MPIPKS Dresden)
Motivation: fractional quantum Hall regime 2D charged particle in magnetic field Landau levels Flux density n φ = qb h Density of States qb/h per unit area hω C E
Motivation: fractional quantum Hall regime 2D charged particle in magnetic field Landau levels Flux density n φ = qb h Density of States qb/h per unit area hω E C e-e repulsion fractional quantum Hall states, at certain ν n 2D n φ
Motivation: fractional quantum Hall regime 2D charged particle in magnetic field Landau levels Flux density n φ = qb h Density of States qb/h per unit area hω E C e-e repulsion fractional quantum Hall states, at certain ν n 2D n φ Bosons? (contact repulsion)
Motivation: fractional quantum Hall regime 2D charged particle in magnetic field Landau levels Flux density n φ = qb h Density of States qb/h per unit area hω E C e-e repulsion fractional quantum Hall states, at certain ν n 2D n φ Bosons? (contact repulsion) ν > ν c 6: vortex lattice (BEC) [NRC, Wilkin & Gunn 01; Sinova, Hanna & MacDonald 02; Baym 04] ν < ν c : FQH states (incl. non-abelian ) [NRC, Wilkin & Gunn 01]
Synthetic Magnetic Field: Rotation a Stir the atomic gas Rotating frame, angular velocity Ω Coriolis Force b Lorentz Force n φ qb h = c 2MΩ h 2D Cloud [Desbuquois et al. (2012)] Objective PBS Imaging Stirring beam [K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)] 10 µ
Synthetic Magnetic Field: Rotation a Stir the atomic gas Rotating frame, angular velocity Ω Coriolis Force b Lorentz Force n φ qb h = c 2MΩ h 2D Cloud [Desbuquois et al. (2012)] PBS Objective Ω < 2π 100Hz n < φ 2 10 7 cm 2 Stirring beam [K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)] Imaging 10 µ
Synthetic Magnetic Field: Optically Dressed States [Y.-J. Lin, R.L. Compton, K. Jiménez-García, J.V. Porto & I.B. Spielman, Nature 462, 628 (2009)]
Synthetic Magnetic Field: Optically Dressed States [Y.-J. Lin, R.L. Compton, K. Jiménez-García, J.V. Porto & I.B. Spielman, Nature 462, 628 (2009)] But... n φ < 1 Rλ 2 107 cm 2 [R cloud size]
Outline Optically Dressed States Design Principles
Optically Dressed States [J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, RMP 83, 1523 (2011)] Coherent optical coupling of N internal atomic states [e.g. 1 S 0 and 3 P 0 for Yb or alkaline earth atom] laser coupling Ω R e g ω 0 Forms the local dressed state of the atom 0 r = α r g + β r e = ( αr β r )
optical coupling ˆV (r) laser frequency ω e ω 0 coupling strength Ω R g Rotating Wave Approximation ω, Ω R ˆV (r) ( ΩR (r) 2 Ω R (r) )
In general ˆV = 2 Optically Dressed States ( (r) ΩR (r) Ω R (r) (r) ) E 1 (r), 1 r E 0 (r), 0 r
In general ˆV = 2 Optically Dressed States ( (r) ΩR (r) Ω R (r) (r) ) E 1 (r), 1 r E 0 (r), 0 r Adiabatic motion in lower energy dressed state, Ψ = ψ 0 (r) 0 r
In general ˆV = 2 Optically Dressed States ( (r) ΩR (r) Ω R (r) (r) ) E 1 (r), 1 r E 0 (r), 0 r Adiabatic motion in lower energy dressed state, Ψ = ψ 0 (r) 0 r [ ] p 2 H eff ψ 0 = 0 r 2M + (p qa)2 ˆV ψ 0 0 r H eff = + V 0 (r) 2M [ Berry connection vector potential qa = i 0 r 0 r ] [J. Dalibard, F. Gerbier, G. Juzeliūnas & P. Öhberg, RMP 83, 1523 (2011)]
Experimental Implementation Rubidium BEC [Lin, Compton, Jiménez-García, Porto & Spielman, Nature 462, 628 (2009)] beam 1 B z ω 2 beam 2 ω 1 gµ B B z k = k 1 ( k 2 ) 2 2π λ ˆV = ( 2 Ω R e i k x Ω R e i k x ) field gradient B z (y) y
Experimental Implementation Rubidium BEC [Lin, Compton, Jiménez-García, Porto & Spielman, Nature 462, 628 (2009)]
Experimental Implementation Rubidium BEC [Lin, Compton, Jiménez-García, Porto & Spielman, Nature 462, 628 (2009)] But... n φ < 1 Rλ 2 107 cm 2 [R cloud size]
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 2 107 cm 2 A dr < 1 λ (2πR) [R 10µm λ 0.5µm]
Maximum flux density: Carefully this time! Vector potential qa = i 0 r 0 r 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 (re)moved by a gauge transformation. [cf. Dirac strings ]
Gauge-Independent Approach (two-level system) Bloch vector n(r) = 0 r ˆ σ 0 r Flux density n φ = 1 8π ɛ ijkɛ µν n i µ n j ν n k r Region A n Solid Angle Ω 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.
Design Principles Optical flux lattices [NRC, Phys. Rev. Lett. 106, 175301 (2011)] Spatially periodic light fields for which the Bloch vector wraps the sphere a nonzero integer number, N φ, times in each unit cell. n φ = N φ A cell 1 λ 2 109 cm 2 vectors (n x, n y ) contours n z N φ = 2 ( sin κx sin κy cos κx i cos κy ˆV = V cos κx + i cos κy sin κx sin κy )
Optically Dressed States Design Principles Lorentz Force: Semiclassical Picture y ei 6 λ/2? X XX XXX g i x Optical Artificial Flux Lattices Magnetism for Cold foratom Cold Atom GasesGases Colle ge de Fran
Optically Dressed States Design Principles Lorentz Force: Semiclassical Picture y ei 6 λ/2? X XX XXX g i x px (~k) y λ/2 Fx p x ~k vy λ/2 Optical Artificial Flux Lattices Magnetism for Cold foratom Cold Atom GasesGases Colle ge de Fran
Optically Dressed States Design Principles Lorentz Force: Semiclassical Picture y ei 6 λ/2? X XX XXX g i x px (~k) y λ/2 Lorentz force, with ~k vy λ/2 qb 2 nφ 2 h λ Fx p x qb 2h λ2 Optical Artificial Flux Lattices Magnetism for Cold foratom Cold Atom GasesGases Colle ge de Fran
Design Principles Example: Implementation for Hyperfine Levels [NRC & Jean Dalibard, EPL 95, 66004 (2011)] θ θ ω 2 ω 2 θ ω 2 ω 1 J g = 1/2 ω 1 σ pol. ω 2
Design Principles Example: Implementation for Hyperfine Levels [NRC & Jean Dalibard, EPL 95, 66004 (2011)] θ θ ω 2 ω 2 θ ω 2 ω 1 J g = 1/2 ω 1 σ pol. ω 2 ( E + ˆV 2 2 E 2 2 E2 z E 1 E2 z E 1 E 2 + 2 + E 2 2 ) N φ = 1 (two level system)
Bandstructure: Optically Dressed States p 2 2M Î + ˆV (r) Design Principles
Bandstructure: Optically Dressed States p 2 2M Î + ˆV (r) Design Principles J g = 1/2 (e.g. 171 Yb, 199 Hg, 6 Li) [E R = h2 2Mλ 2 ] DoS (arb.) x 1/10 V = 2E R, θ = π/4, ɛ = 1.3 3 3.5 4 4.5 5 5.5 E/E R
Design Principles J g = 1/2 (e.g. 171 Yb, 199 Hg, 6 Li) [E R = h2 2Mλ 2 ] DoS (arb.) x 1/10 V = 2E R, θ = π/4, ɛ = 1.3 3 3.5 4 4.5 5 5.5 E/E R J g = 1 (e.g. 23 Na, 39 K, 87 Rb) LLL } V = 8E R, ɛ = 0.8, θ = 0.2, γ 1 = 0, γ 2 = 0.15E R DoS 0 8.2 8.4 8.6 8.8 9 E/E R Narrow topological bands: analogous to lowest Landau level
Design Principles Designing [NRC & Roderich Moessner, PRL 109 215302 (2012)]
Design Principles Designing Ĥ = p2 2M Î + ˆV (r) [NRC & Roderich Moessner, PRL 109 215302 (2012)] Optical lattices are conveniently defined in reciprocal space Couplings V α α k k α, k ˆV α, k of internal states α = 1,..., N
Design Principles Designing Ĥ = p2 2M Î + ˆV (r) [NRC & Roderich Moessner, PRL 109 215302 (2012)] Optical lattices are conveniently defined in reciprocal space Couplings V α α k k α, k ˆV α, k of internal states α = 1,..., N 1 Vκ 12 1 Vκ 21 2 2 1 2 Reciprocal lattice {G} G 1, q 2, q + κ 1 1, q + G G = κ 1 + κ 2
Design Principles Bloch s theorem ψ nq = α,g c nq αg α, q + g α + G E nq c nq αg = 2 q + G + g α 2 c nq 2M αg + V αα G+g α G g cnq α α G α,g
Design Principles Bloch s theorem ψ nq = α,g c nq αg α, q + g α + G E nq c nq αg = 2 q + G + g α 2 c nq 2M αg + V αα G+g α G g cnq α α G α,g Adiabatic limit (K.E. 0) E nq c nq αg = α,g V αα G+g α G g α cnq α G Tight-binding model in reciprocal space
Design Principles Tight-binding model in reciprocal space E nq c nq αg = α,g V αα G+g α G g α cnq α G Bandstructure determines the dressed states in real space [Ĥ = ˆV (r)]
Design Principles Tight-binding model in reciprocal space E nq c nq αg = α,g V αα G+g α G g α cnq α G Bandstructure determines the dressed states in real space [Ĥ = ˆV (r)] conserved momentum real space position, r Brillouin zone real space unit cell Bloch wavefunction dressed state, n r band energies local dressed state energies, E n (r) Berry curvature local flux density, n φ Chern number, C flux through unit cell, N φ
Design Principles Tight-binding model in reciprocal space E nq c nq αg = α,g V αα G+g α G g α cnq α G Bandstructure determines the dressed states in real space [Ĥ = ˆV (r)] conserved momentum real space position, r Brillouin zone real space unit cell Bloch wavefunction dressed state, n r band energies local dressed state energies, E n (r) Berry curvature local flux density, n φ Chern number, C flux through unit cell, N φ For an optical flux lattice, the lowest energy band of the reciprocal-space tight-binding model has non-zero Chern number
Design Principles OFLs with uniform Magnetic Field and Scalar Potential DoS (arb. units) x 1/10 N = 4 V = E R -3-2.5-2 -1.5-1 -0.5 0 E/E R Low energy spectrum closely analogous to Landau levels Narrow bands strongly correlated phases N = 3 scheme for 87 Rb shows robust Laughlin, CF/hierarchy and Moore-Read phases of bosons, even for weak two-body repulsion [NRC & Jean Dalibard, PRL 110, 185301 (2013)]
Outlook Strong magnetic field, n φ 1 λ 2 Novel FQH states of 2D bosons [NRC & J. Dalibard, PRL (2013)] Correlated bosonic phases in 3D? [NRC, van Lankvelt, Reijnders & Schoutens, PRA 05] Strong-coupling superconductivity vs. FQH, ξ pair ā n 1/2 φ [for a cuprate superconductor, would need B > 10 5 T!]
Outlook Strong magnetic field, n φ 1 λ 2 Novel FQH states of 2D bosons [NRC & J. Dalibard, PRL (2013)] Correlated bosonic phases in 3D? [NRC, van Lankvelt, Reijnders & Schoutens, PRA 05] Strong-coupling superconductivity vs. FQH, ξ pair ā n 1/2 φ [for a cuprate superconductor, would need B > 10 5 T!] Other Topological Bandstructures Chern insulators with C > 1 Z 2 Topological Insulators in 2D and 3D Exploration of fractional topological insulators
Summary Coherent optical coupling of internal states provides a powerful way to create topological bands for cold atoms. Ĥ = p2 2M Î + ˆV (r) Simple laser set-ups lead to optical flux lattices : periodic magnetic flux with very high mean density, n φ 1/λ 2. The bandstructure can be designed with significant control. These lattices offer a practical route to the study of novel strongly correlated topological phases in ultracold gases.