(Abelian) Anyon-Hubbard Models in 1D (Optical) Lattices I. - PowerPoint PPT Presentation

(Abelian) Anyon-Hubbard Models in 1D (Optical) Lattices I. (Floquet) Engineering with cold atomic quantum gases: Interaction modulation & Raman assisted hopping II. Lattice shaking & Properties (ground state phase diagram and dynamics) Sebastian Greschner - Anyon Workshop - December 2018 1 / 70

Outline - Day I: I. (Floquet) Engineering with cold atomic quantum gases Digest of cold atom physics Fermions, Bosons, Anyons - Jordan-Wigner-Transformation in 1D and 2D Floquet-Engineering Modulated Interactions and Experiments in Chicago Assisted Hopping Schemes II. Properties Digest of 1D physics Anyon “Interferometer” on a ring Simple ladder model of braiding anyons 1D (Pseudo) Anyon Hubbard model 2 / 70

Digest of Cold Atom Ultracold Gases in Optical Lattices Ultracold quantum gases in optical lattices provide an excellent toolbox for... ... strongly correlated many-body systems in and out of equilibrium ... quantum simulation of condensed matter paradigms, high energy physics ... quantum simulation of interacting synthetic gauge field theories M. Greiner, PhD thesis ▸ clean, scalable lattice system ▸ control, adjustable (in real time) ▸ observable (momentum distribution, measurements with single site resolution, ...) 3 / 70

Digest of Cold Atom Bose Hubbard model Hamiltonian for interacting bosonic particles in a trapping potential Tight binding approximation: Expand bosonic field operator in basis of wannier Ψ ( x ) = ∑ i ˆ b i w ( x − x i ) functions ˆ H BH = −∑ b j + ∑ ( ǫ i − µ ) ˆ n i + ∑ n i ( ˆ n i − 1 ) U ˆ J ij ˆ b † i ˆ 2 ˆ ⟨ ij ⟩ i i Bose Hubbard Hamiltonian with effective parameters for hopping J ij and interaction U 4 / 70

Digest of Cold Atom Density-Dependent Gauge Fields ⇐ � � � � � ⇒ static flux models dynamical LGT Anyons, density dependent fields, ... Dynamic feedback of the particles on the gauge field - “Moving particles create magnetic Mancini, ... Fallani Science 2015 field” Stuhl, ... Spielman, Science 2015, Banerjee, ... Zoller, PRL 2012, Aidelsburger, ... Bloch, PRL 2013, Kasper, ... Berges New J. Phys 2017 ... Wiese, Ann. Phys 2013, ... Struck... Sengstock, Science 2011, Jotzu... Esslinger, Nature ... 2014 Martinez, ... Blatt, Nature 2016 5 / 70

Jordan-Wigner-Transformation Bosons, Fermions, Anyons 6 / 70

Jordan-Wigner-Transformation Jordan-Wigner-Transformation in 1D Consider strong interaction side of the Bose-Hubbard phase diagram U → ∞ , at some fixed filling say 0 < n < 1. Assume hard-core bosons b j , i.e. b 2 j = 0 Can we describe hard-core bosons b j by fermions c j ? j = a † j ( e i α ∑ l < j ˆ n l ) j = b † j ( e − i α ∑ l < j ˆ n l ) b † a † b j = ( e − i α ∑ l < j ˆ n l ) a j a j = ( e i α ∑ l < j ˆ n l ) b j This keeps density operators invariant n j = b † j b j = a † j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j ˆ n l ) a j = a † j a j 7 / 70

Jordan-Wigner-Transformation Jordan-Wigner-Transformation in 1D Consider strong interaction side of the Bose-Hubbard phase diagram U → ∞ , at some fixed filling say 0 < n < 1. Assume hard-core bosons b j , i.e. b 2 j = 0 Can we describe hard-core bosons b j by fermions c j ? j = a † j ( e i α ∑ l < j ˆ n l ) j = b † j ( e − i α ∑ l < j ˆ n l ) b † a † b j = ( e − i α ∑ l < j ˆ n l ) a j a j = ( e i α ∑ l < j ˆ n l ) b j This keeps density operators invariant n j = b † j b j = a † j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j ˆ n l ) a j = a † j a j What does this mean to commutations? j > k n l ) b k = a k a j { e − i α , a j a k = b j ( e i α ∑ l < j ˆ n l )( e i α ∑ l < k ˆ j < k e i α , 7 / 70

Jordan-Wigner-Transformation Jordan-Wigner-Transformation in 1D Consider strong interaction side of the Bose-Hubbard phase diagram U → ∞ , at some fixed filling say 0 < n < 1. Assume hard-core bosons b j , i.e. b 2 j = 0 Can we describe hard-core bosons b j by fermions c j ? j = a † j ( e i α ∑ l < j ˆ n l ) j = b † j ( e − i α ∑ l < j ˆ n l ) b † a † b j = ( e − i α ∑ l < j ˆ n l ) a j a j = ( e i α ∑ l < j ˆ n l ) b j This keeps density operators invariant n j = b † j b j = a † j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j ˆ n l ) a j = a † j a j What does this mean to commutations? j > k n l ) b k = a k a j { e − i α , a j a k = b j ( e i α ∑ l < j ˆ n l )( e i α ∑ l < k ˆ j < k e i α , What about the nearest neighbor hopping term? j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j + 1 ˆ n l ) a j + 1 = a † j ( e i α ˆ n j ) a j + 1 = a † j b j + 1 = a † j a j + 1 = c † b † j c j + 1 7 / 70

Jordan-Wigner-Transformation Jordan-Wigner-Transformation in 1D Consider strong interaction side of the Bose-Hubbard phase diagram U → ∞ , at some fixed filling say 0 < n < 1. Assume hard-core bosons b j , i.e. b 2 j = 0 Can we describe hard-core bosons b j by fermions c j ? j = a † j ( e i α ∑ l < j ˆ n l ) j = b † j ( e − i α ∑ l < j ˆ n l ) b † a † b j = ( e − i α ∑ l < j ˆ n l ) a j a j = ( e i α ∑ l < j ˆ n l ) b j This keeps density operators invariant n j = b † j b j = a † j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j ˆ n l ) a j = a † j a j What does this mean to commutations? j > k n l ) b k = a k a j { e − i α , a j a k = b j ( e i α ∑ l < j ˆ n l )( e i α ∑ l < k ˆ j < k e i α , What about the nearest neighbor hopping term? j ( e i α ∑ l < j ˆ n l )( e − i α ∑ l < j + 1 ˆ n l ) a j + 1 = a † j ( e i α ˆ n j ) a j + 1 = a † j b j + 1 = a † j a j + 1 = c † b † j c j + 1 We have solved analytically strongly interacting many-body problem! We have introduced 1D anyons! (and showed that they have a trivial spectrum...) 7 / 70

Jordan-Wigner-Transformation Anyons Certainly not trivial - e.g. momentum distribution n ( k ) = ∑ ⟨ a † j a j ′ ⟩ j , j ′ e i k ( j − j ′ ) � j e − i Φ ( j ) e i Φ ( j ′) b j ′ ⟩ ⟨ b † Idea: Engineer anyon model in quantum gas experiment by local modification of the hopping. Allow for exchange of anyons: 1D ring G. Tang, S. Eggert, A. Pelster, New J. Phys. 17, 123016 (2015) Particles may pass through each other: “Pseudo” anyons 2D system, ladder, ... 8 / 70

Jordan-Wigner-Transformation Hard core anyons on a ring 1D system with PBC, i.e. add a “long bond” L a 1 = c † L e i α ∑ L l = 1 n l c 1 a † = e i α ( N − 1 ) c † L c 1 Fermions with flux α N / L depending on total particle number For finite system energy changes, time dynamics in the thermodynamic limit same as OBC system 9 / 70

Jordan-Wigner-Transformation 1D Pseudo anyons - Correlated Tunneling Assume bosons on-site anyonic/deformed exchange statistics k − F j , k a † k a j = δ j , k a j a † ⎧ ⎪ j > k , ⎪ ⎪ e − i α , j = k , F j , k ∶= ⎨ ⎪ ⎪ 1 , ⎪ j < k , ⎩ e i α , Correlated/Density dependent hopping model for bosons H = − t ∑ ( b † j e − i α n j b j + 1 + H.c. ) ˆ j similar: two component anyons H = − t ∑ ( a † j ,σ a j + 1 ,σ + H.c. ) ˆ j ,σ k ,σ ′ − F j , k a † k ,σ ′ a j ,σ = δ j , k δ σ,σ ′ again a j ,σ a † 10 / 70

Jordan-Wigner-Transformation Anyons on a ladder ladder spins anyons × × (↓ , 0 ) → ( 0 , ↓ ) ○ × 1 ○ × ( 2 , 0 ) → ( ↑ , ↓ ) ○ × e i α ○ × ( 2 , 0 ) → ( ↑ , ↓ ) Choose some order of lattice sites ○ × 1 H = − t ∑ j ,σ a j + 1 ,σ + H . c . + ○ ○ ˆ a † ( 2 , ↑ ) → ( ↑ , 2 ) ○ × e i α j ,σ = 0 , 1 − t ⊥ ∑ j , ↑ a j , ↓ + H . c . ○ × a † ( 2 , 0 ) → ( ↓ , ↑ ) ○ × 1 j particles ”overcrossing“ e − i α and ”undercrossing“ ○ × ( 2 , 0 ) → ( ↓ , ↑ ) e − i α × ○ e + i α H = − t ∑ j , ↑ e i α n j b j + 1 , ↑ + H . c . + ˆ b † ⋯ ⋯ j − t ∑ j , ↓ e − i α n j + 1 b j + 1 , ↓ + H . c . + ⋯ + Hermitian conjugate ( ← processes) b † + rung exchange j 11 / 70

Jordan-Wigner-Transformation Jordan-Wigner-Transformation in 2D (Fradkin, 1989) There are several ways to generalize the Jordan Wigner transformation to 2D a r = e i α ∑ k θ k , r n k c r θ k , r is the angle between k − r and some direction. The resulting particles are indeed anyons a r a r ′ = e i α ( θ r , r ′ − θ r ′ , r ) a r ′ a r the hopping becomes r e i A r , r ′ c r ′ + H . c . + ⋯ H = ∑ ˆ c † r , r ′ with A r , r ′ = ∑ k ≠ r , r ′ ( θ k , r − θ k , r ′ ) ˆ n k equivalent to flux “attached” to particle: flux = A r , r + e x + A r + e x , r + e x + e y − A r + e y , r + e x + e y − A r , r + e y = α 2 ( n r + n r + e x − n r + e x + e y − n r + e y ) 12 / 70