Laboratoire Kastler Brossel Collège de France, ENS, UPMC, CNRS Introduction to Ultracold Atoms Superfluid – Mott insulator transition Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr ) Advanced School on Quantum Science and Quantum Technologies, ICTP Trieste September 4, 2017
1 Wannier functions and tight-binding limit 2 Bose-Hubbard model 3 Ground state : Superfluid -Mott insulator transition Phase coherence Dynamics and transport Shell structure 4 A glance at fermions Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Very deep lattices In a deep lattice potential, atoms are tightly trapped around the potential minima. Harmonic approximation for each well : V lat ( x ≈ x i ) ≈ 1 2 m a ω 2 lat ( x − x i ) 2 , � � ω lat = 2 V 0 E R . The bands are centered around the energy E n ≈ ( n + 1 / 2) � ω lat . First correction : quantum tunneling across the potential barriers, as in tight-binding methods used in solid-state physics (Linear Combination of Atomic Orbitals) This is best handled using a new basis, formed by so-called Wannier functions . Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Wannier functions Wannier functions : discrete Fourier transforms with respect to the site locations of the Bloch wave functions, 1 � e − iqx i φ n,q ( x ) . w n ( x − x i ) = √ N s q ∈ BZ 1 • All Wannier functions can be deduced from w n ( x ) by translation of x i = id . • There are exactly N s such functions per band (as many as Bloch functions). • Wannier functions form a basis of Hilbert space ( not an eigenbasis of ˆ H ). V 0 = 4 E R V 0 = 10 E R V 0 = 20 E R Cautionary note: Bloch functions are defined up to a q − dependent phase which needs to be fixed to obtain localized Wannier functions [W. Kohn, Phys. Rev. (1959)] . Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Hamiltonian in the Wannier basis Second quantized formalism (useful for the interacting case). Bloch basis : � ε n ( k )ˆ b † n,k ˆ H = b n,k . n,k ∈ BZ 1 ˆ b n,k : annihilation operator for Bloch state ( n, k ) . Wannier basis : � a † H = − J n ( i − j )ˆ n,i ˆ a n,j , n,i,j ˆ a n,i : annihilation operator for Wannier state w n ( x − x i ) . Tunneling energies : � � 2 � � dx w ∗ J n ( i − j ) = n ( x − x j ) 2 M ∆ − V lat ( x ) w n ( x − x i ) . (also called hopping parameters) Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Hamiltonian in the Wannier basis � a † H = − J n ( i − j )ˆ n,i ˆ a n,j , n,i,j Tunneling energies : � � 2 � � 1 � e − i ( qx i − q ′ x j ) dx u ∗ J n ( i − j ) = 2 M ∆ − V lat ( x ) n,q ( x ) u n,q ′ ( x ) , N s q,q ′ ∈ BZ 1 � �� � = − ε n ( q ) δ n,n ′ δ q,q ′ = − 1 � ε n ( q ) e − iq · ( x i − x j ) . N s q ∈ BZ 1 J n ( i − j ) depend only on the relative distance x i − x j between the two sites. • On-site energy ( i = j ): Mean energy of band n J n (0) = − 1 � ε n ( q ) = − E n N s q ∈ BZ 1 • Nearest-neighbor tunneling ( j = i ± 1 ): J n (1) = − 1 � ε n ( q ) e iqx = − J n N s q ∈ BZ 1 Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Tight-binding limit For deep lattices (roughly V 0 ≫ 5 E R ), the tunneling energies fall off very quickly with distance: 0 10 J 0 (1) � Wannier function for V 0 = 10 E R : J 0 (2) J 0 (3) − 2 10 R ] Tunnel Energies [E ] − Tunnel Energies [E − 4 10 − − 6 10 − 0 5 10 15 20 V 0 [E R ] Two useful approximations : • Tight-binding approximation : keep only the lowest terms • Single-band approximation : keep only the lowest band–drop band index and let J 0 (1) ≡ J � � a † a † ˆ H T B = E 0 ˆ i ˆ a i − J ˆ i ˆ a j , i � i,j � Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Three-dimensional lattices Cubic lattice potential : � sin( k α x α ) 2 V lat = V 0 α Dispersion relation : � ǫ n ( q ) = ǫ n α ( q α ) , α = x,y,z • ǫ n ( q ) : 1d dispersion relation, • n : a triplet of integers indexing the various bands • q : quasi momentum ∈ BZ1 : ] − π/d, π/d ] 3 . Bloch functions : φ n , q ( r ) = e i q · r u n x ,q x ( x ) u n y ,q y ( y ) u n z ,q z ( z ) . Wannier functions : W n ( r − r n ) = w n x ( x − n x d x ) w n y ( y − n y d y ) w n z ( z − n z d z ) . Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
1 Wannier functions and tight-binding limit 2 Bose-Hubbard model 3 Ground state : Superfluid -Mott insulator transition Phase coherence Dynamics and transport Shell structure 4 A glance at fermions Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Interacting atoms in a deep optical lattice Basic Hamiltonian for bosons interacting via short-range forces : H = ˆ ˆ H 0 + ˆ H int , � − � 2 � � ˆ Ψ † ( r ) ˆ ˆ H 0 = d r 2 M ∆ + V lat ( r ) Ψ( r ) , � H int = g ˆ d (3) r ˆ Ψ † ( r )ˆ Ψ † ( r )ˆ Ψ( r )ˆ Ψ( r ) . 2 • ˆ Ψ( r ) : field operator annihilating a boson a position r , • V lat ( r ) : lattice potential, • g = 4 π � 2 a : coupling constant, M • scattering length a >0 : repulsive interactions. • Not simpler in the Bloch basis. • Can be drastically simplified in the Wannier basis Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Interacting bosons in the Wannier basis Basis of Wannier functions W ν ( r − r i ) : • r i : position of site i , � ˆ W ν ( r − r i )ˆ Ψ( r ) = a ν,i . • ν : band index ν,i • ˆ a ν,i : annihilation operator Single-particle Hamiltonian : • Tight-binding approximation : keep only the lowest terms • Single-band approximation : keep only the lowest band–drop band index, J 0 (1) ≡ J � a † ˆ → ˆ H 0 − H T B = − J ˆ i ˆ a j � i,j � Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Interacting bosons in the Wannier basis Basis of Wannier functions W ν ( r − r i ) : • r i : position of site i , � ˆ Ψ( r ) = W ν ( r − r i )ˆ a ν,i . • ν : band index ν,i • ˆ a ν,i : annihilation operator Interaction Hamiltonian : H int = 1 � ˆ → ˆ a † a † H int − U ijkl ˆ i ˆ j ˆ a k ˆ a l 2 ijkl � d r W ∗ ( r − r i ) W ∗ ( r − r j ) W ( r − r k ) W ( r − r l ) U ijkl = g log( | W ( x, y, 0) | 2 ) for V 0 = 5 E R : In the tight binding regime, strong localization of Wannier function W ( r − r i ) around r i . On-site interactions ( i = j = k = l ) are strongly dominant: H int ≈ 1 � a † a † ˆ a i + · · · U iiii ˆ i ˆ i ˆ a i ˆ 2 i Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Bose Hubbard model 1 Single band approximation 2 Tight-binding approximation 3 On-site interactions Bose-Hubbard model : a j + U � � a † H BH = − J ˆ i ˆ n i (ˆ ˆ n i − 1) . 2 � i,j � i a † n i = ˆ ˆ i ˆ a i : operator counting the number of particles at site i . y • Tunneling energy : √ 2 J J = max ε ( q ) − min ε ( q ) 2 z J z = 6 : number of nearest neighbors U • On-site interaction energy : � d r w ( r ) 4 . U = g 3 U x Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Parameters of the Bose Hubbard model Calculation for 87 Rb atoms [a=5.5 nm] in a lattice at λ L = 820 nm: U/E r 10 0 J/E R U nn /E R Energy [ E R ] 10 − 1 J nnn /E R 10 − 2 10 − 3 10 − 4 5 10 15 20 25 30 35 40 V 0 [ E R ] Harmonic oscillator approximation : � � V 0 � � 3 / 4 ∆ band ≈ � ω lat 2 V 0 U 8 = , ≈ π k L a . E R E R E R E R E R 1 Single band approximation : • V 0 ≫ E R � ER � 1 / 4 • U ≪ ∆ band : k L a ≪ V 0 2 Tight-binding approximation : V 0 ≫ 5 E R 3 On-site interactions : V 0 ≫ E R Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
1 Wannier functions and tight-binding limit 2 Bose-Hubbard model 3 Ground state : Superfluid -Mott insulator transition Phase coherence Dynamics and transport Shell structure 4 A glance at fermions Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Non-interacting limit U = 0 BEC in the lowest energy Bloch state q = 0 : � � N 1 � � N 1 1 � b † a † ˆ | Ψ � N = √ | ∅ � = √ √ N s ˆ | ∅ � q =0 i N ! N ! i • Fixed number of particles N : canonical ensemble Probability to find n i atoms at one given site i : � 1 � p ( n i ) ≈ e − n n n i N , 1 n i ! + O N s √ Poisson statistics, mean n , standard deviation ∼ n In the thermodynamic limit N → ∞ , N s → ∞ , one finds the same result as for a coherent state with the same average number of particles N : √ √ � � N ˆ b † � a † n ˆ i | ∅ � | Ψ � coh = N e q =0 | ∅ � = N i e i • Fluctuating number of particles N : grand canonical ensemble H BH → G = H BH − µN Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Weakly-interacting limit U ≪ J Coherent state wavefunction in the grand canonical ensemble : ∞ α n i � � i | Ψ � coh = | α i � , | α i � = N i √ n i ! | n i � i , ˆ a i | α i � = α i | α i � i n i =0 with { α i } i =1 , ··· ,N s the variational parameters. One can relate the presence of the condensate to a non-zero expectation value of the matter wave field α i = � ˆ a i � , playing the role of an order parameter : � N N s e iφ • Condensate wavefunction : α i = � ˆ a i � = • Mean density : n = | α i | 2 = condensate density Spontaneous symmetry breaking point of view. Starting point to formulate a Gross-Pitaevskii (weakly interacting) theory : variational ansatz with self-consistent α i determined by the total (single-particle + interaction)Hamiltonian. “Adiabatic continuation” from the ideal Bose gas. Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
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