Exact solutions for inhomogeneous 1D quantum gases Anna Minguzzi Laboratoire de Physique et Mod´ elisation des Milieux Condens´ es, Grenoble – p.1/36
1D quantum gases Quasi-1D geometry: ultracold atoms in tight transverse confinement µ, k B T ≪ � ω ⊥ 2D deep optical lattices, chip traps z x y – p.2/36
Experimental results 1D bosons in the strongly interacting regime density profiles, momentum distribution, correlation functions, collective modes, transport... [T Kinoshita et al, 2005] [T Kinoshita et al (2004)] [B. Paredes et al, 2004] Atomic density [arb. units] τ =200 µ s τ =1400 µ s τ =2000 µ s Distance z [ µ m] [S. Palzer et al, 2009] [E Haller et al, 2009] – p.3/36
The model ultracold dilute bosonic gases: binary interactions through s -wave collisions for atoms in a tight waveguide [Olshanii, 1998] v ( x ) = gδ ( x ) with g = 2 a s � ω ⊥ (1 − 0 . 4602 a s /a ⊥ ) − 1 model Hamiltonian [Lieb and Liniger, 1963] − � 2 ∂ 2 � � H = + V ( x i ) + g δ ( x i − x j ) ∂x 2 2 m i i i<j Lieb-Liniger model with external potential coupling strength: γ = gn/ ( � 2 n 2 /m ) note: strong coupling at weak densities – p.4/36
g From quasicondensate to TG Bose-Einstein condensation in 3D: off-diagonal long range order for | x − x ′ | → ∞ [Penrose and Onsager, 1965] � Ψ † ( x )Ψ( x ′ ) � → n 0 – p.5/36
From quasicondensate to TG quantum fluctuations: important in one-dimension in 1D quasi -long range order for | x − x ′ | → ∞ [Haldane, 1981] 1 � Ψ † ( x )Ψ( x ′ ) � → | x − x ′ | 1 / 2 K 5 Luttinger parameters K 4 K : Luttinger parameter 3 2 depends on interactions v s / v 1 F 0 0.1 1 10 100 g Regimes of quantum degeneracy at T = 0 : γ ≪ 1 “quasicondensate” condensate with fluctuating phase, K ≫ 1 γ ≫ 1 “Tonks-Girardeau” gas impenetrable-boson limit, K = 1 – p.5/36
Impenetrable bosons: special features For g → ∞ the many-body wavefunction vanishes at contact Ψ( ...x j = x ℓ ... ) = 0 Exact solution by mapping onto noninteracting fermions [MD Girardeau, 1960] 1 √ Ψ( x 1 ...x N ) = Π 1 ≤ j<ℓ ≤ N sign(x j − x ℓ ) det( ψ l ( x k )) N ! with ψ l ( x ) single particle orbitals for arbitrary external potential, also time dependent fermionization ⇒ impenetrable bosons are robust to two- and three-body particle losses – p.6/36
Plan exact solutions for strongly interacting 1D gases: external confinement and full quantum dynamics TG gases in equilibrium: 0.9 0.9 0.9 BBFF BFBF BFFB n ( x ) a ho 0.6 0.6 0.6 0.3 0.3 0.3 extensions of the model, Bose- 0 0 0 -3 0 3 -3 0 3 -3 0 3 0.9 0.9 0.9 FBBF FBFB FFBB n ( x ) a ho 0.6 0.6 0.6 0.3 0.3 0.3 Fermi mixtures 0 0 0 -3 0 3 -3 0 3 -3 0 3 x/a ho x/a ho x/a ho TG gases out-of-equilibrium: 20 n � L 0.6 10 k y L � Π sudden stirring of bosons on a 0 � 10 0 ring � 20 � 20 � 10 0 10 20 k x L � Π – p.7/36
New solvable models : the Bose-Fermi mixture – p.8/36
1D spinors and mixtures Optical trapping allow for the study of multicomponent systems spinor bosons [J. Kronjaeger et al PRL 105, 090402 (2010)] Extensions of the Girardeau solution for the strongly repulsive limit of Bose-Fermi mixtures [M. Girardeau and A. Minguzzi PRL 99, 230402 (2007)] , spin-1 bosons [F . Deuretzbacher et al, PRL 100, 160405 (2008)] , spin-1/2 fermions [Liming Guan et al, PRL 102, 160402 (2009)] – p.9/36
1D Bose-Fermi mixtures with repulsive BB and BF interactions mean-field and Luttinger liquid analysis at weak coupling: instability towards demixing Homogeneous system with equal coupling constants and equal masses: Bethe Ansatz solution – no demixing [C.K. Lai and C.N. Yang, PRA 3, 393 (1971), A. Imambekov and E. Demler Ann. Phys. 321, 2390 (2006)] mixture in harmonic trap: partial demixing at intermediate interactions Fermi Bose Bose Fermi [A. Imambekov, E. Demler, ibid. (2006)] x x x x b f b f ⇒ exact spatial structure in the trap at large interactions? ⇐ – p.10/36
A symmetric model with a large degeneracy Model: N B bosons, N F fermions with coupling constants g BB = g BF and m B = m F , in harmonic trap BF mixture with small relative mass difference: 173 Yb- 174 Yb In the TG limit g BB , g BF → ∞ : large degeneracy of the ground state ( N B + N F )! /N B ! /N F ! 3,5 3 Energy levels for N B = 1 , Energy/ h w 2,5 N F = 1 : at increasing 2 1,5 interactions, the even and 1 odd levels approach 0,5 0 0 10 5 15 interaction strength – p.11/36
A basis set for the manifold We want to determine the wavefunction Ψ in each of the N ! coordinate sectors x P (1) < x P (2) < ... < x P ( N ) with P a permutation, P ∈ S N TG limit: Ψ = 0 at each BB and BF contact ⇒ in a given coordinate sector, Ψ ∝ Ψ F Constraint: satisfy bosonic and fermionic symmetry under particle exchange : N B ! N F ! conditions note! degeneracy left: N ! /N B ! N F ! = ways you can order in a row N B bosons and N F fermions, eg BBFF, BFBF, BFFB, FBBF, FBFB, FFBB – p.12/36
A basis set for the manifold BBFF, BFBF, BFFB, FBBF, FBFB, FFBB Starting point: the snippet orthonormal basis √ � x 1 ..x N | P � = N ! | Ψ F ( x 1 ..x N ) | nonvanishing only in the coordinate sector P – p.13/36
A basis set for the manifold BBFF, BFBF, BFFB, FBBF, FBFB, FFBB Starting point: the snippet orthonormal basis √ � x 1 ..x N | P � = N ! | Ψ F ( x 1 ..x N ) | nonvanishing only in the coordinate sector P idea! combine the snippets which correspond to the same BBFF sequence ⇒ orthonormal basis (since each snippet is used only once) – p.13/36
A basis set for the manifold BBFF, BFBF, BFFB, FBBF, FBFB, FFBB Starting point: the snippet orthonormal basis √ � x 1 ..x N | P � = N ! | Ψ F ( x 1 ..x N ) | nonvanishing only in the coordinate sector P idea! combine the snippets which correspond to the same BBFF sequence ⇒ orthonormal basis (since each snippet is used only once) Example: x 1 , x 2 bosons; x 3 , x 4 fermions; coordinate sectors associated to BBFF : x 1 < x 2 < x 3 < x 4 x 2 < x 1 < x 3 < x 4 x 1 < x 2 < x 4 < x 3 x 2 < x 1 < x 4 < x 3 Ψ BBFF = � x 1 ..x N | ( e + (12))( e − (34)) � – p.13/36
A basis set for the manifold BBFF, BFBF, BFFB, FBBF, FBFB, FFBB Starting point: the snippet orthonormal basis √ � x 1 ..x N | P � = N ! | Ψ F ( x 1 ..x N ) | nonvanishing only in the coordinate sector P idea! combine the snippets which correspond to the same BBFF sequence ⇒ orthonormal basis (since each snippet is used only once) Example: x 1 , x 2 bosons; x 3 , x 4 fermions; coordinate sectors associated to BFBF : x 1 < x 3 < x 2 < x 4 x 2 < x 3 < x 1 < x 4 x 1 < x 4 < x 2 < x 3 x 2 < x 4 < x 1 < x 3 Ψ BFBF = � x 1 ..x N | (23)( e + (12))( e − (34)) � – p.13/36
Density profiles for the BBFF basis BBFF, BFBF, BFFB, FBBF, FBFB, FFBB Analogous to a system of distinguishable particles: 0.9 0.9 0.9 BBFF BFBF BFFB n ( x ) a ho 0.6 0.6 0.6 0.3 0.3 0.3 0 0 0 -3 0 3 -3 0 3 -3 0 3 0.9 0.9 0.9 FBBF FFBB FBFB n ( x ) a ho 0.6 0.6 0.6 0.3 0.3 0.3 0 0 0 -3 0 3 -3 0 3 -3 0 3 x/a ho x/a ho x/a ho [B. Fang, P . Vignolo, M. Gattobigio, C. Miniatura, A. Minguzzi PRA 84, 023626 (2011)] – p.14/36
A special solution start from the Bethe Ansatz solution for the homogeneous system [Lai and Yang (1971), Imambekov and Demler (2006)] introduce y 1 , ...y N B = P − 1 (1) ..., P − 1 ( N B ) relative positions of the bosons in a sequence TG limit of the Bethe Ansatz solution: decoupling Ψ BA = det[ e i 2 π N κ i y j ]Ψ F ( x 1 , ...x N ) where κ = {− ( N B − 1) / 2 + N/ 2 , ..., N/ 2 , ... ( N B − 1) / 2 + N/ 2 } Generalize to the inhomogeneous case: use Ψ F ( x 1 , ...x N ) for harmonic trap Conjecture: this solution is the one connected to the (nondegenerate) solution at finite interactions (with g BB = g BF ) – p.15/36
Intermezzo : particle exchange symmetry Two possible Young tableaus B B F B B Y= Y’= F F F The ground state at finite interactions has the Y symmetry [Lai, Yang (1971)] to each tableau is associated a value of the Casimir ˆ C = � invariant: i<j ( i, j ) with ( i, j ) particle permutation c Y = ( N B ( N B + 1) − N F ( N F − 1)) / 2 c Y ′ = ( N B ( N B − 1) − N F ( N F + 1)) / 2 – p.16/36
Casimir operator Representation of the Casimir operator on the BBFF basis for N B = 2 , N F = 2 : 0 1 − 1 1 − 1 0 1 0 1 1 0 − 1 − 1 1 0 0 1 − 1 1 1 0 0 1 1 − 1 0 1 1 0 1 0 − 1 − 1 1 1 0 similar structure for N B = 3 N F = 3 – p.17/36
Symmetry check Use the Casimir to “test” the symmetry of a wavefunction � Ψ | ˆ C | Ψ � � Ψ | Ψ � Check for N B = 3 N F = 3 : the “BA” solution has the Y symmetry � Ψ BA | ˆ C | Ψ BA � = 3 � Ψ BA | Ψ BA � F B B B Ψ BA has the symmetry of the F ground state F – p.18/36
Spatial structure of the BF mixture The BA solution yields a non-demixed density profile: connection with partial demixing at intermediate interactions? – p.19/36
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