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Dipolar Molecules in Optical Lattices Can [10] K Dipolar Molecules in Optical Lattices Tomasz Sowiński1,2, Omjyoti Dutta2, Philipp Hauke2, Luca Tagliacozzo2, Maciej Lewenstein2,3 1Institute of Physics of the Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland 2 ICFO –The Institute of Photonic Sciences, Av. Carl Friedrich Gauss, num. 3, 08860 Castelldefels (Barcelona), Spain 3 ICREA – Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, E-08010 Barcelona, Spain (Dated: July 22, 2018) We study the extended Bose–Hubbard model describing an ultracold gas of dipolar molecules in an optical lattice, taking into account all on-site and nearest-neighbor interactions, including occupation-dependent tunneling and pair tunneling terms. Using exact diagonalization and the multiscale entanglement renormalization ansatz, we show that these terms can destroy insulating phases and lead to novel quantum phases. These considerable changes of the phase diagram have to be taken into account in upcoming experiments with dipolar molecules. PACS numbers: 37.10Jk,67.85.Hj,75.40.Cx Trapping and manipulating ultracold gases in optical creasing dipolar interaction, the system enters from the lattices has allowed the realization of many-body physics CB phases to a novel state which has a one-particle su- in a controlled environment. For atoms interacting via perfluid (SF) and pair-superfluid (PSF) properties. Par- contact interaction, a quantum phase transition from a ticularly we find a region where both of them coexists superfluid (SF) to a Mott insulator (MI) has been pre- with the SF order parameter has alternating sign at con- dicted and observed [1]. In the simplest case, these sys- secutive sites. tems can be theoretically described by the Bose-Hubbard Our system consists of dipolar bosons polarized by an (BH) model, which has two parameters: a tunneling J external electric field along the z direction and confined and an on-site interaction U [2, 3]. A natural extension in a square optical lattice. The corresponding Hamilto- 2 of the Bose–Hubbard model comes from including long- 3r † r ~ 2 r r nian reads H = d Ψ ( ) h− 2m ∇ + Vlatt( )i Ψ( )+ range interactions between particles. Experiments on ul- R 1 d3r d3r′Ψ†(r)Ψ†(r′)V(r − r′)Ψ(r)Ψ(r′), where tracold polar molecules have renewed interest in extended 2 RR Ψ†(r) (Ψ(r)) are the bosonic creation (annihilation) field Bose-Hubbard models which can model such systems in operators. V (r)= V sin2 2π x + sin2 2π y +mΩ2z2/2 optical lattices [4–7]. Because of the strong electric dipole latt 0 λ λ z is an external lattice potential of lattice depth V , gen- moment of polar molecules, long-range interactions play 0 erated by a laser field of wave-length λ, with Ω charac- a crucial role in the collective behavior of the system, z terizing the external harmonic potential in z direction. leading to the appearance of states with long-range or- The dipole–dipole interaction is denoted by V(r). By ex- der, like various structured insulating states, supersolids, 2 −κz /2 panding the field operator Ψ(r)= Wi(x, y)e aˆi Wigner crystals, pair-supersolids, etc. [9–15]. Pi in lowest Bloch-band Wannier-functions Wi(x, y), and by In this Letter, we study the ground-state of dipolar restricting ourselves to on-site and NN terms, we arrive molecules in a 2D square optical lattice with a harmonic at the extended BH model trapping along the polarization direction of the dipoles. U We derive a modified BH model which includes addi- H = −J aˆ†aˆ + nˆ (ˆn − 1)+ V nˆ nˆ X i j 2 X i i X i j tional occupation-dependent nearest-neighbor (NN) hop- {ij} i {ij} ping processes arising from long-range dipolar interac- P − T aˆ† (ˆn +ˆn )ˆa + aˆ†aˆ†aˆ aˆ , (1) tions in the lowest Bloch band. Usually, interaction- X i i j j 2 X i i j j induced hopping terms are neglected when discussing {ij} {ij} dipolar bosonic molecules. In this Letter, we show that † arXiv:1109.4782v3 [cond-mat.quant-gas] 5 Apr 2012 these terms considerably change the physics of dipolar where aˆi (aˆi ) annihilates (creates) a particle on lattice † soft-core bosons. Soft-core bosons in square and one- site i, nˆi =ˆai aˆi is the corresponding density operator, J dimensional lattices have been discussed in the litera- the standard tunneling coefficient, U the on-site interac- ture within the extended Hubbard model, focusing on tion, and V the NN interaction, arising from a truncation the presence of stable supersolidity [17, 18]. In the of the dipolar interactions to the dominating term. Dipo- usual case with only NN interaction, at sufficient dipo- lar interactions lead to two novel terms in Eq. (1): The lar strength, the ground states at half- and unit-filling term proportional to T describes one-particle tunneling are checkerboard (CB) insulating states. Using exact di- to a neighboring site induced by the occupation of that agonalization (ED) and multiscale entanglement renor- site, and the term proportional to P is responsible for malization ansatz (MERA), we solve the one-dimensional NN pair tunneling [19–21]. extended Hubbard model including the novel occupation- The matrix elements U, V , T , and P are given by a dependent NN hopping processes. We find that with in- sum of dipolar and δ-like contact interactions, V(r−r′)= 2 150 0 0.5 1 1.5 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 70 (c) 8 12 50 -1 1 16 TD /J 40 P /J (d) -50 -2 D χ 0.5 UD /J 10 Parameters -3 -150 V /J D (d) π/8 N = 16 π contribution (a) -4 0 /4 Checkerboard 0.2 3π/8 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 1 π/2 κ κ (b) 5π/8 Lattice flattening Lattice flattening 3π/4 0.5 7π/8 π S(q) 0.1 0 functions FIG. 1. Dependence of the dipolar part (subscript D) of Correlation U, V , T , and P on the lattice flattening κ for lattice depth 0 0.5 1 1.5 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 Electric dipole d (D) V0 = 6ER and γ = 52. Electric dipole d (D) FIG. 2. We plot various properties of the exact ground state of ′ 2 (3) ′ 1 (z−z ) a half-filled system as a function of dipole moment d. Fig. (a) gδ (r − r )+ γ ′ 3 − 3 ′ 5 . We measure all h |r−r | |r−r | i shows the contribution of the CB states to the ground state lengths in units of the laser wave length λ and all en- of the system for N = 8. In Fig. (b) we plot the one-particle 2 2 2 ergies in recoil energies ER = 2π ~ /(mλ ), where m and two-particle correlation functions φi and Φi. The dotted is the bosonic mass. Additionally, we define the lattice line shows φi when we neglect the terms T and P . When T, P =6 0, the solid line and dash-dotted line shows φi and Φi flattening κ = ~Ωz/2ER as well as the dimensionless cou- pling constants describing contact and dipolar interac- respectively as a function of dipole moment d. In Fig. (c) we plot the fidelity susceptibility χ(d) for the half-filled system tion, g = 16π2a /λ and γ = md2/(~2ε λ) (where a is s 0 s for different system sizes. In Fig. (d) we have shown the the s-wave scattering length, ε0 is the vacuum permittiv- structure-factor S at different ordering wave vectors for the ity, and d is the electric dipole moment of the bosons). half-filled system with 16 sites. For concreteness, we consider an ultracold gas of dipo- lar molecules confined in a optical lattice with lattice depth V0 = 6ER, mass m = 220a.m.u and λ = 790 nm tion to an insulating state. In the half-filled system, the [22]. We also assume that the s-wave scattering length transition occurs because for large enough V the parti- of the molecules, as ≈ 100a0. For these parameters, cles can decrease their energy by avoiding every second g ≈ 1.06 is approximately constant. We consider dipole site. If we neglect T and P , the situation will not change moments d up to ∼ 3 D (γ up to ∼ 470), which can be by further increasing d (dotted lines in Fig. 2), since achievable for molecules like bosonic RbCs, KLi [8] etc. this only increases V even more. However, the situa- To illustrate the relative strengths of different parame- tion changes significantly when we take into account the ters, in Fig. 1, we compare for γ = 52 the tunneling J density-induced tunneling T and the pair tunneling P . In with the dipolar contribution (subscript D) to the pa- this case, for d ≈ 1.1 D, a second phase transition occurs, rameters U, V , T and P . For the parameters chosen, TD destroying the CB order [solid lines in Fig. 2(a)]. Pre- and PD are 1 orders of magnitude smaller than VD where vious studies have completely neglected such a possible as UD/TD can be tuned by changing κ. On the other destruction of CB order at large d. At the transition, the hand, TD can dominate over J for large γ. In addition, contribution of the CB state to the ground state decreases T and J can have opposite sign as seen in Fig.
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