Multimode Bose-Hubbard model for quantum dipolar gases in confined geometries

We theoretically consider ultracold polar molecules in a wave guide. The particles are bosons: They experience a periodic potential due to an optical lattice oriented along the wave guide and are polarized by an electric field orthogonal to the guide axis. The array is mechanically unstable by opening the transverse confinement in the direction orthogonal to the polarizing electric field and can undergo a transition to a double-chain (zigzag) structure. For this geometry we derive a multimode generalized Bose-Hubbard model for determining the quantum phases of the gas at the mechanical instability, taking into account the quantum fluctuations in all directions of space. Our model limits the dimension of the numerically relevant Hilbert subspace by means of an appropriate decomposition of the field operator, which is obtained from a field theoretical model of the linear-zigzag instability. We determine the phase diagrams of small systems using exact diagonalization and find that, even for tight transverse confinement, the aspect ratio between the two transverse trap frequencies controls not only the classical but also the quantum properties of the ground state in a nontrivial way. Convergence tests at the linear-zigzag instability demonstrate that our multimode generalized Bose-Hubbard model can catch the essential features of the quantum phases of dipolar gases in confined geometries with a limited computational effort.

Figure 1

(a) Ultracold dipolar gases in an optical lattice along $x$ form an array when the confinement in the $y-z$ plane is sufficiently tight. (b) The dipoles form a zigzag chain when the trap frequency along $y$ is below a critical value and the dipoles are aligned along the $z$ axis orthogonal to the plane where the transition occurs. Starting from the array in the $x$ direction we develop a multimode Bose-Hubbard model which describes the onset of this classical structure, thus treating the transverse displacement as a continuous variable, while systematically accounting for quantum fluctuations along all directions in spaces.

Figure 2

Coefficients of the multimode EBH Hamiltonian are reported as a function of ${\omega }_y$ (in units of ${\omega }_y^{\left(c\right)}$) for different values of ${\sigma }_z$: (a) reports the on-site interaction coefficients $U_m^x$, (b) the density-assisted tunneling terms $T_m^x$, and (c) the pair tunneling terms $P_m^x$ (in units of $|J^x|$) for ${\sigma }_z=0.3375a$ (solid lines) and ${\sigma }_z=0.18a$ (dashed lines) and $m=0$ (blue line), $m=1$ (red line), $m=2$ (yellow line). The optical lattice depth and period are $V_L=10E_R$ (with $E_R$ the recoil energy) and $a=395$ nm, the scattering length is $a_S=a/50$, the dipole moment is $p=1.15$ Debye, consistent with ${}^{85}\mathrm{Rb}-{}^{133}\mathrm{Cs}$ bosonic molecules.

Figure 3

Real part of (a) single-particle correlation $\varphi$, Eq. (26), (b) two-particle correlations $\mathrm{\Phi }$, Eq. (27), and (c) $S\left(\pi \right)\equiv S_x(\pi /a)$, Eq. (26), as a function of the dipole moment $p$ (in Debye) for a lattice of 12 sites filled with 6 particles and periodic boundary conditions. The blue curve (i) is obtained for the ground state of Hamiltonian $H_0^x$, Eq. (11), the black curve (ii) is found after setting $T^x=P^x=0$ in Hamiltonian $H_0^x$. The parameters are $V_L=6E_R, a_S=a/100$, and ${\sigma }_z=0.2279a$. The other parameters are the same as in Fig. 2. Note that for each value of $p$ we modify the trap frequency ${\omega }_y$ according to the prescription $\int dx{x}^2{w}^2\left(x\right)=\int dy{y}^2{\varphi }_0^2\left(y\right)$, see text.

Figure 4

Contour plot of (a) the real part of the single-particle correlations $\varphi$, (b) the two-particle correlations $\mathrm{\Phi }$, and (c) the structure form factor $S_x(\pi /a)$ as a function of the width ${\sigma }_z$ (in units of $a$) and of the dipole moment $p$ (in units of $p_0=1.15$ Debye) for unit filling. The red area denotes the region where the on-site interaction $U_0^x$ is negative. The parameters are $V_L=20E_R, {\omega }_y=1.45{\omega }_y^{\left(c\right)}$, and $a_S=a/50$.

Figure 5

Contour plot of (a) the real part of the single-particle correlations $\varphi$, (b) the two-particle correlations $\mathrm{\Phi }$, and (c) the structure form factor $S_x(\pi /a)$ as a function of the width ${\sigma }_z$ (in units of $a$) and of the on-site scattering strength $g$ (in units of $g_0=4\pi a_S/M$ with $a_S=a/50$) and for unit filling. The red area denotes the region where the on-site interaction $U_0^x$ is negative. The parameters are $V_L=20E_R, {\omega }_y=1.45{\omega }_y^{\left(c\right)}$, and $p=1.15$ Debye.

Figure 6

Occupation $n_m=\langle n_{j,m}\rangle$ of the local Hamiltonian eigenstate $m$ as a function of the trap frequency ${\omega }_y$ for $m=0$ (orange line), $m=1$ (red dotted), $m=2$ (blue dashed), and $m=3$ (green dashed-dotted). The calculation has been performed for $N=6$ particles over six sites. The parameters are $V_L=6E_R, {\sigma }_z=0.1a, g=g_0$, and $p=p_0$.

Figure 7

The structure factor $\xi$, Eq. (28), indicating zigzag order for ${\sigma }_z=0.1a$ at an optical lattice depth of $V_L=15E_R$ for unit filling with four sites (red points) and six sites (green) as a function of ${\omega }_y/{\omega }_y^{\left(c\right)}$. The calculations were done with exact diagonalization using the first four orbitals. The other parameters are the same as in Fig. 6.