Tunable Superfluidity and Quantum Magnetism with Ultracold Polar Molecules

By selecting two dressed rotational states of ultracold polar molecules in an optical lattice, we obtain a highly tunable generalization of the $t\mathrm{\text{−}}J$ model, which we refer to as the $t\mathrm{\text{−}}J\mathrm{\text{−}}V\mathrm{\text{−}}W$ model. In addition to $XXZ$ spin exchange, the model features density-density interactions and density-spin interactions; all interactions are dipolar. We show that full control of all interaction parameters in both magnitude and sign can be achieved independently of each other and of the tunneling. As a first step towards demonstrating the potential of the system, we apply the density matrix renormalization group method to obtain the 1D phase diagram of the simplest experimentally realizable case. Specifically, we show that the tunability and the long-range nature of the interactions in the $t\mathrm{\text{−}}J\mathrm{\text{−}}V\mathrm{\text{−}}W$ model enable enhanced superfluidity. Finally, we show that Bloch oscillations in a tilted lattice can be used to probe the phase diagram experimentally.

Figure 1

(a) Eigenenergies of $H_0=B{\mathbf{N}}^2-d_{0}E$ as a function of $E$ (the top axis uses $d$ and $B$ of KRb). Only $M\ge 0$ states are labeled. (b),(c) Examples of level configurations $\{|m_0⟩,|m_1⟩\}$: (b) $\{\sqrt{a}|0⟩+\sqrt{1-a}|\overline{1}⟩,|1⟩\}$; (c) $\{\sqrt{a}|\widehat{2}⟩+\sqrt{1-a}|\overline{2}⟩,\sqrt{b}|1⟩+\sqrt{c}|\overline{1}⟩+\sqrt{1-b-c}|2⟩\}$. Red (blue) levels make up the dressed rotor state $|m_0⟩$ ($|m_1⟩$).

Figure 2

Phase diagrams of the $t\mathrm{\text{−}}{J}_{\perp }$ chain with (a) nearest-neighbor and (b) dipolar $J_{\perp }$. We identify a metallic (repulsive Luttinger liquid) phase with dominant spin-density-wave correlations (SDW), a gapless superfluid with dominant triplet and singlet superfluid correlations (TS/SS), a singlet superfluid with a spin gap ($\mathrm{SS}+\mathrm{SG}$), and phase separation (PS). The spin gap is $\lesssim 0.35t$ in (a) and $\lesssim 0.7t$ in (b). Solid lines indicate phase transitions (green, SG closes; red, inverse compressibility becomes zero); dashed lines are extrapolations. The numbers show the value of $K_{\rho }$ on the dotted lines. The line $K_{\rho }=2$ is a crossover line within the $\mathrm{SS}+\mathrm{SG}$ phase [32]. The shaded region in (b) displays $K_{\rho }=1\pm 0.15$ as an estimate of the numerical accuracy.

Figure 3

Bloch oscillations in a dipolar $t\mathrm{\text{−}}{J}_{\perp }$ chain of 20 sites with open boundary conditions at fillings $n=0.1$ and 0.2, and tilting field $E_{\mathrm{tilt}}/t=1$ per site. The main plot shows the difference between the center-of-mass position $x_{\mathrm{CM}}$ at time zero and at the first minimum. The vertical line indicates the approximate value of $J_{\perp }/t$, at which the spin gap closes for $n=0.2$. The inset shows the time evolution of $x_{\mathrm{CM}}$ at $n=0.2$ at three indicated values of $J_{\perp }/t$.