Quantum Fluids of Self-Assembled Chains of Polar Molecules

We study polar molecules in a stack of strongly confined pancake traps. When dipole moments point perpendicular to the planes of the traps and are sufficiently strong, the system is stable against collapse but attractive interaction between molecules in different layers leads to the formation of dipolar chains, analogously to the chaining phenomenon in classical rheological electro- and magnetofluids. We analyze properties of the resulting quantum liquid of dipolar chains and show that only the longest chains undergo Bose-Einstein condensation with a strongly reduced condensation temperature. We discuss several experimental methods for studying chains of polar molecules.

Figure 1

(a) Schematic figure of a stack of pancake traps. The external electric field polarizes molecules in the direction perpendicular to the layers. Vertical ellipses denote self-assembled chains of molecules. (b) A typical shape of a dipolar chain. Shaded ellipses denote single molecule wave functions in individual layers [see Eq. (1)] with radius $R_j$.

Figure 2

Binding energy (note the minus sign) per molecule as a function of dipole strength, $U_0$, for chain length $M=2$ and $M=81$. Results are obtained using the variational wave function in Eq. (1), with $\omega =0.01E_0$. $E_0\equiv {\hslash }^2/m{d}^2=\frac{2}{{\pi }^2}{E}_R$ with $E_R$ being the recoil energy of the 1D optical lattice. Inset: Optimal values of the variational parameters for a single chain, $R_0$(solid curve) and $\xi$ (dashed curve), as a function of $U_0$ for $M=81$. Note that the value of $\xi$ has been multiplied by ${10}^3$ to fit the scale of the vertical axes.

Figure 3

Number of chains in a multilayer system ($L=81$) as a function of the chain length, $M$. Other parameters are $N_{\mathrm{tot}}={10}^5$, $U_0=2$, and $\hslash \omega =0.01E_0$. Condensate $T_c$ is $0.11E_0$ for such parameters. Solid, dashed, and dotted lines are for temperature $T=0.1$, 0.3, and 0.5 $E_0$, respectively, and the upward arrow indicates the condensation of the longest chain ($M=L=81$) Inset: Condensate $T_c$ as a function of dipole strength, $U_0$, for multilayer system $L=2$ and $L=81$. The number of dipoles per layer for these two systems are set the same so that they have the same $T_c$ in the noninteracting regime. Solid lines are calculated within the noninteracting chain approximation (discussed in the text). Dotted lines are schematic since reliable theory in the vicinity of $U_{0,L}^{*}$ should include interactions between chains.