Single-particle versus pair superfluidity in a bilayer system of dipolar bosons

We consider the ground state of a bilayer system of dipolar bosons, where dipoles are oriented by an external field in the direction perpendicular to the parallel planes. Quantum Monte Carlo methods are used to calculate the ground-state energy, the one-body and two-body density matrix, and the superfluid response as a function of the separation between layers. We find that by decreasing the interlayer distance for fixed value of the strength of the dipolar interaction, the system undergoes a quantum phase transition from a single-particle to a pair superfluid. The single-particle superfluid is characterized by a finite value of both the atomic condensate and the super-counterfluid density. The pair superfluid phase is found to be stable against formation of many-body cluster states and features a gap in the spectrum of elementary excitations.

Figure 1

Energy per particle with half of the dimer binding energy subtracted as a function of the reduced interlayer distance $h/r_0$ for $n{r}_0^2=1$. Red solid and blue open symbols correspond to the results obtained using the guiding wave functions in Eqs. (2) and (3), respectively. The horizontal lines correspond to the energies of a single layer of dipoles with effective interaction strength $\tilde{n}{\tilde{r}}_0^2=0.5$ and $\tilde{n}{\tilde{r}}_0^2=32$. (Inset) Energy per particle ($n{r}_0^2=1$) and ${\epsilon }_b/2$ (solid line) as a function of the interlayer separation. In both cases the dashed line corresponds to the large $h/r_0$ limit.

Figure 2

Atomic condensate $n_0$ and molecular condensate $n_0^{\text{mol}}$ as a function of $h/r_0$ at the density $n{r}_0^2=1$. Arrows correspond to the condensate fraction of a single layer of dipoles at the effective interaction strength $\tilde{n}{\tilde{r}}_0^2=0.5$ (red arrow) and $\tilde{n}{\tilde{r}}_0^2=32$ (blue arrow). The dashed green line and symbols correspond to the super-counterfluid density of Eq. (6).

Figure 3

Excitation gap $\Delta$ as a function of $h/r_0$ at the density $n{r}_0^2=1$. (Red symbols and line) DMC results and guide to the eye. The dashed blue line corresponds to ${\epsilon }_b/2$. (Inset) Energy $E\left(P\right)$ for two values of $h/r_0$ in the pair and single-particle superfluid phase with the corresponding linear and quadratic fit.

Figure 4

Schematic phase diagram featuring the single-particle (upper region) and the pair superfluid (lower region). The dots correspond to the transition points as obtained from DMC simulations. The two arrows show the freezing density of a single layer of particles (right) and of dimers (left). The line separates the region where $|{\epsilon }_b|/2<\mu$ (weak pairing) from the region where $|{\epsilon }_b|/2>\mu$ (strong pairing).