Tilted dipolar bosons in the quasi-two-dimensional regime: From liquid stripes to droplets

TL;DR

This work analyzes tilted dipolar bosons in a quasi-two-dimensional geometry with realistic confinement along $z$ to map the finite-density phase diagram. A variational approach, benchmarked against the extended Gross-Pitaevskii equation, reveals three thermodynamic phases—gas, unmodulated liquid, and striped liquid—with stripe formation governed by the tilt angle $\alpha$ and the ratio $a/a_{dd}$, and modulated phases tied to the density threshold $n_{\rm lim}$. Beyond-mean-field corrections via $\epsilon_{\rm BMF}(n)$ and the quasi-2D/3D scaling crossover are crucial for accurate phase boundaries, particularly since the maximum $a/a_{dd}$ for stripe emergence is around $0.76$ and the striped phase acquires a 3D character when $n_{\rm peak}>n_{\rm lim}$. In finite systems, self-bound droplets arise above a critical particle number $N_{\rm crit}$ and stripe order can produce supersolid-like states within experimentally accessible parameters, providing guidance for observing novel dipolar phases and highlighting future quantum Monte Carlo work to quantify the superfluid fraction and thermal effects in reduced dimensions.

Abstract

We characterize a system of tilted dipoles in a quasi two-dimensional (flattened) geometry and in the thermodynamic limit. We consider a finite trapping in the z-axis achievable in current experiments. We compute the phase diagram of the system at its equilibrium density for high tilting angles, where it becomes self-bound, and a striped liquid state emerges. To characterize the system, we perform a variational calculation, which is benchmarked with the solution of the extended Gross-Pitaevskii equation. We connect the phenomenology in the thermodynamic limit to the physics of the finite-size system, provide parameters for the realization of potentially supersolid striped states and study the critical number for dipolar droplet formation. Our results are helpful to guide potential experiments in the study of dipolar atoms in quasi two-dimensional geometries in the dipole-dominated regime.

Figures (12)

• Figure 1: Schematic representation of the system of dipoles in a flattened geometry. Left: isodensity surfaces in the homogeneous liquid phase. The different colors correspond to surfaces where $n = 0.5 n_{\rm peak}$ (red), $n = 0.8 n_{\rm peak}$ (light blue) and $n = n_{\rm peak}$ (dark blue), with $n$ the 3D density and $n_{\rm peak}$ the peak density. Center: isodensity surface ($n = 0.5 n_{\rm peak}$ (dark blue)) of the system in the liquid stripe phase. Right: graphical representation of a polarized dipole (thick black arrow) with tilting angle $\alpha$.
• Figure 2: Energy per particle for the senoidal (top, see Eq. \ref{['senoidal']}) and gaussian (bottom, see Eq. \ref{['gaussian']}) ansatz, respectively. The value of the scattering length is set to $a/a_{\rm dd} = 0.7$. We have subtracted the energy of the non-interacting system, $E_{\rm NI}/(N E_0) = \hbar \omega/(2 E_0)$.
• Figure 3: Phase diagram of the infinite dipolar system in a flattened geometry at the equilibrium 2D density.
• Figure 4: 2D equilibrium density (top) and contrast of the density ($C = \frac{n_{\rm max.} - n_{\rm min.}}{n_{\rm max.} + n_{\rm min.}}$) (center, bottom) as a function of the tilting angle. The scattering length is set to $a/a_{\rm dd} = 0.7$ in the top and bottom figures while it is set to $a/a_{\rm dd} = 0.6$ in the center figure.
• Figure 5: $a)$ 2D column density $n(x,y) = \int dz \abs{\psi({\bf r})}^2$ of a single stripe of the eGPE solution. $b)$,$c)$ Column densities $n(y) = \int dx dz \abs{\psi({\bf r})}^2$ ($b)$) and $n(z) = \int dx dy \abs{\psi({\bf r})}^2$ ($c)$) from the ground state solution of the eGPE (Eq. \ref{['eGPE']}) and the gaussian ansatz at the variational energy minimum (Eq. \ref{['gaussian']}). The scattering length and tilting angle are $a/a_{\rm dd} = 0.7$ and $\alpha = 75.92^o$, respectively, while the 2D density is set to $n_{\rm 2D} r_0^2 = 20$.