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arXiv:2606.13502 · Physical Review A

Quasi-two-dimensional trapped tilted dipoles at zero and finite temperatures in the strongly dipolar regime

Juan Sánchez-Baena

Published Sep 8, 2026• 0 citations
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Abstract

Motivated by the recent experimental observation of dipolar supersolid stripes in a quasi two-dimensional geometry [arXiv:2512.13280 (2025)], we study a trapped system of fully polarized dipoles in a strongly axially confined geometry, both at zero and finite temperatures, by means of Bogoliubov theory. The dipoles are strongly harmonically trapped along the z axis and subjected to a box trap in the x-y plane. We characterize the physics of the trapped system at zero and finite temperatures as a function of the tilting angle of the dipoles, the number of particles and the scattering length, restricting ourselves to the experimentally relevant regime of large condensate fractions. We also illustrate the influence of the aspect ratio of the box trap in the liquid character of the system and its structure. We observe a remarkable promotion of spatial modulations when temperature is increased while keeping the total particle number constant for specific configurations, in qualitative agreement with previous Monte Carlo results in a 3D geometry. Our results are useful to understand the zero temperature physics of the trapped dipolar system in the quasi-2D limit and in the strongly dipolar regime. In addition, they allow to assess the effect of temperature in its equilibrium properties in experimentally relevant conditions, which may be useful for thermometry applications.

DOI
10.1103/xbdg-r5g3

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Motivated by the recent experimental observation of dipolar supersolid stripes in a quasi two-dimensional geometry [arXiv:2512.13280 (2025)], we study a trapped system of fully polarized dipoles in a strongly axially confined geometry, both at zero and finite temperatures, by means of Bogoliubov theory. The dipoles are strongly harmonically trapped along the z axis and subjected to a box trap in the x-y plane. We characterize the physics of the trapped system at zero and finite temperatures as a function of the tilting angle of the dipoles, the number of particles and the scattering length, restricting ourselves to the experimentally relevant regime of large condensate fractions. We also illustrate the influence of the aspect ratio of the box trap in the liquid character of the system and its structure. We observe a remarkable promotion of spatial modulations when temperature is increased while keeping the total particle number constant for specific configurations, in qualitative agreement with previous Monte Carlo results in a 3D geometry. Our results are useful to understand the zero temperature physics of the trapped dipolar system in the quasi-2D limit and in the strongly dipolar regime. In addition, they allow to assess the effect of temperature in its equilibrium properties in experimentally relevant conditions, which may be useful for thermometry applications.
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.
Two-dimensional (2D) dipolar atomic gases present unique opportunities for exploring novel quantum phases due to their anisotropic and long-range interactions. However, the behavior of strongly dipolar Bose gases in 2D remains unclear, especially when dipoles are tilted. Here, we demonstrate the creation and characterization of strongly dipolar 2D condensates in a quasi-2D harmonic trap with tunable dipole orientation. By investigating scale invariance properties through breathing collective mode measurements, we observe significant breaking of scale invariance when dipoles are tilted in-plane indicating the dominance of the nonlocal dipole-dipole interactions (DDIs) in this regime. Interestingly, the breaking of the scale invariant dynamics is accompanied by an increase in quantum fluctuations, as shown by comparison with mean-field and beyond mean-field theoretical studies. Our experiments also reveal that at critical tilt angles around 70°, stripe-type density modulations emerge, suggesting the presence of a roton spectrum in 2D, while the system still shows hydrodynamic nature with the phase-locking breathing behavior. This observation elucidates the many-body effect induced by DDIs in 2D, thus marking a crucial step toward realizing 2D supersolids and other exotic quantum phases.
Recent experimental advances in creating stable dipolar bosonic systems, including polar molecules with large electric dipole moments, have led to vigorous theoretical activities. Recent reporting of observation of roton feature in dipolar erbium has provided added impetus to theoretical and experimental work. Here we discuss our mean-field theory work on 2D and quasi-2D dipolar bosons with dipoles oriented at an angle to the direction perpendicular to the confining 2D plane, i.e. for {\it non-zero tilt angles}. Using Bogoliubov-de Gennes equations, we present results on a number of T=0 properties of both 2D and quasi-2D systems, such as excitation spectra, structure functions, sound velocities, quantum depletion, etc. We explore instabilities at varying tilt angle, density and dipolar coupling. We map out phase diagrams as a function of tilt angle, dipole strength and density. We find the development of maxon-roton behavior leading to roton instabilities at large densities for small tilt angles, and at low densities for large tilt angles. The behavior is anisotropic in k-space; accordingly the roton instabilities occur first in the kyk_yky​ direction, suggestive of inhomogeneity and stripe phase, with density mode becoming soft in the yyy-direction. Beyond a critical tilt angle, at any density, the dipolar system collapses owing to a phonon instability. We discuss similarities and differences between the properties of 2D and quasi-2D dipolar systems at non-zero tilt angles.
Motivated by a recent experimental work arXiv:2608.20327, we theoretically investigate the nature of phase transitions in a trapped dipolar quantum gas upon tuning the contact interaction strength and dipole orientation. Using extended Gross-Pitaevskii simulations including thermal and quantum effects, we map the phase diagram comprising unmodulated superfluids, triangular droplet arrays, and stripe states. We further characterize the order of the transitions separating them through imaginary-time hysteresis analysis. The unmodulated-to-modulated transition evolves from discontinuous to continuous with increasing dipole tilt before becoming discontinuous again at large tilts. The degree of continuity is further affected by particle number, temperature, and confinement. While continuous behavior generally correlates with the emergence of stripe states, this correspondence is not systematic, and finite-size effects also influence the continuity of the transition. The triangular-to-stripe transition is found to be discontinuous at small contact interaction strengths and becomes increasingly continuous upon approaching the unmodulated-to-modulated phase boundary. Our results reveal a tunable transition order in finite dipolar quantum gases and provide guidance for its experimental investigation.
Motivated by recent experiments with ultracold magnetic atoms trapped in optical lattices where the orientation of atomic dipoles can be fully controlled by external fields, we study, by means of quantum Monte Carlo, the ground-state properties of dipolar bosons trapped in a two-dimensional lattice with density-induced hopping and where the dipoles are tilted along the xzxzxz plane. We present ground-state phase diagrams of the above system at different tilt angles. We find that as the dipolar interaction increases, the superfluid phase at half-filling factor is destroyed in favor of either a checkerboard or stripe solid phase for tilt angle $\ensuremath{\theta}\ensuremath{\lesssim}{30}^{\ensuremath{\circ}}$ or $\ensuremath{\theta}\ensuremath{\gtrsim}{30}^{\ensuremath{\circ}}$, respectively. More interesting physics happens at tilt angles $\ensuremath{\theta}\ensuremath{\gtrsim}{58}^{\ensuremath{\circ}}$, where we find that as the dipolar interaction strength increases, solid phases first appear at filling factor lower than 0.5. Moreover, unlike what is observed at lower tilt angles, we find that at half filling, a stripe supersolid intervenes between the superfluid and stripe solid phase.
The recent advances in creating nearly degenerate quantum dipolar gases in optical lattices are opening the doors for the exploration of equilibrium physics of quantum systems with anisotropic and long-range dipolar interactions. In this paper we study the zero- and finite-temperature phase diagrams of a system of hard-core dipolar bosons at half-filling, trapped in a two-dimensional square optical lattice. The dipoles are aligned parallel to one another and tilted out of the optical lattice plane by means of an external electric field. At zero-temperature, the system is a superfluid (SF) at all tilt angles θ provided that the strength of dipolar interaction is below a critical value . Upon increasing the interaction strength while keeping θ fixed, the SF phase is destabilized in favor of a checkerboard or a stripe solid (SS) depending on the tilt angle. We explore the nature of the phase transition between the two solid phases, identifying a region of metastable emulsion states intervening between the two solid lobes. Additionally, we study the stability of these quantum phases against thermal fluctuations and find that the SS is the most robust, making it the best candidate for experimental observation.