Thermally Assisted Supersolidity in a Dipolar Bose-Einstein Condensate

TL;DR

This work addresses the challenge of stabilizing supersolid order in dipolar Bose–Einstein condensates by incorporating finite-temperature effects through a temperature-dependent extended Gross–Pitaevskii equation that includes Lee–Huang–Yang (LHY) and thermal fluctuation contributions. By mapping the finite-temperature phase diagram for a $^{162}$Dy BEC in an oblate trap, the authors demonstrate that temperature can constructively shift the uniform-to-modulated phase boundary toward larger scattering lengths and lower densities, enabling supersolid, honeycomb, and labyrinthine morphologies under more accessible conditions. Real-time temperature ramps reveal path-dependent dynamics: heating induces crystallization into a supersolid, while cooling melts it back to a uniform BEC, illustrating temperature as a versatile control knob for accessing and stabilizing modulated quantum phases. In addition, moderate thermal fluctuations stabilize isolated droplets that are unstable at zero temperature, enriching the experimentally reachable parameter space and suggesting concrete diagnostics such as Bragg peaks, interference fringes, and arrested expansion. Collectively, the study reframes finite temperature from merely a decoherence source into a resource for engineering and sustaining supersolid order in dipolar quantum gases, with broad implications for experimental exploration and theory benchmarking.

Abstract

Supersolidity in a dipolar Bose-Einstein condensate (BEC), which is the coexistence of crystalline density modulation and global phase coherence, emerges from the interplay of contact interactions, long-range dipole-dipole forces, and quantum fluctuations. Although realized experimentally, stabilizing this phase at zero temperature often requires high peak densities. Here we chart the finite-temperature phase behavior of a harmonically trapped dipolar BEC using an extended mean-field framework that incorporates both quantum (Lee-Huang-Yang) and thermal fluctuation effects. We find that finite temperature can act constructively: it shifts the supersolid phase boundary toward larger scattering lengths, lowers the density threshold for the onset of supersolidity, and broadens the stability window of modulated phases. Real-time simulations reveal temperature-driven pathways (crystallization upon heating and melting upon cooling) demonstrating the dynamical accessibility and path dependence of supersolid order. Moreover, moderate thermal fluctuations stabilize single-droplet states that are unstable at zero temperature, expanding the experimentally accessible parameter space. These results identify temperature as a key control parameter for engineering and stabilizing supersolid phases, offering realistic routes for their observation and control in dipolar quantum gases.

Figures (5)

• Figure 1: Finite-temperature phase diagram at $T=50$ nK for a $^{162}$Dy BEC in a harmonic trap with $(\omega_x,\omega_y,\omega_z) = 2 \pi \times (125,125,250)$ Hz. (a) Phase boundaries in the $(N, a_s)$ plane. (b) Representative column-density profiles $\rho_{2\text{D}} (x,y) = \int dz,|\Psi(\bm{r})|^{2}$ at selected points, illustrating the uniform BEC, SSD, honeycomb, and labyrinthine phases.
• Figure 2: Effect of temperature on the supersolid–BEC transition. (a) and (b) Density contrast $C$ versus scattering length $a_s$ at $T=0$ and $T=200$ nK, respectively. Insets: representative column-density maps illustrating the concurrent reduction of modulation.
• Figure 3: Effect of temperature on the BEC–honeycomb transition. (a) Phase boundary in the temperature–particle-number ($T$–$N$) plane at fixed $a_s=88.85 a_0$. (b), (c) Density contrast $C$ versus particle number $N$ at $T=0$ and $T=100$ nK, respectively, for fixed $a_s=90 a_0$. Insets: concurrent growth of modulation in the column density.
• Figure 4: Real-time response to linear temperature ramps at fixed $a_s=85 a_0$ and $N=1.3 \times 10^{4}$ with ramp duration $\tau_r = 100$ ms. (a) Cooling: $T(t)$ from $150$ to $0$ nK; the supersolid melts into a uniform BEC. (b) Heating: $T(t)$ from $0$ to $150$ nK; the system crystallizes into a supersolid. Insets: time evolution of the column density $\rho_{2 \text{D}} (x, y, t)$ illustrating the loss (a) or growth (b) of spatial modulation.
• Figure 5: (a) Line cuts of the real-space density along $x$ at $y=0$ for a $T=0$ BEC (dashed blue) and a $T=150$ nK single-droplet state (solid orange). (b), (c) Real-time evolution after trap release at $T=0$ and $T=150$ nK, respectively, highlighting rapid ballistic/hydrodynamic expansion at $T=0$ and arrested expansion for the self-bound droplet at finite temperature.