From elastic to inelastic deformation of a dipolar supersolid

TL;DR

This work addresses how a finite-size dipolar supersolid responds to dilation and compression by modeling a quasi-1D dipolar Bose-Einstein condensate in a box trap with an extended Gross-Pitaevskii equation that includes contact, dipolar, Lee-Huang-Yang, and three-body terms. It identifies stationary supersolid states with discrete unit-cell counts $N_{ ext{uc}}$ and leverages roton-instability analysis to map an elastic region bounded by critical widths $w_{c, ext{AB}}$ and $w_{c, ext{CD}}$, forming an elastic-to-inelastic phase diagram. Through dynamical simulations of box-width quenches, it shows that the system tracks excited stationary states and preserves crystal structure up to the thresholds, after which unit cells bifurcate or merge, signaling inelastic deformation as quantified by the overlap $\\mathcal{O}(t)$. These results provide a controlled framework for the mechanical properties of quantum solids and potential applications in quantum thermodynamics and materials science.

Abstract

Due to its peculiar superfluid-crystal duality feature, supersolid has received great research interest. Recently, researchers have paid much attention to its elastic response properties; however, the inelastic deformation has barely been explored. In this work, we study the transition from elastic to inelastic deformation of a dipolar supersolid Bose-Einstein condensate trapped in a box potential (i.e., a dipolar supersolid with finite size). We obtained the stationary supersolid states (both ground and excited) of the system, and examined the relation between the supersolid size and the number of unit cells it can accommodate, which can essentially help us to understand the dynamical responses of the supersolid during a dilation or compression process. We found that within a certain dilation or compression extent, the supersolid can retain its original crystal structure, that is, it endures an elastic deformation; however, when the extent exceeds a critical threshold, the original crystal structure of the supersolid will be disrupted, which signifies an inelastic deformation. Furthermore, we both analytically and numerically determined the critical point of the transition from elastic to inelastic deformation, and mapped out a phase diagram. These results open up new territory in the research of supersolid mechanical properties, and may find applications in quantum material science and quantum-based technologies.

Figures (7)

• Figure 1: Collective excitation spectra of uniform dipolar BECs with different dipole-dipole interaction strength, Eq. (\ref{['eq:collectiveExcitationSpectrum']}). When $g_{dd}=1.70$, the curve of $\Omega\left(k\right)$ shows a roton minimum (dashed violet line). Around $g_{dd}=1.55$, the roton minimum drops to zero (dash-dotted green line). When $g_{dd}=1.41$, $\Omega\left(k\right)$ becomes complex valued in the gray color region $\left|k\right|\in\left[k_{\mathrm{RI,min}},k_{\mathrm{RI,max}}\right]$, which means a roton instability (RI, solid cyan line). Other parameters are $\rho_{0}=40$, $g=-1$, $\gamma=0.001$ and $p=0.006$.
• Figure 2: Stationary supersolid states in a box potential with width $w=12$. (a) The ground state which contains $N_{uc}=23$ unit cells (solid violet line), and the excited state which contains $N_{uc}=22$ unit cells (dashed green line). (b) The stationary energy as a function of unit cell numbers, with the zero energy point shifted to the ground state energy. (c) The discrete collective excitation spectrum $\Omega\left(k_{n}\right)$ with $k_{n}=\left(n-1/2\right)\pi/w$ and $\rho_0 = N/\left(2w\right)$. The roton instability (RI) region is highlighted in gray color. In panels (b,c), the solid line just connects the data points for guiding the eye. The parameters used are $N=1000$, $g=-1$, $g_{dd}=1.41$, $\gamma=0.001$, $p=0.006$, $V_{0}=100$, and $\sigma=0.1$ (these parameters will remain unchanged in the following contents).
• Figure 3: Ground supersolid states in box potential as the width changes. (a) Ground state energy $E_{0}$ (left axis, solid violet line) and unit cell number $N_{\mathrm{uc}}$(right axis, dashed green line) as a function of the box width $w$. (b) The ground supersolid state $\phi_{0}\left(x\right)$ as a function of the box width $w$. The inset enlarges the small boxed area, to clearly show the abrupt change of the wavefunction.
• Figure 4: Excited supersolid states in box potential as the width changes. (a) Energy spectrum. The lowest few $E_{i}-E_{0}$ values are plotted as a function of the box width $w$. The color of these points represents the number of unit cells contained in the corresponding supersolid state. The points for $N_{\mathrm{uc}}=23$ and $30$ are connected with a black line for guiding the eye. (b,c) Wavefunctions of the supersolid state with $N_{\mathrm{uc}}=23$ (b) and $30$ (c) unit cells, as a function of box width $w$. These two panels terminate at $w=15.8$ and $14.7$ respectively, because when $w>15.8$, the supersolid state with $N_{\mathrm{uc}}=23$ no longer exists; and when $w<14.7$, the supersolid state with $N_{\mathrm{uc}}=30$ also does not exist any more.
• Figure 5: Elastic and inelastic deformation phase diagram of a finite-size supersolid. The maximum and minimum number of unit cells of a supersolid trapped in a box well with width $w$ are plotted. The two violet solid lines are the numerical (NUM) results, while the two green dashed lines are the results obtained by analytical roton instability analysis (RI). The pink area can be seen as the elastic deformation region, while quenching out this region will lead to an inelastic deformation. The two arrows AB and CD indicate two quenching processes, with $w_{c,\mathrm{AB}}=15.8$ and $w_{c,\mathrm{CD}}=14.7$ being the corresponding elastic to inelastic transition critical points.