Infinite dipolar droplet: A simple theory for the macrodroplet regime

In this paper we develop a theory for an infinitely long droplet state of a zero-temperature dipolar bosonic gas. The infinite droplet theory yields simpler equations to solve for the droplet state and its collective excitations. We explore the behavior of infinite droplets using numerical and variational solutions, and we demonstrate that it can provide a quantitative description of large finite droplets of the type produced in experiments. We also consider the axial speed of sound and the thermodynamic limit of a dipolar droplet.

Figure 1

(Left) Density isosurface of a finite free-space droplet with $N=2.5\times {10}^4{}^{164}\mathrm{Dy}$ atoms, $a_{\mathrm{dd}}=130.8a_0$, and $a_s=80a_0$. (Right) Schematic of a section of the infinite dipolar droplet we develop here.

Figure 2

Variational theory effective interaction ${\tilde{U}}_0\left(k_z\right)$ with the low-$k_z$ and high-$k_z$ limits shown. Results for a ${}^{164}\mathrm{Dy}$ system with $n=2.5\times {10}^3/\mu \mathrm{m}, a_s=80a_0, a_{\mathrm{dd}}=130.8a_0$, and $l=0.383 \mu \mathrm{m}$.

Figure 3

(a)–(c) The dynamic structure factor $S(k_z,\omega )$ for an infinite droplet state of ${}^{164}\mathrm{Dy}$ atoms with $n=2.5\times {10}^3/\mu \mathrm{m}$ and for (a,d) $a_s=115a_0$, (b,e) $a_s=100a_0$, and (c,f) $a_s=80a_0$. Cases (a)–(c) are in free-space (${\omega }_{\rho }=0$) and cases (d)–(f) are confined with ${\omega }_{\rho }/2\pi =150$ Hz. The variational result for $E_{k_z}^{\text{var}}$ is shown with magenta solid lines. The lower bound of the continuum ${\epsilon }_{k_z}^{\text{cont}}$ is shown by the blue dashed lines in panels (a)–(c), and in panels (d)–(f) the yellow dash-dot line shows the free-particle dispersion ${\epsilon }_{k_z}$. We broadened the $\delta$ function in the dynamic structure factors by setting $\delta \left(\omega \right)=e^{-{(\omega /{\omega }_B)}^2}/\sqrt{\pi }{\omega }_B$, with ${\omega }_B/2\pi =15$ Hz.

Figure 4

Mapping and comparison of finite and infinite droplets. The $z=0$ transverse density profile for a finite droplet with (a) $N=2.5\times {10}^3$ and (b) $N=25\times {10}^3$ atoms compared to the infinite droplet results. (c) Comparison of the $1/e$-density half-width ${\sigma }_{\rho }$ in the $z=0$ plane as the number of atoms in the droplet varies (for the variational theory this is $l$). The finite droplet (d) aspect ratio and (e) linear density (17) as $N$ varies. Here ${\sigma }_z$ is the $1/e$-density half-width of the droplet on the $z$ axis. These results are for a free-space droplet of ${}^{164}\mathrm{Dy}$ atoms with $a_{\mathrm{dd}}=130.8a_0$ and $a_s=80a_0$. The droplet is only self-bound for $N$ exceeding the critical value of $N_c\approx 1899$.

Figure 5

Excitation spectrum of a free-space droplet of $N={10}^5{}^{164}\mathrm{Dy}$ atoms with $a_s=80a_0$. The BdG excitation energies of a finite droplet are plotted as $\times$ symbols against a $z$-wave-vector assignment (these results are from Ref. [21] and assignment details are discussed therein). Results are sorted by the $z$ component of angular momentum of the excitation [blue ($m=0$), red ($m=1$), green ($m=2$), and black ($m=3$)]. Solid lines give the infinite droplet results for comparison using the linear density $n=4231.6/\mu \mathrm{m}$. Black dash-dotted lines give the variational result, Eq. (14), and the black dashed line shows ${\epsilon }_{k_z}$ for reference. The inset shows the infinite droplet result including the continuum excitations that develop at higher energies. Note that the $E_{k_{z}00}$ band (blue dots) becomes imaginary for $k_z\lesssim 0.04 \mu {\mathrm{m}}^{-1}$.

Figure 6

(a) $M{c}_z^2$ [16] as a function of $a_s$ for fixed linear density $n=2.5\times {10}^3/\mu \mathrm{m}$ from EGPE (blue lines) and variation (red line) calculations. Results are shown for infinite free-space droplets (solid line), transversally confined infinite droplets (dashed line) and a transversally confined condensate with ${\gamma }_{\text{QF}}=0$ (dashed-dotted line). The upper inset shows how $k_z^{*}$ varies with $n$ for $a_s/a_0=75$ (black line, top), 80 (red line, middle), and 85 (blue line, bottom). The lower inset shows the imaginary part of the $E_{k_{z}00}$ spectrum for a free-space droplet with $a_s=80a_0$. (b) The radial $1/e$-density widths of the states. These results use $a_{\mathrm{dd}}=130.8a_0$, and for trapped cases ${\omega }_{\rho }/2\pi =150$ Hz is used.