Fingerprinting Rotons in a Dipolar Condensate: Super-Poissonian Peak in the Atom-Number Fluctuations

We demonstrate that measurements of atom-number fluctuations in a trapped dipolar condensate can reveal the presence of the elusive roton excitation. The key signature is a super-Poissonian peak in the fluctuations as the size of the measurement cell is varied, with the maximum occurring when the size is comparable to the roton wavelength. The magnitude of this roton feature is enhanced with temperature. The variation in fluctuations across the condensate demonstrates that the roton excitations are effectively confined to propagate in the densest central region, realizing a density trapped roton gas. While our main results are based on full numerical solutions of the mean-field equations, we also develop and validate a simple local density theory. Finally, we consider fluctuations measured within a washer-shaped cell which filters out the contribution of modes with nonzero angular momentum and provides a signal sensitive to individual roton modes.

Figure 1

Schematic geometry of the system we consider: a radially symmetric pancake condensate realized by tight confinement along $z$; cylinder and washer-shaped cells in which number fluctuations are measured. The cylindrical cell is parameterized by the radius $R$ from the trap center to the cylinder axis and its diameter $D$. The washer-shaped cell is centered on the trap center and is parameterized by the midradius $R$ and width $W$. For both cells the $z$ extent is taken to be larger than the condensate thickness.

Figure 2

(a) Stability phase diagram in interaction parameter space for $N_0=2.5\times {10}^4$ ${}^{164}\mathrm{Dy}$ atoms in a $\lambda =20$ trap with ${\omega }_{\rho }=2\pi \times 10.8{\mathrm{s}}^{-1}$. The shaded region indicates where a stable solution exists, and the gray portions of the boundary mark the boundary of density oscillation islands [6]. Crosses indicate the three sets of parameters we examine in this Letter with $a_s/a_0=0$, 9.04, 18.1 as the cases we label as $\alpha$, $\beta$, $\gamma$, respectively, with $a_0$ the Bohr radius. The dipole length is $a_{\mathrm{dd}}\equiv m{g}_{\mathrm{dd}}/4\pi {\hslash }^2\approx 132.6a_0$ for ${}^{164}\mathrm{Dy}$. (b) Quasiparticle energies of $m_z=0$ modes as $a_s$ is varied along the dashed path in (a). Vertical lines mark the cases $\alpha$, $\beta$, and $\gamma$, and the crossed circle indicates the lowest energy roton mode. The contours of the density fluctuation $\delta n_i$ for the roton are shown in the inset. Contour spacing is $3\times {10}^{11}{\mathrm{cm}}^{-3}$, and solid (dot-dashed) lines represent positive (negative) contours, while nodes are shown as thick black lines.

Figure 3

$\delta N_{\sigma }^2$ as a function of cylinder position $R$ for various diameters $D$ for case $\alpha$. (a) $T=0$ with a depletion of $\tilde{N}=171$. (b) $T=5.2\mathrm{nK}$ with $\tilde{N}=1570$. Insets: Condensate, depletion, and fluctuation density. Densities are evaluated at $z=0$, and $n_F$ is calculated by using modes with ${ϵ}_j<20\hslash {\omega }_{\rho }$.

Figure 4

Peak fluctuations (at $R=0$) within a cylindrical cell (a) as a function of temperature at diameter $D=1.67\mu \mathrm{m}$ and (b) as a function of cylinder diameter at $T=5.2\mathrm{nK}$. Each graph has results for three different values of contact interaction (see Fig. 2). Solid line, full calculations; dashed line, LDA result (see the text).

Figure 5

Fluctuations versus washer radius $R$ and for various washer widths $W$ for $a_s=0$. (a) $T=0$ results. Inset: $\delta n_i$ for the lowest roton mode [repeated from the inset in Fig. 2b]. (b) $T=5.2\mathrm{nK}$ results. Inset: The dashed curve shows the number fluctuations calculated by using only the lowest roton mode, compared to the full calculation (solid curve).