Roton spectroscopy in a harmonically trapped dipolar Bose-Einstein condensate

We study a harmonically trapped Bose-Einstein condensate with dipole-dipole interactions in a regime where a roton spectrum emerges. We show that the roton spectrum is clearly revealed in the static and dynamic structure factors which can be measured using Bragg spectroscopy. We develop and validate a theory based on the local-density approximation for the dynamic structure factor.

Figure 1

Stability phase diagram and related excitation properties of a quasi-2D uniform dipolar BEC. White and light-gray regions indicate where the BEC is dynamically stable. In the light-gray region the spectrum has a roton minimum. In the dark-gray and black regions the system is dynamically unstable. This can arise from modes at zero momentum (phonon instability, black region) or finite momentum (roton instability, dark-gray region) developing imaginary parts. Subplots A–D show cases of the spectrum (3), with the real (solid line) and imaginary (dashed line) parts shown.

Figure 2

(a) Spectrum of $m_z=0$ quasiparticle modes demonstrating the roton softening. (b)–(d) Contour plots of modes at $D=700$. (b) Condensate density. (c),(d) Density perturbation [$\delta {\rho }_j={\psi }_0(u_j+v_j)$] for the first (c) and third (roton) (d) $m_z=0$ quasiparticles [these modes are indicated by circles in (a)]. Solid (dotted) contours indicate positive (negative) density perturbations. (e) $S\left(k\right)$ for $D$ values corresponding to the vertical dashed lines in (a). Inset: A comparison of full calculations for $S\left(k\right)$ (symbols) to the LDA results [Eq. (6)] (lines) for $D=200,600,725$ (lowest to higher curves). Results for $\lambda =40$ and $C=0$.

Figure 3

Characterization of the roton properties of a $\lambda =40$ trapped dipolar BEC using $S\left(k\right)$. (a) Peak value of $S\left(k\right)$. (b) Peak wave vector $k_{\mathrm{peak}}$ (none shown when $\max \left[S\right(k\left)\right]<1.05$). Circles mark the parameters of states analyzed in Fig. 2e. Gray boundary line indicates when the system is unstable due to softening of excited modes, with black segments indicating that the BEC is in a biconcave state at the boundary (see [3]).

Figure 4

(a)–(f) $S(k,\omega )$ for indicated values of $D$. A Gaussian of width $\Delta \omega =0.5{\omega }_{\rho }$ is used to smooth the $\delta$ functions in $S(k,\omega )$. The white dashed line shows the free particle dispersion $\epsilon \left(k\right)/\hslash$. The solid black and gray lines are the mean response $\overline{\omega }\left(k\right)$ obtained from GPE-based and quasi-2D approximation LDA calculations, respectively (see text and below). White ellipse in (f) identifies a roton response feature. LDA calculations of $S(k,\omega )$ for the parameters in (f) are shown in (f1) and (f2). (f1) GPE-based LDA approach of Eq. (6). (f2) quasi-2D approximation LDA approach (see text). The mean response $\overline{\omega }\left(k\right)\equiv \int d\omega \omega S(k,\omega )/S\left(k\right)$ is shown for each result. Other parameters: $C=0$ and $\lambda =40$.