Excitations and number fluctuations in an elongated dipolar Bose-Einstein condensate

We study the properties of a magnetic dipolar Bose-Einstein condensate in an elongated (cigar shaped) confining potential in the beyond quasi-one-dimensional regime. In this system the dipole-dipole interactions develop a momentum dependence related to the transverse confinement and the polarization direction of the dipoles. This leads to density fluctuations being enhanced or suppressed at a length scale related to the transverse confinement length, with local atom number measurements being a practical method to observe these effects in experiments. We use mean-field theory to describe the ground state, excitations, and the local number fluctuations. Quantitative predictions are presented based on full numerical solutions and a simplified variational approach that we develop. In addition to the well-known roton excitation, occurring when the dipoles are polarized along a tightly confined direction, we find an “antiroton” effect for the case of dipoles polarized along the long axis: A nearly noninteracting ground state that experiences strongly repulsive interactions with excitations of sufficiently short wavelength.

Figure 1

Schematic diagram: A one-dimensional Gaussian cell of length $L$ (shown by magenta surface profile) along the axial direction ($z$ axis) is used to measure atom number in an elongated dipolar BEC. The dipole magnetic moment ${\mathit{\mu }}_m$ makes angle $\alpha$ with the axial direction, and we show cases where the dipoles are polarized along ($\alpha =0$) and transverse ($\alpha =\pi /2$) to the long axis.

Figure 2

Comparison of analytical and numerical results for the $k_z$-space interaction for the variational Gaussian function and $\alpha =0$. Results are shown for several values of $\eta$ and $g_s=0$. Since the results for $\eta$ and $1/\eta$ are identical we only show cases with $\eta \le 1$. The analytic result is given by Eqs. (21) and (22) with $q_0\left(\eta \right)={\left(\frac{2\eta }{1+{\eta }^2}\right)}^{4/5}$. The numerical result is obtained by evaluating ${\tilde{U}}_0^{\text{num}}\left(k_z\right)=\int \frac{d{\mathbf{k}}_{\mathit{\rho }}}{{\left(2\pi \right)}^2}\tilde{U}\left(\mathbf{k}\right){\left|{\mathcal{F}}_{\mathit{\rho }}\left\{{\left({\chi }^{\text{var}}\right)}^2\right\}\right|}^2$.

Figure 3

Properties of excitations for a ${}^{164}\mathrm{Dy}$ elongated dipolar system with $\alpha =0, a_s=140a_0$, and $a_{dd}=130.8a_0$. (a) The BdG excitation energy $E_{k_{z}j}$ for even parity modes (blue dots), variational $E_{k_z}^{\text{var}}$ (solid black line), and free-particle ${\epsilon }_{k_z}$ (black dashed line) dispersion relations. The red dashed line indicates the linear phonon spectrum. The inset shows ${\left|\chi \right|}^2$ with the black dotted line indicating where the $|{\chi }^{\text{var}}{|}^2$ is 1/$e$ of its peak value. (b) $S\left(k_z\right)$ at $T=0$ obtained from BdG and variational theories. (Inset) $k_z$-space interaction (21). (c) $S(k_z,\omega )$ obtained from BdG results. (d) $S\left(k_z\right)$ as in (b), but over a larger $k_z$ range. In (b) and (d) the static structure factor for $a_{dd}=0$ and $a_s=9.17a_0$ (gray line) is also shown. Results are for isotropic transverse confinement ${\omega }_{x,y}/2\pi =150\mathrm{Hz}$ and $n=2.5\times {10}^3\mu \mathrm{m}$. The $\delta$ functions in the definition of $S(k_z,\omega )$ are frequency broadened by Gaussian of width $50\mathrm{Hz}$.

Figure 4

Properties of excitations for a ${}^{164}\mathrm{Dy}$ elongated dipolar system with $\alpha =\pi /2, a_s=140a_0$, and $a_{dd}=130.8a_0$. (a) The BdG excitation energy $E_{k_{z}j}$ for even parity modes (blue dots), with the variational $E_{k_z}^{\text{var}}$ (solid black line) and free-particle ${\epsilon }_{k_z}$ (black dashed line) dispersion relations shown. The red dashed line indicates the linear phonon spectrum. The inset shows ${\left|\chi \right|}^2$ with the black dotted line indicating where the $|{\chi }^{\text{var}}{|}^2$ is 1/$e$ of its peak value. (b) Zero temperature static structure factor obtained from BdG and variational theories. (Inset) Variational $k_z$-space interaction. (c) Dynamic structure factor obtained from BdG results. Other parameters as in Fig. 3.

Figure 5

Properties of excitations for a ${}^{164}\mathrm{Dy}$ elongated dipolar system with $\alpha =\pi /2$ and $a_s=115a_0$. (a) The BdG excitation energy $E_{k_{z}j}$ for modes that are of even parity in $x$ and $y$ (blue dots), with the variational $E_{k_z}^{\text{var}}$ (solid black line) and the free-particle dispersion ${\epsilon }_{k_z}$ (black dashed line) dispersion relations shown. The red dashed line indicates the linear phonon spectrum using the variational speed of sound. The local minimum of the dispersion relations identifies the roton excitation (filled circles). The inset shows ${\left|\chi \right|}^2$ with the black dotted line indicating where the $|{\chi }^{\text{var}}{|}^2$ is 1/$e$ of its peak value. (b) Zero temperature static structure factor obtained from BdG and variational theories. (Inset) Variational $k_z$-space interaction. (c) Dynamic structure factor obtained from BdG results. (d) The roton energy from the variational and BdG calculation as a function of $a_s$, shown from when it appears until it goes to zero energy. Other parameters as in Fig. 3.

Figure 6

Speed of sound and stability. The speed of sound versus $a_s$ for an elongated dipolar system at $\alpha =0$ (lower curves) and $\alpha =\pi /2$ (upper curves). We show both the BdG and variational results [see Eqs. (13), (24), and (23)]. The left-hand termination point of the curves marks where the system is unstable. Other parameters as in Fig. 3.

Figure 7

Relative number fluctuations of an elongated dipolar condensate versus linear cell size for (a) $T=0$, (b) $T=5\mathrm{nK}$, and (c) $T=50\mathrm{nK}$. Solid lines are calculated using the GPE ground state and BdG excitations for the three cases considered in Sec. 3 and a contact interactions only case with $a_s=140a_0$. The dashed lines are approximate large-cell results based on the variational speed of sound (see text) plotted for $L>10\mu \mathrm{m}$ and the filled circles in (b) and (c) indicate ${\lambda }_T$. In all subplots the transverse confinement length is shown for reference. The inset of (a) shows the (unscaled) number fluctuations calculated at $T=0$ using the GPE ground state and BdG excitations for large cells. The inset of (b) shows the full results (solid lines) compared to the numerically integrated variational result (dotted) (see text).

Figure 8

Peak and dip features in the relative number fluctuations. (a) The maximum (peak) and minimum (dip) value of $\mathrm{\Delta }{N}_L^2/N_L$ for $\alpha =\pi /2$ and $\alpha =0$, respectively. Results shown for $T=50\mathrm{nK}$ from the BdG (blue lines) and variational (black lines) theories. The small magenta bar indicates the $a_s$ range over which the roton occurs in the spectrum of the $\alpha =\pi /2$ system. (b) The cell size $L_{\text{peak}}$ and $L_{\text{dip}}$ where the peak and dip occur, respectively, corresponding to the results in (a). The inset to (b) uses the $a_s=140a_0$ results from Fig. 7 to illustrate the peak and dip in the relative number fluctuations for the $\alpha =\pi /2$ and $\alpha =0$ cases, respectively.