Beyond mean-field low-lying excitations of dipolar Bose gases

We theoretically investigate various beyond mean-field effects on Bose gases at zero temperature featuring the anisotropic and long-range dipole-dipole interaction in addition to the isotropic and short-range contact interaction. Within the realm of the Bogoliubov–de Gennes theory, we consider static properties and low-lying excitations of both homogeneous and harmonically trapped dipolar bosonic gases. For the homogeneous system, the condensate depletion, the ground-state energy, the equation of state, and the speed of sound are discussed in detail. Making use of the local density approximation, we extend these results in order to study the properties of a dipolar Bose gas in a harmonic trap and in the regime of large particle numbers. After deriving the equations of motion for the general case of a triaxial trap, we analyze the influence of quantum fluctuations on important properties of the gas, such as the equilibrium configuration and the low-lying excitations in the case of a cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation modes, we also discuss the radial quadrupole mode. We find that the latter acquires a quantum correction exclusively due to the dipole-dipole interaction. As a result, we identify the radial quadrupole as a reasonably accessible source for the signature of dipolar many-body effects and stress the enhancing character that dipolar interactions have for quantum fluctuations in the other oscillation modes.

Figure 1

Functions ${\mathcal{Q}}_3\left(x\right)$ (lower, red curve) and ${\mathcal{Q}}_5\left(x\right)$ (upper, blue curve), which govern the dependence of the condensate depletion and the ground-state energy correction on the relative dipolar interaction strength $x={\epsilon }_{\mathrm{dd}}$, respectively.

Figure 2

Comparison between the sound velocities in the mean-field approximation (22), plotted in dashed curves, and its quantum corrected version (34), represented here in solid curves. The light gray (red) and the dark gray (blue) curves are for ${\epsilon }_{\mathrm{dd}}=0.6$ and ${\epsilon }_{\mathrm{dd}}=1$, respectively.

Figure 3

Relative correction to the aspect ratio $\delta \kappa$ in units of $\widetilde{\delta \kappa }$ as a function of ${\epsilon }_{\mathrm{dd}}$. The lower (blue) curve corresponds to $\lambda =1.00$, while the middle (red) one stands for $\lambda =0.75$ and the upper (green) curve is for $\lambda =0.5$.

Figure 4

Relative correction to the aspect ratio $\delta \kappa$ in units of $\widetilde{\delta \kappa }$ as a function of $\lambda$. The lower (green) curve corresponds to ${\epsilon }_{\mathrm{dd}}=0.89$, while the middle (blue) one represents ${\epsilon }_{\mathrm{dd}}=0.95$, and the upper (red) curve is for ${\epsilon }_{\mathrm{dd}}=0.97$.

Figure 5

Quantum correction to the frequencies of the low-lying excitations in units of $\widetilde{\delta \Omega }$ as a function of the trap aspect ratio $\lambda$. The curves displayed in black, light gray (red), and dark gray (blue) correspond, in turn, to the monopole, quadrupole, and radial quadrupole oscillations. Moreover, continuous curves are for the parameter values $a_s=100a_0$ and ${\epsilon }_{\mathrm{dd}}=0.69$, whereas dashed curves represent ${\epsilon }_{\mathrm{dd}}=0$ [60]. In the latter case, the curves do not depend on $a_s$.

Figure 6

Thomas-Fermi radii in units of the nondipolar radius in the $Ox$ direction $R_x^{0,\mathrm{g}}$ as functions of time in units of ${\omega }_x^{-1}$ during monopole oscillations. The dark gray (blue) and the light gray (red) curves correspond to $R_x$ and $R_z$, respectively. We adopt the values ${\epsilon }_{\mathrm{dd}}=0.89$ and $a_s=100a_0$. Dashed curves represent mean-field, while full curves represent quantum-corrected results. Here we show the oscillations from ${\omega }_x^{}t=0$ until ${\omega }_x^{}t=5$.

Figure 7

Thomas-Fermi radii in units of the nondipolar radius in the $Ox$ direction $R_x^{0,\mathrm{g}}$ as functions of time in units of ${\omega }_x^{-1}$ during monopole oscillations. The dark gray (blue) and the light gray (red) curves correspond to $R_x$ and $R_z$, respectively. We adopt the values ${\epsilon }_{\mathrm{dd}}=0.89$ and $a_s=100a_0$. Dashed curves represent mean-field, while full curves represent quantum-corrected results. Here the plots go from ${\omega }_x^{}t=40$ to ${\omega }_x^{}t=45$.

Figure 8

Thomas-Fermi radii in units of the nondipolar radius in the $Ox$ direction $R_x^{0,\mathrm{g}}$ as functions of time in units of ${\omega }_x^{-1}$ during quadrupole oscillations. The dark gray (blue) and the light gray (red) curves correspond to $R_x$ and $R_z$, respectively. We adopt the values ${\epsilon }_{\mathrm{dd}}=0.89$ and $a_s=100a_0$. Dashed curves represent mean-field, while full curves represent quantum-corrected results.