Static-response theory and the roton-maxon spectrum of a flattened dipolar Bose-Einstein condensate

Important information for the roton-maxon spectrum of a flattened dipolar Bose-Einstein condensate is extracted by applying a static perturbation exhibiting a periodic in-plane modulation. By solving the Gross-Pitaevskii equation in the presence of the weak perturbation, we evaluate the linear density response of the system and use it, together with sum rules, to provide a Feynman-like upper-bound prediction for the excitation spectrum, finding excellent agreement with the predictions of full Bogoliubov calculations. By suddenly removing the static perturbation, while still maintaining the trap, we find that the density modulations—as well as the weights of the perturbation-induced side peaks of the momentum distribution—undergo an oscillatory behavior with double the characteristic frequency of the excitation spectrum. The measurement of the oscillation periods could provide an easy determination of dispersion relations.

Figure 1

(a) GPE prediction for the static density response [Eq. (3)] versus in-plane momentum, and (b) energy dispersion for (a1),(b1) a dipolar and (a2),(b2) a nondipolar condensate. The BdG energy bands appear as blue solid lines, while the sum-rule upper bound—based on the the density response of the GPE ground state to a perturbing lattice potential with $k_{\rho }=\left|{\mathbf{k}}_{\mathrm{L}}\right|$ (3, 4, 5)—is marked with plus symbols. (b1) The sum rule's ability to provide an almost exact prediction for the roton-maxon spectrum of the lowest band is highly nontrivial given that our 3D GPE calculations inherently account for contributions from higher bands. (b2) A similar sum-rule prediction for a nondipolar BEC exhibits good agreement with the lowest BdG band only for $k_{\rho }{l}_z\lesssim 1$. Both regimes have $\mu =9.5\hslash {\omega }_z$ and compressibility $\hslash {\omega }_z\chi (k_{\rho }\to 0)=0.15$, and for the static-response calculations, we use $V_{\mathrm{L}}=0.0025\hslash {\omega }_z$. Vertical dashed lines represent the cases in Figs. 2 and 3.

Figure 2

In-trap dynamics of the density contrast $\mathrm{\Delta }n$ [Eq. (4)] and momentum response $N_{{\mathbf{k}}_{\mathrm{L}}}$ after sudden lattice removal at $t=0$. Lattices with (a) a maxon and (b) a roton wavelength are considered. The single-mode predictions (7) and (8)—with $\epsilon$ extracted from Fig. 1—are shown as solid blue lines, while the time-dependent GPE results appear as symbols.

Figure 3

In-trap oscillation maxima of the density contrast and the momentum response (see Fig. 2), as a function of the static lattice strength prior to its sudden removal. As in Fig. 2, here we consider (a) a maxon and (b) a roton. The linear responses (7) and (8)—using BdG energies—are shown as solid blue lines, while GPE results appear as symbols. The vertical dashed lines represent the $V_{\mathrm{L}}$ considered in Fig. 2.