Stability of the density-wave state of a dipolar condensate in a pancake-shaped trap

We study a dipolar boson-fermion mixture in a pancake-shaped geometry at absolute zero temperature, generalizing our previous work on the stability of polar condensates and the formation of a density-wave state in cylindrical traps. After examining the dependence of the polar condensate stability on the strength of the fermion-induced interaction, we determine the transition point from a ground-state Gaussian to a hexagonal density-wave state. We use a variational principle to analyze the stability properties of those density-wave state.

Figure 1

(Color online) Energy landscape $E_g$ obtained by the variational Gaussian ansatz Eq. (9) as a function of $\eta =d_z∕d$ and $d_z∕{\mathcal{l}}_z$ at a constant dipolar interaction strength $g_{3\mathrm{d}}=30$. In these figures the energy is cut off at $E_g=1.5$ from above and $E_g=-0.5$ from below for viewability. (a) Behavior of $E_g$ in the absence of the fermion-induced interaction $g_{\mathrm{ind}}=0$. (b) Energy $E_g$ for $g_{\mathrm{ind}}=0.02$. The plot is characterized by presence of two minima: (i) The true, narrow Gaussian ground state located at $(\eta ,d_z∕{\mathcal{l}}_z)\simeq (3,2.3)$; and (ii) a pancake-shaped metastable state located at $(\eta ,d_z∕{\mathcal{l}}_z)\simeq (0.2,1.4)$. (c) Energy landscape for $g_{\mathrm{ind}}=0.06$. The energies of the ground and metastable states approach each other. (d) The broad Gaussian metastable state at $(\eta ,d_z∕{\mathcal{l}}_z)\simeq (0.2,1.4)$ eventually becomes the true ground state for $g_{\mathrm{ind}}=0.07$. In the figures the global (ground state) and local minima (metastable state) are indicated by a circle and cross, respectively.

Figure 2

Critical number of bosonic ${}^{52}\mathrm{Cr}$ atoms needed for the transition from the Gaussian state to a hexagonal density-wave state for the trap aspect ratios $\lambda =0.25$ (solid curve) and $\lambda =0.22$ (dashed curve). Here ${\mathcal{l}}_z=0.15\mu \mathrm{m}$.

Figure 3

(a) Interaction energy ${\mathcal{E}}_{\mathrm{int}}$ of Eq. (17) as a function of $\xi ∕{\mathcal{l}}_z$ at $\lambda =0.25$, ${\mathcal{l}}_z=0.15\mu \mathrm{m}$, $d_z=1.5{\mathcal{l}}_z$, and $g^{<}=0.18$. For $g_{bf}=0$ the energy is unbounded from below, but for $g_{bf}=70a_0$, a minimum appears at $\xi ∕{\mathcal{l}}_z\approx 0.1$. (b) Log-scale plot of the normalized width $\xi ∕{\mathcal{l}}_z$ that minimizes ${\mathcal{E}}_{\mathrm{int}}$ as a function of $g^{<}$.

Figure 4

Effective potential $V_{\mathrm{eff}}$ as a function of $k_{\perp }{d}_z$ for the parameters of Fig. 3b for $g^{<}=0.2$ (solid curve), and the critical value of $g^{<}\approx 0.14$ (dashed curve).