Dipolar condensates with tilted dipoles in a pancake-shaped confinement

The effect of dipolar orientation with respect to the condensate plane on the mean-field dynamics of dipolar Bose-Einstein condensates in a pancake-shaped confinement is discussed. The stability of a quasi-two-dimensional condensate, with respect to the tilting angle, is found to be different from a two-dimensional layer of dipoles, indicating the relevance of the transverse extension while characterizing two-dimensional dipolar systems. An anisotropic excitation spectrum exhibiting a highly tunable, rotonlike minimum can arise entirely from the dipole-dipole interactions, by tilting the dipoles. At the magic angle and in the absence of contact interactions, the long-wavelength excitations are not phononlike and always unstable. The post-roton-instability dynamics, in contrast to phonon instability, in a uniform condensate, is featured by a transient, defect-free, stripe pattern, which eventually undergoes local collapses, and driving the condensate back into the stable regime can make them sustained for longer. Hopping between stripes has been observed before it melts into a uniform state in the presence of dissipation. Finally, we discuss a class of solutions, in which a quasi-two-dimensional condensate is self-trapped in one direction, as well as a regime of interaction parameters, including attractive short-range interactions, at which a two-dimensional anisotropic soliton can be stabilized, and we show that a chromium condensate with a relatively small number of atoms is well suited for this.

Figure 1

(a) The dipoles are oriented in the $xz$ plane forming an angle $\alpha$ with the $z$ axis. (b) For ${\alpha }_m<\alpha <\pi /2$ a purely Q2D dipolar condensate is unstable against collapse, indicated by the smaller (yellow) shaded region, but an ideal 2D layer of dipoles is unstable when $\pi /2-{\alpha }_m<\alpha <\pi /2$, indicated by the larger (gray) shaded region.

Figure 2

(a) 3D numerical results for the Bogoliubov excitations along the $q_y$ axis, i.e., ${\epsilon }_y=\epsilon (q_x=0,q_y)$ for different $\alpha$'s for $\tilde{g}=0$ and ${\tilde{g}}_d=0.05$. When $\alpha ={\alpha }_m$ the condensate is unstable against long-wavelength excitations (LWI), when $\alpha =0.92$ rad there is RI, and for $\alpha =0.91$ rad the spectrum becomes stable and has a maxon-roton behavior. Since the parameters well satisfy the 2D criteria ${\mu }_{\text{2D}}\ll \hslash {\omega }_z$, the 3D calculations (dashed red line) are in excellent agreement with the 2D analytical results (solid lines). The imaginary part of ${\epsilon }_y$ is shown in the negative $y$ axis. (b) The 3D results for the roton momentum $q_r$ as a function of ${\tilde{g}}_d$ for different $\tilde{g}$ values. For a fixed $\tilde{g}$ and ${\tilde{g}}_d$, the value of $\alpha$ is adjusted such that $\epsilon (q_x=0,q_y=q_r)=0$. The curves are terminated at a value of $g_d$ such that no stable rotons are possible for $0\le \alpha <{\alpha }_m$. In (c) and (d) we show the stability and instability regimes of a pancake dipolar condensate with a uniform density in the $xy$ plane, as a function of $\alpha$ and ${\tilde{g}}_d$. Figure (c) is for $\tilde{g}=0$ and (d) is for $\tilde{g}=0.1$. The abbreviations in the figures (c) and (d) stand for the following: PI, phonon instability; RI, roton instability; R, stable roton; and S, stable spectrum without rotons. In (c), at $\alpha ={\alpha }_m$, the vertical line indicates the LWI.

Figure 3

The stability and instability regions of the pancakelike dipolar condensates with a uniform density in the $xy$ plane, as a function of the tilting angle $\alpha$ and the $s$-wave scattering length $a$ (in units of Bohr radius $a_0$) for Cr, Er, and Dy BECs, with ${\omega }_z=2\pi \times 800\mathrm{Hz}$ and the condensate density $n_{\text{2D}}/\left(\sqrt{2\pi }{l}_z\right)={10}^{20}{\text{m}}^{-3}$, based on the Bogoliubov modes obtained by solving Eqs. (6) and (7). The dipolar coupling constant ${\tilde{g}}_d$ is calculated using the intrinsic magnetic moments of each of the atoms. The abbreviations in the figure stand for the following: PI, phonon instability; RI, roton instability; R, stable roton; and S, stable spectrum without rotons.

Figure 4

(a) The post-PI transient stripe pattern with dislocation defects in a Q2D homogeneous condensate and (b) the stripe pattern without defects in post-RI dynamics. The roton softening happens along the $q_y$ axis and hence the stripes are parallel to the $x$ axis. The formation of defect-free stripes can be identified as the trait of anisotropic RI in dipolar BECs. For the simulation we have considered a Cr BEC with density $n_{\text{2D}}/\sqrt{2\pi }{l}_z=5\times {10}^{19}{\text{m}}^{-6}, {\omega }_z=2\pi \times 800$ Hz and for (a) $a=0$ and $\alpha ={\alpha }_m+0.14$ rad and (b) $a=5a_0$ and $\alpha ={\alpha }_m-0.14$ rad with $a_0$ being the Bohr radius. The snapshots are taken at (a) $t=0.1\text{s}$ and (b) $t=0.016\text{s}$.

Figure 5

(a) The contour plot for the energy $E(L_x,L_y,L_z)$ as a function of $L_x$ and $L_y$ with $L_z=1$ for a chromium condensate with number of particles $N=1000, \alpha ={\alpha }_m$, and $a=0$. The trapping frequencies are taken as ${\omega }_z=2\pi \times 800\mathrm{Hz}$ and ${\omega }_y=2\pi \times 80\mathrm{Hz}$. The self-trapping along the $x$ axis is evident from the plot. (b) The corresponding ground-state density ${\left|\psi (x,y)\right|}^2$ in the $xy$ plane. (c) The low-lying $xy$ modes as a function of the tilting angle ($\alpha$) and other parameters are the same as that of (a) and (b). (d) The low-lying $xy$ modes as a function of the number of atoms in the condensate ($N$) and other parameters are the same as that of (a) and (b)

Figure 6

The stability and instability regimes of self-trapped solutions in a Q2D dipolar condensate as a function of $a$ and $\alpha$. We consider a Cr condensate with ${\omega }_z=2\pi \times 800\mathrm{Hz}, {\omega }_y=2\pi \times 80\mathrm{Hz}$, and $N=1000$ for (a) and $N=5000$ for (b). The abbreviations in the figure stand for the following: EI, expansion instability; CI, collapse instability; 1D, 1D self-trapping; and 2D, 2D anisotropic soliton.