Ground states for 3D dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses

Abstract

We consider ground states of three-dimensional dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses, which can be described equivalently by positive $L^2$-constraint critical point of the Gross-Pitaevskii energy functional \[E(u)\!=\!\frac{1}{2}\int_{{\mathbb{R}^3}} {|\nabla u|}^2dx+\frac{λ_{1}}{2}\int_{{\mathbb{R}^3}} {| u|}^4dx+\frac{λ_{2}}{2} \int_{\mathbb{R}^{3}}\left(K \star|u|^{2}\right)|u|^{2} d x+\frac{2λ_{3}}{p}\int_{{\mathbb{R}^3}} {|u|}^{p}dx,\] where $2<p<\frac{10}{3}$, ${\lambda }_3<0$, $\star$ is the convolution, $K(x)=\frac{1-3{\cos }^2\theta (x)}{{|x|}^3}$, $\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector $x$. If ${\lambda }_1<\frac{4\pi }{3}{\lambda }_2\le 0$ or ${\lambda }_1<-\frac{8\pi }{3}{\lambda }_2\le 0$, $E(u)$ is unbounded on the $L^2$-sphere $S_c:=\{u\in H^1({\mathbb{R}}^3):{\int }_{{\mathbb{R}}^3}{|u|}^{2}dx=c^2\}$, so we turn to study a local minimization problem $$m(c,R_0):=\underset{u\in V_{R_0}^c}{\inf }E(u)$$ for a suitable $R_0>0$ with $V_{R_0}^c:=\left\{u\in S_c:({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx{)}^{\frac{1}{2}}<R_0\right\}$. We show that $m(c,R_0)$ is achieved by some $u_c>0$, which is a stable ground state. Furthermore, by refining the upper bound of $m(c,R_0)$, we provide a precise description of the asymptotic behavior of $u_c$ as the mass $c$ vanishes, i.e. $$[\frac{p|{\lambda }_3|}{2{\gamma }_c}{]}^{\frac{1}{p-2}}{u}_c(\frac{x+y_c}{\sqrt{2{\delta }_p{\gamma }_c}})\to W_p\mathrm{in}{H}^1({\mathbb{R}}^3)\mathrm{forsome}{y}_c\in {\mathbb{R}}^3\mathrm{as}c\to 0^{+}.$$

References in corpus (2)

• Coherent collapse of a dipolar Bose-Einstein condensate for different trap geometries
• Existence of minimizers in generalized Gross-Pitaevskii theory with the Lee-Huang-Yang correction