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arXiv:2011.00804

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

Xiao Luo, Tao Yang

Published Nov 2, 2020• 1 citation
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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 $2\!0$ with $V^c_{R_0} \!:=\!\left\{u \!\in\! S_c : \big(\int_{{\mathbb{R}^3}} {{|\nabla u|}^2dx}\big)^{\frac{1}{2}} \!0$, 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. $${[\frac{{p|{λ_3}|}}{2{γ_c}}]^{\frac{1}{p - 2}}}{u_c}(\frac{x + {y_c}}{{\sqrt {2{δ_p}{γ_c}} }}) \to {W_p}\;\;\;\;{\rm{in}}\;\;\;\;{H^1}({\mathbb{R}^3})\;\;\;\;{\rm{for some}}\;\;\;\;{y_c} \in {\mathbb{R}^3}\;\;\;\;{\rm{as}}\;\;\;\;c \to {0^ + }.$$

DOI
10.48550/arxiv.2011.00804

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Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator (−Δ)1/2(-\Delta)^{1/2}(−Δ)1/2 is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.
We prove existence and qualitative properties of standing wave solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation (GPE), the latter being currently the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates. Using a mountain pass argument on spheres in L2L^2L2 and constructing appropriately localized Palais-Smale sequences we are able to prove existence of real positive ground states as saddle points of the energy. The analysis is deployed in the set of possible states, thus overcoming the problem that the energy is unbounded below. We also prove a corresponding nonlocal Pohozaev identity with no rest term, a crucial part of the analysis.
Abstract We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/N\mathfrak {a}/Na/N a / N , moving in the three-dimensional unit torus Λ\Lambda Λ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.
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