Spin-stripe state of a dipolar Bose-Einstein condensate

Abstract

This work concerns a theoretical study of a two-component ultra-cold bosonic gas of polarised magnetic dipolar atoms. The atoms interact by short-ranged contact interactions and long-ranged dipole-dipole interactions. The two components can correspond to two different spin-states of a single atomic species or two different atomic species, as realised in recent experiments. Here, we develop a general theory for the ground states and excitations of a two-component dipolar Bose-Einstein condensate. The ground states are described by dipolar Gross-Pitaevskii equations and Bogoliubov-de Gennes equations describe the excitations. We apply these theories to systems in 3D harmonic traps and a uniform planar potential. Specialised numerical methods are developed to provide accurate solutions of these equations, and to deal with the singular dipole-dipole interaction kernel. These methods involve spectral representations based on plane waves and Bessel functions. We also discuss the methods we used for obtaining ground state solutions of the nonlinear Gross-Pitaevskii energy functional. Dipole-dipole interactions can favour spatial structure, as demonstrated in the recent observation of supersolid states of matter in a single-component dipolar gas. In the two-component system these interactions can also favour a ground state with a periodic order in the pseudo-spin density. A two-component system can be either miscible (components overlap) or immiscible (components are separated). Our focus here is in the spin-stripe state, which occurs in a flattened trap geometry and involves an immiscible stripe pattern along one of the weakly trapped or untrapped directions. We examine the dynamical stability conditions of the uniform miscible state in planar confinement, and produce an associated stability phase diagram. We identify the various instabilities and analytical criteria for estimating these. We focus on the occurrence of, and conditions that favour, density-roton or spin-roton excitations, anticipating that the softening of these excitations will lead to structured ground states. To characterize these excitations we introduce density and spin dynamical structure factors, which can be measured in experiments using Bragg spectroscopy. Building on the stability phase diagram we compute ground states where the immiscible state is unstable to spin-rotons, finding triangular and stripe immiscible states with a length scale set by the confinement and interactions. We find that the transition to the stripe state is continuous and coincides with the spin-roton softening in the uniform miscible state. We study the excitations of the spin stripe state which exhibits three gapless excitation branches arising from the spontaneous broken symmetries. We identify a balanced parameter regime, where the excitations exhibit a nonsymmorphic symmetry (a symmetry is nonsymmorphic when it is a combination of point-group operations with nonprimitive lattice translations, i.e. not an integer multiple of lattice constants), which is revealed in the density and spin dynamical structure factors. We also consider the dynamics of the system following a quench from the uniform to immiscible stripe states. In the early time dynamics, a spin-stripe state is formed in a such way that the stripe order only extends over small domains. In subsequent dynamics, these domain sizes grow as the system phase-orders towards its equilibrium state. We identify the relevant topological defects such as disclinations, dislocations and grain boundaries. Using ensembles of large scale simulations we show that dynamical scaling holds for the growth of order.