We investigate the stability properties and the dynamics of Bose–Einstein condensates with axial symmetry, especially with dipolar long-range interaction, using both simulations on grids and variational calculations. We present an extended variational ansatz which is applicable for axial symmetry and show that this ansatz can reproduce the lowest eigenfrequencies of the Bogoliubov spectrum, and also the corresponding eigenfunctions. Our variational ansatz is capable of describing the roton instability of pancake-shaped dipolar condensates for arbitrary angular momenta. After investigating the linear regime we apply the ansatz to determine the dynamics and show how the angular collapse is correctly described within the variational framework.
We investigate the stability properties and the dynamics of Bose–Einstein condensates with axial symmetry, especially with dipolar long-range interaction, using both simulations on grids and variational calculations. We present an extended variational ansatz which is applicable for axial symmetry and show that this ansatz can reproduce the lowest eigenfrequencies of the Bogoliubov spectrum, and also the corresponding eigenfunctions. Our variational ansatz is capable of describing the roton instability of pancake-shaped dipolar condensates for arbitrary angular momenta. After investigating the linear regime we apply the ansatz to determine the dynamics and show how the angular collapse is correctly described within the variational framework.
We develop theory for a flattened dipolar Bose–Einstein condensate produced by harmonic confinement along one direction. The role of both short-ranged contact interactions and long-ranged dipole–dipole interactions is considered, and the dipoles are allowed to be polarized along an arbitrary direction. We discuss the symmetry properties of the condensate and the part of the excitation spectrum determining stability, and introduce two effective interaction parameters that allow us to provide a general description of the condensate properties, rotons, and stability. We diagonalize the full theory to obtain benchmark results for the condensate and quasiparticle excitations, and characterize the exact mean field stability of the system. We provide a unified formulation for a number of approximate schemes to describe the condensate and quasiparticles, including the standard quasi-two-dimensional approximation, two kinds of variational ansatz, and a Thomas–Fermi approximation. Some of these schemes have been widely used in the literature despite not being substantiated against the exact theory. We provide this validation and establish the regimes where the various theories perform well.
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.
In this work several aspects of dipolar Bose-Einstein condensates are investigated. These condensates have attracted much interest since their experimental realization and offer the chance to explore the novel, fascinating, and sometimes unexpected physics originating from the dipole-dipole interaction. For the theoretical description the time-dependent variational principle with an ansatz of coupled Gaussian wave packets is applied to the analysis of this form of matter. In order to describe physical systems, where the shape of the condensate wave function cannot be approximated adequately by origin-centered Gaussian functions, translational and rotational degrees of freedom of the particular wave packets are included. This provides not only a full-fledged alternative method to full-numerical grid calculations that are demanding in computation time, used for the determination of ground states and investigation of the real-time dynamics, but also allows for the calculation of excited states and the examination of stability properties and offers the possibility for a deeper physical understanding of dipolar condensates. The variational method is applied to three different physical systems. In the first one anisotropic quasi-two-dimensional solitons are investigated. Ground states and small excitations are presented and the full flexibility of the approach is utilized in the simulation of the collision of two solitons. Thereby simulations of collisions are performed, where initially the solitons are in the repelling side-by-side configuration and move towards each other with a specific momentum. It is shown that the results are in excellent agreement with full-numerical grid calculations. Collisions with various initial parameters such as the initial momentum and impact factor are performed, allowing to classify the qualitative interactions of two solitons and illuminating the effects of the solitons' two-dimensional nature on the collision process. In the second system dipolar Bose-Einstein condensates in a triple-well potential are investigated. This constitutes a well-suited model system for periodic optical potentials with important contributions of the non-local and anisotropic dipole-dipole interaction, which show a variety of effects such as self-organization and formation of patterns. Here, a macroscopic sample of dipolar bosons in the mean-field limit is addressed. The analysis goes beyond the calculation of ground states and clarifies the role of excited and metastable states in such systems. In particular, the formation of phases is found to originate from the interplay of several states with distinct stability properties. As some of the phases are formed by metastable states, special attention is paid to the characteristics of phase transitions in real time and the dynamical stabilization of the condensate. The dynamical simulations that support these results, constitute a way for an experimental observation of these findings. The third physical application is an open quantum system. Here, an external potential models gain and loss of particles and features parity with simultaneous time-reversal symmetry. For these dipolar condensates real eigenvalues of the nonlinear Hamiltonian are found, preserving the PT-symmetry of the external potential, as well as states that break this symmetry. A detailed stability analysis of these, in general complex states, is performed. The comparison with a similar system featuring a non-dipolar condensate demonstrates the role of the dipolar interaction for novel effects. Furthermore, the real-time dynamics is investigated, which reveals, among other effects, long-living oscillations of the two wells' population.
The calculation of properties of Bose-Einstein condensates with dipolar interactions has proven a computationally intensive problem due to the long range nature of the interactions, limiting the scope of applications. In particular, the lowest lying Bogoliubov excitations in three-dimensional harmonic trap with cylindrical symmetry were so far computed in an indirect way, by Fourier analysis of time-dependent perturbations, or by approximate variational methods. We have developed a very fast and accurate numerical algorithm based on the Hankel transform for calculating properties of dipolar Bose-Einstein condensates in cylindrically symmetric traps. As an application, we are able to compute many excitation modes by directly solving the Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in different trap geometries. We use these results to calculate the quantum depletion of the condensate by a combination of a computation of the exact modes and the use of a local density approximation.
We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.
Bose-Einstein condensates with an attractive 1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.