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arXiv:2610.27878

Computing excited states of dipolar Bose--Einstein condensates by constrained gentlest ascent dynamics

Fu Zheng, Wei Liu, Xu Qian

cond-mat.quant-gasmath.NAquant-phSubmitted Oct 20, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

Dipolar Bose--Einstein condensates (BECs), i.e., BECs with long-range dipole-dipole interactions (DDI), exhibit extraordinarily rich physics and serve as a versatile experimental platform for exploring exotic quantum states of matter. Within the mean-field framework, a dipolar BEC can be described by the Gross--Pitaevskii (GP) model with a DDI term. Its stationary states consist of the ground state, which minimizes the GP energy functional under the normalization constraint, and excited states, which are higher-energy critical points that typically correspond to constrained saddle points. Excited states encode collective excitation modes and play a key role in the dynamical stability and quantum phase transitions of cold-atom systems. However, in contrast to the extensively studied ground state, they remain far less explored numerically, especially in dipolar systems. In this work, we develop a numerical method based on the constrained gentlest ascent dynamics (CGAD) that systematically computes excited states of dipolar BECs by stably targeting constrained saddle points with any prescribed Morse index. Specifically, a semi-implicit sine-pseudospectral discretization with an orthonormalization correction step is adopted, and the nonlocal DDI terms are evaluated efficiently and accurately via the nonuniform fast Fourier transform (NUFFT). The resulting scheme is applied to one-, two-, and three-dimensional dipolar BECs, enabling a detailed numerical investigation of their excited states. Remarkably, our simulations reveal that excited states can persist even in regimes where the ground state of the condensate becomes mean-field unstable. This work provides a computational framework for studying excited states in dipolar quantum gases and offers insights for future theoretical and experimental investigations.

Comments
26 pages, 8 figures

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Dipolar Bose--Einstein condensates (BECs), i.e., BECs with long-range dipole-dipole interactions (DDI), exhibit extraordinarily rich physics and serve as a versatile experimental platform for exploring exotic quantum states of matter. Within the mean-field framework, a dipolar BEC can be described by the Gross--Pitaevskii (GP) model with a DDI term. Its stationary states consist of the ground state, which minimizes the GP energy functional under the normalization constraint, and excited states, which are higher-energy critical points that typically correspond to constrained saddle points. Excited states encode collective excitation modes and play a key role in the dynamical stability and quantum phase transitions of cold-atom systems. However, in contrast to the extensively studied ground state, they remain far less explored numerically, especially in dipolar systems. In this work, we develop a numerical method based on the constrained gentlest ascent dynamics (CGAD) that systematically computes excited states of dipolar BECs by stably targeting constrained saddle points with any prescribed Morse index. Specifically, a semi-implicit sine-pseudospectral discretization with an orthonormalization correction step is adopted, and the nonlocal DDI terms are evaluated efficiently and accurately via the nonuniform fast Fourier transform (NUFFT). The resulting scheme is applied to one-, two-, and three-dimensional dipolar BECs, enabling a detailed numerical investigation of their excited states. Remarkably, our simulations reveal that excited states can persist even in regimes where the ground state of the condensate becomes mean-field unstable. This work provides a computational framework for studying excited states in dipolar quantum gases and offers insights for future theoretical and experimental investigations.
We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.
In this paper, we propose an efficient and robust numerical method to study the elementary excitation of dipolar Bose--Einstein condensates, which is governed by the Bogoliubov--de Gennes equations (BdGEs) with nonlocal dipole-dipole interaction, around the mean field ground state. Analytical properties of the BdGEs are investigated, which could serve as benchmarks for the numerical methods. To evaluate the nonlocal interactions accurately and efficiently, we propose a new simple Fourier spectral convolution (SFSC) method. Then, integrating SFSC with the standard Fourier spectral method for spatial discretization and implicitly restarted Arnoldi methods (IRAM) for the eigenvalue problem, we derive an efficient and spectrally accurate method, named the SFSC-IRAM, for the BdGEs. Ample numerical tests are provided to illustrate the accuracy and efficiency. Finally, we apply the new method to study systematically the excitation spectrum and Bogoliubov amplitudes around the ground state with different parameters in different spatial dimensions.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
We numerically study the nonlocal Gross–Pitaevskii equation (NGPE) which describes the dynamics of Bose–Einstein condensates (BEC) with dipole–dipole interaction at extremely low temperature. In preparation for the numerics, first we reformulate the dimensionless NGPE into a Schrödinger–Poisson system. Then, we discretize the three-dimensional Schrödinger–Poisson system in space by a sixth-order compact finite difference method and in time by a splitting technique. By means of three-dimensional discrete fast Sine transform, we develop a fast solver for the resulting discretized system. Finally, we present numerical examples in three dimensions to demonstrate the power of the numerical methods and to discuss some physics of dipolar BEC. The merits of the proposed method for the NGPE are that it is fast and unconditionally stable. Moreover, the method is of spectral-like accuracy in space, and conserves the particle number and the energy of the system in the discretized level.
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.
Abstract. In this paper, we propose a regularized Newton method for computing ground states of Bose-Einstein condensates (BECs), which can be formulated as an energy minimization problem with a spherical constraint. The energy functional and constraint are discretized by either the fi-nite difference, or sine or Fourier pseudospectral discretization schemes and thus the original infinite dimensional nonconvex minimization problem is approximated by a finite dimensional constrained nonconvex minimization problem. Then an initial solution is first constructed by using a feasible gra-dient type method, which is an explicit scheme and maintains the spherical constraint automatically. To accelerate the convergence of the gradient type method, we approximate the energy functional by its second-order Taylor expansion with a regularized term at each Newton iteration and adopt a cascadic multigrid technique for selecting initial data. It leads to a standard trust-region subprob-lem and we solve it again by the feasible gradient type method. The convergence of the regularized Newton method is established by adjusting the regularization parameter as the standard trust-region strategy. Extensive numerical experiments on challenging examples, including a BEC in three di-mensions with an optical lattice potential and rotating BECs in two dimensions with rapid rotation and strongly repulsive interaction, show that our method is efficient, accurate and robust. Key words. Bose-Einstein condensation, Gross-Pitaevskii equation, ground state, energy func-tional, spherical constraint, gradient type method, regularized Newton method. 1. Introduction. Since