Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2301.08304 · Physics of Fluids

Numerical model of the Gross–Pitaevskii equation for rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

Satori Tsuzuki

Published Apr 1, 2023• 3 citations
arXivHTMLPDF

Abstract

This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.

DOI
10.1063/5.0143556

Related papers

Three-dimensional analysis of vortex-lattice formation in rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

Satori Tsuzuki, Eri Itoh, Katsuhiro Nishinari

Abstract Recently, we presented a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit time-integrating scheme; our SPH scheme could reproduce the vortex lattices and their formation processes in rotating quasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In this study, we have successfully demonstrated a simulation of rotating 3D BECs trapped in a 3D harmonic potential forming ‘cigar-shaped’ condensates. We have found that our scheme can reproduce the following typical behaviors of rotating 3D BECs observed in the literature: (i) the characteristic shape of the lattice formed in the cross-section at the origin and its formation process, (ii) the stable existence of vortex lines along the vertical axis after reaching the steady state, (iii) a ‘cookie-cutter’ shape, with a similar lattice shape observed wherever we cut the condensate in a certain range in the vertical direction, (iv) the bending of vortex lines when approaching the inner edges of the condensate, and (v) the formation of vortex lattices by vortices entering from outside the condensate. Therefore, we further validated our scheme by simulating rotating 3D BECs.

arXiv:2307.08667•Journal of Physics Communications•Dec 1, 2023•1 citation

Vortex-lattice in a uniform Bose–Einstein condensate in a box trap

S K Adhikari

We study numerically the vortex-lattice formation in a rapidly rotating uniform quasi-two-dimensional Bose-Einstein condensate (BEC) in a box trap. We consider two types of boxes: square and circle. In a square-shaped 2D box trap, when the number of generated vortices is the square of an integer, the vortices are found to be arranged in a perfect square lattice, although deviations near the center are found when the number of generated vortices is arbitrary. In case of a circular box trap, the generated vortices in the rapidly rotating BEC lie on concentric closed orbits. Near the center, these orbits have the shape of polygons, whereas near the periphery the orbits are circles. The circular box trap is equivalent to the rotating cylindrical bucket used in early experiment(s) with liquid He II. The number of generated vortices in both cases is in qualitative agreement with Feynman's universal estimate. The numerical simulation for this study is performed by a solution of the underlying mean-field Gross-Pitaevskii (GP) equation in the rotating frame, where the wave function for the generated vortex lattice is a stationary state. Consequently, the imaginary-time propagation method can be used for a solution of the GP equation, known to lead to an accurate numerical solution. We also demonstrated the dynamical stability of the vortex lattices in real-time propagation upon a small change of the angular frequency of rotation, using the converged imaginary-time wave function as the initial state.

arXiv:1903.08672•Journal of Physics Condensed Matter•Mar 29, 2019•13 citations

Instabilities and vortex-lattice formation in rotating conventional and dipolar dilute-gas Bose-Einstein condensates

A. Martin, Nicholas G. Parker, Rick van Bijnen +2 more

A theoretical study of vortex-lattice formation in atomic Bose-Einstein condensates confined by a rotating elliptical trap is presented. For the conventional case of purely s-wave interatomic interactions, this is done through a consideration of both hydrodynamic equations and time-dependent simulations of the Gross-Pitaevskii equation. We discriminate three distinct, experimentally testable regimes of instability: ripple, interbranch , and catastrophic . Additionally, we generalize the classical hydrodynamical approach to include long-range dipolar interactions, showing how the static solutions and their stability in the rotating frame are significantly altered. This enables us to examine the routes towards unstable dynamics, which, in analogy to conventional condensates, may lead to vortex-lattice formation.

Laser Physics•Mar 1, 2008•5 citations

Multi-vortex crystal lattices in Bose–Einstein condensates with a rotating trap

Shuangquan Xie, P. G. Kevrekidis, Théodore Kolokolnikov

We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.

arXiv:1708.04336•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•May 1, 2018•5 citations

Vortex synchronization in Bose–Einstein condensates: a time-dependent Gross–Pitaevskii equation approach

Ryan Barnett, Edward H. Chen, Gil Refael

In this work, we consider vortex lattices in rotating Bose–Einstein condensates composed of two species of bosons having different masses. Previously (Barnett et al 2008 New J. Phys. 10 043030), it was claimed that the vortices of the two species form bound pairs and the two vortex lattices lock. Remarkably, the two condensates and the external drive all rotate at different speeds owing to the disparity of the masses of the constituent bosons. In this paper, we study the system by solving the full two-component Gross–Pitaevskii equations numerically. Using this approach, we verify the stability of the putative locked state that is found to exist within a disc centered on the axis of rotation and that depends on the mass ratio of the two bosons. We also derive a refined estimate for the locking radius tailored to the experimentally relevant case of a harmonic trap and show that this agrees with the numerical results. Finally, we analyze in detail the rotation rates of the different components in the locked and unlocked regimes.

arXiv:0909.4072•New Journal of Physics•Apr 1, 2010•13 citations

Vortex lattice formation in a rotating Bose-Einstein condensate

Makoto Tsubota, Kenichi Kasamatsu, Masahito Ueda

We study the dynamics of vortex lattice formation of a rotating trapped Bose-Einstein condensate by numerically solving the two-dimensional Gross-Pitaevskii equation, and find that the condensate undergoes elliptic deformation, followed by unstable surface-mode excitations before forming a quantized vortex lattice. The origin of the peculiar surface-mode excitations is identified to be phase fluctuations at the low-density surface regime. The obtained dependence of a distortion parameter on time and that on the driving frequency agree with the recent experiments by Madison et al. [Phys. Rev. Lett. 86, 4443 (2001)].

arXiv:cond-mat/0104523•Physical Review A•Jan 10, 2002•322 citations
This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross–Pitaevskii equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the fourth-order Runge–Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the hexagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs.
Abstract Recently, we presented a new numerical scheme for vortex lattice formation in a rotating Bose–Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit time-integrating scheme; our SPH scheme could reproduce the vortex lattices and their formation processes in rotating quasi-two-dimensional (2D) BECs trapped in a 2D harmonic potential. In this study, we have successfully demonstrated a simulation of rotating 3D BECs trapped in a 3D harmonic potential forming ‘cigar-shaped’ condensates. We have found that our scheme can reproduce the following typical behaviors of rotating 3D BECs observed in the literature: (i) the characteristic shape of the lattice formed in the cross-section at the origin and its formation process, (ii) the stable existence of vortex lines along the vertical axis after reaching the steady state, (iii) a ‘cookie-cutter’ shape, with a similar lattice shape observed wherever we cut the condensate in a certain range in the vertical direction, (iv) the bending of vortex lines when approaching the inner edges of the condensate, and (v) the formation of vortex lattices by vortices entering from outside the condensate. Therefore, we further validated our scheme by simulating rotating 3D BECs.
We study numerically the vortex-lattice formation in a rapidly rotating uniform quasi-two-dimensional Bose-Einstein condensate (BEC) in a box trap. We consider two types of boxes: square and circle. In a square-shaped 2D box trap, when the number of generated vortices is the square of an integer, the vortices are found to be arranged in a perfect square lattice, although deviations near the center are found when the number of generated vortices is arbitrary. In case of a circular box trap, the generated vortices in the rapidly rotating BEC lie on concentric closed orbits. Near the center, these orbits have the shape of polygons, whereas near the periphery the orbits are circles. The circular box trap is equivalent to the rotating cylindrical bucket used in early experiment(s) with liquid He II. The number of generated vortices in both cases is in qualitative agreement with Feynman's universal estimate. The numerical simulation for this study is performed by a solution of the underlying mean-field Gross-Pitaevskii (GP) equation in the rotating frame, where the wave function for the generated vortex lattice is a stationary state. Consequently, the imaginary-time propagation method can be used for a solution of the GP equation, known to lead to an accurate numerical solution. We also demonstrated the dynamical stability of the vortex lattices in real-time propagation upon a small change of the angular frequency of rotation, using the converged imaginary-time wave function as the initial state.
A theoretical study of vortex-lattice formation in atomic Bose-Einstein condensates confined by a rotating elliptical trap is presented. For the conventional case of purely s-wave interatomic interactions, this is done through a consideration of both hydrodynamic equations and time-dependent simulations of the Gross-Pitaevskii equation. We discriminate three distinct, experimentally testable regimes of instability: ripple, interbranch , and catastrophic . Additionally, we generalize the classical hydrodynamical approach to include long-range dipolar interactions, showing how the static solutions and their stability in the rotating frame are significantly altered. This enables us to examine the routes towards unstable dynamics, which, in analogy to conventional condensates, may lead to vortex-lattice formation.
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.
In this work, we consider vortex lattices in rotating Bose–Einstein condensates composed of two species of bosons having different masses. Previously (Barnett et al 2008 New J. Phys. 10 043030), it was claimed that the vortices of the two species form bound pairs and the two vortex lattices lock. Remarkably, the two condensates and the external drive all rotate at different speeds owing to the disparity of the masses of the constituent bosons. In this paper, we study the system by solving the full two-component Gross–Pitaevskii equations numerically. Using this approach, we verify the stability of the putative locked state that is found to exist within a disc centered on the axis of rotation and that depends on the mass ratio of the two bosons. We also derive a refined estimate for the locking radius tailored to the experimentally relevant case of a harmonic trap and show that this agrees with the numerical results. Finally, we analyze in detail the rotation rates of the different components in the locked and unlocked regimes.
We study the dynamics of vortex lattice formation of a rotating trapped Bose-Einstein condensate by numerically solving the two-dimensional Gross-Pitaevskii equation, and find that the condensate undergoes elliptic deformation, followed by unstable surface-mode excitations before forming a quantized vortex lattice. The origin of the peculiar surface-mode excitations is identified to be phase fluctuations at the low-density surface regime. The obtained dependence of a distortion parameter on time and that on the driving frequency agree with the recent experiments by Madison et al. [Phys. Rev. Lett. 86, 4443 (2001)].