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AIP conference proceedings

Gross–Pitaevskii equation with noncommuting coordinates and momenta on oscillation Boson system

Mohd Faudzi Umar, Nurisya Mohd Shah, Mohamad Nazri Abdul Halif

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Bose-Einstein condensates, formed at ultracold temperatures, provide a unique platform for exploring quantum phenomena at a macroscopic scale. In this paper, we explore the effect of noncommutativity in both coordinates and momenta on the energy spectrum of the Gross-Pitaevskii equation for a Bose-Einstein condensate that is harmonically trapped and interacting. This study is conducted within the framework of noncommutative quantum mechanics, focusing on a first-order approximation in the noncommutative parameters, given their very small magnitude. The result shows that the introduction of noncommutative effects leads to the emergence of vortex formation in the condensate. By comparing our theoretical findings with experimental data, we estimate the value of the noncommutative parameter.

DOI
10.1063/5.0312698

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Bose-Einstein condensates, formed at ultracold temperatures, provide a unique platform for exploring quantum phenomena at a macroscopic scale. In this paper, we explore the effect of noncommutativity in both coordinates and momenta on the energy spectrum of the Gross-Pitaevskii equation for a Bose-Einstein condensate that is harmonically trapped and interacting. This study is conducted within the framework of noncommutative quantum mechanics, focusing on a first-order approximation in the noncommutative parameters, given their very small magnitude. The result shows that the introduction of noncommutative effects leads to the emergence of vortex formation in the condensate. By comparing our theoretical findings with experimental data, we estimate the value of the noncommutative parameter.
We study the stability of vortices in a binary system of Bose-Einstein condensat es, with their wave functions modeled by a set of coupled, time-dependent Gross-Pitaevskii equations. Beginning with an effective two-dimensional system, we identify miscible and immiscible regimes characterized by the inter- and intra-atomic interactions and the initial configuration of the system. We then consider a binary system of Bose-Einstein condensates placed in a rotating harmonic trap and study the single united vortex state in this system. We derive an approximate form for the energy of a single united vortex in the binary system and the critical angular velocity for the global stability of a vortex at the center of the trap. We also compute the metastability onset angular velocity for the local stability of a vortex at the center of the trap. Numerical solutions to the Gross-Pitaevskii equations support these expressions. These rotational results inform us of a novel bimodal formation within the miscible regime of the binary condensate system. We thus demonstrate the nontrivial aspects of vortex stability in interacting binary Bose-Einstein condensates as a result of their nonlinear interactions.
The recent realization of Bose–Einstein condensation in atomic gases opens new possibilities for the observation of macroscopic quantum phenomena. There are two important features of these systems — weak interaction and significant spatial inhomogeneity. Because of this a non-trivial 'zeroth-order' theory exists, compared to the 'first-order' Bogolubov theory. The zeroth-order theory is based on the mean-field Gross–Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. Elementary excitations define the thermodynamic behavior of the system and result in a Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in quantum collapse-revival of the collective oscillations.
In this thesis we solve the Gross-Pitaevskii equation numerically in order to model the response of trapped Bose-Einstein condensed gases to perturbations by electromagnetic fields. First, we simulate output coupling of pulses from the condensate and compare our results to experiments. The excitation and separation of eigenmodes on flow through a constriction is also studied. We then move on to the main theme of this thesis: the important subject of quantised vortices in Bose condensates, and the relation between Bose-Einstein condensation and superfluidity. We propose methods of producing vortex pairs and rings by controlled motion of objects. Full three-dimensional simulations under realistic experimental conditions are performed in order to test the validity of these ideas. We link vortex formation to drag forces on the object, which in turn is connected with energy transfer to the condensate. We therefore argue that vortex formation by moving objects is intimately related to the onset of dissipation in superfluids. We discuss this idea in the context of a recent experiment, using simulations to provide evidence of vortex formation in the experimental scenario. Superfluidity is also manifest in the property of persistent currents, which is linked to vortex stability and dynamics. We simulate vortex line and ring motion, and find in both cases precessional motion and thermodynamic instability to dissipation. Strictly speaking, the Gross-Pitaevskii equation is valid only for temperatures far below the BEC transition. We end the thesis by describing a simple finite-temperature model to describe mean-field coupling between condensed and non-condensed components of the gas. We show that our hybrid Monte-Carlo/FFT technique can describe damping of the lowest energy excitations of the system. Extensions to this model and future research directions are discussed in the conclusion. (author)
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
We studied the evolutionary patterns of two-dimensional Bose-Einstein condensates incorporating higher-order nonlinear interactions in harmonic potential. Using the Gross-Pitaevskii equation model with higher-order nonlinear corrections, we derived the analytical solitary vortex solutions via the variational method. The impact of the higher-order nonlinear interaction on the vortex dynamics is quantitatively analyzed, showing its key nonlinear feature contribution in the asymmetric vortex evolution with more precise evolutionary pattern generated. We found that, for the circular symmetric solution, if the nonlinear strength is not high, the higher-order nonlinear corrections essentially have only a tiny perturbative effect on the system’s quasi-static oscillation state, whereas for asymmetric evolution of the solitary vortex, incorporating higher-order corrections will generate an evolution pattern that better matches the results of numerical simulation. The theoretical results derived here can be used to guide relevant experimental studies of higher-order nonlinear effects in ultracold atomic systems.
In this study, we investigate the evolution of vortex in harmonically trapped two-component coupled Bose–Einstein condensate with quintic-order nonlinearity. We derive the vortex solution of this two-component system based on the coupled quintic-order Gross–Pitaevskii equation model and the variational method. It is found that the evolution of vortex is a metastable state. The radius of vortex soliton shrinks and expands with time, resulting in periodic breathing oscillation, and the angular frequency of the breathing oscillation is twice the value of the harmonic trapping frequency under infinitesimal nonlinear strength. At the same time, it is also found that the higher-order nonlinear term has a quantitative effect rather than a qualitative impact on the oscillation period. With practical experimental setting, we identify the quasi-stable oscillation of the derived vortex evolution mode and illustrated its features graphically. The theoretical results developed in this work can be used to guide the experimental observation of the vortex phenomenon in ultracold coupled atomic systems with quintic-order nonlinearity.