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arXiv:2501.18787 · Nonlinearity

Quantitative derivation of the two-component Gross–Pitaevskii equation in the hard-core limit with uniform-in-time convergence rate

Jacky J. Chong, Jinyeop Lee, Zhiwei Sun

Published Aug 25, 2026• 1 citation
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Abstract

Abstract We derive, in the Gross–Pitaevskii (GP) limit, the time-dependent two-component GP equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose–Einstein condensate. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number N . This improves upon the works of Michelangeli and Olgiati 2014 J. Nonlinear Math. Phys. 24 466–4; Olgiati 2017 Effective Non-Linear Dynamics of Binary Condensates and Open Problems pp 239–56) by providing a sharper, N -dependent, time-independent convergence rate, and also implies a uniform-in-time bound for the one-component case. Furthermore, our approach extends the framework of Benedikter et al (2015 Commun. Pure Appl. Math., 68 1399–482) to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime.

DOI
10.1088/1361-6544/ae7ccf

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Abstract We derive, in the Gross–Pitaevskii (GP) limit, the time-dependent two-component GP equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose–Einstein condensate. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number N . This improves upon the works of Michelangeli and Olgiati 2014 J. Nonlinear Math. Phys. 24 466–4; Olgiati 2017 Effective Non-Linear Dynamics of Binary Condensates and Open Problems pp 239–56) by providing a sharper, N -dependent, time-independent convergence rate, and also implies a uniform-in-time bound for the one-component case. Furthermore, our approach extends the framework of Benedikter et al (2015 Commun. Pure Appl. Math., 68 1399–482) to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime.
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.
Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross‐Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one‐particle reduced density, the form of the initial data is preserved by the many‐body evolution, up to a small error that vanishes as N−1/2 in the limit of large N.© 2015 Wiley Periodicals, Inc.
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N\mathit{N}N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2\mathit{N}^{-1/2}N−1/2 in the limit of large N\mathit{N}N.
We investigate the dynamics of short-range interacting Bose gases with varying degrees of diluteness and interaction strength. By applying a combined mean-field and semiclassical space-time rescaling to the dynamics in both the Gross--Pitaevskii and hard-core limit regimes, we prove that the local one-particle mass, momentum, and energy densities of the many-body system can be quantitatively approximated by solutions to the compressible Euler system in the strong sense, up to the first blow-up time of the fluid description, as the number of particles tends to infinity. In the hard-core limit regime, two novel results are presented. First, we rigorously prove, for the first time, that the internal energy of the fluid takes the form 4πc0ρ24π\mathfrak{c}_{0}ρ^{2}4πc0​ρ2 (equivalently, pressure P=2πc0ρ2P=2π\mathfrak{c}_{0}ρ^{2}P=2πc0​ρ2), arising solely from the kinetic energy density of the many-body system, rather than the interaction energy density, marking a fundamental difference from the Gross--Pitaevskii and other mean-field regimes. Second, the newly discovered coupling constant c0\mathfrak{c}_{0}c0​ is the electrostatic capacity of the interaction potential, corresponding to the scattering length of the hard-core potential. Furthermore, in other limiting regimes, including those beyond the Gross--Pitaevskii regime, we find that the limiting equation is described by an eikonal system, offering a rigorous first-principle justification for using the ``geometric optics approximation'' to describe the dynamics of ultracold Bose gases.
We investigate dynamics in two-component Bose-Einstein condensates in the context of coupled Gross-Pitaevskii equations and derive results for the evolution of the total density fluctuations. Using these results, we show how, in many cases of interest, the dynamics can be accurately described with an effective one-component Gross-Pitaevskii equation for one of the components, with the trap and interaction coefficients determined by the relative differences in the scattering lengths. We discuss the model in various regimes, where it predicts breathing excitations, and the formation of vector solitons. An effective nonlinear evolution is predicted for some cases of current experimental interest. We then apply the model to construct quasistationary states of two-component condensates.