Numerical Bogoliubov approximation of bosonic many-body quantum dynamics
Abstract
We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac--Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.
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BibTeX
@misc{henaff2026numerical,
title = {{Numerical Bogoliubov approximation of bosonic many-body quantum dynamics}},
author = {Yoann Le Hénaff and Christian Lubich and Peter Pickl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08956},
eprint = {2610.08956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical Bogoliubov approximation of bosonic many-body quantum dynamics
AU - Yoann Le Hénaff
AU - Christian Lubich
AU - Peter Pickl
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08956
UR - https://arxiv.org/abs/2610.08956
AB - We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac--Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.
ER -