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Frontiers of Physics

Solving 1D harmonic trapped three contacting bosons by the generator coordinate method

Hao-Rong Feng, Jing-An Sun, Guang-Jie Guo, Wang Zhang, Bo Zhou, Y. G.

Published Jul 15, 2025• 0 citations
Publisher

Abstract

We study the quantum system of three ultracold one-dimensional identical bosons with δ -contact interaction in a harmonic trap by proposing a method termed the generator coordinate method (GCM)-polynomial ansatz (PA). Based on the asymptotic property of our system, we describe the wave function as a (pseudo-)polynomial multiplied by the asymptotic Gaussian function, then apply the GCM to this PA description to solve the system. Our results include not only the ground and first excited states, which are in agreement with previous calculations, but also a dozen unexplored excited states. We present and discuss the eigenenergy spectra and eigenstates, including periodic patterns and degeneracies. Additionally, we reproduce the states and properties at extreme interaction limits, such as Bose–Einstein (BE) condensate, fermionization at Tonks–Girardeau (TG) gas limit and TG/super-TG mapping.

DOI
10.15302/frontphys.2026.012201

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We study the quantum system of three ultracold one-dimensional identical bosons with δ -contact interaction in a harmonic trap by proposing a method termed the generator coordinate method (GCM)-polynomial ansatz (PA). Based on the asymptotic property of our system, we describe the wave function as a (pseudo-)polynomial multiplied by the asymptotic Gaussian function, then apply the GCM to this PA description to solve the system. Our results include not only the ground and first excited states, which are in agreement with previous calculations, but also a dozen unexplored excited states. We present and discuss the eigenenergy spectra and eigenstates, including periodic patterns and degeneracies. Additionally, we reproduce the states and properties at extreme interaction limits, such as Bose–Einstein (BE) condensate, fermionization at Tonks–Girardeau (TG) gas limit and TG/super-TG mapping.
Abstract We propose a new method, termed generator coordinate method (GCM)-correlated pair wave function (CPWF), for studying one-dimensional bosons confined in harmonic potentials with contact repulsive interactions. Our approach involves using the effective CPWF as a basis, combined with the GCM to handle complex many-particle correlations accurately. We demonstrate the reliability of our GCM-CPWF wave functions by comparing ground energy and one-body density with those obtained by other numerical methods. Moreover, we present the energy spectrum up to six particles and the occupation number on the harmonic oscillator state. Utilizing these GCM-CPWF wave functions, we explore the properties of the ground and excited states of the many-particle system. Our GCM-CPWF framework is highly flexible and can be generalized to investigate more complex many-particle systems.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
We consider a small number of identical bosons trapped in a two-dimensional isotropic harmonic potential and also the NNN-boson system when it is feasible. The atom-atom interaction is modeled by means of a finite-range Gaussian interaction. The spectral properties of the system are scrutinized and, in particular, we derive analytic expressions for the degeneracies and their breaking for the lower-energy states at small but finite interactions. We demonstrate that the degeneracy of the low-energy states is independent of the number of particles in the noninteracting limit and also for sufficiently weak interactions. In the strongly interacting regime, we show how the many-body wave function develops holes whenever two particles are at the same position in space to avoid the interaction, a mechanism reminiscent of the Tonks-Girardeau gas in one dimension. The evolution of the system as the interaction is increased is studied by means of the density profiles, pair correlations, and fragmentation of the ground state for N=2N=2N=2, 3, and 4 bosons.
We consider an ultracold atomic gas trapped in a one-dimensional harmonic oscilla- tor potential. We study the transition from a Bose-Einstein condensate, in the small atom-atom interaction limit, to a strongly interacting Tonks-Girardeau gas, as the interaction is varied from zero to large values. The response of the system against a monopolar perturbation is characterized by the moments of the dynamic structure function. The calculations have been done by direct diagonalization of the second quantized many-body Hamiltonian in a truncated Fock space. The corresponding virial theorem for the many-body states has been employed to cross-check the validity of our numerical techniques.
We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the TG and STG gas show the same profiles at the strongly interacting point −1/γ=0-1/γ=0−1/γ=0, and in the weakly interacting limit γ=0γ=0γ=0 the ground states TG and BS join to each other smoothly.
We study ground state properties of spinless, quasi one-dimensional bosons which are confined in a harmonic trap and interact via repulsive delta-potentials. We use the exact diagonalization method to analyze the pair correlation function, as well as the density, the momentum distribution, different contributions to the energy and the population of single-particle orbitals in the whole interaction regime. In particular, we are able to trace the fascinating transition from bosonic to fermi-like behavior in characteristic features of the momentum distribution which is accessible to experiments. Our calculations yield quantitative measures for the interaction strength limiting the mean-field regime on one side and the Tonks-Girardeau regime on the other side of an intermediate regime.