We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-range Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. We also explicitly verify by a variational treatment that in the zero-range limit the positive $\ensuremath{\delta}$ potential in two dimensions only reproduces the noninteracting results, if the Hilbert space in not truncated, and demonstrate that an extremely large Hilbert space is required to approach the ground state when one is to tackle the limit of zero-range interaction numerically. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature. The present results indicate that a finite-range interparticle potential is numerically amenable for treating the statics and the nonequilibrium dynamics of interacting many-particle systems (bosons) in two dimensions.
We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-range Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. We also explicitly verify by a variational treatment that in the zero-range limit the positive \ensuremathδ potential in two dimensions only reproduces the noninteracting results, if the Hilbert space in not truncated, and demonstrate that an extremely large Hilbert space is required to approach the ground state when one is to tackle the limit of zero-range interaction numerically. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature. The present results indicate that a finite-range interparticle potential is numerically amenable for treating the statics and the nonequilibrium dynamics of interacting many-particle systems (bosons) in two dimensions.
We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum ∣m∣ state with various values of the two-body interaction range σ and the two-body interaction strength g2 to study the resulting energy spectrum and analyze the role of Hilbert space dimensions N. We compare results for both attractive and repulsive interaction for δ-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given ∣m∣ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength g2>0, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength g2V(r)≤−1, ground state energy diverges to physically unacceptable negative-infinity for ∣m∣=0 state. Further, for ∣m∣=1, the ground state energy becomes independent of the interaction strength. For a ∣m∣, we present a comparative study between the Gaussian interaction potential and the δ-function potential. Further, we observe that for a given g2 and ∣m∣, for δ-function potential {\it i.e.} σ→0, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with σ→1, the ground state energy converges for a considerably small critical Hilbert space.
We consider a small number of identical bosons trapped in a two-dimensional isotropic harmonic potential and also the N-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=2, 3, and 4 bosons.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.
We describe the properties of a pair of ultracold bosonic atoms in a one-dimensional harmonic trapping potential with a tunable zero-ranged barrier at the trap center. The full characterization of the ground state is done by calculating the reduced single-particle density, the momentum distribution, and the two-particle entanglement. We derive several analytical expressions in the limit of infinite repulsion (Tonks-Girardeau limit) and extend the treatment to finite interparticle interactions by numerical solution. As pair interactions in double wells form a fundamental building block for many-body systems in periodic potentials, our results have implications for a wide range of problems.
We investigate a system of N spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive two-body finite-range Gaussian interaction potential of large s-wave scattering length. Exact diagonalization of the Hamiltonian matrix is carried out to obtain the N-body ground state as well as low-lying excited states, using Davidson algorithm in beyond lowest-Landau-level approximation. We examine the finite-range effects of the interaction potential on the many-body ground state energy as also the degree of condensation of the Bose-condensed gas. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential. We further analyze the effect of finite-range interaction potential on the breathing mode collective excitation. Our theoretical results may be relevant for experiments currently conducted on quasi-two-dimensional Bose gas with more realistic interaction potential.
We investigate a system of bosons in a two-dimensional harmonic trap. In the limit of strong attractive interactions, the bosons make a droplet insensitive to external confinement. For weak interactions, in contrast, the ground state is given by the harmonic trap. In this work, we conduct a variational study of the transition between these two limits. We find that this transition occurs at the critical interaction strength whose value is universal if scaled appropriately with the number of particles. To connect the change in the properties of the system to the classical description of phase transitions, we analyze the static response of the Bose gas related to the isothermal compressibility. Finally, we perform numerically exact calculations for a few particles to demonstrate the effects of finite-range interactions on this transition. We conclude that finite-range effects wash out the point of transition.