Two rotating particles interacting via two-body Gaussian potential harmonically confined in two spatial dimensions

TL;DR

This work analyzes two rotating, spinless bosons in a two-dimensional harmonic trap with a tunable Gaussian interparticle potential. By separating center-of-mass motion and expanding the relative motion in a harmonic-oscillator basis, the authors derive a transcendental equation for the relative-energy eigenvalue in a given angular-momentum sector, capturing the effects of the interaction range σ and strength g_2. The Gaussian potential yields off-diagonal couplings that ensure convergence within a finite critical Hilbert space, in contrast to the δ-function limit which fails to converge in a finite basis; the l = 0 sector can form bound states for attractive interactions, while the l = 1 sector remains non-binding. The results illuminate how interaction range and rotation sculpt the ground-state and low-lying spectra and decompose the energy into kinetic, trap, and interaction contributions, providing a tractable framework for exploring few-body physics in rotating quasi-2D systems.

Abstract

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 $g_{2}$ to study the resulting energy spectrum and analyze the role of Hilbert space dimensions $\widetilde{N}$. 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 $g_{2}>0$, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength $g_{2}V(r)\leq -1$, 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 $g_{2}$ and $|m|$, for $δ$-function potential {\it i.e.} $σ\to 0$, 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 $σ\to 1$, the ground state energy converges for a considerably small critical Hilbert space.

Figures (9)

• Figure 1: (Colour online) Plots for the ground-state energy $E_{0}$ for the relative motion of the pair vs size of the Hilbert space $N$ for relative angular momentum $l=0$ and interaction strength $g_{2}=+1$ for interaction range $\sigma=0.0001$.
• Figure 2: (Colour online) Plots for the ground-state energy $E_{0}$ of the pair vs the size $N$ of the Hilbert space for relative angular momentum $l=0$, interaction strength $g_{2}=+1$ for various values of interaction range $\sigma=0.1,0.2,0.4,0.5$.
• Figure 3: (Colour online) The ground-state energy $E_{0}$vs size of the Hilbert space $N$, for relative angular momentum $l=1$ with interaction strength $g_{2}=+1$ and various values of interaction range $\sigma=0.1,0.2,0.4,0.5$. It is observed that the size $N_{c}$ of the critical Hilbert space, at which the ground-state energy converges, decreases with increase in interaction range $\sigma$.
• Figure 4: (Colour online) Plot for the ground-state energy $E_{0}$ of the pair vs number of basis-states $N$ for both attractive and repulsive interaction with $g_{2}=\mp1$ and interaction range $\sigma=0.1$, for relative angular momentum $l=0$.
• Figure 5: (Colour online) Plot for the ground-state energy $E_{0}$vs number of basis-states $N$ for both attractive and repulsive interaction with $g_{2}=\mp1$ and interaction range $\sigma=0.1$, for relative angular momentum $l=1$.