We study the ground-state correlations between two atoms in a two-dimensional isotropic harmonic trap. We consider a finite-range soft-core interaction that can be applied to simulate various atomic systems. We provide detailed results on the dependence of the correlations on the parameters of the system. Our investigations show that in the hardcore limit, the wave function can be approximated as the product of the radial and angular components regardless of the interaction range. This implies that the radial and angular correlations are independent of one another. However, correlations within the radial and angular components persist and are heavily influenced by the interaction range. The radial correlations are generally weaker than the angular correlations. When soft-core interactions are considered, the correlations exhibit more complex behavior.
We study the ground-state correlations between two atoms in a two-dimensional isotropic harmonic trap. We consider a finite-range soft-core interaction that can be applied to simulate various atomic systems. We provide detailed results on the dependence of the correlations on the parameters of the system. Our investigations show that in the hardcore limit, the wave function can be approximated as the product of the radial and angular components regardless of the interaction range. This implies that the radial and angular correlations are independent of one another. However, correlations within the radial and angular components persist and are heavily influenced by the interaction range. The radial correlations are generally weaker than the angular correlations. When soft-core interactions are considered, the correlations exhibit more complex behavior.
Abstract We study the ground-state entanglement between two atoms in a two-dimensional isotropic harmonic trap. We consider a finite-range soft-core interaction that can be applied to simulate various atomic systems. We provide detailed results on the dependence of the correlations on the parameters of the system. Our investigations show that in the hardcore limit, the wave function can be approximated as the product of the radial and angular components regardless of the interaction range. This implies that the radial and angular correlations are independent of one another. However, correlations within the radial and angular components persist and are heavily influenced by the interaction range. The radial correlations are generally weaker than the angular correlations. When soft-core interactions are considered, the correlations exhibit more complex behavior.
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.
We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas-Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.
The exactly solvable model of two indistinguishable quantum particles (bosons or fermions) confined in a one-dimensional harmonic trap and interacting via finite-range soft-core interaction is presented and many properties of the system are examined. Particularly, it is shown that independently on the potential range, in the strong interaction limit bosonic and fermionic solutions become degenerate. For sufficiently large ranges a specific crystallization appears in the system. The results are compared to predictions of the celebrated Busch et al. model and those obtained in the Tonks-Girardeau limit. The assumed inter-particle potential is very similar to the potential between ultra-cold dressed Rydberg atoms. Therefore, the model can be examined experimentally.
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 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.