Abstract
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.
Highlights
In the following we study properties of the system of two identical quantum particles of mass m confined in a one-dimensional harmonic trap of frequency Ω and interacting via soft-core finite-range rectangular potential
We present properties of the exactly solvable model of two interacting particles confined in a harmonic trap
Inter-particle forces are modeled by a square wall controlled by two independent parameters: potential range and its strength
Summary
In the following we study properties of the system of two identical quantum particles of mass m confined in a one-dimensional harmonic trap of frequency Ω and interacting via soft-core finite-range rectangular potential. Depending on the situation we consider symmetric (for bosons) or antisymmetric (for fermions) wave functions with respect to an exchange of particles’ positions. Our aim is to give a straightforward and analytical prescription for the eigenstates of the Hamiltonian (1) as a function of the potential depth V and its range a. With these solutions, we examine different properties of a few of the lowest eigenstates in the bosonic and fermionic cases. We consider different single-particle system characteristics (density profile, momentum distribution) as well as inter-particle correlations reflected in a reduced single-particle density matrix
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