Abstract
The system of two interacting bosons in a two-dimensional harmonic trap is compared with the system consisting of two noninteracting fermions in the same potential. In particular, we discuss how the properties of the ground state of the system, e.g., the different contributions to the total energy, change as we vary both the strength and range of the atom–atom interaction. In particular, we focus on the short-range and strong interacting limit of the two-boson system and compare it to the noninteracting two-fermion system by properly symmetrizing the corresponding degenerate ground state wave functions. In that limit, we show that the density profile of the two-boson system has a tendency similar to the system of two noninteracting fermions. Similarly, the correlations induced when the interaction strength is increased result in a similar pair correlation function for both systems.
Highlights
The Bose–Fermi mapping [1] theorem establishes a relation between the ground state energy and the wave function of strongly interacting bosons with those corresponding to noninteracting fermions in the same trapping potential
The ground state energy of two strongly interacting bosons in a harmonic trap for a short-range interaction is not equal to the noninteracting two-fermion system in the same potential, as it was in one dimension
The wave function resulting from symmetrizing the corresponding noninteracting fermionic ground state is found to be a very good variational trial wave function. It provides an upper-bound very close to the ground state energy obtained by exact diagonalization in the strong interacting and short-range limit
Summary
The Bose–Fermi mapping [1] theorem establishes a relation between the ground state energy and the wave function of strongly interacting bosons with those corresponding to noninteracting fermions in the same trapping potential. Both systems have the same energy, and the ground state wave function of the interacting bosonic system can be obtained by symmetrizing the noninteracting fermionic one by taking the absolute value. We compare the numerical calculations for the ground state of two interacting bosons in two dimensions with a short-range interaction with the properties obtained from the analytical wave functions that describe two noninteracting bosons, two noninteracting fermions, and the corresponding symmetrized wave function.
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