For a quantum system of $N$ identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding one-particle reduced density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ analytically. In case of bosons ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all, the ``gap'' at the ``Fermi level'' decreases with increasing coupling strength but remains well pronounced even for strong interaction.For a quantum system of $N$ identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding one-particle reduced density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ analytically. In case of bosons ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all, the ``gap'' at the ``Fermi level'' decreases with increasing coupling strength but remains well pronounced even for strong interaction.

arXiv:1307.6858Physical Review AOct 9, 201338 citations

One-dimensional delta -interacting bosons in a harmonic trapping potential

Vanja Dunjko

Jan 1, 20140 citations

Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap

Marvin D. Girardeau, Ewan M. Wright, J. M. Triscari

The exact N-particle ground-state wave function for a one-dimensional condensate of hard-core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as ${N}^{0.59},$ in contrast to ${N}^{0.5}$ for a spatially uniform system, and N for a true Bose-Einstein condensate.The exact N-particle ground-state wave function for a one-dimensional condensate of hard-core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as ${N}^{0.59},$ in contrast to ${N}^{0.5}$ for a spatially uniform system, and N for a true Bose-Einstein condensate.

arXiv:cond-mat/0008480Physical Review AFeb 5, 2001223 citations

Quantum Entanglement of Two Harmonically Trapped Dipolar Particles

Przemysław Kościk

We study systems of two identical dipolar particles confined in quasi one-dimensional harmonic traps. Numerical results for the dependencies of the entanglement on the control parameters of the systems are provided and discussed in detail. In the limit of a strong interaction between the particles, the occupancies and the von Neumann entropies of the bosonic and fermionic ground states are derived in closed analytic forms by applying the harmonic approximation. The strong correlation regimes of the system with the dipolar bosons and the system with the charged ones are compared with each other in regard to aspects of their entanglement.We study systems of two identical dipolar particles confined in quasi one-dimensional harmonic traps. Numerical results for the dependencies of the entanglement on the control parameters of the systems are provided and discussed in detail. In the limit of a strong interaction between the particles, the occupancies and the von Neumann entropies of the bosonic and fermionic ground states are derived in closed analytic forms by applying the harmonic approximation. The strong correlation regimes of the system with the dipolar bosons and the system with the charged ones are compared with each other in regard to aspects of their entanglement.

arXiv:1410.0578Few-Body SystemsJan 11, 20156 citations

For a quantum system of

N

identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding one-particle reduced density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ analytically. In case of bosons ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all, the ``gap'' at the ``Fermi level'' decreases with increasing coupling strength but remains well pronounced even for strong interaction.

{N}^{0.59},

in contrast to

{N}^{0.5}

for a spatially uniform system, and N for a true Bose-Einstein condensate.