We present a method for solving few-body problems for trapped particles and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e., the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${C}_{6v}$ . We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.We present a method for solving few-body problems for trapped particles and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e., the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${C}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.

arXiv:1209.1398Physical Review ANov 30, 201262 citations

Analysis of Classical and Quantum Interacting Particle Systems

Bogusław Zegarliński

QP-PQ, quantum probability and white noise analysisJul 1, 200226 citations

Equilibration and approximate conservation laws: Dipole oscillations and perfect drag of ultracold atoms in a harmonic trap

Robert Bamler, Achim Rosch

The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum-generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning $\ensuremath{\delta}\ensuremath{\omega}$ of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by interspecies interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of interspecies interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, $1/\ensuremath{\tau}\ensuremath{\propto}{(\ensuremath{\delta}\ensuremath{\omega})}^{2}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is a single-particle scattering rate.The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum-generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning $\ensuremath{\delta}\ensuremath{\omega}$ of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by interspecies interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of interspecies interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, $1/\ensuremath{\tau}\ensuremath{\propto}{(\ensuremath{\delta}\ensuremath{\omega})}^{2}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is a single-particle scattering rate.

arXiv:1504.03375Physical Review AJun 5, 20158 citations

Spectroscopy for a few atoms harmonically trapped in one dimension

Nathan L. Harshman

Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3--5$ , are provided.Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3--5$, are provided.

arXiv:1312.6107Physical Review AMar 28, 201443 citations

{C}_{6v}

symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of

{C}_{6v}

. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.

N=3--5

, are provided.