Spectral properties and lifetimes of neutral fermions and bosons in a magnetic quadrupole trap

We investigate the motion of neutral fermions and bosons in a three-dimensional magnetic quadrupole trap. Inspecting the underlying Hamiltonian a variety of symmetries is revealed which give rise to degeneracies of the resonance energies. Our numerical approach which involves the eigenvalue problem resulting from a Sturmian basis set together with the complex scaling method enables us to calculate several hundred resonance states. The distributions of the energies and decay widths of the resonances are analyzed for both spin-$\frac{1}{2}$ fermions and spin-1 bosons. We also investigate under what conditions quasibound states with long lifetimes can be achieved. An effective scalar Schrödinger equation describing such states is derived. The results are applied to the cases of the alkali-metal atoms ${}^{87}\mathrm{Rb}$ and ${}^6\mathrm{Li}$ trapped in a hyperfine ground state.

Figure 1

Energies $E$ and decay widths $\Gamma$ (logarithmic scale) of resonances in the magnetic quadrupole trap for four different values of the $J_z$ quantum number. The structure of the resonance energy pattern is significantly changed with increasing values of the quantum number $m$. The triangular distribution reverses. At the same time a transition to a more regular pattern is observed.

Figure 2

Decay width of the energetically lowest resonance shown as a function the quantum number $m$ of the $z$ component of the total angular momentum (spin-$\frac{1}{2}$ particle). An exponential decrease of the decay width is observed. The increased stability of high $m$ states originates from a localization of such states far away from the center of the trap where transitions to unbound states take place.

Figure 3

Resonances in the $m=\frac{37}{2}$ subspace. The resonance states can be characterized by the two quantum numbers $(n_{\rho },n_z)$ which denote the number of nodes in the $\rho$ and $z$ directions, respectively. The right-hand side shows the probability density of the wave functions $\mid {\Phi }_{\mathit{qb}}⟩$. The wave functions are approximately centered at the coordinates $\mathbf{(}{z}_0=0,{\rho }_0={\left(2m\right)}^{2∕3}\mathbf{)}$.

Figure 4

Energies $E$ and decay widths $\Gamma$ (logarithmic scale) of resonances in the magnetic quadrupole trap for four values of the $J_z$ quantum number (spin-1 particle). The structure of the resonance energy pattern undergoes a significant alteration with increasing $m$ quantum number.

Figure 5

Decay width of the energetically lowest resonance plotted against the quantum number $m$ of the $z$ component of the total angular momentum. An exponential decrease of the decay width is observed. The increased stability of high-$m$ states originates from a localization of such states far away from the center of the trap.