Spectral properties and lifetimes of neutral spin-$\frac{1}{2}$ fermions in a magnetic guide

We investigate the resonant motion of neutral spin-$\frac{1}{2}$-fermions in a magnetic guide. A wealth of unitary and antiunitary symmetries is revealed in particular giving rise to a twofold degeneracy of the energy levels. To compute the energies and decay widths of a large number of resonances the complex scaling method is employed. We discuss the dependence of the lifetimes on the angular momentum of the resonance states. In this context the existence of so-called quasibound states is shown. In order to approximately calculate the resonance energies of such states a radial Schrödinger equation is derived which improves the well-known adiabatic approximation. The effects of an additionally applied homogeneous Ioffe field on the resonance energies and decay widths are also considered. The results are applied to the case of the ${}^6\mathrm{Li}$ atom in the $F=\frac{1}{2}$ hyperfine ground state.

Figure 1

Dependence of the resonance energies (upper part) and decay widths (lower part) on the complex scaling angle $\theta$ as observed in the numerical calulations. The stationary region is located in between the dashed lines.

Figure 2

Energies $E$ and decay widths $\Gamma$ (logarithmic scale) for resonances in a magnetic guide for $\kappa =1$. An increasing Ioffe field strength leads to a global decrease of the decay widths and therefore has a stabilizing effect. At the same time the resonance pattern becomes distorted and a regrouping of the resonances into pairs is observed.

Figure 3

Resonance energies and decay widths in the $\kappa =1$ or $\kappa =-1$ subspaces at $\gamma =1$ (linear scale plot). Due to the finite Ioffe field formally degenerate pairs split up. States with negative quantum number $m$ are shifted towards higher resonance energies and vice versa. The splitting decreases with decreasing decay width.

Figure 4

Resonance energies and ${\Lambda }_z$ eigenvalues of resonances for $\gamma =0$ (left picture) and $\gamma =1$ (right picture). For $\gamma =0$ the energies of the resonances form a symmetric pyramidlike pattern. For finite Ioffe field strength the pattern becomes asymmetric.

Figure 5

Decay widths and ${\Lambda }_z$ eigenvalues of resonances for $\gamma =0$ (left picture) and $\gamma =1$ (right picture). The decay width decreases exponentially with increasing modulus of $m$. With increasing Ioffe field strength one observes an overall decrease of the decay widths and a widening of the distribution.

Figure 6

Energies $E_{\mathrm{lr}}$ of the lowest resonances of each $m$ subspace calculated by using Eq. (28). Their accuracy is higher than the resolution of the plot.

Figure 7

Comparison of the quasibound energies $E_{\mathit{qb}}$ and adiabatic energies $E_{\mathit{ad}}$ to the exact energies $E$. The figure shows the energies of the lowest four states in the $m=\frac{15}{2}$ and $l=8$ subspace, respectively. The discrepancy between $E$ and $E_{\mathit{qb}}$ is hardly visible throughout the complete $\gamma$ interval shown.

Figure 8

Graphical representation of the eigenenergies (39). The values of the quantum number $\kappa$ are calculated using Eq. (7). A pattern of equidistant horizontal lines with alternating $\kappa$ values is formed.