Electronic structure of atoms in magnetic quadrupole traps

We investigate the electronic structure and properties of atoms exposed to a magnetic quadrupole field. The spin spatial as well as generalized time-reversal symmetries are established and shown to lead to a twofold degeneracy of the electronic states in the presence of the field. Low lying as well as highly excited Rydberg states are computed and analyzed for a broad regime of field gradients. The delicate interplay between the Coulomb and various magnetic interactions leads to complex patterns of the spatial spin polarization of individual excited states. Electromagnetic transitions in the quadrupole field are studied in detail thereby providing the selection rules and, in particular, the transition wavelengths and corresponding dipole strengths. The peculiar property that the quadrupole magnetic field induces permanent electric dipole moments of the atoms is derived and discussed.

Figure 1

Vectorial plots of the magnetic quadrupole field (2). The intersections are the $\left(x,y\right)$ plane for $z=0$ (a) and the $\left(y,z\right)$ plane for $x=0$ (b).

Figure 2

(a) Energy-level splitting of the $m=\frac{1}{2}$ states of the $n=20$ multiplet in the quadrupole field. (b) Splitting of the same energies in a homogeneous field.

Figure 3

Energy spectrum of the multiplets $n=33–37$ for the states with $m=\pm \frac{1}{2}$. For low gradients the energy levels split almost linearly followed by a transition region where the diamagnetic term becomes increasingly more important. For high gradients different $n$-multiplets overlap and no symmetries, i.e., approximate quantum numbers, are left.

Figure 4

Energy-level structure of the three lowest $n$ multiplets. The $J_z$ quantum number ranges from $-\frac{5}{2}$ to $\frac{5}{2}$.

Figure 5

Expectation value of the radial coordinate $r$ plotted against the principal quantum number $n$ at different gradients [(a) $b={10}^{-10}$, (b) $b={10}^{-8}$].

Figure 6

Spatial probability density of the ground state in the quadrupole field (a) and a homogeneous field pointing in the $z$ direction (b) for $b=B=10$ and $m=\frac{1}{2}$.

Figure 7

Expectation values of the $z$ component of the spin operator as a function of the quantum number $n$ for different gradients [(a) $b={10}^{-10}$, (b) $b={10}^{-8}$].

Figure 8

(a) Spatial probability density of the $45\text{th}$ excited state for $m=\frac{1}{2}$ and $b={10}^{-8}$ in logarithmic representation. (b) $S_z$ polarization for the same state.

Figure 9

Spatial probability density (a) and $S_z$ polarization (b) for the $1117\text{th}$ excited state for $m=\frac{1}{2}$ and $b={10}^{-8}$. At large $r$ the $S_z$ polarization becomes similar to $W_S^{+}$ indicating an antiparallel alignment of the electronic spin to the magnetic field.

Figure 10

$S_z$ polarization of the two eigenstates of the Hamiltonian (31). The electronic spin either points parallel $\left(W_S^{-}\right)$ or antiparallel $\left(W_S^{+}\right)$ in the local direction of the field.

Figure 11

Dipole strengths of $\Delta m=0$ $\pi$ transitions from the ground state to excited levels. Solid lines denote transitions in the quadrupole field whereas transitions in the homogeneous field are indicated by dashed lines. (a) The line possessing the largest wavelength belongs to the $n=1\to 10$ transition. (b) Magnification of the line structure of the $n=1\to 12$ transitions.

Figure 12

Dipole strengths of $\Delta m=-1$ $\sigma$ transitions (${\sigma }^{-}$ transitions) from the ground state to excited levels. Solid lines denote transitions in the quadrupole field whereas transitions in the homogeneous field are indicated by dashed lines. (a) The line possessing the lowest wavelength belongs to the $n=1\to 10$ transition. (b) Magnification of the line structure of the $n=1\to 12$ transition.

Figure 13

Expectation value of the dipole operator $D_{\pi }$ plotted against the principal quantum number $n$ for different gradients [(a) $b={10}^{-10}$, (b) $b={10}^{-8}$].