Rydberg atoms in a magnetic guide

We investigate electronically excited atoms in a magnetic guide. It turns out that the Hamiltonian describing this system possesses a wealth of both unitary as well as antiunitary symmetries that constitute an uncommon extensive symmetry group. One consequence is the twofold degeneracy of any energy level. The spectral properties are investigated for a wide range of field gradients and the spatial distributions of the spin polarization are analyzed. Wavelengths, oscillator strengths, and selection rules are provided for the corresponding electromagnetic transitions. The effects due to an additional homogeneous bias field constituting a Ioffe-Pritchard trap are explored equally.

Figure 1

Vectorial plots of the magnetic field (3). (a) Intersection for $z=0$. The quadrupolar shape of the field is clearly recognized. (b) Intersection for $x=0$ revealing the translational invariance with respect to the $z$ coordinate.

Figure 2

(a) Splitting of the energy levels belonging to the $n=5$ multiplet ($\kappa =\pm 1$ subspace) with increasing gradient. The level structure is dominated by the linear Zeeman term. The splitting is linear and symmetric around the energy for $b=0$. (b) Intra $n$ manifold mixing of the $n=5$ multiplet in the $\kappa =1$ subspace. Due to the increasing dominance of the diamagnetic term the level splitting becomes nonlinear.

Figure 3

Inter $n$ manifold mixing between the $n=6$ and $n=7$ multiplets in the $\kappa =1$ subspace. The mixing threshold is indicated by the dashed line. A large number of avoided crossings occur.

Figure 4

(a) Expectation value of the $z$ component of the electronic spin operator for several excited states $\left(b={10}^{-7}\right)$. (b) Zoomed view of the $n=15$ multiplet. The magnitude of $⟨S_z⟩$ decreases for states possessing a large energy shift due to the external field.

Figure 5

Tomographic cuts through the spin polarization $W_{SB}$ of the 83rd excited state. The state belongs to the $n=8$ multiplet within the $\kappa =1$ subspace $\left(b={10}^{-7}\right)$. The cuts are made at $z=\pm 20$ and $z=0$. Positive and negative values are indicated white and black, respectively. We observe a rich pattern of different spin polarizations around the origin. From $r\approx 60$ on the nodal structure is replaced by a regular striped pattern varying periodically with the azimuthal angle $\varphi$.

Figure 6

Graphical representation of allowed dipole transitions between $\kappa$ subspaces. The arrows point from $\kappa$ to ${\kappa }^{\prime }$.

Figure 7

(a) Dipole strenghts for $\pi$ transition from the ground state of the $\kappa =1$ subspace to excited states belonging to the $\kappa =-1$ subspace $\left(b={10}^{-7}\right)$. The line for smallest $\lambda$ belongs to the $n=1\to 7$ transition. (b) Zoomed view of the line belonging to the transition to the $n=12$ multiplet. The line center is dominated by two sublines. The two bunches accompanying the line center at its left and right hand sides spossess a much smaller dipole strength.

Figure 8

(a) Dipole strenghts for ${\sigma }^{+}$ transition from the ground state of the $\kappa =-i$ subspace to excited states belonging to the $\kappa =-1$ subspace $\left(b={10}^{-7}\right)$. The line for smallest $\lambda$ belongs to the $n=1\to 7$ transition. (b) Zoomed view of the line belonging to the transition to the $n=12$ multiplet. The line consists of three bunches each of which consisting of a number of sublines. The line is dominated by two sublines, one each located in the left and right hand side bunches.

Figure 9

Expectation values of the $z$ component of the electronic spin at a finite homogeneous field strength $\left(B_I={10}^{-5}\right)$ and a gradient of $b={10}^{-7}$. At low degree of excitation the homogeneous field dominates the electronic states. In this regime $S_z$ becomes an approximate constant of motion admitting $⟨S_z⟩$ to possess one of the two possible values $\pm 1∕2$. States lying above the critical principal quantum number $n_c$ become increasingly dominated by the quadrupole field. As a result the expecation values tend toward $⟨S_z⟩=0$.

Figure 10

Tomographic cuts through the spin polarization $W_{SB}$ [Eq. (36)] of the 83rd excited state at a finite Ioffe field strength $B_I={10}^{-5}$. The state belongs to the $n=8$ multiplet inside the $\kappa =1$ subspace $\left(b={10}^{-7}\right)$. The cuts are made at $z=\pm 20$ and $z=0$. Positive and negative values are indicated by white and black, respectively. One observes large gray regions with $W_{SB}\approx 0$.

Figure 11

(a) Dipole strengths for $\pi$ transition from the ground state of the $\kappa =1$ subspace to excited states belonging to the $\kappa =-1$ subspace ($b={10}^{-7}$ and $B_I={10}^{-5}$). The line for smallest $\lambda$ belongs to the $n=1\to 7$ transition. (b) Zoomed view of the line belonging to the transition to the $n=12$ multiplet. Several additional lines appear at finite homogeneous field strength (some are marked by an arrow). The line center is dominated by a single subline emerging from a transition which is induced by the external homogeneous field.