Steady-state solutions for atomic multipole moments in an arbitrarily oriented static magnetic field

We derive algebraic expressions for the atomic multipole moments $m_{k,q}\left(F\right)$ describing the steady-state polarization of an atomic ensemble with angular momentum $F$ in a static magnetic field of arbitrary direction and general multipole relaxation rates ${\Gamma }_q^{\left(k\right)}$. The longitudinal moments $m_{k,0}$ are given in terms of truncated continued fractions, while the transverse moments $m_{k,q\ne 0}$, representing the ensemble coherence, are given by products of truncated continued fractions. The special case of isotropic relaxation leads to particularly simple and elegant expressions. Our results are relevant for all domains of physics that consider the evolution of a spin system interacting with vector-type perturbations.

Figure 1

Atomic multipole moments—visualized following Refs. [8, 9, 10]—that determine the absorption of unpolarized (a), linearly polarized (b), and circularly polarized (c) light. Unpolarized light “feels” only the longitudinal alignment $m_{2,0}$ along $\vec{k}$. Linearly polarized light probes only the longitudinal alignment $m_{2,0}$ along the light's polarization vector $\widehat{\varepsilon }$. Circularly polarized light senses both the longitudinal orientation $m_{1,0}$ and the longitudinal alignment $m_{2,0}$ along $\vec{k}$. For all three cases, the two upper rows represent multipole moments of opposite sign. The graphs in the lower row show the axes of rotational symmetry (chosen as quantization axis) and the planes of reflection symmetry of both the incident light and the relevant multipole moments of the medium.

Figure 2

Dependence of the $m_{1,0}$ resonance width (a) and contrast (b) on the orientation $\theta$ of the scanned magnetic field with respect to the quantization axis.