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Physical Review A

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

G. Bevilacqua, Evelina Breschi, Antoine Weis

Published Mar 7, 2014• 23 citations
Publisher

Abstract

We derive algebraic expressions for the atomic multipole moments ${m}_{k,q}(F)$ 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 ${\ensuremath{\Gamma}}_{q}^{(k)}$. The longitudinal moments ${m}_{k,0}$ are given in terms of truncated continued fractions, while the transverse moments ${m}_{k,q\ensuremath{\ne}0}$, 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.

DOI
10.1103/physreva.89.033406

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We derive algebraic expressions for the atomic multipole moments mk,q(F){m}_{k,q}(F)mk,q​(F) describing the steady-state polarization of an atomic ensemble with angular momentum FFF in a static magnetic field of arbitrary direction and general multipole relaxation rates \ensuremathΓq(k){\ensuremath{\Gamma}}_{q}^{(k)}\ensuremathΓq(k)​. The longitudinal moments mk,0{m}_{k,0}mk,0​ are given in terms of truncated continued fractions, while the transverse moments mk,q\ensuremath≠0{m}_{k,q\ensuremath{\ne}0}mk,q\ensuremath=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.
A theoretical treatment of the relaxation and transfer of atomic multipole moments due to interatomic collisions is presented, applicable to systems of emitters and perturbers with any angular momenta. Expressions are obtained for the rate constants, within the impact approximation, in terms of the elements of the scattering matrix evaluated as a function of impact parameter for an emitter-perturber collision. Processes in which the total electronic excitation energy is not conserved are excluded and collisions are assumed to occur isotropically. Explicit evaluation of rate constants has been carried out using the first-order dipole-dipole (resonance) interaction, assuming the classical straight line path approximation. Four special cases have been treated by numerical integration of the Schrodinger equation in the interaction representation. These are: J g (ground state angular momentum)=0, J e (excited state)=1; j g = 1 / 2 , J e = 1 / 2 ; J g = 1 / 2 , J e = 3 / 2 ; J g =1, J e =1. The ( 1 / 2 , 1 / 2 ) rate constants are shown to be simply related to those for the (0, 1) case. Within the context of the various approximations, numerical accuracy is expected to be better than 1%.
We present a theoretical study of double-resonance alignment magnetometers using linearly polarized light, in which the effect of atomic high-order multipole moments is considered. Starting from the effective master equation of our system obtained by eliminating the excited state adiabatically, we derive the full evolution equations of the atomic multipole moments. The analytic solutions of resonance signals involving the four-order multipole moments effect are obtained by using the perturbation approach. We present that the four-order multipole moments effect is negligible in the weak laser field, and the results reduce to that obtained by a three-step approach. However, the role of four-order multipole moments coupled by two-order tensor moments is more significant with the increasing Rabi frequency of light, which cannot be ignored. Meanwhile, the analytic expressions of relaxation processes are also studied, which are a linear combination of the laser-induced equivalent relaxation rate Γ L and the spin-exchange collision rate Γ g . The expected domain of validity of the three-step approach on light power is roughly given by Γ L < 1 2 Γ g . In addition, the steady-state results of resonance signals are presented in a strong radio-frequency magnetic field; in that case, the physical mechanism of the splitting of resonance signals is discussed. These results are valid for arbitrary light power and for an arbitrarily oriented static magnetic field.
The spontaneous relaxation rates for an atom in free space and close to an absorbing surface are calculated to various orders of the electromagnetic multipole expansion. The spontaneous decay rates for dipole, quadrupole, and octupole transitions are calculated in terms of their respective primitive electric multipole moments and the magnetic relaxation rate is calculated for the dipole and quadrupole transitions in terms of their respective primitive magnetic multipole moments. The theory of electromagnetic-field quantization in magnetoelectric materials is used to derive general expressions for the decay rates in terms of the dyadic Green's function. We focus on the decay rates in free space and near an infinite half space. For the decay of atoms near to an absorbing dielectric surface, we find a hierarchy of scaling laws depending on the atom-surface distance zzz.
Interpretation of linearly polarized parameters of the spectral lines of the second solar spectrum permits to obtain information about the magnetic field vector of the medium where the spectral lines are formed, provided suitable theoretical and numerical methods be applied to extract the physical information. This linear polarization is modified by the magnetic field and by collisions. In this context, with a few approximations, we provide in the present paper general and simple formulae giving the coefficients of the atomic master equation (depolarization, polarization transfer, population transfer and relaxation coefficients) for the case of an atom in any (but not spherically symmetric) level, perturbed by collisions with hydrogen atoms. It would be possible to apply this method for fast calculation of multipole rates for any level having an external shell l ≠q 0 .
The choice of an axis of quantization automatically selects those species from a bulk which could become stationary in a field-free environment with related spatial orientations of their coupling angular momenta. But if this axis fails to coincide with the direction of a uniform external magnetic field, a field-imposed transverse torque forces these atoms' total angular momenta into steady nutation that cannot be abolished by coupling-related internal torques. Making any stationary state impossible for these atoms, it also causes the contributions to the energy expectation value to oscillate. This is the result of a rigorous solution to the time-dependent Schrödinger equation describing the coupling dynamics of two spins 1/2 in an external field. Demanding to solve a four-dimensional system of coupled linear differential equations of first order, the bulkiness of even this simplest case already brings the calculations to the verge of becoming infeasible analytically.