In this work,
π,
σ+ and
σ− transitions between magnetic sublevels of the
D1 line of all alkali atoms are considered analytically. General block Hamiltonian matrices in presence of a magnetic field for the ground and excited states are built in order to describe all the transitions. Eigenvalues and eigenkets describing ground and excited levels are calculated, "modified" and unperturbed transfer coefficients as a function of the nuclear spin
I, the magnetic quantum number
m and the magnetic field magnitude
B are defined. Transition cancellations are observed only for some
π transitions of each isotope. The main result is that we obtain one single formula which expresses the magnetic field values cancelling these transitions. These values also correspond to the case when some of other transitions intensity have their maximum. In addition, we examine the derivative of
π transition "modified" transfer coefficients in order to find the magnetic field values which correspond to the intensities maximum. The accuracy of the magnetic field
B values is only limited by the uncertainty of the involved physical quantities.
In this work, $\ensuremath{\pi}, {\ensuremath{\sigma}}^{+}$, and ${\ensuremath{\sigma}}^{\ensuremath{-}}$ transitions between magnetic sublevels of the
D1 line of all alkali-metal atoms are considered analytically. General block Hamiltonian matrices in the presence of a magnetic-field for the ground and excited states are built in order to describe all the transitions. Eigenvalues and eigenkets describing ground and excited levels are calculated and modified and unperturbed transfer coefficients as a function of the nuclear spin
I, the magnetic quantum number
m, and the magnetic-field magnitude
B are defined. Transition cancellations are observed only for some $\ensuremath{\pi}$ transitions of each isotope. The main result is that we obtain a single formula which expresses the magnetic-field values canceling these transitions. These values also correspond to the case when the intensities of some of the other transitions reach their maximum. In addition, we examine the derivative of $\ensuremath{\pi}$ transition modified transfer coefficients in order to find the magnetic-field values which correspond to the intensities' maximum. The accuracy of the magnetic-field
B values is only limited by the uncertainty of the physical quantities involved.
We have analyzed the magnetic field dependencies of the intensities of all the optical transitions between magnetic sublevels of hyperfine levels, excited with σ + , π , and σ − polarized light, for the D 1 and D 2 lines of 87 R b and 85 R b atoms. Depending on the type of transition and the quantum numbers of the involved levels, the Hamiltonian matrices are of 1 × 1 , 2 × 2 , 3 × 3 , or 4 × 4 dimension. As an example, analytical expressions are presented for the case of 2 × 2 dimension matrices for the D 1 line of both isotopes. Eigenvalues and eigenkets are given, and the expression for the transition intensity as a function of B has been determined. It is found that some π transitions of 87 R b and 85 R b get completely canceled for certain, extremely precise, values of B . No cancellation occurs for σ + or σ − transitions of the D 1 line. For matrices with a size over 2 × 2 , analytical formulas are heavy, and we have performed numerical calculations. All the B values canceling σ + , π , and σ − transitions of the D 1 and D 2 lines of 87 R b and 85 R b are calculated, with an accuracy limited by the precision of the involved physical quantities. The experimental implementation feasibility and its possible outcome are addressed. We believe the experimental realization will allow an increase in the precision of the physical quantities involved, in particular the upper state atomic level energy.
In a previous work the
π transitions of the
D1 line of alkali-metal atoms were shown to cancel at magnetic-field values given by a single closed-form expression. The
D2 line has until now resisted the same treatment, because the fixed-
m Hamiltonian blocks of the
2P3/2 manifold reach the dimension
4×4, and the corresponding formulas were considered to be too heavy to be useful. In this work we show that this difficulty is only apparent. Since the Zeeman interaction couples only levels with
ΔF=±1, every block, whatever its dimension, is tridiagonal; its characteristic polynomial therefore obeys a three-term recursion, and the components of its eigenvectors are polynomials in the eigenvalue. Using these two properties we obtain the eigenvalues in closed form by Ferrari's and Cardano's formulas, the eigenvectors without any further diagonalization, and finally a single relation which gives the magnetic-field value canceling a
D2 transition as an explicit function of the excited-state eigenvalue, the nuclear spin
I, the ground-state hyperfine splitting and the magnetic quantum number. A necessary condition on
m and on the polarization is derived, which replaces the selection rule known for the
D1 line. All the magnetic-field values canceling
π,
σ+ and
σ− transitions of
23Na,
39K,
40K,
41K,
85Rb,
87Rb and
133Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the
D1 line, the
σ± transitions of the
D2 line do cancel, and they account for the majority of the values found. The accuracy of the calculated
B values is limited only by the uncertainties of the excited-state hyperfine constants.
The theory of how excited-state hyperfine structure can be determined using a magnetic field to decouple the nuclear and electronic angular momentum is presented. The initial excited state is assumed to have an anisotropic Zeeman sublevel population distribution, which can be conveniently created by a laser-induced multiphoton excitation of an unpolarized atom. The Zeeman sublevel populations are subsequently mixed by the hyperfine interaction. The degree of mixing is controlled by a magnetic field applied along the quantization axis. The relative Zeeman sublevel populations can be monitored by detecting the polarization of fluorescence emitted when the excited state radiatively decays. It is convenient to take the ratio of two time-integrated fluorescent signals measured for orthogonal linear polarizations, since this is independent of the excited-state number density. The plot of this ratio versus magnetic field is very well approximated by a Lorentzian curve whose half-width is proportional to the magnetic dipole hyperfine coupling constant. This method was used to determine the magnitude of the magnetic dipole constant of the cesium 8
D3/2 state to be 3.92\ifmmode\pm\else\textpm\fi{}0.10 MHz, which is in excellent agreement with previous results.
It has been shown that the D 1 line of atomic vapors of alkali metals excited by π-polarized radiation in a strong transverse magnetic field includes specific “guiding” (indicating) atomic transitions between the magnetic sublevels of the hyperfine structure. The dependence of the frequency shift of the guiding transitions on the magnetic field, as well as on their dipole moments, is asymptotic for all other transitions. An experiment with a nanocell with Rb vapor with a thickness of half the wavelength (λ/2 method) for ensuring a sub-Doppler spectral resolution has completely confirmed the presence of guiding transitions. Two groups of six transitions for 85 Rb and two groups of four transitions for 87 Rb have been detected in the absorption spectra in magnetic fields above 4 kG. A guiding transition has been identified in each of four groups. Four transitions forbidden at B = 0 have been also detected; with an increase in the magnetic field, their probabilities also approach the probabilities of the guiding transitions.