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 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 alkali atoms hyperfine structure
D1 line all
π,
σ+ and
σ− transitions between magnetic sublevels are considered analytically. General
2×2 block matrices for ground and excited states were built in order to describe every transition. Eigenvalues and eigenkets for the mentioned above matrices are calculated and ``modified transfer coefficients as a function of nuclear angular momentum
I, magnetic quantum number
m and magnetic field magnitude
B are defined. Transition cancellations are observed only for
π transitions of each isotope. All the magnetic field values, which cancel transitions are calculated with an accuracy limited by the involved quantities uncertainties.
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.
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.
Abstract The circular dichroism effect has been investigated for atomic transitions of the Rb D1 line in magnetic fields of up to 3 kG using circularly polarized σ+ and σ− radiation. The process of selective reflection from a 350-nm-thick nanocell has been used, which makes it possible to form narrow atomic lines and observe separately the behavior of individual transitions. Two groups consisting of six (85Rb atoms) and four (87Rb atoms) transitions are formed in magnetic fields B > 0.5 kG upon σ+ and σ− laser excitation. All transitions have been identified. It is shown that the strongest transitions for 87Rb and 85Rb atoms in magnetic fields of up to several kG are formed under σ− irradiation. A further increase in the magnetic field makes it possible to attain the Paschen–Back regime on a hyperfine structure, for which the probabilities of transitions upon σ+ and σ− excitation become identical. The theoretical model and experiment are in good agreement.
The evolution of the hyperfine structure magnetic sublevel populations of 87 Rb and 133 Cs atoms in resonant linearly polarized laser fields is theoretically analyzed. Analytical expressions are obtained for stationary populations of the magnetic sublevels of the ground state 5 2 S 1/2 F g = 1, F f = 2 in 87 Rb and 6 2 S 1/2 F g = 3, F f = 4 in 133 Cs under optical pumping at the F g ↔ F e = F g and F f ↔ F e = F f transitions of the D 2 lines ( n 2 S 1/2 n 2 P 3/2 ) versus the initial populations. The characteristic population stabilization time are obtained by numerical solution for the density matrix equations, depending on the intensity of the laser fields and the detuning of the optical resonance. It is shown that the alternation of the polarization direction of the laser field at the transition F f ↔ F e = F f almost completely transfers atoms to the lower sublevel of the "clock" M1 transition F g M = 0 ↔ F f M = 0 in rubidium and cesium frequency standards, which proportionally increases the signal of the recording system.