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Physical Review A

Cancellation of D1 line transitions of alkali-metal atoms by magnetic-field values

Artur Aleksanyan, Rodolphe Momier, E. A. Gazazyan, A. Papoyan, Claude Leroy

Published Apr 15, 2022• 2 citations
Publisher

Abstract

In this work, $\ensuremath{\pi}, {\ensuremath{\sigma}}^{+}$, and ${\ensuremath{\sigma}}^{\ensuremath{-}}$ transitions between magnetic sublevels of the ${D}_{1}$ 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.

DOI
10.1103/physreva.105.042810

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In this work, \ensuremathπ,\ensuremathσ+\ensuremath{\pi}, {\ensuremath{\sigma}}^{+}\ensuremathπ,\ensuremathσ+, and \ensuremathσ\ensuremath−{\ensuremath{\sigma}}^{\ensuremath{-}}\ensuremathσ\ensuremath− transitions between magnetic sublevels of the D1{D}_{1}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 III, the magnetic quantum number mmm, and the magnetic-field magnitude BBB are defined. Transition cancellations are observed only for some \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ transition modified transfer coefficients in order to find the magnetic-field values which correspond to the intensities' maximum. The accuracy of the magnetic-field BBB values is only limited by the uncertainty of the physical quantities involved.
All magnetic field values cancelling D1D_1D1​ line transitions of alkali metal atoms
In this work, πππ, σ+σ^+σ+ and σ−σ^-σ− transitions between magnetic sublevels of the D1D_1D1​ 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 III, the magnetic quantum number mmm and the magnetic field magnitude BBB are defined. Transition cancellations are observed only for some πππ 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 πππ transition "modified" transfer coefficients in order to find the magnetic field values which correspond to the intensities maximum. The accuracy of the magnetic field BBB values is only limited by the uncertainty of the involved physical quantities.
All alkali atoms D1D_1D1​ line magnetic field values cancelling transitions
In this work alkali atoms hyperfine structure D1D_1D1​ line all π\piπ, σ+\sigma^+σ+ and σ−\sigma^-σ− transitions between magnetic sublevels are considered analytically. General 2×22 \times 22×2 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 III, magnetic quantum number mmm and magnetic field magnitude BBB are defined. Transition cancellations are observed only for π\piπ transitions of each isotope. All the magnetic field values, which cancel transitions are calculated with an accuracy limited by the involved quantities uncertainties.
Cancellation of D2D_2D2​ line transitions of alkali-metal atoms by magnetic-field values
In a previous work the πππ transitions of the D1D_1D1​ line of alkali-metal atoms were shown to cancel at magnetic-field values given by a single closed-form expression. The D2D_2D2​ line has until now resisted the same treatment, because the fixed-mmm Hamiltonian blocks of the 2P3/2^2P_{3/2}2P3/2​ manifold reach the dimension 4×44\times44×4, 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=±1ΔF=\pm1ΔF=±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 D2D_2D2​ transition as an explicit function of the excited-state eigenvalue, the nuclear spin III, the ground-state hyperfine splitting and the magnetic quantum number. A necessary condition on mmm and on the polarization is derived, which replaces the selection rule known for the D1D_1D1​ line. All the magnetic-field values canceling πππ, σ+σ^+σ+ and σ−σ^-σ− transitions of 23^{23}23Na, 39^{39}39K, 40^{40}40K, 41^{41}41K, 85^{85}85Rb, 87^{87}87Rb and 133^{133}133Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the D1D_1D1​ line, the σ±σ^{\pm}σ± transitions of the D2D_2D2​ line do cancel, and they account for the majority of the values found. The accuracy of the calculated BBB values is limited only by the uncertainties of the excited-state hyperfine constants.
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.
When atoms are placed in a static magnetic field, they undergo shifts of their energy levels and changes in their transition probabilities. These two effects must be taken into account when considering absorption profiles of alkali-metal D lines, which result from the contribution of many transitions influenced by the laser spectrum and Doppler broadening. The model presented here gives the D-line absorption coefficients of alkali-metal vapors in the presence of an arbitrary static magneitc field. They are evaluated considering various laser polarizations. Experimental measurements of D-line absorption profiles for 85Rb^{85}\mathrm{Rb}85Rb, 87Rb^{87}\mathrm{Rb}87Rb, and 133Cs^{133}\mathrm{Cs}133Cs isotopes show excellent agreement with theoretical predictions.
Atomic transitions in alkali metals that have zero probability in the absence of a magnetic field but have large probabilities in the presence of a magnetic field are called magnetically induced (MI). They are of interest because of their large probabilities, which exceed the probabilities of usual transitions in a wide magnetic field range. Magnetically induced transitions are classified as type-1 (MI1) and type-2 (MI2) and their total number is about 100. In this work, MI2 transitions are examined between ground Fg and excited levels Fe of the hyperfine structure satisfying the condition Fe−Fg=ΔF=±2{{F}_{{\text{e}}}} - {{F}_{{\text{g}}}} = \Delta F = \pm 2Fe​−Fg​=ΔF=±2 , which are forbidden in zero magnetic field but have large probabilities in the presence of a magnetic field. The probabilities of the MI2 transitions with ΔF=+2\Delta F = + 2ΔF=+2 and the MI transitions with ΔF=−2\Delta F = - 2ΔF=−2 are maximal in the case of optical radiation with the σ+ and σ– polarizations, respectively. This difference is called type-1 magnetically induced circular dichroism (MICD1). It has been shown for the first time that the probability of the strongest MI2 transition in the 85Rb atom corresponding to the D2 line in magnetic fields >100 G in the case of σ+{{\sigma }^{ + }}σ+ radiation is larger than the probability of the strongest MI2 transition in the case of σ– radiation by a factor of 2.5. This difference is called type-2 magnetically induced circular dichroism (MICD2). It has been shown how to determine the strongest MI transition for any alkali metal atom, which is important for its application in magneto-optical processes. Theoretical curves reproduce well experimental results.