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Physical Review

Excited-State Mixing in the Optical Pumping of Alkali-Metal Vapors

F. A. Franz, Judy R. Franz

Published Aug 5, 1966• 61 citations
Publisher

Abstract

The effect of excited-state mixing on the optical pumping of alkali-metal vapors is considered for the case where the energy separation between the $^{2}P_{\frac{1}{2}}$ and $^{2}P_{\frac{3}{2}}$ first excited states is relatively large, allowing the isolation of mixing effects within a particular $J$ level. Two models for the mixing process are proposed, one a random reorientation of the total electronic angular momentum J, and the other a reorientation of J subject to the selection rule $\ensuremath{\Delta}{m}_{J}=0, \ifmmode\pm\else\textpm\fi{}2$. The probabilities for mixing transitions among $J$-state sublevels have been calculated for an alkali atom of nuclear spin $\frac{3}{2}$, and can be used to calculate the optical-pumping transition probabilities for a vapor subject to any degree of excited-state mixing. Several possible experiments suggested by these calculations are discussed.

DOI
10.1103/physrev.148.82

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Cross sections for random collisional reorientation of J within the sublevels of the $6^{2}P_{\frac{3}{2}}$ excited state of cesium have been measured using the ${D}_{2}$ optical-pumping technique. The measured cross sections ${{\ensuremath{\sigma}}_{M}}^{2}$ are: Cs-He (3.2\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$, Cs-Ne (2.9\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$, Cs-Ar (6.0\ifmmode\pm\else\textpm\fi{}1.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$, Cs-Kr (9.2\ifmmode\pm\else\textpm\fi{}1.9)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$, Cs-Xe (11.3\ifmmode\pm\else\textpm\fi{}2.3)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ ${\mathrm{cm}}^{2}$. Simplified equations for ${\mathrm{D}}_{1}$ and ${\mathrm{D}}_{2}$ pumping with zero, partial, and complete excited-state mixing are derived for a hypothetical alkali atom with zero nuclear spin. The calculations are extended to cesium atoms with nuclear spin $\frac{7}{2}$ pumped by ${\ensuremath{\sigma}}^{+}$ ${\mathrm{D}}_{2}$ radiation. The equilibrium value ${〈{S}_{z}〉}_{0}$ and the time dependence $〈{S}_{z}(t)〉$ of the electronic spin polarization are computed for various degrees of excited-state mixing. $〈{S}_{z}(t)〉$ is found to be nearly single-exponential in the zero- and the complete-mixing cases, but shows a distinct double-exponential behavior in the intermediate region. ${〈{S}_{z}〉}_{0}$ is shown to be dependent on the degree of mixing, and to change sign as the degree of mixing increases. Measurements were performed with a new transmission technique by passing a separate ${\ensuremath{\sigma}}^{+}$-polarized ${\mathrm{D}}_{2}$ pumping beam and an alternately ${\ensuremath{\sigma}}^{+}$- and ${\ensuremath{\sigma}}^{\ensuremath{-}}$-polarized ${\mathrm{D}}_{1}$ testing beam through the cesium absorption cell. The cross sections for collisionally induced mixing were evaluated by measuring the buffer pressure at which ${〈{S}_{z}〉}_{0}$ passed through zero.

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•Jan 1, 1963•0 citations
The effect of excited-state mixing on the optical pumping of alkali-metal vapors is considered for the case where the energy separation between the 2P12^{2}P_{\frac{1}{2}}2P21​​ and 2P32^{2}P_{\frac{3}{2}}2P23​​ first excited states is relatively large, allowing the isolation of mixing effects within a particular JJJ level. Two models for the mixing process are proposed, one a random reorientation of the total electronic angular momentum J, and the other a reorientation of J subject to the selection rule \ensuremathΔmJ=0,\ifmmode±\else\textpm\fi2\ensuremath{\Delta}{m}_{J}=0, \ifmmode\pm\else\textpm\fi{}2\ensuremathΔmJ​=0,\ifmmode±\else\textpm\fi2. The probabilities for mixing transitions among JJJ-state sublevels have been calculated for an alkali atom of nuclear spin 32\frac{3}{2}23​, and can be used to calculate the optical-pumping transition probabilities for a vapor subject to any degree of excited-state mixing. Several possible experiments suggested by these calculations are discussed.
The distribution of population in the ground-state sublevels of an optically pumped alkali-metal vapor has been found to be strongly dependent on the mechanism assumed responsible for relaxation. Two modes of optical pumping zero mixing and complete mixing in the excited state, and three modes of alkali relaxation, uniform, Zeeman, and electron randomization, are considered. Calculations of the sublevel relative populations are made in terms of experimentally measurable parameters for all six combinations of pumping and relaxation modes. Optical-pumping transient signals are then used to demonstrate that alkali relaxation due to wall collisions is of the low-frequency "Zeeman" variety, while relaxation due to alkali-rare-gas collisions takes place through randomization of the spin of the alkali valence electron. Equations for the rate of change of the electronic spin polarization are derived for all cases, and are found to be nonexponential, in contrast with the case of an alkali of zero nuclear spin.
Rate equations for the optical pumping of alkali-metal vapors simultaneously subject to pumping, relaxation in the excited state, and relaxation in the ground state are shown to yield analytic solutions for the alkali-metal electronic and nuclear-spin polarizations in the limit of weak pumping. Nuclear-spin and hyperfine-interaction effects are properly included. Analysis of the double-exponential form of the pumping transient of the ground-state electronic-spin polarization is shown to yield new methods for the measurement of cross sections for collisional relaxation in both the ground and excited states. Formulas are derived that relate relaxation cross sections to experimentally measurable parameters. Formulas describing the influence of nuclear spin on signals obtained from standard depolarization of resonance-radiation experiments also are provided. Ambiguities in earlier optical-pumping experiments and calculations arising from oversimplified descriptions of relaxation processes are discussed.
Cross sections for random collisional reorientation of J within the sublevels of the 62P326^{2}P_{\frac{3}{2}}62P23​​ excited state of cesium have been measured using the D2{D}_{2}D2​ optical-pumping technique. The measured cross sections ${{\ensuremath{\sigma}}_{M}}^{2}$ are: Cs-He (3.2\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ cm2{\mathrm{cm}}^{2}cm2, Cs-Ne (2.9\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ cm2{\mathrm{cm}}^{2}cm2, Cs-Ar (6.0\ifmmode\pm\else\textpm\fi{}1.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ cm2{\mathrm{cm}}^{2}cm2, Cs-Kr (9.2\ifmmode\pm\else\textpm\fi{}1.9)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ cm2{\mathrm{cm}}^{2}cm2, Cs-Xe (11.3\ifmmode\pm\else\textpm\fi{}2.3)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$ cm2{\mathrm{cm}}^{2}cm2. Simplified equations for D1{\mathrm{D}}_{1}D1​ and D2{\mathrm{D}}_{2}D2​ pumping with zero, partial, and complete excited-state mixing are derived for a hypothetical alkali atom with zero nuclear spin. The calculations are extended to cesium atoms with nuclear spin 72\frac{7}{2}27​ pumped by ${\ensuremath{\sigma}}^{+}$ D2{\mathrm{D}}_{2}D2​ radiation. The equilibrium value 〈Sz〉0{〈{S}_{z}〉}_{0}〈Sz​〉0​ and the time dependence 〈Sz(t)〉〈{S}_{z}(t)〉〈Sz​(t)〉 of the electronic spin polarization are computed for various degrees of excited-state mixing. 〈Sz(t)〉〈{S}_{z}(t)〉〈Sz​(t)〉 is found to be nearly single-exponential in the zero- and the complete-mixing cases, but shows a distinct double-exponential behavior in the intermediate region. 〈Sz〉0{〈{S}_{z}〉}_{0}〈Sz​〉0​ is shown to be dependent on the degree of mixing, and to change sign as the degree of mixing increases. Measurements were performed with a new transmission technique by passing a separate ${\ensuremath{\sigma}}^{+}$-polarized D2{\mathrm{D}}_{2}D2​ pumping beam and an alternately ${\ensuremath{\sigma}}^{+}$- and ${\ensuremath{\sigma}}^{\ensuremath{-}}$-polarized D1{\mathrm{D}}_{1}D1​ testing beam through the cesium absorption cell. The cross sections for collisionally induced mixing were evaluated by measuring the buffer pressure at which 〈Sz〉0{〈{S}_{z}〉}_{0}〈Sz​〉0​ passed through zero.
Relaxation constants T1{T}_{1}T1​ and T2{T}_{2}T2​ have been computed for experiments involving optical pumping and optical detection in alkali metal vapor. The calculations have been performed for several possible spin relaxation mechanisms; namely, magnetic dipole, electric quadrupole, spin exchange, and the optical pumping process itself. For all of these mechanisms a reorientation experiment will approximately predict a spin resonance line width (equivalent to the statement ${T}_{1}\ensuremath{\approx}{T}_{2}$ for spin-\textonehalf{} particles.) However, a spin reorientation experiment of the type originally performed by Dehmelt, employing circularly polarized light, gives approximate results because of the nonexponential character of the reorientation. A more suitable experiment is one employing hyperfine population differences and unpolarized light.
We present a comprehensive theory of nuclear spin polarization of 3He{}^{3}\mathrm{He}3He and 129Xe{}^{129}\mathrm{Xe}129Xe gases by spin-exchange collisions with optically pumped alkali-metal vapors. The most important physical processes considered are (1) spin-conserving spin-exchange collisions between like or unlike alkali-metal atoms; (2) spin-destroying collisions of the alkali-metal atoms with each other and with buffer-gas atoms; (3) electron-nuclear spin-exchange collisions between alkali-metal atoms and 3He{}^{3}\mathrm{He}3He or 129Xe{}^{129}\mathrm{Xe}129Xe atoms; (4) spin interactions in van der Waals molecules consisting of a Xe atom bound to an alkali-metal atom; (5) optical pumping by laser photons; (6) spatial diffusion. The static magnetic field is assumed to be small enough that the nuclear spin of the alkali-metal atom is well coupled to the electron spin and the total spin is very nearly a good quantum number. Conditions appropriate for the production of large quantities of spin-polarized 3He{}^{3}\mathrm{He}3He or 129Xe{}^{129}\mathrm{Xe}129Xe gas are assumed, namely, atmospheres of gas pressure and nearly complete quenching of the optically excited alkali-metal atoms by collisions with N2{\mathrm{N}}_{2}N2​ or H2{\mathrm{H}}_{2}H2​ gas. Some of the more important results of this work are as follows: (1) Most of the pumping and relaxation processes are sudden with respect to the nuclear polarization. Consequently, the steady-state population distribution of alkali-metal atoms is well described by a spin temperature, whether the rate of spin-exchange collisions between alkali-metal atoms is large or small compared to the optical pumping rate or the collisional spin-relaxation rates. (2) The population distributions that characterize the response to sudden changes in the intensity of the pumping light are not described by a spin temperature, except in the limit of very rapid spin exchange. (3) Expressions given for the radio-frequency (rf) resonance linewidths and areas can be used to make reliable estimates of the local spin polarization of the alkali-metal atoms. (4) Diffusion effects for these high-pressure conditions are mainly limited to thin layers at the cell surface and at internal resonant surfaces generated by radio-frequency magnetic fields when the static magnetic field has substantial spatial inhomogeneities. The highly localized effects of diffusion at these surfaces are described with closed-form analytic functions instead of the spatial eigenmode expansions that are appropriate for lower-pressure cells.
The effects of collisions on the thermal mixing of atomic spin levels in excited states of alkali-metal atoms are studied. The separation in energy is sufficiently small, in the case of the resonance doublet in these atoms, that the probability for energy transfer by shocks of the second kind is appreciable at the temperatures required in order that a vapor pressure exist that is high enough for a signal to be detected. Polarized and unpolarized emission is considered, and the case of Na is considered numerically. (T.F.H.)