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Physical Review A

Effect of Spin-Exchange Collisions on Density Matrices Which Include Nuclear Spin

R. H. Lambert

Published Jun 1, 1970• 5 citations
Publisher

Abstract

The equation of motion for the electronic polarization of a ground-state alkali atom undergoing spin-exchange collisions is derived. It is shown that the equation of motion for the electronic polarization is unaffected by the inclusion of nuclear spin. A simple interpretation of the spin-exchange frequency shift is also given.
DOI
10.1103/physreva.1.1841

Related papers

Nuclear-Spin Effects Are Important in Analyzing Spin-Exchange Experiments

Hyatt M. Gibbs

Lambert's proof that the inclusion of nuclear spin does not alter the equation of motion for the electronic polarization is shown to be valid only when the hyperfine interaction is negligible. Usually the hyperfine frequency is much greater than the frequency of spin-exchange collisions; then the effect of the hyperfine interaction between collisions invalidates Lambert's proof. In general, spin exchange with a second depolarized species relaxes the electronic polarization as the sum of two exponentials, as shown by Gibbs and Grosset\^ete.

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Atomic spin-exchange collisions in magnetic fields

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Spin-exchange collisions among alkali-metal atoms affect the coherence time and gyromagnetic ratio of atoms by redistributing the ground-state population. We theoretically study the interactions between magnetic fields and atoms in the presence of spin-exchange collisions. Using the modified Bloch equations derived from the density-matrix equation in a low spin-polarization limit, we obtain explicit formulas describing the role of spin-exchange collisions in the interactions between atoms and magnetic fields, especially the nonresonant radio-frequency fields, which are widely used to manipulate atoms. The results derived from these equations agree well with density-matrix simulations and previous experimental results, and provide a simple and precise way to quantitatively analyze the effect of spin-exchange collisions on atoms. Our paper is helpful to gain physical insight into the spin-exchange collisions and optimize the performance of atom-based precision measurement instruments.

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Interaction between spin polarized alkali atoms

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Collisions of alkali-metal atoms in the ground state with the electron spin S = 1/2 are accompanied by exchange of electron coordinates between colliding particles, which leads to polarization transfer between them (i.e., to the well-known phenomenon of spin exchange). In addition, along with the polarization transfer from one partner to another, the magnetic resonance lines of colliding atoms are broadened and shifted in spin-exchange collisions. In particular the last two processes depend on the complex spin-exchange cross section. The real part of the cross section determines the so-called "spin-exchange cross section," which is responsible for the broadening of magnetic resonance lines, while the imaginary part determines the shift cross section, which governs the frequency shift of magnetic resonance. In order to describe the spin-exchange process one have to know complex spin-exchange cross sections. Complex spin-exchange cross sections are calculated on the basis of the data on the singlet (X 1 Σ + ) and triplet (a 3 Σ + ) potentials describing the interaction alkali-metal atoms in the ground state. Collisions of alkali-metal atoms Cs, Rb, K, Na in the ground state are considered in the energy interval of 10 −4 −10 −2 au.

Journal of Physics Conference Series•Dec 1, 2018•0 citations

Frequency Shifts in Spin-Exchange Optical Pumping Experiments

L. C. Balling, R. J. Hanson, F. M. Pipkin

A theory is derived to predict the density matrices describing two atoms after a spin-exchange collision from the density matrices before the collision and the scattering amplitude for binary collisions. This theory is then applied to a system of quasifree electrons interacting with optically pumped rubidium atoms. For calculating the change in the electron density matrix, the ${\mathrm{Rb}}^{85}$ and ${\mathrm{Rb}}^{87}$ are replaced by a fictitious rubidium isotope with no nuclear spin. Expressions are derived for the change in the light transmission when a radiofrequency field is present at the electron frequency. In addition to the linewidth for the electron signal, the spin-exchange theory predicts a frequency shift the magnitude of which depends upon the two-body scattering amplitude, the rubidium polarization and the rubidium density. Experiments performed to test various aspects of the theory are then reported. The measurements were made on a system of quasifree electrons interacting with rubidium atoms. Measurements of the electron linewidth as a function of temperature indicated that spin-exchange collisions dominated the electron relaxation. The predicted frequency shift was measured by observing the electron resonance frequency first with left circularly polarized light and then with right circularly polarized light. For electrons colliding with rubidium atoms the shift is negative when the rubidium polarization is positive and the ratio of the shift to the linewidth is $\frac{\ensuremath{\delta}{\ensuremath{\nu}}_{0}}{\ensuremath{\Delta}\ensuremath{\nu}}=\ensuremath{-}0.025\ifmmode\pm\else\textpm\fi{}0.005$. From the measured values of the shift and the linewidth, a value is derived for the electron-rubidium spin-flip cross section. In one Appendix the replacement of the ${\mathrm{Rb}}^{85}$ and ${\mathrm{Rb}}^{87}$ by the fictitious rubidium isotope with no nuclear spin is justified. In a second Appendix the calculations for the simple electron-rubidium system are generalized and applied to more complicated systems. Results are reported for the change in the density matrix of hydrogen atoms when they collide with polarized electrons, the change in the density matrix of ${\mathrm{Rb}}^{87}$ atoms when they collide with polarized electrons, and the change in the density matrix of hydrogen atoms when they collide with hydrogen atoms.

Physical Review•Feb 3, 1964•131 citations

Collision-induced spin exchange of alkali-metal atoms with3He

Timur V. Tscherbul, P Zhang, H. R. Sadeghpour +1 more

Collisions of atoms and molecules in external electromagnetic fields can depolarize electronic and nuclear spins, leading to decoherence of quantum superposition states. The fundamental microscopic processes giving rise to spin depolarization are spin exchange and spin relaxation. In this work, we present an analysis of spin exchange transitions in collision of alkali metal atoms with 3He. We use accurate ab initio calculations to evaluate the electron spin density of Li-He and the interaction energy of Rb-He. The Fermi contact interaction constants for the complexes of Na, K, and Rb atoms with 3He are extracted from the measured frequency shift enhancement factors using the ab initio interaction potentials. The spin density calculations also yield the hyperfine pressure shift of Li-He.

Journal of Physics Conference Series•Nov 1, 2009•0 citations
Lambert's proof that the inclusion of nuclear spin does not alter the equation of motion for the electronic polarization is shown to be valid only when the hyperfine interaction is negligible. Usually the hyperfine frequency is much greater than the frequency of spin-exchange collisions; then the effect of the hyperfine interaction between collisions invalidates Lambert's proof. In general, spin exchange with a second depolarized species relaxes the electronic polarization as the sum of two exponentials, as shown by Gibbs and Grosset\^ete.
Spin-exchange collisions among alkali-metal atoms affect the coherence time and gyromagnetic ratio of atoms by redistributing the ground-state population. We theoretically study the interactions between magnetic fields and atoms in the presence of spin-exchange collisions. Using the modified Bloch equations derived from the density-matrix equation in a low spin-polarization limit, we obtain explicit formulas describing the role of spin-exchange collisions in the interactions between atoms and magnetic fields, especially the nonresonant radio-frequency fields, which are widely used to manipulate atoms. The results derived from these equations agree well with density-matrix simulations and previous experimental results, and provide a simple and precise way to quantitatively analyze the effect of spin-exchange collisions on atoms. Our paper is helpful to gain physical insight into the spin-exchange collisions and optimize the performance of atom-based precision measurement instruments.
Collisions of alkali-metal atoms in the ground state with the electron spin S = 1/2 are accompanied by exchange of electron coordinates between colliding particles, which leads to polarization transfer between them (i.e., to the well-known phenomenon of spin exchange). In addition, along with the polarization transfer from one partner to another, the magnetic resonance lines of colliding atoms are broadened and shifted in spin-exchange collisions. In particular the last two processes depend on the complex spin-exchange cross section. The real part of the cross section determines the so-called "spin-exchange cross section," which is responsible for the broadening of magnetic resonance lines, while the imaginary part determines the shift cross section, which governs the frequency shift of magnetic resonance. In order to describe the spin-exchange process one have to know complex spin-exchange cross sections. Complex spin-exchange cross sections are calculated on the basis of the data on the singlet (X 1 Σ + ) and triplet (a 3 Σ + ) potentials describing the interaction alkali-metal atoms in the ground state. Collisions of alkali-metal atoms Cs, Rb, K, Na in the ground state are considered in the energy interval of 10 −4 −10 −2 au.
A theory is derived to predict the density matrices describing two atoms after a spin-exchange collision from the density matrices before the collision and the scattering amplitude for binary collisions. This theory is then applied to a system of quasifree electrons interacting with optically pumped rubidium atoms. For calculating the change in the electron density matrix, the Rb85{\mathrm{Rb}}^{85}Rb85 and Rb87{\mathrm{Rb}}^{87}Rb87 are replaced by a fictitious rubidium isotope with no nuclear spin. Expressions are derived for the change in the light transmission when a radiofrequency field is present at the electron frequency. In addition to the linewidth for the electron signal, the spin-exchange theory predicts a frequency shift the magnitude of which depends upon the two-body scattering amplitude, the rubidium polarization and the rubidium density. Experiments performed to test various aspects of the theory are then reported. The measurements were made on a system of quasifree electrons interacting with rubidium atoms. Measurements of the electron linewidth as a function of temperature indicated that spin-exchange collisions dominated the electron relaxation. The predicted frequency shift was measured by observing the electron resonance frequency first with left circularly polarized light and then with right circularly polarized light. For electrons colliding with rubidium atoms the shift is negative when the rubidium polarization is positive and the ratio of the shift to the linewidth is $\frac{\ensuremath{\delta}{\ensuremath{\nu}}_{0}}{\ensuremath{\Delta}\ensuremath{\nu}}=\ensuremath{-}0.025\ifmmode\pm\else\textpm\fi{}0.005$. From the measured values of the shift and the linewidth, a value is derived for the electron-rubidium spin-flip cross section. In one Appendix the replacement of the Rb85{\mathrm{Rb}}^{85}Rb85 and Rb87{\mathrm{Rb}}^{87}Rb87 by the fictitious rubidium isotope with no nuclear spin is justified. In a second Appendix the calculations for the simple electron-rubidium system are generalized and applied to more complicated systems. Results are reported for the change in the density matrix of hydrogen atoms when they collide with polarized electrons, the change in the density matrix of Rb87{\mathrm{Rb}}^{87}Rb87 atoms when they collide with polarized electrons, and the change in the density matrix of hydrogen atoms when they collide with hydrogen atoms.
Collisions of atoms and molecules in external electromagnetic fields can depolarize electronic and nuclear spins, leading to decoherence of quantum superposition states. The fundamental microscopic processes giving rise to spin depolarization are spin exchange and spin relaxation. In this work, we present an analysis of spin exchange transitions in collision of alkali metal atoms with 3He. We use accurate ab initio calculations to evaluate the electron spin density of Li-He and the interaction energy of Rb-He. The Fermi contact interaction constants for the complexes of Na, K, and Rb atoms with 3He are extracted from the measured frequency shift enhancement factors using the ab initio interaction potentials. The spin density calculations also yield the hyperfine pressure shift of Li-He.