Measurement of the occupation number of metastable atoms in the hyperfine-substate β
3 in an atomic hydrogen beam
After its discovery in 1947 by Willis Eugene Lamb and Robert C. Retherford the Lamb shift was used to create Lamb shift polarimeter to separate the
2S1/2 α1 and
α2 hyperfine substates of hydrogen as well as the
α3 substate of deuterium. But for a new project at the Technical University of Munich, investigation of the bound-beta decay of a neutron into a hydrogen atom and a neutrino, a Lamb shift polarimeter is needed that is also capable of separating the
β3 substate of hydrogen. Unfortunately, our first attempt to use a Sona transition unit to exchange the occupation numbers between
α1 and
β3 failed, because of the unexpected complexity of the transitions in this unit [1]. The second idea of using a new kind of spinfilter which uses two radio frequencies to separate all four hyperfine substates of hydrogen also failed. Our third attempt is now to build a transition unit that can induce magnetic dipole transitions between
α2 and
β3 as well as between
α2 and
β4 (
π transitions, i.e. an exchange of the occupation numbers of these states). This is a similar transition to what is used in atomic beam sources, in this case not for ground state atoms but for metastable atoms, which requires a much lower radio frequency. Another difference of this new idea is the smaller interaction time of the atoms with the photons inside the transition unit due to their much higher velocity of roughly
2⋅105 m/s compared to velocities of about
103 m/s in an atomic beam source.
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 and
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 and
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 atoms when they collide with polarized electrons, and the change in the density matrix of hydrogen atoms when they collide with hydrogen atoms.
Ultracold gases of fermionic alkaline-earth(-like) atoms are hopeful candidates for the quantum simulation of many-body physics induced by magnetic impurities (e.g., the Kondo physics), because there are spin-exchange interactions (SEIs) between two atoms in the electronic ground (
1S0) and metastable (
3P) state, respectively. Nevertheless, this SEI cannot be tuned via magnetic Feshbach resonance. In this paper, we propose three methods to control the SEI between one atom in the
1S0 state and another atom in the
3P2 states or $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ dressed states, with one or two laser beams. These methods are based on the spin-dependent ac-Stark shifts of the
3P2 states or the $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ Raman coupling. We show that due to the structure of alkaline-earth (like) atoms, the heating effects induced by the laser beams of our methods are very weak. For instance, for ultracold Yb atoms, ac-Stark-shift difference of variant spin states of the
3P2(F=3/2) level, or the strength of the $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ Raman coupling, could be of the order of $(2\ensuremath{\pi})\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}$, while the heating rate (photon scattering rate) is only of the order of Hz. As a result, the Feshbach resonances, with which one can efficiently control the SEI by changing the laser intensity, may be induced by the laser beams with a low-enough heating rate, even if the scattering lengths of the bare interatomic interaction are so small that they are comparable with the length scale associated with the van der Waals interaction.