1. Introduction Optical pumping and the Zeeman Effect The Hamiltonian of an atom with a single electron outside filled shells (as for rubidium) in a magnetic field is HH = HH 0 + ηηii JJ μμ JJ BB JJ μμ II BB II, where II is the spin of the nucleus and JJ is the total spin of all the electrons in the atom. We have JJ = 1, since JJ ssheeeeee = 0 for filled shells, and LL tttttttttt = 0 (figure 1). The term ηηii JJ expresses the interaction between the electron and nuclear magnetic momenta and is called the hyperfine interaction because it splits degenerate energy levels, as does the fine structure interaction. This form resembles the spin-orbit interaction LL SS, but now both operators are acting in abstract spin spaces. It splits levels with JJ = ± 1 and we take it as given, being equal to 3036 MMMMMM and 6835 MMMMMM, for Rb-85 and Rb-87 D 1 lines, respectively (figure ). The contribution of this term is easiest to calculate from II JJ = 1 (FF(FF + 1) II(II + 1) JJ(JJ + 1)) where FF = II + JJ is the total angular momentum. Fig.1: Grotrian diagram for Rb-85 and the D-doublet fine structure (similar to the famous Na D-doublet), with the DD 1 line at 795 nnnn PP1 PP3 SS1, and the DD line at 780 nnnn SS1. We will be using the DD 1 line only and, unlike in the atomic fluorescence experiment, zoom in much more on the line structure. Terms μμ JJ BB JJ, μμ II BB II describe the interaction with the external magnetic field. The μμ II II BB term can be neglected (even though the angular momenta II JJ) because μμ II μμ JJ = μμ BB. The shifts with magnetic field of the energy levels are then due to HH iiiiii = μμ JJ BB JJ only. Page 1
The shifts of the Zeeman Effect in the small field limit can be obtained with 1 st order perturbation theory. The dependence is linear for small fields as in our case, with ΔEE = μμ BB gg FF mmmm, where mm = FF, FF, the Lande gyromagnetic factors are gg FF = gg JJ FF(FF+1) II(II+1)+JJ(JJ+1) FF(FF+1) with gg JJ = for the D 1 line, and μμ BB is the Bohr magneton (see Appendix for full calculation). JJ = 1, II = 5, LL = 0 JJ = 1, II = 3, LL = 0 FF = 3 gg FF = 1 3 FF = gg FF = 1 3036 MMMMMM 6835 MMMMMM gg FF = 1 FF = gg FF = 1 3 FF = 1 Fig.: DD 1 line hyperfine structure of Rb-85 (left) and Rb-87 (right) in no magnetic field shows how the lines in figure 1 are split by the hyperfine interaction. The lines are further split into FF + 1 levels when a magnetic field is applied.. Experimental setup The maximum energy shift is 35 MMMMMM for a 50 GG maximum applied field and μμ BB = 1.4 MMMMMM/GG. Even at the highest field we apply, we only slightly move the energies compared to the hyperfine splitting. The Zeeman shift is only 0.001 ccmm 1 in usual spectroscopy units, too small to measure with grating spectrometers of typical resolution ~0.1 1 ccmm 1. A different, high-resolution spectroscopy, method must be applied..1. Optical pumping A cell containing Rb atomic vapor (with a natural isotope mixture of 7 % Rb-85 and 8 % Rb- 87) is placed between the Rb lamp light source and a detector (Fig. 3). The hot water sleeve keeps the cell at a temperature > 40 0 CC (Rb evaporation point), to maintain the Rb atoms in the gas phase. Exercise: Rubidium is used because it has only two very strong and sharp resonances in the infrared. Estimate the Rb resonance cross section and compare to the sum rule σσ(ωω)dddd = ππ ee mmmm 5 mm = 1.7 10. ss Question: why do we go through the trouble of heating with hot water, when heating tapes are available? 0 Page
Rb lamp Coil pair for DC and BB(tt) Nulling coils Hot water circulator Rb cell RF coils Photodiode Fig.3: the experimental setup. Polarizing optics is not shown. Optical pumping introduces an inversion between the occupancy of the atomic energy levels, by applying the restrictions of selection rules for transition that occur when the light beam passes through the cell. It is possible to interpret the optical pumping as a classical macro-polarization PP = ii,aaaaaaaaaa from adding the individual atomic polarizations. Optical pumping polarizes the atomic vapor and creates a macroscopic magnetic moment pointing along the horizontal axis... Zeeman Effect Several magnetic fields are applied: A nulling field BB NN along the vertical axis. Since a few GG are important, it is necessary to remove the background contribution of Earth s magnetic field. This is done with a small vertical coil pair that compensates the vertical background component. There is no background transverse (or horizontal across the optical axis) field if the setup is oriented with its optical axis along the horizontal component of the background field. The remaining background longitudinal field (along the optical axis) is compensated with the Zeeman DC field coils (next). A DC and time-dependent field BB ZZ,0 + BB ZZ (tt) oriented along the optical axis. This field splits the state energies with the Zeeman Effect. This requires a large pair, capable of 50 G uniform field across the cell s relatively large volume that sets the DC bias field BB ZZ,0, and a smaller variable field, modulated at a frequency ff 0. In addition, an RF oscillation along the cell axis at frequency ff RRRR is applied to the sample with a coil pair and a function generator. The oscillation can be written as two counter propagating linearly-polarized waves, each of which is a sum of RCP and LCP waves. pp ii Page 3
This moment (or alternatively, inverted microscopic distribution) is changed by the RF oscillating magnetic field at a fixed DC bias field. As the total field applied to the cell varies, we are moving sideways on the energy level diagram. When the separation between levels is equal to the applied RF frequency, the optical pumping inversion is disrupted; the cell absorbs more strongly, and transmission changes are detected by the photodiode. This allows inferring the separation between atomic energy levels with a resolution much higher than possible with a spectrometer because the RF oscillation is induced by a current, the frequency of which can be accurately determined. 3. Measurements To observe a resonance, the RF frequency ff RRRR is kept constant and the horizontal variable field BB zz (tt) is applied Connect ff 0 to the oscilloscope reference and the PD to its input Observe the dip in the intensity on the oscilloscope as the resonance condition is met 4. Conclusion Measurements confirm the quantum calculations. The Rb HF structure is applied in atomic clocks on GPS satellites. The Cs HF structure is applied in the even more-precise fountain atomic clock, where the atoms fall back through a region of RF fields. Locking the electronics microwave frequency to the more stable atomic microwave frequency gives a precise time reference. It is also possible to start a precession of the total moment of the cell with changes of applied BB in time (for instance, by turning off the RF field). These transient effects can be detected in the variations of the PD intensity. 5. Appendix Since parts of the Hamiltonian do not commute, no set of eigenkets (of II or JJ ) will diagonalize to total HH. We can choose any basis set we like, obtain the representation of HH in this basis set and then diagonalize the matrix to obtain the exact solution to the energy levels. When choosing the FF, mm FF basis set (eigenkets of FF = II + JJ and its projection along the zz axis), we have to calculate FF, mm FF ηηii JJ μμ JJ BB JJ FF, mm FF. Although the calculation looks similar, we are not doing PT and our matrix elements are not 1 st order energy shifts. To make the matrix small take the simpler case II = 1, JJ = 1 FF = 0,1. The basis set is FF, mm FF = { 1,1, 1,0, 1, 1, 0,0 }. Page 4
The result is ΔEE = ηη 4 μμμμ 0 0 ηη 4 ηη 0 0 0 μμμμ (exercise: check this). The evolution of the 0 0 + μμμμ 0 0 μμμμ 4 0 3ηη 4 eigenvalues with applied magnetic field is shown in Fig. 4. This is the simplest case. Our Rb cell has a mixture of Rb-85 and Rb-87 isotopes with II = 5 and II = 3, respectively. The size of the matrix (II + 1) (JJ + 1) or 8 and 1, respectively, is the number of states and is unwieldy in these cases. Figure shows the solution for II = 5, JJ = 1 obtained with the Breit Rabi equation 1 EE FF = II ± 1, mm FF = ± Δww 4mmmm 1 + + II+1 xx where Δww = ηη, xx = gg JJμμ BB BB Δww positive gg JJ in the B-R equation (verified in class)]. [the plus sign and Exercise: obtain and plot the result for the 8 energies for II = 3, JJ = 1 nnnnmm FF nnnnmm FF almost good Is there an exactly0good basis Set for small but finite fields? Analog of nnnnmm jj for the Z Effect? nnmm IImm JJ mm II = 5, mm JJ = 1 mm II = 3, mm JJ = 1 mm II = 1, mm JJ = 1 mm II = 1, mm JJ = 1 mm II = 1, mm JJ = 1 mm II = 1, mm JJ = 1 FF = 3, mm FF = 3,,1,0, 1,, 3 mm II = 3, mm JJ = 1 FF = 1, mm FF = 1,0, 1 mm II = 5, mm JJ = 1 FF = 0, mm FF = 0 mm II = 1, mm JJ = 1 FF =, mm FF =,1,0, 1, mm II = 5, mm JJ = 1 mm II = 3, mm JJ = 1 At low fields, the magnetic interaction is a perturbation on top of the hyperfine splitting. FF and mm FF are almost good quantum numbers It is remarkable that, in this case, there is a closed0from solution over the entire range of the magnetic field, including the middle range, where PT does not apply: this solution is called the Breit Rabi equation mm II = 1, mm JJ = 1 At high fields, the hyperfine splitting is a perturbation of the levels split by the magnetic field. mm II and mm JJ are good quantum numbers JJ = 1 mm JJ = ± 1 and II = 1 mm II = ± 1 mm II = 1, mm JJ = 1 mm II = 1, mm JJ = 1 mm II = 3, mm JJ = 1 mm II = 5, mm JJ = 1 Fig.4: Left panel: eigenvalues for an atom with II = 1, JJ = 1 in an applied external magnetic field. Right panel: the 1 energy levels for an atom with II = 5, JJ = 1 (Rb-85) in a magnetic field. The applied fields are relatively small (only the left-hand-side edge of the plots is measured), where the shifts are approximately linear. It is also possible to fit the curvature of results such as shown and obtain the hyperfine interaction constant ηη, even when relatively small magnetic fields are available. Page 5
Name Phys-60 Quantum Mechanics Laboratory Optical pumping and the Zeeman Effect lab report Dates of measurements: Page 6