Light-induced polarization effects in atoms with partially resolved hyperfine structure and applications to absorption, fluorescence, and nonlinear magneto-optical rotation

The creation and detection of atomic polarization is examined theoretically through the study of basic optical-pumping mechanisms and absorption and fluorescence measurements and the dependence of these processes on the size of ground- and excited-state hyperfine splittings is determined. The consequences of this dependence are studied in more detail for the case of nonlinear magneto-optical rotation in the Faraday geometry (an effect requiring the creation and detection of rank-two polarization in the ground state) with alkali-metal atoms. Analytic formulas for the optical rotation signal under various experimental conditions are presented.

Figure 1

Doppler-free (solid line) and Doppler-broadened (dashed line) absorption spectra for the ${}^{85}\text{R}\text{b}$ $D_2$ line. A Maxwellian velocity distribution at room temperature is assumed. If the incident light frequency is tuned near the center of the $F_g=2\to F_e$ transition group, the condition discussed in the text is fulfilled. Namely, the light detuning from each resonance frequency is either much less than or much greater than the Doppler width. The condition holds somewhat less rigorously for light tuned to the center of the $F_g=3\to F_e$ transition group.

Figure 2

(Color online) Excitation with $z$-polarized light on the (a) $F_g=1\to F_e=0$ and (b) $F_g=1\to F_e=1$ transitions of a totally resolved $J_g=1/2\to J_e=1/2$ transition with $I=1/2$. Alignment is produced in the $F_g=1$ hyperfine level in both cases. Relative atomic populations are indicated by the number of dots displayed above each ground-state level. Relative transition strengths are indicated by the widths of the arrows—here the transition strengths are all the same.

Figure 3

(Color online) As Fig. 2 but with excited-state hfs unresolved; light is resonant with the $F_g=1\to F_e$ transition group. No alignment is produced in the $F_g=1$ ground state.

Figure 4

(Color online) Excitation with $z$-polarized light on the $F_g=2\to F_e$ transition group of a $D_1$ transition with unresolved excited-state hfs. The nuclear spin is $I=3/2$. No alignment is produced in the $F_g=2$ hyperfine level. The width of each arrow represents the relative transition strength, which can be obtained from terms of the sum in Eq. (3).

Figure 5

(Color online) Level diagram for a $D_1$ transition with resolved hfs for an atom with $I=1/2$ showing (a) optical excitation and (b) spontaneous decay with linearly polarized light resonant with the $F_g=1\to F_e=1$ transition. The branching ratio for each allowed decay is the same, leading to an excess of atoms in the $|F_g=1,m=0⟩$ sublevels over the populations of the $|F_g=1,m=\pm 1⟩$ sublevels by a ratio of 2:1.

Figure 6

(Color online) As Fig. 5, but with light tuned to the $F_g=0\to F_e=1$ transition. In this case an excess of atoms results in the $m=\pm 1$ states so that the polarization has the opposite sign as that in Fig. 5.

Figure 7

(Color online) As Fig. 5 but with unresolved ground-state hfs; light is tuned to the $F_g\to F_e=1$ transition group. The contributions to the ground-state polarization illustrated in Figs. 5, 6 cancel so that no ground-state polarization is produced.

Figure 8

(Color online) As Fig. 5 but with unresolved excited-state hfs; light is tuned to the $F_g=1\to F_e$ transition group. Three decay channels transfer atoms to the $|F_g=1,m=0⟩$ sublevels, while the $|F_g=1,m=\pm 1⟩$ sublevels are each fed by two decay channels. Since all the branching ratios are the same, the resulting population imbalance is 3:2.

Figure 9

(Color online) $D_1$ transition for an atom with $I=1/2$ subject to linearly polarized light resonant with the $F_g=1\to F_e=1$ transition. In part (a) the $F_g=1$ ground state is unpolarized and there is light absorption. In part (b) the $F_g=1$ state has the same total population but is aligned, and there is no absorption.

Figure 10

(Color online) As Fig. 9, but with unresolved excited-state hfs. In this case there is no difference in the absorption seen for an (a) unpolarized and (b) aligned $F_g=1$ ground state.

Figure 11

The transit and wall optical-rotation effects. Both effects occur in an anti-relaxation-coated vapor cell and can be distinguished by their greatly different magnetic resonance widths.

Figure 12

(Color online) Illustration of the rotating polarizer model for optical rotation. (a) Optical pumping on a $F_g=1\to F_e=0$ or $F_g=1\to F_e=1$ transition causes the medium to act as a polarizing filter with transmission axis along the input light polarization $\hat{\mathbf{e}}$. (b) When the transmission axis rotates due to Larmor precession, the output light polarization ${\hat{\mathbf{e}}}^{\prime }$ follows the transmission axis and so rotates in the same sense as the Larmor precession. (c) If polarization produced by pumping on one transition is probed on the other, the polarization functions as a polarizing filter with transmission axis perpendicular to the input light polarization. (Attenuation of the light beam is not indicated.) (d) When the medium polarization rotates, the output light polarization tends to rotate toward the transmission axis, in the opposite sense to the Larmor precession in this case.

Figure 13

Dependence on light detuning from the $F_g=1\to F_e=1$ transition of (top) the components of ground-state alignment due to depopulation (dashed) and repopulation (solid) [Eq. (24)] and (bottom) optical rotation for a given amount of alignment [Eq. (26)]. Gray vertical lines show $F_g\to F_e$ transition resonance frequencies. Parameter values in units of $\Gamma$ are $\gamma ⪡1$, $A_g=10$, $A_e=5$.

Figure 14

Spectra of normalized optical rotation ${\ell }_0/\left({\kappa }_s{x}_1\right)\left(d\phi /d\ell \right)$ for the Doppler-free transit effect. Left column: components due to polarization produced by depopulation (dashed) and repopulation (solid); right column: total signal. Parameter values in units of $\Gamma$ are $\gamma ⪡1$, $A_g⪢1,A_e$.

Figure 15

Maximum of the spectrum of the Doppler-free nonlinear magneto-optical rotation transit effect as a function of excited-state hyperfine splitting. Plotted are the component due to polarization produced by depopulation (dash-dotted) due to polarization produced by repopulation (dashed) and the total signal (solid). Parameters as in Fig. 14.

Figure 16

As Fig. 14, with $A_g$ and $A_e$ varied simultaneously $\left(A_g=20A_e\right)$.

Figure 17

Maximum of the spectrum of the Doppler-broadened effect as a function of excited-state hyperfine splitting. Plotted are the component due to polarization produced by depopulation (dash-dotted), due to polarization produced by repopulation (dashed), and the total signal (solid). Equation (36) is used for $A_e<10$, and Eq. (34) is used for $A_e>10$. Parameter values in units of $\Gamma$ are ${\Gamma }_D=100$, $\gamma ⪡1$, $A_g⪢{\Gamma }_D$.

Figure 18

Spectra, as Fig. 14, but for the Doppler-broadened transit effect. Parameter values in units of $\Gamma$ are ${\Gamma }_D=100$, $\gamma ⪡1$, $A_g⪢{\Gamma }_D$.

Figure 19

As Fig. 18, with $A_g$ and $A_e$ varied simultaneously $\left(A_g=20A_e\right)$.

Figure 20

Spectra, as Fig. 18, but for the wall effect.

Figure 21

As Fig. 15, but for the wall effect. Parameters are the same as in Fig. 17.

Figure 22

As Fig. 19, but for the wall effect.

Figure 23

Maximum of the normalized optical rotation spectra ${\ell }_0/\left({\kappa }_s{x}_{F_g}\right)d\phi /d\ell$ for the Doppler-free transit, Doppler-broadened transit, and wall effects on the $D_1$ line for $I=1/2$, 3/2, and 5/2. We assume ${\Gamma }_D=100$, $\gamma ⪡1$, $A_g⪢{\Gamma }_D$ in units of $\Gamma$. The maxima for the $F_g=I\pm 1/2\to F_e$ transitions are plotted separately. Each plot shows rotation due to polarization produced by depopulation (dot-dashed line), rotation due to polarization produced by repopulation (dashed line), and total rotation (solid line). The sharp features seen in some of the plots occur when two terms contributing to the largest resonance cancel. In general, since all resonances are not canceled at the same time, the maximum of the spectrum does not go to zero.

Figure 24

Maximum of the normalized optical rotation spectra, as in Fig. 23, but for the $D_2$ line. For the $I=3/2$, $F_g=2$ and the $I=5/2$, $F_g=3$ systems for the Doppler-broadened transit effect, the two contributions to optical rotation nearly cancel, with the consequence that the approximations used in obtaining the analytic formulas for the total Doppler-broadened signal begin to break down. Numerical convolution is employed in these cases.

Figure 25

Maximum of the spectrum of normalized optical rotation for the (a) Doppler-broadened transit effect and (b) wall effect for various alkali-metal atoms. Circles indicate the $D_1$ line and triangles indicate the $D_2$ line. Room temperature Maxwellian velocity distributions are assumed. Normalized rotation is defined here as ${\ell }_0/\left({\kappa }_s{x}_{I+J_g}\right)\left(d\phi /d\ell \right)$, where ${\ell }_0$ in this case is the unsaturated absorption length at the detuning that gives maximum absorption. The normalized magnitude of unsuppressed optical rotation is nominally on the order of unity; however, this is to some degree dependent on the normalization convention chosen. For example, if the maximum matrix element of $d_z$ is used in the definition of ${\kappa }_s$ rather than the reduced matrix element, the values in this plot are increased by a factor of $\sim 6$.