Role of transverse magnetic fields in electromagnetically induced absorption for elliptically polarized light

Using the closed $F_g\to F_e=F_g+1$ $D2$ line transition of Rb atoms, we report experimental and numerical evidence of influence of additional transverse magnetic field when examining electromagnetically induced absorption (EIA) in the Hanle configuration. The effect was analyzed with two directions of additional magnetic field and in the cases of excitation with linear, elliptical, and circular laser light polarization. The transverse magnetic field brings substantial differences in the resonance line shape, amplitude, and width. Our theoretical model includes Doppler broadening. Numerical solutions of our theoretical model are in good agreement with experimental results. This analysis points out the importance of the presence of stray magnetic fields when studying EIA.

Figure 1

Level diagrams for the $F_g=2\to F_e=3$ transition of ${}^{87}\text{R}\text{b}$ (up) and $F_g=3\to F_e=4$ transition of ${}^{85}\text{R}\text{b}$ (down). Lines connecting Zeeman sublevels stand for both optical pumping (by ${\sigma }^{\pm }$ and $\pi$ light) and spontaneous emission.

Figure 2

Two directions of ${\vec{B}}_{\text{trans}}$ which were taken into consideration: (a) ${\vec{B}}_{\text{trans}}^x$, along the $x$ axis; (b) ${\vec{B}}_{\text{trans}}^y$, along the $y$ axis. The laser light propagates along the $z$ axis. Note that laser electric field $\vec{E}$ for linear polarization and major axis of polarization ellipse are always directed along the $x$ axis, but we change direction of ${\vec{B}}_{\text{trans}}$.

Figure 3

Experimental setup. External cavity diode laser (ECDL), optical insulator (OI), Doppler-free dichroic atomic vapor laser lock (DDAVLL), variable neutral density filter (VNDF), polarizer $\left(P\right)$, detector $\left(D\right)$.

Figure 4

Mutual orientations of ${\vec{B}}_{\text{trans}}$ and $\vec{E}$ considered in our analysis: (a) and (b) linear polarization; (c) and (d) elliptical polarization $\chi =10°$.

Figure 5

Calculated (a) and measured (b) Hanle EIA resonances for linearly polarized laser light, when ${\vec{B}}_{\text{trans}}\parallel \vec{E}$ for ${}^{87}\text{R}\text{b}$. The laser intensity is $I=0.6\text{mW}/{\text{cm}}^2$; solid curves, $B_{\text{trans}}^x=0$; dashed curves, $B_{\text{trans}}^x=200\text{mG}$; and dotted curves, $B_{\text{trans}}^x=460\text{mG}$.

Figure 6

Calculated (curves) and measured (points) amplitudes [(a) and (c)] and widths [(b) and (d)], of the EIA for ${}^{87}\text{R}\text{b}$ [(a) and (b)] and for ${}^{85}\text{R}\text{b}$ [(c) and (d)]. ${\vec{B}}_{\text{trans}}\parallel \vec{E}$. The laser intensities were $I=0.6\text{mW}/{\text{cm}}^2$ [(a) and (b)] and $I=1\text{mW}/{\text{cm}}^2$ [(c) and (d)]. Theoretical results for amplitudes were scaled from amplitudes in Fig. 5 by the constant number.

Figure 7

Calculated (a) and measured (b) Hanle EIA resonances for linearly polarized laser light, for ${}^{87}\text{R}\text{b}$. ${\vec{B}}_{\text{trans}}\perp \vec{E}$. Solid curves, $B_{\text{trans}}^y=0$; dashed curves, $B_{\text{trans}}^y=50\text{mG}$; dotted curves, $B_{\text{trans}}^y=200\text{mG}$; and dashed-dotted curves, $B_{\text{trans}}^y=300\text{mG}$. The laser intensity is $I=0.6\text{mW}/{\text{cm}}^2$.

Figure 8

Calculated (curves) and measured (points) amplitudes (a) and widths (b), of the EIA for ${}^{87}\text{R}\text{b}$. The laser polarization is linear, ${\vec{B}}_{\text{trans}}\perp \vec{E}$, and intensity is $I=0.6\text{mW}/{\text{cm}}^2$. Theoretical results for amplitudes were scaled from amplitudes in Fig. 7 by the constant number.

Figure 9

Calculated populations of ground-state sublevels for ${}^{87}\text{R}\text{b}$. Laser light polarization is linear and intensity is $I=0.6\text{mW}/{\text{cm}}^2$; population of (a) $m_g=0$, (b) sum of $m_g=1$ and $m_g=-1$, (c) sum of $m_g=2$ and $m_g=-2$; solid curves, $B_{\text{trans}}=0\text{mG}$; dashed curves, $B_{\text{trans}}^x=200\text{mG}$; and dotted curves, $B_{\text{trans}}^y=200\text{mG}$.

Figure 10

Calculated (a) and measured (b) Hanle EIA resonances for elliptically polarized laser light, $\chi =10°$, when ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_x$ (i.e., parallel to the major axis of polarization ellipse) for ${}^{85}\text{R}\text{b}$. The laser intensity is $I=0.6\text{mW}/{\text{cm}}^2$; solid curves, $B_{\text{trans}}^x=0$; dashed curves, $B_{\text{trans}}^x=50\text{mG}$; and dotted curves, $B_{\text{trans}}^x=100\text{mG}$.

Figure 11

Calculated (curves) and measured (points) amplitudes (a) and widths (b), of the EIA for ${}^{85}\text{R}\text{b}$. The laser ellipticity $\chi =10°$, ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_x$, and intensity is $I=0.6\text{mW}/{\text{cm}}^2$. Theoretical results for amplitudes were scaled from amplitudes in Fig. 10 by the constant number.

Figure 12

Calculated (a) and measured (b) Hanle EIA resonances for elliptically polarized laser light, for ${}^{85}\text{R}\text{b}$. ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_y$ is parallel to the minor axis of the polarization ellipse. The laser ellipticity is $\chi =10°$, and the intensity is $I=0.6\text{mW}/{\text{cm}}^2$. Solid curves, $B_{\text{trans}}^y=0$; dashed curves, $B_{\text{trans}}^y=50\text{mG}$; dotted curves, $B_{\text{trans}}^y=100\text{mG}$; and dashed-dotted curves, $B_{\text{trans}}^y=120\text{mG}$.

Figure 13

Calculated (curves) and measured (points) amplitudes (a) and widths (b) of the EIA for ${}^{85}\text{R}\text{b}$. The laser polarization is elliptical $\chi =10°$, ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_y$, and intensity is $I=0.6\text{mW}/{\text{cm}}^2$. Theoretical results for amplitudes were scaled from amplitudes in Fig. 12 by the constant number.

Figure 14

Mutual orientations of ${\vec{B}}_{\text{trans}}$ and $\vec{E}$ considered in our analysis: (a) elliptical polarization $\chi =35°$ and (b) circular polarization.

Figure 15

Calculated [(a) and (c)] and measured [(b) and (d)] Hanle transmission data for elliptically polarized laser light, $\chi =35°$. Laser intensity is $I=0.6\text{mW}/{\text{cm}}^2$. Results are for ${}^{85}\text{R}\text{b}$ and ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_x$ [(a) and (b)], and for ${}^{87}\text{R}\text{b}$ and ${\vec{B}}_{\text{trans}}\parallel {\vec{e}}_y$ [(c) and (d)]. Solid curves, $B_{\text{trans}}^x=0\text{mG}$; dashed curves, $B_{\text{trans}}^x=100\text{mG}$.

Figure 16

Calculated populations of ground sublevels for ${}^{87}\text{R}\text{b}$. $B_{\text{trans}}^y=100\text{mG}$ and intensity is $I=0.6\text{mW}/{\text{cm}}^2$; (a) $\chi =0°$, (b) $\chi =10°$, (c) $\chi =35°$; solid curves, population of $m_g=0$; dashed curves, sum of $m_g=1$ and $m_g=-1$; and dotted curves, sum of $m_g=2$ and $m_g=-2$.

Figure 17

Calculated (a) and measured (b) Hanle transmission data for circularly polarized laser light, for ${}^{87}\text{R}\text{b}$ and for the laser intensity of $I=0.6\text{mW}/{\text{cm}}^2$; solid curves, $B_{\text{trans}}=0\text{mG}$; dashed curves, $B_{\text{trans}}=30\text{mG}$; and dotted curves, $B_{\text{trans}}=100\text{mG}$.