Atomic four-level $N$ systems

We investigate the atomic four-level $N$ configuration both analytically and numerically, for various pump and probe intensities, with and without transfer of coherence (TOC) and Doppler broadening, and compare the results obtained to those of realistic atomic systems. We find that TOC affects the whole spectrum, in addition to producing an electromagnetically induced absorption (EIA) peak at line center. We show that the EIA peak splits as the pump intensity increases. These results are compared with those of realistic systems. When the pump is ${\sigma }^{+}$ polarized and the probe is $\pi$ polarized, the results are similar to those of the $N$ configuration. When the pump and probe polarizations are both linear with perpendicular polarizations, various $N$-like subsystems contribute to the spectrum. Consequently, the splitting of the EIA peak only occurs at very high pump intensities. We also discuss the influence of the probe on the pump absorption and refraction and find that both the pump and probe show EIA peaks when the pump intensity is low, and complementary behavior when the pump is intense. At both low and high pump intensity, the pump and probe dispersions are of opposite sign.

Figure 1

Energy level scheme for the four-level $N$ configuration atom interacting with two resonant pumps, $V_1$ and $V_2$ and a tunable probe, $V_p$.

Figure 2

Calculated probe absorption spectra, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$, with $V_1=0.816V_2$ and $\gamma ∕\Gamma =0.001$. In the left column [(a), (c), and (e)] TOC is included and in the right column [(b), (d), and (f)] TOC is not included. The dotted curve is the probe absorption in the absence of pump lasers. In (a) and (b), $V_2∕\Gamma =0.1$, in (c) and (d), $V_2∕\Gamma =0.5$, and in (e) and (f), $V_2∕\Gamma =2$.

Figure 3

Doppler broadened spectra for the same parameters as in Fig. 2. The Doppler width is $D∕\Gamma =10$.

Figure 4

$\Lambda$ system interacting with a resonant pump: (a) calculated probe absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (solid line), and refraction, $\text{Re}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (dashed line); (b) calculated pump absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_1}\right)\Gamma ∕V_1$ (solid line), and refraction, $\text{Re}\left({\rho }_{e_1{g}_1}\right)\Gamma ∕V_1$ (dashed line). Parameters are $V_1∕\Gamma =1$, $V_2=0$, $V_p∕\Gamma =0.1$, and $\gamma ∕\Gamma =0.001$. The spectra are plotted as a function of $\delta ∕\Gamma$.

Figure 5

$V$ system interacting with a resonant pump: (a) and (c), calculated probe absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (solid line), and refraction, $\text{Re}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (dashed line); (b) and (d), calculated pump absorption, $\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)\Gamma ∕V_2$ (solid line), and refraction, $\text{Re}\left({\rho }_{e_2{g}_2}\right)\Gamma ∕V_2$ (dashed line). Parameters for (a) and (b) are $V_1=0$, $V_2∕\Gamma =0.25$ and $V_p∕\Gamma =0.025$, and for (c) and (d), $V_1=0$, $V_2∕\Gamma =1$ and $V_p∕\Gamma =0.1$. $\gamma ∕\Gamma =0.001$ for all cases. For clarity, the refractions in (b) and (d) which are centered at zero, have been shifted upwards. In addition the refraction in (d) has been divided by 10.

Figure 6

$N$ system interacting with two resonant pumps: (a) calculated probe absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (solid line) and refraction, $\text{Re}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (dashed line); (b) calculated pump absorption, $\Gamma \left[\mathrm{Im}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (solid line) and refraction, $\Gamma \left[\text{Re}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\text{Re}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (dashed line); (c) real (solid line) and imaginary (dashed line) parts of $\left({\rho }_{e_1{g}_1}\right)\Gamma A^2∕V_1$, and (d) real (solid line) and imaginary (dashed line) parts of $\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)\Gamma ∕V_2$. Parameters are $V_2∕\Gamma =0.1$, $V_p∕\Gamma =0.01$, $A=0.816$, and $\gamma ∕\Gamma =0.001$. For clarity, the refractions in (b), (c), and (d), which are centered at zero, have been shifted upwards. In addition the refraction in (c) and (d) have been multiplied by 10.

Figure 7

$N$-system interacting with two resonant pumps: (a) calculated probe absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (solid line) and refraction, $\text{Re}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (dashed line); (b) calculated pump absorption, $\Gamma \left[\mathrm{Im}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (solid line) and refraction, $\Gamma \left[\text{Re}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\text{Re}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (dashed line); (c) real (solid line) and imaginary (dashed line) parts of $\left({\rho }_{e_1{g}_1}\right)\Gamma A^2∕V_1$, and (d) real (solid line) and imaginary (dashed line) parts of $\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)\Gamma ∕V_2$. Parameters are $V_2∕\Gamma =0.5$, $V_p∕\Gamma =0.05$, $A=0.816$ and $\gamma ∕\Gamma =0.001$. For clarity, the refractions in (b) and (d), which are centered at zero, have been shifted upwards.

Figure 8

$N$ system interacting with two resonant pumps: (a) calculated probe absorption, $\mathrm{Im}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$ (solid line) and refraction, $\text{Re}\left({\rho }_{e_1{g}_2}\right)\Gamma ∕V_p$; (b) calculated pump absorption, $\Gamma \left[\mathrm{Im}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (solid line) and refraction, $\Gamma \left[\text{Re}\left({\rho }_{e_1{g}_1}\right)A^2∕V_1+\text{Re}\left({\rho }_{e_2{g}_2}\right)∕V_2\right]$ (dashed line); (c) real (solid line) and imaginary (dashed line) parts of $\left({\rho }_{e_1{g}_1}\right)\Gamma A^2∕V_1$, and (d) real (solid line) and imaginary (dashed line) parts of $\mathrm{Im}\left({\rho }_{e_2{g}_2}\right)\Gamma ∕V_2$. Parameters are $V_2∕\Gamma =2$, $V_p∕\Gamma =0.2$, $A=0.816$, and $\gamma ∕\Gamma =0.001$. For clarity, the refractions in (b) and (d), which are centered at zero, have been shifted upwards.

Figure 9

Energy-level scheme for the $F_g=2\to F_e=3$ transition interacting with a $\sigma$ polarized resonant pump laser and a tunable $\pi$ polarized probe laser.

Figure 10

Realistic atomic system: probe absorption for the cycling $F_g=2\to F_e=3$ transition of the $D_2$ line in ${}^{87}\mathrm{Rb}$, interacting with a $\sigma$ polarized resonant pump laser and a tunable $\pi$ polarized probe laser, with (a) $\Omega ∕\Gamma =2$, and (b) $\Omega ∕\Gamma =15$. In (c), (e), and (g) and in (d), (f), and (h), we plot the individual contributions to the absorption of (a) and (b), respectively, marked the same way as in Fig. 9. The atomic density was taken to be ${10}^{12}\text{atoms}∕{\text{cm}}^3$ and $\gamma ∕\Gamma =0.001$.

Figure 11

Realistic atomic system: probe absorption for the cycling $F_g=2\to F_e=3$ transition, of the $D_2$ line in ${}^{87}\mathrm{Rb}$, interacting with a ${\sigma }^{+}$ polarized resonant pump laser and a tunable $\pi$ polarized probe laser, with (a) $\Omega ∕\Gamma =2$, and (b) $\Omega ∕\Gamma =15$. The atomic density was taken to be ${10}^{12}\text{atoms}∕{\text{cm}}^3$ and $\gamma ∕\Gamma =0.001$.

Figure 12

The numerator of the second term of Eq. (A9) as a function of increasing $V_2$, with (solid line) and without (dashed line) TOC. When the numerator is positive, EIA occurs.