Figure 1

The interference of an absorptive resonance and an amplifying resonance can lead to an enhanced refractive index with vanishing absorption. (a) Shows the most straightforward way to achieve such an interference. Due to various difficulties, the scheme in (a) is not practical. (b) Shows an equivalent scheme using Raman transitions induced by two control lasers, $E_{c1}$ and $E_{c2}$. By changing the excitation configuration from the ground level, a Raman resonance can be made amplifying or absorptive for the probe beam $E_p$.Reuse & Permissions

Figure 2

The real part ${\chi }^{\prime }$ (solid line) and the imaginary part ${\chi }^{\prime \prime }$ (dashed line) of the susceptibility of the probe laser for Raman transitions separated by $\Delta =10\gamma$, $5\gamma$, and $\gamma$, respectively. At the midpoint between the resonances, there is destructive interference in the imaginary part of the susceptibility yielding vanishing absorption. At the same point, the real part of the susceptibility obtains a large value due to constructive interference, which simultaneously yields an enhanced refractive index.Reuse & Permissions

Figure 3

Simplified experimental schematic. The experiment is performed in a magnetically shielded, natural abundance Rb vapor cell with a length of $L=1$ mm. The three experimental beams, $E_p$, $E_{c1}$, and $E_{c2}$, are obtained by appropriate frequency shifting and amplifying the output of a single master external cavity diode laser. After the cell, we separate the probe laser beam with a high extinction polarizer and perform our measurements. Some experiments utilize optical pumping lasers, which propagate in a direction opposite to the experimental laser beams.Reuse & Permissions

Figure 4

Refractive index enhancement using two Raman transitions in two isotopes of Rb. We start this experiment by pumping both of the atomic species to the $F=2$ hyperfine level. This is achieved using two optical pump lasers locked to appropriate transitions. With the atoms pumped, we electromagnetically induce two Raman transitions, one in each isotope. A key issue with this scheme is the cross-coupling of the two optical pumping processes which results in a low value of the relative index.Reuse & Permissions

Figure 5

Refractive index enhancement in ${}^{85}$Rb. We optically pump to the $F=3$ level and induce two Raman transitions. The spacing between the $F=2$ and $F=3$ levels is 3.035 GHz. This configuration has significant advantages over that of Fig. 4.Reuse & Permissions

Figure 6

Normalized probe transmission through the cell as its frequency is scanned across the D${}_2$ line (one unit indicates 100$\%$ transmission). The cell temperatures are $T=96,115,144,$ and 163 ${}^{\circ }$C for plots (a) through (d), respectively. The vapor pressure model gives $N=4.4\times {10}^{12},1.5\times {10}^{13},7.1\times {10}^{13}$, and $1.8\times {10}^{14}$ cm${}^{-3}$ at these temperatures. The solid lines are numerical calculations without any adjustable parameters that use these density values and known dipole matrix elements, and Doppler- and pressure-broadened linewidths of the excited levels. The good agreement between the data and the calculation shows that the estimated values of the density are reasonably accurate.Reuse & Permissions

Figure 7

Normalized probe transmission through the cell with (solid circles) and without (solid diamonds) an optical pumping laser for cell temperatures of (a) $T=96$ ${}^{\circ }$C ($N=4.4\times {10}^{12}$ cm${}^{-3}$) and (b) $T=163$ ${}^{\circ }$C ($N=1.8\times {10}^{14}$ cm${}^{-3}$). The optical pumping laser pumps both isotopes to their lower energy hyperfine level resulting in stronger higher frequency features of the spectrum.Reuse & Permissions

Figure 8

The percentage of atoms pumped from $F=2$ to $F=1$ for ${}^{87}$Rb as the vapor density is increased. The optical pumping laser is locked to the $F=2\to F^{\prime }=3$ transition of the D${}_2$ line in ${}^{87}$Rb. We pump a large fraction of atoms for densities as high as $1.2\times {10}^{14}$ cm${}^{-3}$, above which the pumping efficiency drops sharply. We observe similar results for ${}^{85}$Rb, and also for pumping from lower hyperfine to upper hyperfine levels (not shown).Reuse & Permissions

Figure 9

(a) The normalized transmitted intensity $(I_{\mathrm{out}}/I_{\mathrm{in}})$ of the probe laser as its frequency is scanned across the gain resonance for a cell length of $L=1$ mm. We observe a peak gain of factor of 10. The probe ($E_p$) and control laser ($E_{c1}$) configuration is as shown in Fig. 5. (b) The transmitted intensity as the probe laser is scanned across the absorption resonance showing a peak absorption of a factor of 13. For this particular data, the $E_p$ and $E_{c2}$ configuration is slightly different than that shown in Fig. 5. (c) The inferred real (solid line) and imaginary (dashed line) parts of the relative index due to the Raman transitions only if the two transitions were combined while retaining their original strength.Reuse & Permissions

Figure 10

The peak intensity gain for the probe laser beam as the vapor temperature is varied. The highest gain is obtained for a temperature of $T\approx 155$ ${}^{\circ }$C, which corresponds to an atomic density of $N\approx 1.4\times {10}^{14}$ cm${}^{-3}$.Reuse & Permissions

Figure 11

The peak intensity gain for the probe laser beam as the single-photon detuning from the D${}_2$ line is varied. The highest gain is obtained for a detuning of about 7 GHz.Reuse & Permissions

Figure 12

The normalized transmitted probe intensity $(I_{\mathrm{out}}/I_{\mathrm{in}})$ as its frequency is scanned across the gain resonance for a control laser ($E_{c1}$) power of 18 mW (solid line), 39 mW (dashed line), 64 mW (crosses), 93 mW (pluses), 120 mW (open triangles), 143 mW (solid circles), and 165 mW (solid diamonds). As the control laser power is increased, the strength of the Raman transition increases up to a certain control power level and then saturates. The shift and the broadening of the Raman transition are also clearly observed. The shift is due to the AC stark shift of the hyperfine levels, which results from the high intensity of the control laser.Reuse & Permissions

Figure 13

(a) The peak probe gain that is observed as the control laser power is increased. The gain increases up to a control laser power value of about 100 mW and then saturates. (b) The Raman resonance shift vs the control laser power. The shift is a result of the difference in AC Stark shifts of the lower and upper hyperfine levels. The solid line is a numerical calculation without any adjustable parameters. We observe reasonable agreement between the calculation and the experimental results. The discrepancy is likely due to a slight misalignment of the probe laser beam from the center of the control lasers. (c) The full width at half maximum (FWHM) of the Raman transition as the control laser power is varied. The solid line is a numerical calculation without any adjustable parameters. The observed power broadening of the Raman transition is much larger than what the calculation indicates.Reuse & Permissions

Figure 14

The real (solid lines) and the imaginary (data points and dashed lines) parts of the refractive index when the two Raman resonances are combined. The dashed lines are fits to the data that assumes each Raman resonance to have a Lorentzian lineshape. The solid lines are the calculated real part of the refractive index based on these fits. In plots (a) and (b) the gain resonance occurs before the absorption resonance whereas in plot (c) the situation is reversed. We observe a change in refractive index of $\left|\Delta n\right|\approx 0.4\times {10}^{-4}$ with low absorption.Reuse & Permissions

Figure 15

The maximum relative index for a hot Rb atomic vapor. The index increases linearly up to a density of about 10${}^{15}$ cm${}^{-3}$ and then saturates at a value of $\Delta n\approx 0.4$.Reuse & Permissions

Figure 16

The probe transmission as its frequency is scanned across the D${}_2$ line for a cell length of $L=30$ $\mu$m and an atomic density of $2\times {10}^{14}$ cm${}^{-3}$. The two outer features are due to ${}^{87}$Rb, whereas the two inner features are due to ${}^{85}$Rb. The solid line is a calculation without any adjustable parameters.Reuse & Permissions