Coherent control of refractive index in far-detuned $\Lambda$ systems

Enhancement and control of the index of refraction in a mixture of two three-level atomic species that form a pair of far-detuned $\Lambda$ schemes under two-photon resonance has been studied. We employ the density-matrix approach to properly take population relaxation into account and to describe the interaction of each $\Lambda$ system with the electromagnetic fields. Both $\Lambda$ systems are driven by a corresponding far-detuned coherent field at one atomic transition and are probed by the same weak field. In the dressed-state basis, it represents a superposition of effective two-level subsystems with the positions, widths, and amplitudes of the resonances controlled by the driving fields and allows for efficient control of the susceptibility of the total system; leading to refractive index (RI) enhancement with vanishing absorption in the absence of amplification. We analyze the experimental implementation of such a system in a cell of Rb atoms with a natural abundance of isotopes. An upper limit estimate of the RI enhancement is obtained.

Figure 1

Mixture of two three-level $\Lambda$ systems. The initially populated level is indicated by the dots.

Figure 2

An equivalent representation of a mixture of two three-level subsystems driven by coherent off-resonant fields.

Figure 3

Combined real (solid) and imaginary (dashed) parts of the susceptibility from two three-level systems where the $x$ axis is normalized to ${\gamma }_{ab}^{\mathrm{s}1}={\gamma }_{ab}^{\mathrm{s}2}={\gamma }_{ab}$ and the $y$ axis is normalized to $\eta =3N_{\mathrm{s}1}{\lambda }_{\mathrm{pr}}^3{\gamma }_{\mathrm{r}}^{\mathrm{s}1}/\left(8{\pi }^2\right)=3N_{\mathrm{s}2}{\lambda }_{\mathrm{pr}}^3{\gamma }_{\mathrm{r}}^{\mathrm{s}2}/\left(8{\pi }^2\right)=1$ with ${\Omega }_{\mathrm{s}1}={\Omega }_{\mathrm{s}2}=2{\gamma }_{ab}$, ${\Delta }_{\mathrm{s}1}={\Delta }_{\mathrm{s}2}=20{\gamma }_{ab}$, and ${\gamma }_{cb}^{\mathrm{s}1}={\gamma }_{cb}^{\mathrm{s}2}=0.016{\gamma }_{ab}$. Resonances have equal strength and width but are shifted by FWHM. The obtained maximum at zero absorption is $0.5\eta$.

Figure 4

Combined real (solid) and imaginary (dashed) parts of the susceptibility from two three-level systems. With ${\Omega }_{\mathrm{s}1}=4{\gamma }_{ab}$ and ${\Omega }_{\mathrm{s}2}=1.5{\gamma }_{ab}$ and all other parameters, the same as in Fig. 3. Amplification is weaker and is narrower than absorption. The relative shift is adjusted to get zero absorption and no gain. The obtained maximum at zero absorption is $0.33\eta$.

Figure 5

The peak real part of the susceptibility for the $D_2$ line of ${}^{85}\mathrm{Rb}$ as a function of absolute temperature.

Figure 6

The ratio of hyperfine-to-optical decoherence rates for the $D_2$ line of ${}^{85}\mathrm{Rb}$ plotted as a function of temperature for the case where there is no buffer gas (solid), with a neon buffer gas at 10 Torr (dashed line), and at 300 Torr (dot-dashed line).

Figure 7

The maximum real part of the two-level susceptibility for the Rb $D_2$ line plotted as a function of the buffer-gas pressure. Plotted at a temperature of 550 K.

Figure 8

The real (solid) and imaginary (dashed) parts of the susceptibility as a function of the detuning plotted for the scheme without gain and with ${\Omega }_1=2\pi \times 150\mathrm{MHz}$ described in the text.

Figure 9

The real (solid) and imaginary (dashed) parts of the susceptibility as a function of the detuning plotted for the scheme with ${\Omega }_1={\Omega }_2=2\pi \times 1.5\mathrm{GHz}$ as described in the text.

Figure 10

The expected RI enhancement for the experiment described in Ref. [7] plotted as a function of two-photon detuning, using the experimental numbers as reported in the text.