Figure 1

(a) Three-level interaction scheme of three laser fields with ${}^{87}\mathrm{Rb}$ atoms: the long-lived coherence is created on hyperfine ground-state sublevels with strong driving field $E_d$ and weak probe (anti-Stokes) field $E_p$; the probe and Stokes fields $E_s$ are generated by electro-optic modulation. (b) Hanle configuration: the ground-state coherence is created on magnetic sublevels with two circularly polarized components $E_{\pm }$ of a monochromatic linearly polarized laser field. In both cases $\Delta$ is the one-photon detuning of the laser(s) from atomic resonance, and $\delta$ is the two-photon detuning due to frequency mismatch in case (a) or an external magnetic field in case (b).Reuse & Permissions

Figure 2

Schematic of the experimental setup.Reuse & Permissions

Figure 3

Transmission of the probe field as a function of two-photon detuning $\delta$ for various one-photon detunings $\Delta$. The values of $\Delta$ are shown on the upper right corner of each graph. These data are recorded in the presence of (columns, left to right) $100\mathrm{Torr}$ of $\mathrm{Ne}$, $30\mathrm{Torr}$ of $\mathrm{Ne}$, $0.12\mathrm{Torr}$ of $\mathrm{Kr}$, and no buffer gas. All signals are normalized to the nonabsorbed transmission. The asymmetry of the resonance curves for $\Delta =828$ and $1138\mathrm{MHz}$ in the cell with $0.12\mathrm{Torr}$ $\mathrm{Kr}$ and for $\Delta =517$ and $724\mathrm{MHz}$ in the vacuum cell is due to the slope of the one-photon absorption contour. The intensities of the laser fields are $I_d\simeq 1.20\mathrm{mW}∕{\mathrm{cm}}^2$ and $I_p\simeq 0.13\mathrm{mW}∕{\mathrm{cm}}^2$.Reuse & Permissions

Figure 4

Angle $\varphi$ of two-photon resonance for ${}^{87}\mathrm{Rb}$ cells with different amounts of a buffer gas.Reuse & Permissions

Figure 5

Amplitude $D$ of the two-photon resonance. For easier comparison, the values of $D$ are normalized to the resonance amplitude at zero detuning.Reuse & Permissions

Figure 6

Dressed-states picture for three-level $\Lambda$ scheme. State $\mid D_1⟩={\xi }_1[\mid a⟩+\left({\Omega }_d∕\Delta \right)\mid c⟩]$ has approximately the same energy as bare state $\mid a⟩$, and state $\mid D_2⟩={\xi }_2\left[\mid c⟩-\left({\Omega }_d^{*}∕\Delta \right)\mid a⟩\right]$ is close to the two-photon resonance. The normalization coefficients are ${\xi }_{1,2}\approx 1+o\left({\mid {\Omega }_d\mid }^2∕{\Delta }^2\right)$. EIT is observed for $\delta =0$ (maximum quantum interference), and absorption resonance corresponds to $\delta =-{\mid {\Omega }_d\mid }^2∕\Delta$ (probe is resonant with $\mid D_2⟩$).Reuse & Permissions

Figure 7

Three-level $\Lambda$ system.Reuse & Permissions

Figure 8

Coefficients $A$, $B$, and $C$ describing the line shape of the coherent resonance Eq. (1): extracted from the experimental data for ${}^{87}\mathrm{Rb}$ cell with $30\mathrm{Torr}$ (left column), calculated using Eqs. (18, 19, 20) (middle column), and obtained by numerical modeling (right column).Reuse & Permissions

Figure 9

Calculated probe field transmission spectra for the conditions corresponding to the experimental data in Fig. 3. We use the following parameters: ${\gamma }_{\mathrm{deph}}=450\mathrm{MHz},{\gamma }_{bc}=0.5\mathrm{kHz}$ for the cell with $100\mathrm{Torr}$ of $\mathrm{Ne}$; ${\gamma }_{\mathrm{deph}}=150\mathrm{MHz},{\gamma }_{bc}=0.7\mathrm{kHz}$ for the cell with $30\mathrm{Torr}$ of $\mathrm{Ne}$; and ${\gamma }_{\mathrm{deph}}=0,{\gamma }_{bc}=30\mathrm{kHz}$ for the cell without a buffer gas. These values are in good agreement with published results for collisional dephasing [47, 49]. For all graphs $ku=250\mathrm{MHz}$, ${\Omega }_d=2.5\mathrm{MHz}$, and ${\Omega }_p=0.5\mathrm{MHz}$. These values are reasonably close to the estimated actual Rabi frequencies ${\Omega }_d=2.1\mathrm{MHz}$ and ${\Omega }_p=0.7\mathrm{MHz}$. The small deviation between them can be attributed to the nonuniform spatial distribution of the laser intensities.Reuse & Permissions

Figure 10

Numerical calculation of the resonance strength $\left(D\right)$ vs one-photon detuning for a probe field propagating in a medium with $\mathrm{Ne}$ buffer gas and with vacuum. One-photon detuning and resonance width are given in units of ${\gamma }_r$.Reuse & Permissions

Figure 11

Numerical calculation of angle $\left(\varphi \right)$ vs one-photon detuning for a probe field propagating in a medium with $\mathrm{Ne}$ buffer gas and with vacuum. One-photon detuning and resonance width are given in units of ${\gamma }_r$.Reuse & Permissions

Figure 12

Two-photon resonance shift $\left({\delta }_0\right)$ as a function of one-photon detuning for the vacuum cell (squares), cell with $0.12\mathrm{Torr}$ of $\mathrm{Kr}$ (stars), and cell with $30\mathrm{Torr}$ of $\mathrm{Ne}$ (crosses). Inset: The result of the numerical simulation for the $30\mathrm{Torr}$ cell.Reuse & Permissions

Figure 13

Angle $\left(\varphi \right)$ and amplitude $\left(D\right)$ of the two-photon resonance for the generated Stokes field.Reuse & Permissions

Figure 14

Width of the two-photon resonance $\tilde{\gamma }$ as a function of one-photon detuning $\Delta$ for ${}^{87}\mathrm{Rb}$ cell with $30\mathrm{Torr}$ of $\mathrm{Ne}$ (cross), $0.12\mathrm{Torr}$ of $\mathrm{Kr}$ $(\times )$, and without buffer gas (squares, inset). The minimum width of the EIT resonance in the vacuum cell with no buffer gas is $\tilde{\gamma }(\Delta =0)=17\mathrm{kHz}$.Reuse & Permissions

Figure 15

Numerical calculation of resonance width $\left(2\tilde{\gamma }\right)$ vs one-photon detuning for a probe field propagating in a medium with $\mathrm{Ne}$ buffer gas and with vacuum (inset).Reuse & Permissions

Figure 16

Width of the two-photon resonance $\left(\gamma \right)$ for the probe and generated Stokes field.Reuse & Permissions

Figure 17

Left column: Transmission through ${}^{87}\mathrm{Rb}$ cells with (a) $1\mathrm{Torr}$; (b) $3\mathrm{Torr}$ and (c) $10\mathrm{Torr}$ of $\mathrm{Ne}$ buffer gas as the laser frequency is swept across the $F=2\to F^{\prime }$ transitions. Solid line: Transmission at zero magnetic field. In this case the ground-state coherence is unperturbed and we denote the transmission as ${\mathrm{T}}_{\mathrm{coh}}$. Dashed line: Transmission at large magnetic field $\left(\delta ⪢{\gamma }_0\right)$. In this case the coherence is destroyed, and we denote the transmission as ${\mathrm{T}}_{\mathrm{lin}}$. Right column: The peak contrast in transmission as a function of magnetic field (two-photon detuning) for fixed laser frequency (shown as a dotted line on the previous graphs) for the same cells. The normalization of the output signal is $C=({\mathrm{T}}_{\mathrm{coh}}-{\mathrm{T}}_{\mathrm{lin}})∕{\mathrm{T}}_{\mathrm{coh}}$. Thus $C>0$ is a manifestation of EIT shown in part (d), and $C<0$ indicates enhanced absorption as seen in (e) and (f). Because of the difference in geometrical size of the cells, the atomic density is adjusted for each case so that ${\mathrm{T}}_{\mathrm{lin}}\sim 20\%$. Laser power is $2\mathrm{mW}$.Reuse & Permissions