Effects of temperature and ground-state coherence decay on enhancement and amplification in a $\Delta$ atomic system

We study phase-sensitive amplification of electromagnetically induced transparency in a warm ${}^{85}\mathrm{Rb}$ vapor wherein a microwave driving field couples the two lower-energy states of a $\Lambda$ energy-level system thereby transforming into a $\Delta$ system. Our theoretical description includes effects of ground-state coherence decay and temperature effects. In particular, we demonstrate that driving-field-enhanced electromagnetically induced transparency is robust against significant loss of coherence between ground states. We also show that for specific field intensities, a threshold rate of ground-state coherence decay exists at every temperature. This threshold separates the probe-transmittance behavior into two regimes: probe amplification vs probe attenuation. Thus, electromagnetically induced transparency plus amplification is possible at any temperature in a $\Delta$ system.

Figure 1

Level scheme showing $\Delta$ atomic level configuration in ${}^{85}\mathrm{Rb}$ formed by two optical fields and a microwave field. The coupling field at frequency ${\omega }_{\mathrm{c}}$, with strength ${\Omega }_{\mathrm{c}}$ (thick red), connects the $5S_{1/2},F=3\left(\right|2\rangle )\leftrightarrow 5P_{3/2},F=3^{\prime }\left(\right|3\rangle )$ transition. The probe field at frequency ${\omega }_{\mathrm{p}}$ (thin red), with strength ${\Omega }_{\mathrm{p}}$, connects the $5S_{1/2},F=2\left(\right|1\rangle )\leftrightarrow 5P_{3/2},F=3^{\prime }\left(\right|3\rangle )$ transition. The microwave drive field at frequency ${\omega }_{\mu \mathrm{w}}$, with strength ${\Omega }_{\mu \mathrm{w}}$ connects levels $5S_{1/2}F=2\left(\right|1\rangle )\leftrightarrow 5S_{1/2},F=3\left(\right|2\rangle )$. The detunings of the coupling and probe fields from $|3\rangle$ are denoted by ${\delta }_{\mathrm{c}}$ and ${\delta }_{\mathrm{p}}$ respectively. The decay rate ${\gamma }_{ij}$ from level $|j\rangle$ to level $|i\rangle$ assumes values ${\gamma }_{12}=0.001$ MHz and ${\gamma }_{23}=5$ MHz.

Figure 2

Probe absorption $\mathrm{Im}\left({\rho }_{13}\right)$ vs probe detuning ${\delta }_{\mathrm{p}}$ for a $\Delta$ system in ${}^{85}\mathrm{Rb}$ gas with ${\Omega }_{\mathrm{c}}=6.4$ MHz, ${\Omega }_{\mathrm{p}}=1.0$ MHz, and ${\Omega }_{\mu \mathrm{w}}=0.8$ MHz. (a) Probe absorption with no ground-state coherence decay (${\gamma }_{\mathrm{c}}=0$ MHz) at $T=0$ K is given. The various values of $\varphi$ are indicated in the legend. (b) Probe absorption with finite ground-state coherence decay values and at finite temperatures given for a fixed value of $\varphi =\pi /2$. The various cases are: $T=333$ K and ${\gamma }_{\mathrm{c}}=1.0$ MHz (solid black line), $T=333$ K and ${\gamma }_{\mathrm{c}}=0.1$ MHz (dot-dashed red line), and $T=233$ K and ${\gamma }_{\mathrm{c}}=1.0$ MHz (dashed blue line).

Figure 3

Contour plot showing the change from probe-field input intensity to transmitted intensity for temperatures $T=0$ K to $T=400$ K and for ${\gamma }_{\mathrm{c}}$ ranging from $0.001$ MHz to $3.000$ MHz. The Rabi frequencies are fixed at ${\Omega }_{\mathrm{c}}=6.4$ MHz, ${\Omega }_{\mathrm{p}}=1.0$ MHz, and ${\Omega }_{\mu \mathrm{w}}=0.8$ MHz. “Reference Line” denotes a reference contour along which the probe-field transmitted intensity is identical whether the microwave drive is on or off. The symbols A, B, and C refer to regions of amplification, enhancement and absorption respectively.