Slowing the probe field in the second window of double-double electromagnetically induced transparency

For Doppler-broadened media operating under double-double electromagnetically induced transparency (EIT) conditions, we devise a scheme to control and reduce the probe-field group velocity at the center of the second transparency window. We derive numerical and approximate analytical solutions for the width of EIT windows and for the group velocities of the probe field at the two distinct transparency windows, and we show that the group velocities of the probe field can be lowered by judiciously choosing the physical parameters of the system. Our modeling enables us to identify three signal-field strength regimes (with a signal-field strength always higher than the probe-field strength), quantified by the Rabi frequency, for slowing the probe field. These three regimes correspond to a weak signal field, with the probe-field group velocity and transparency-window width both smaller for the second window compared to the first window, a medium-strength signal field, with a probe-field group velocity smaller in the second window than in the first window but with larger transparency-window width for the second window, and the strong signal field, with both group velocity and transparency-window width larger for the second window. Our scheme exploits the fact that the second transparency window is sensitive to a temperature-controlled signal-field nonlinearity, whereas the first transparency window is insensitive to this nonlinearity.

Figure 1

Four-level tripod electronic structure with high-energy state $|4\rangle$ and lower-energy levels $|3\rangle$, $|2\rangle$, and $|1\rangle$ in order of decreasing energy. Transitions are driven by probe (p), coupling (c), and signal (s) fields with frequencies ${\omega }_{\text{x}}$ and detunings ${\delta }_{\text{x}}$ with x $\in \left\{\text{p,c,s}\right\}$. Dephasing rates are ${\gamma }_{\varphi i}$ for $i\in \{2,3\}$.

Figure 2

(a) Absorption and (b) dispersion as a function of the probe detuning ${\delta }_{\text{p}}$, with numerical (dotted line), analytical (solid line), and approximate linear equations (dashed line) for ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\Omega }_{\text{c}}={\gamma }_4$, ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\Omega }_{\text{p}}=0.05{\gamma }_4$, ${\delta }_{\text{s}}=9$ MHz, and ${\delta }_{\text{c}}=0$.

Figure 3

Plot of Lorentzian function (dashed line) and Gaussian function (dotted line) vs normalized atomic velocity.

Figure 4

Plots of $\text{Im}\left[{\chi }_{\text{p}}^{\left(1\right)}\right]$ and $\text{Re}\left[{\chi }_{\text{p}}^{\left(1\right)}\right]$ vs probe detuning ${\delta }_{\text{p}}$ at different temperatures for ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\Omega }_{\text{c}}={\gamma }_4$, ${\Omega }_{\text{s}}=0.35{\gamma }_4$, ${\delta }_s=9$ MHz, and ${\delta }_c=0$. (a), (c), and (e) are $\text{Im}\left[{\overline{\chi }}_{\text{p}}\right]$ and (b), (d), and (f) are $\text{Re}\left[{\overline{\chi }}_{\text{p}}\right]$. We set $T$= (1, 10, 100) K for (a), (b), (c), (d), and (e), (f), respectively, which is equivalent to $W_{\text{L}}=(34.8,110,348)$ MHz, respectively. The dotted line corresponds to the analytical solution using the Lorentzian line-shape function, whereas the dashed line is the numerical solution using the Maxwell-Boltzmann distribution function.

Figure 5

Numerical (dashed line) and analytical (dotted line) solutions of the HWHM ($\overline{\daleth }$) for the (a) first and (b) second EIT transparency windows vs Doppler width $W_{\text{L}}$ for ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\Omega }_{\text{c}}={\gamma }_4$, ${\Omega }_{\text{s}}=0.35{\gamma }_4$, ${\delta }_s=9$ MHz, and ${\delta }_c=0$. Inset: numerical (dashes) and analytical (dots) HWHM of the second EIT window evaluated for the gain term eliminated.

Figure 6

Plots of the numerical (dashed line) and analytical (dotted line) results for the partial derivative of dispersion with respect to Doppler width for (a) the first window and (b) the second window for ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\Omega }_{\text{c}}=1.5{\gamma }_4$, ${\Omega }_{\text{s}}=0.5{\gamma }_4$, ${\delta }_s=13.5$ MHz, and ${\delta }_c=0$. Inset: numerical (dashed line) and analytical (dotted line) results for the partial derivative of dispersion vs Doppler width at the second window for $I_2\equiv 0$.

Figure 7

Plot of the numerical (dotted line) and analytical (dashed line) results for the partial derivative of dispersion vs coupling field at the first window for Doppler width $W_{\text{L}}=409\text{MHz}$, ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz.

Figure 8

Plot of the partial derivative of dispersion (dotted-dashed line) and HWHM (dashed line) for the second EIT window and HWHM (upper horizontal dotted line) and partial derivative of dispersion (lower horizontal dotted line) for the first EIT window vs normalized signal-field Rabi frequency with ${\Omega }_{\text{c}}={\gamma }_4$, $W_{\text{L}}=700\text{MHz}$, ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, and ${\gamma }_2=40$ kHz.

Figure 9

Populations of states $|1\rangle$ and $|2\rangle$ represented by ${\rho }_{11}$ (dotted line) and ${\rho }_{22}$ (dashed line), respectively, and absorption of the probe field represented by $5\text{Im}\left[{\rho }_{14}\right]$ (solid line) as a function of time $t$ evaluated numerically by solving the master equation. (a) Coupling field is stronger than the probe field with ${\Omega }_{\text{c}}={\gamma }_4$ and ${\Omega }_{\text{p}}=0.3{\gamma }_4$. (b) Coupling and probe fields have the same strength with ${\Omega }_{\text{c}}={\Omega }_{\text{p}}={\gamma }_4$. The system is initially prepared with ${\rho }_{11}^{\left(0\right)}=1$ and ${\rho }_{22}^{\left(0\right)}={\rho }_{33}^{\left(0\right)}={\rho }_{44}^{\left(0\right)}=0$. The chosen parameters are ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\delta }_{\text{s}}=0.5{\Omega }_{\text{c}}$, ${\delta }_{\text{p}}={\delta }_{\text{c}}=0$.

Figure 10

Populations of levels $|1\rangle$ and $|3\rangle$ represented by ${\rho }_{11}$ (dotted line) and ${\rho }_{33}$ (dashed line), respectively, as a function of time $t$ evaluated numerically by solving the master equation. (a) Signal- and probe-field strengths are of equal magnitude less than coupling field, with ${\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.5{\gamma }_4$ while ${\Omega }_{\text{c}}={\gamma }_4$. (b) Coupling-, signal-, and probe-field strengths are of equal magnitude ${\Omega }_{\text{c}}={\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.35{\gamma }_4$. (c) Signal-field strength is stronger than the probe field, with ${\Omega }_{\text{c}}={\gamma }_4$, ${\Omega }_{\text{s}}=0.5{\gamma }_4$, and ${\Omega }_{\text{p}}=0.15{\gamma }_4$. Other parameters are ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\delta }_{\text{s}}={\delta }_{\text{p}}=0.5{\Omega }_{\text{c}}$, and ${\delta }_{\text{c}}=0$. Initial populations are ${\rho }_{11}^{\left(0\right)}=1$ and ${\rho }_{22}^{\left(0\right)}={\rho }_{33}^{\left(0\right)}={\rho }_{44}^{\left(0\right)}=0$.

Figure 11

Populations of levels $|1\rangle$, $|2\rangle$, and $|3\rangle$ represented by ${\rho }_{11}$ (dotted line), ${\rho }_{22}$ (dotted-dashed line), and ${\rho }_{33}$ (dashed line), respectively, as a function of time $t$ evaluated numerically by solving the master equation. The conditions are (a) ${\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.3{\gamma }_4$ and ${\Omega }_{\text{c}}={\gamma }_4$, (b) ${\Omega }_{\text{p}},{\Omega }_{\text{c}}\gg {\Omega }_{\text{s}}$ with ${\Omega }_{\text{c}}={\Omega }_{\text{p}}={\gamma }_4$ and ${\Omega }_{\text{s}}=0.3{\gamma }_4$, and (c) ${\Omega }_{\text{s}},{\Omega }_{\text{c}}\gg {\Omega }_{\text{p}}$ with ${\Omega }_{\text{c}}={\Omega }_{\text{s}}=1{\gamma }_4$ and ${\Omega }_{\text{p}}=0.3{\gamma }_4$. The initial population is ${\rho }_{11}^{\left(0\right)}=1$ and ${\rho }_{22}^{\left(0\right)}={\rho }_{33}^{\left(0\right)}={\rho }_{44}^{\left(0\right)}=0$. Other parameters are ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, ${\delta }_{\text{p}}={\delta }_{\text{s}}={\delta }_{\text{c}}=0$. Insets (a), (b), and (c) are the absorptions represented by $\text{Im}{\rho }_{14}$.

Figure 12

Numerically evaluated steady-state populations ${\rho }_{11}$ (dotted line), ${\rho }_{22}$ (dotted-dashed line), ${\rho }_{33}$ (dashed line), and ${\rho }_{44}$ (solid line) vs probe-field detuning ${\delta }_{\text{p}}$ evaluated (a), (b) at zero temperature and (c), (d) at 100 K. Parameter choices are (a) ${\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.3{\gamma }_4$. (b) ${\Omega }_{\text{s}}\gg {\Omega }_{\text{p}}$, ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\Omega }_{\text{p}}=0.01{\gamma }_4$, (c) ${\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.3{\gamma }_4$, and (d) ${\Omega }_{\text{s}}\gg {\Omega }_{\text{p}}$, ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\Omega }_{\text{p}}=0.01{\gamma }_4$. Other parameters are ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, and ${\Omega }_{\text{c}}={\gamma }_4$, ${\delta }_{\text{s}}=0.5{\Omega }_{\text{c}}$, ${\delta }_{\text{c}}=0$.

Figure 13

Steady-state populations ${\rho }_{11}$ (dotted line), ${\rho }_{22}$ (dotted-dashed line), ${\rho }_{33}$ (dashed line), and ${\rho }_{44}$ (solid line) vs probe-field detuning ${\delta }_{\text{p}}$ for ${\Omega }_{\text{s}}\gg {\Omega }_{\text{p}}$ in the presence of incoherent pumping with constant rate $r$ evaluated numerically by solving the master equation. The solutions correspond to (a) low temperature for $r=1$ MHz and (b) for two temperatures 100 and 400 K with $r=0.01$ MHz. Other parameters are ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\Omega }_{\text{p}}=0.01{\gamma }_4$, ${\gamma }_{41}={\gamma }_{42}=6$ MHz, ${\gamma }_{43}=12$ MHz ${\gamma }_3=10$ kHz, ${\gamma }_2=40$ kHz, and ${\Omega }_{\text{c}}={\gamma }_4$, ${\delta }_{\text{s}}=13.5$ MHz, ${\delta }_{\text{c}}=0$.

Figure 14

Numerically evaluated steady-state populations ${\rho }_{11}$ (dotted line), ${\rho }_{22}$ (dotted-dashed line),${\rho }_{33}$ (dashed line), and ${\rho }_{44}$ (solid line) vs probe-field detuning ${\delta }_{\text{p}}$ at zero temperature and incorporating dephasing ${\gamma }_{\varphi 41}={\gamma }_{\varphi 43}=150\text{MHz}$ and ${\gamma }_{\varphi 2}=0.8$ MHz. The plots show (a) ${\Omega }_{\text{s}}={\Omega }_{\text{p}}=0.3{\gamma }_4$ and (b) ${\Omega }_{\text{s}}\gg {\Omega }_{\text{p}}$, ${\Omega }_{\text{s}}=0.3{\gamma }_4$, ${\Omega }_{\text{p}}=0.01{\gamma }_4$. Other parameters are ${\gamma }_4=18$ MHz, ${\gamma }_3=10$ kHz, and ${\Omega }_{\text{c}}={\gamma }_4$, ${\delta }_{\text{s}}=0.5{\Omega }_{\text{c}}$, ${\delta }_{\text{c}}=0$.