Doppler-free adiabatic self-induced transparency

We demonstrate that a Doppler-broadened two-level medium can be made transparent to a laser pulse by an appropriate adiabatic variation in the laser field amplitude and its nominal detuning. This technique of adiabatic self-induced transparency (ASIT) is compared with the well-known self-induced transparency phenomenon, showing that the adiabatic method is much more robust against variations in the system parameters. We also discuss a possible experimental implementation of ASIT using ${}^{87}\text{R}\text{b}$ atoms.

Figure 1

(Color online) (a) Two-level system under consideration where $\Omega$ and $\Delta$ are the Rabi frequency and the field-atom detuning; (b) three-level system interacting with two fields with Rabi frequencies ${\Omega }_P$ and ${\Omega }_S$; (c) equivalence between the density-matrix variables of the two-level system and the three-level picture. Schematic representation, for the two-level case, of the Bloch vector evolution on the Bloch sphere of the $\left(U,V,W\right)$ variables, under (d) SIT with $2\pi$ pulses, (e) STIRAP, and (f) ASIT processes.

Figure 2

(Color online) [(a), (c), (e), and (g)] Temporal profiles of the EM (black line) and $D$ [magenta (light gray) line] pulses. [(b), (d), (f), and (h)] Time evolution of the population difference $\left(W\right)$ and the medium coherence $\left(U,V\right)$ in a homogeneously broadened two-level system under the [(a) and (b)] SIA, [(c) and (d)] SIT, and [(e)–(h)] ASIT processes. The areas of the pulses are equal to [(a), (b), (e), and (f)] $A=A_{\Omega }=A_{\Delta }=9\pi$ and [(c), (d), (g), and (h)] $=10\pi$. The $D$ pulses and the rise and fall parts of the EM pulses have Gaussian profile. The temporal width of the $D$ pulses is ${\tau }_{\Delta }=0.003{\gamma }^{-1}$, ${\Delta }_0=0$, and [(e) and (f)] $\overline{\Delta }/\gamma =5318$ and [(g) and (h)] $=5909$. [(a), (b), (e), and (f)] ${\Omega }_0/\gamma =1772$ and [(c), (d), (g), and (h)] $=2363$. The rise ${\tau }_r$ and fall ${\tau }_f$ times of the EM pulses are equal to ${\tau }_{r,f}={\tau }_{\Delta }/2$. The $D$ pulses are shifted away by $\Delta \tau =1.5{\tau }_{\Delta }$ from the top flat part of the EM pulse. The time duration of the top flat part of the EM pulses is set to keep the EM pulse area indicated above. In all cases represented, the detuning of the EM pulses during the top flat part is close to zero value.

Figure 3

(Color online) [(b)–(d)] Distributions of the final value (red solid line) of the atomic population inversion, $W_{\text{fin}}$, and of the maximum (empty circles) and minimum (crosses) values of the mixing angle $\theta$ with respect to the Doppler shift $kv$ at different amplitudes of the EM and $D$ pulses. The areas of the EM and $D$ pulses, defined as $A_{\Omega }={\Omega }_0{\tau }_{\Omega }/\sqrt{2}$ and $A_{\Delta }=\overline{\Delta }{\tau }_{\Delta }/\sqrt{2}$, are equal to each other: (b) $A=A_{\Omega }/\pi =A_{\Delta }/\pi =2$; (c) $=6$; (d) $=10$. The Maxwell distribution of the atomic velocities is shown in (a) (dashed line). The inset of (a) presents the time evolution of the exciting EM $\left(\Omega \right)$ and $D$ $\left(\Delta \right)$ pulses. Abrupt decrease in the ${\theta }_{\text{max}}$ corresponds to the velocity class for which $kv\sim {\Delta }_0+\overline{\Delta }$. The temporal widths of the input pulses are ${\tau }_{\Delta }={\tau }_{\Omega }=0.01{\gamma }^{-1}$, ${\Delta }_0=0$, and $\Delta \tau =1.5{\tau }_{\Omega }$. The Doppler width is fixed at $\Delta {\omega }_D={10}^3\gamma$ and $\Gamma =2\gamma$. Macroscopic response of the inhomogeneously broadened 2L medium is given by the integration of the atomic response [shown by red solid lines in (b)–(d)] weighted by the probability density for the atomic velocities of the Maxwell distribution [shown by the dashed line in (a)].

Figure 4

(Color online) [(a), (c), and (e)] Temporal profiles of the EM pulse (black line) and the total detuning seen by the atom [magenta (light gray) line] for three different atomic velocity classes: [(a) and (b)] $kv=\Delta {\omega }_{D}3/4$, [(c) and (d)] $kv=\Delta {\omega }_D/3$, and [(e) and (f)] $kv=-\Delta {\omega }_D/3$. (b), (d), and (f) represent the time evolution of the atomic variables $W$, $U$, and $V$. The pulse areas are $A_{\Omega }=A_{\Delta }=10\pi$. The other parameters are the same as in Fig. 3.

Figure 5

(Color online) (b) Time evolution of the macroscopic population difference $\left(W\right)$ and the medium coherence $\left(U,V\right)$ in the inhomogeneously broadened two-level system under excitation of EM and $D$ pulses shown in (a). (c) shows the evolution of the mixing angle $\theta$ for atoms with zero projection of the velocity onto the light propagation direction. The pulse areas are $A_{\Omega }=A_{\Delta }=10\pi$. Distribution of the final amplitude for the single-atom variables $U$ and $W$ on the Doppler shift $kv$ is shown in (d) and (e), respectively. Dashed lines in (d) and (e) are plotted to visualize the Maxwell distribution of the atomic velocities. The maxima (upper line) and minima (lower line) for the variable $\theta$ are shown in (f), where dots represent its zero final value. The other parameters are the same as in Fig. 3.

Figure 6

Contour plot of the final values of the macroscopic population inversion, $W_{\text{fin}}$, after (a) SIT and (b) ASIT processes in a Doppler-broadened 2L system as a function of the pulse area and duration. $A_{\Delta }=A_{\Omega }$ in (b) and (d), and $A_{\Delta }=0$ in (a) and (c). Other parameters of the 2L system and of the pulses are the same as in Fig. 3. White regions in (a) and (b) correspond to the domains where $W_{\text{fin}}\to -1$. Plots for time-averaged values of $g^2⟨{\left(\Omega V\right)}^2⟩/{⟨{\Omega }^2⟩}^2$ in $g^2$ units indicate (c) the pure absorptive nature of the SIT phenomenon and (d) the more dominantly dispersive nature of the ASIT process in the domain close to the left vertical axis.

Figure 7

Contour plot of macroscopic variable $W_{\text{fin}}$ under conditions of ASIT in the plane $\left(A_{\Omega }=A_{\Delta },\Delta {\omega }_D\right)$ for a Doppler-broadened 2L system of ${}^{87}\text{R}\text{b}$ atoms excited by EM and $D$ pulses of equal areas for two values of the time duration: (a) ${\tau }_{\Omega }={\tau }_{\Delta }=0.01{\gamma }^{-1}$ and (b) ${\tau }_{\Omega }={\tau }_{\Delta }=0.05{\gamma }^{-1}$. The considered transition of the ${}^{87}\text{R}\text{b}$ atoms is $(|5{{}^{2}S}_{1/2},F=2,m_F=2⟩\phantom{)}\leftrightarrow |5{{}^{2}P}_{3/2},F=3,m_F=3⟩$, with ${\lambda }_0=780\text{nm}$, $\gamma =5.83\times 2\pi \times {10}^6{\text{s}}^{-1}$, and $\Gamma =2\gamma$.