1. Introduction

A coherent superflash [1–4] is an enhanced peak intensity transmission compared to the input pulse. It occurs because of the abrupt phase shift of the total transmitted field ($E_t$) when the incident light ($E_0$) transmitted through a high optical depth (OD) Lorentz dielectric is suddenly turned off. Free induction decay (FID) is a special case of superflash for the on-resonance input carrier frequency, whose peak value is limited to the maximum input intensity. To overcome this limitation, a stack of coherent transient peaks can be produced at the falling edge by sending a series of on-resonant square-modulated pulses through two-level atoms [5] or a three-level EIT system [6]. Most recently, this phenomenon has been revisited as the so-called coherent "superflash" in the two-level system of $^{88}$Sr and the origin is "strong phase rotation and large amplitude" of the forward-scattered field ($E_{s}$) at a certain probe laser detuning $\Delta _{p}$ [1]. The forward-scattered field ($E_{s}$) comprises the field radiated by the dipoles in a high-OD medium. In the steady-state off-resonance case, light is absorbed by the medium such that the transmitted intensity $I_{t} < I_{0}$. This steady-state absorption of the high-OD medium can be interpreted as cancellation because $E_{s}$ is not completely in phase with the incident field $E_{0}$. During the abrupt extinction of $E_{0}$, $E_{s}$ does not disappear instantaneously and further, it is equivalent to the transmitted field immediately after the probe laser extinction $E_{t}(0^{+})$. At a certain $\Delta _{p}$ associated with $|E_{s}| = 2E_{0}$, the transmitted field immediately after extinction becomes $E_{t}(0^{+}) = -2 E_{0}$ with a negative sign, indicating that the phase of $E_{s}$ is opposite to that of $E_{0}$. Consequently, the value of the transmitted intensity immediately after extinction (superflash, $I_{s}\equiv |E_s|^{2}$) asymptotically increases to $4 I_{0}$ (where $I_{0} = |E_{0}|^{2}$) as the OD approaches infinity. With a finite OD, an output intensity up to $3 I_{0}$ was obtained when the detuning of the probe laser was $\Delta _p = 11.2 \gamma _{31}$ [1].

In this study, we extend the superflash to a three-level EIT medium case to provide additional control parameters. We introduce an auxiliary coupling laser along with the probe laser which is incident on a three-level medium [Fig. 1(a)] as a medium-independent optically-tunable parameter. Consequently, we find that the Rabi frequency of coupling laser $\Omega _{c}$ is used to tune the "peak detuning" $\Delta _p \equiv \Delta _{peak}$. We also observed the OD dependence of the superflash in the three-level medium, which is different from that of the two-level medium. We call this new class of superflash as Type II, in contrast to the previously reported class of superflash (Type I). We discuss the temporal profiles of the transmitted intensity $I_{t}(t)$ and investigate the relationship between the corresponding peaks in the $I_{s}$ spectra and the phase shift $\phi _{s}$ of the forward-scattered field.

 

Fig. 1. Square modulated pulse entering a three-level EIT system. (a) Schematic of the square modulated pulse and $^{85}$Rb energy levels, (b) steady-state transmission, and (c) group delay of a square pulse through a two-level ($\Omega _c=0$, black dashed line) or three-level EIT medium ($\Omega _c=4.2\gamma _{13}$, red solid line, $8.4\gamma _{13}$, blue dashed line) [6].

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We briefly review the forward-scattered intensity $I_{s}$ in terms of the medium susceptibility $\chi (\omega )$ of a three-level EIT medium. The forward-scattered intensity spectrum $I_{s}$ can be evaluated from the transmitted field $E_{t}(t, z)$ as a function of the incident probe detuning $\Delta _{p}$ in a three-level EIT medium by utilizing numerical Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) [6–14]. The incident lasers consists of a weak probe laser $E_{p}$ for coupling the transition between $| {1}\rangle$ and $| {3}\rangle$ and strong coupling laser $E_{c}$ for the transition between $| {2}\rangle$ and $| {3}\rangle$, expressed as

In general, the wavenumber is complex: $k(\omega ) = k_{r}(\omega ) + ik_{i}(\omega )$, and its real and the imaginary parts are $k_{r}(\omega ) = \Re {k(\omega )} = \omega _{0}n_{r}(\omega )/c$ and $k_{i}(\omega )= \Im {k(\omega )} = \omega _{0}n_{i}(\omega )/c$, respectively. The real and imaginary parts of the refractive index $n_{r}(\omega )$ and $n_{i}(\omega )$ can be expressed in terms of the susceptibility of the medium as $n_{r}(\omega ) = 1 +\chi _{r}(\omega )/2$ and $n_{i}(\omega ) =\chi _{i}(\omega )/2$, respectively, from $n(\omega )=\sqrt {1+\chi (\omega )} \sim 1+\chi (\omega )/2$ and Eq. (3). The steady-state of the transmitted field amplitude is usually smaller than the incident amplitude of $E_{p}$ owing to the absorption of the incident light by atoms, as described by the transfer function $E_{t}(\omega, z = L) = E_{0}(\omega ) e^{i k(\omega ) \cdot L}$, which follows the Beer-Lambert law of absorption for the steady-state, $I_{0}(\omega ) e^{-a(\Delta _{p})L}$, where the absorption coefficient is given by $\alpha (\Delta _{p}) = 2 k_{i}(\Delta _{p})$.

We assume that our Lorentz dielectric medium is a dilute cold atomic gas of $^{85}$Rb [6]. The energy level configuration of the atoms is a three-level EIT $\Lambda$, as shown in Fig. 1(a), which transforms into a two-level system in the absence of the incident coupling laser ($\Omega _c = 0$). $| {1}\rangle$ and $| {3}\rangle$ denote the ground state and excited state, respectively. $| {2}\rangle$ denotes an auxiliary state associated with the coupling laser, which was absent in the previously studied superflash in a two-level system. For further detailed discussion, we use the EIT medium parameters obtained from [13,6–16]: OD $=62$, medium linewidth $\Gamma = 2\gamma _{31} = 2\pi \cdot 6$ MHz, corresponding to the decay rate of the state $| {3}\rangle$, and ground state dephasing rate $\gamma _{21} = 0.005 \gamma _{31}$. The susceptibility for a three-level EIT $\Lambda$ medium is expressed as follows [6]:

Equation (3) is the complex susceptibility that can be separated into real and imaginary parts. We require only the real part as shown below.

This phase will be used to extract the relation between the normalized Rabi frequency, $\bar {\Omega }_{c}$, and normalized probe detuning, $\bar {\Delta }_{p}$.

2. Temporal transmission

We now consider the case in which the amplitude of the incident probe laser is a square-modulated pulse. The transient peak has a maximum intensity of $I_{0}$ for the on-resonance case ($\Delta _p = 0$) at both the rising and falling edges [6,17]. In particular, the peak at the falling edge is known as coherent flash or FID. The temporally transmitted square pulse through a two- or three-level system can be calculated using Eq. (2). Figure 2 shows the evaluated transmitted field intensities for various coupling Rabi frequencies $\Omega _{c}$ and probe laser detunings $\Delta _{p}$ as a function of time. The total temporal window of FFT/IFFT is $2.5 \mu$s, turned on at $t = 0.12 5\mu$s, and then turned off at $t = 2 \mu$s.

 

Fig. 2. Temporal profile of the transmitted intensities for different values of $\Omega _c$ and $\Delta _{p}$; (a-c) $\Omega _{c} = 0$, (d-f) $\Omega _{c} = 4.2\gamma _{31}$, and (g-i) $\Omega _{c} = 8.4\gamma _{31}$. For each value of $\Omega _{c}$, the probe laser detunings are (a, d, g) $\Delta _{p} \sim 11 \gamma _{31}$, (b, e, h) $\Delta _{p} \sim 5 \gamma _{31}$, and (c, f, i) $\Delta _{p} \sim 0$. For a two-level system, (a-c) $\Omega _{c} = 0$, the superflash occurs only for far off-resonance detuning $\Delta _{p} \sim 11 \gamma _{31}$ as discussed in Ref. [1]. The FID appears at the extinction of the square pulse (falling edge) in (c) for both $\Omega _{c}=0$ and $\Delta =0$. For non-zero Rabi frequency, (d-f) $\Omega _{c} = 4.2\gamma _{31}$ and (g-i) $\Omega _{c} = 8.4\gamma _{31}$, a new feature appears for near on-resonance $\Delta _{p} \sim 0$ in (f) and (i), where EIT occurs. We refer to the new superflash as Type II compared to the conventional Type I for the two-level case in (a). Type II peaks appear in Fig. 3 as sharp peaks in the $I_{s}$ spectrum (red solid line plot).

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The first row in Fig. 2(a–c) shows the two-level cases in the absence of a coupling laser ($\Omega _{c} = 0$). We can retrieve the conventional superflash peak value $3I_{0}$ of the two-level medium when the system is in the off-resonance condition near $\Delta _{p} = 11.2\gamma _{31}$ as shown in Fig. 2(a) [1]. It should be noted that the FID appears at the extinction point of the square pulse (falling edge) in Fig. 2(c) for the two-level on-resonance case $\Delta =0$. The second and third rows show the transmitted field intensities with coupling lasers for $\Omega _{c} = 4.2\gamma _{31}$ [Fig. 2(d–f)] and for $\Omega _{c} = 8.4\gamma _{31}$ [Fig. 2(g–i)], respectively. Both the on- and off-resonance cases undergo FID [12,18] at the falling edge and shares some features in both the steady-state and transient regions. Figure 2(a), 2(d), and 2(g) show the superflash at the falling edge obtained by grouping similar detuning ($\Delta _{p} \sim 11 \gamma _{31}$) for different Rabi frequencies of the coupling laser, namely, $\Omega _c = 0$, $\Omega _c = 4.2\gamma _{31}$, and $\Omega _c = 8.4\gamma _{31}$, respectively. The steady-state value of the temporal transmitted intensity hovers around $0.5 I_{0}$ corresponding to the steady-state absorption level near $\Delta _{p} = 11 \gamma _{31}$. As $\Delta _p$ decreases, Fig. 2(b), 2(e), and 2(h) exhibit interesting phenomena known as the anti-flash ($I_{s} < I_{0}$) at the abrupt-extinction part, as discussed extensively in [1,3]. The principle of anti-flash production is fundamentally identical to that of the superflash and such transient phenomena can be understood using a forward-scattered field, $E_{t} = E_{0} + E_{s} \leq E_{0}$. We observe a normal coherent flash ($I_{s} \approx I_{0}$) at the abrupt-ignition part of the pulse. In the steady-state region, $0\,\mu$s $< t < 2\,\mu$s, the transmitted intensity levels decrease significantly.

Interestingly, Fig. 2(c), 2(f), and 2(i) show the unique EIT features of temporal profiles as the detuning of the probe laser is further decreased close to the resonance ($\Delta _p \sim 0$). At abrupt ignition, a regular coherent flash is observed, as in the other cases. However, more significant differences between the temporal profiles of two-level and three-level EIT systems were observed in both the steady-state and transient (abrupt-extinction) parts of the pulse. Figure 2(c) shows that in the steady state, the transmission intensity is negligible owing to the complete absorption through an optically thick two-level system. In contrast, for the other two cases in EIT [Fig. 2(f) and 2(i)], the steady-state transmission levels increase up to $0.5 I_{0}$. In the transient parts, a coherent flash ($I_{s} \approx I_{0}$) is observed at the abrupt-ignition part of the pulse [18], whereas the formation of the superflash ($I_{s} \approx 3I_{0} > I_{0}$) at the abrupt-extinction part of the pulse occurs as in the far-off detuned two-level, as shown in Fig. 2(a) [1,3]. For the two-level system [$\Omega _c=0$; Fig. 2(c)], we observe a regular coherent flash ($I_{s} \approx I_{0}$) at the abrupt-extinction point. However, with EIT, the peak values of the superflash increase, and thus, $I_{s} \approx 2.75I_{0}$ and $I_{s} \approx 3I_{0}$ for $\Omega _c = 4.2 \gamma _{13}$ [Fig. 2(f)] and $\Omega _c = 8.4 \gamma _{13}$ [Fig. 2(i)], respectively. Although the peak detunings and Rabi frequencies differ significantly, it is remarkable to observe similar values of the maximal coherent flash and superflash peaks, as shown in Fig. 2(a) and Fig. 2(i), respectively. Thus, EIT creates a new feature near resonance, which we define as the Type II superflash and is different from the previously reported superflash (Type I).

3. Spectrally tunable superflash $I_{s}$ in EIT

For further analysis of the superflash in steady-state transmission, we now consider the dependence of the probe laser detuning for four different Rabi frequencies: $\Omega _{c} = 0.0\gamma _{31}$, $\Omega _{c} = 4.2\gamma _{31}$, $8.4\gamma _{31}$, and $16.8\gamma _{31}$, as shown in Fig. 3. The tops of each image in Fig. 3 show the forward-scattered intensities $I_{s}$ (red solid lines) as a function of $\Delta _{p}$, while the bottom of each image in Fig. 3 show the phase shift $\phi _{s}$ (blue solid lines) and steady-state transmission $I_{ss}$ (red dashed lines), respectively. Each data point of the forward-scattered intensities $I_{s}$ (red solid lines) is obtained by extracting the values of the transmitted intensities at the falling edges of the square pulse in Fig. 2. The steady-state transmitted intensities $I_{ss}$ (indicated by the dashed red lines) follow the Beer-Lambert law of absorption.

 

Fig. 3. Spectral properties of superflash and medium; $I_s$ spectrum (red solid lines in each top plot), phase shift $\phi _{s}$ (blue solid lines in each bottom plot), and steady-state transmission $I_{ss}$ (red dashed plots in each bottom plot) as a function of the normalized probe laser detunings $\Delta _p/\gamma _{13}$ for different Rabi frequencies (a) $\Omega _{c} = 0$ (two-level), (b) $\Omega _{c} = 4.2\gamma _{31}$, (c) $\Omega _{c} = 8.4\gamma _{31}$ and (d) $\Omega _{c} = 16.8\gamma _{31}$ (EIT). The OD of the medium is fixed at 62. The two-level case in (a) reproduces the feature in Fig. 1 (d) in Ref [1]. The examples of EIT in (b-d) show a new type of superflash peaks near the edge of the EIT windows.

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For $\Omega _{c} = 0$, the red solid lines $I_{s}$ of the superflash exhibit two global maxima across $\Delta _{p}$, as shown in Fig. 3(a). These superflashes, which we call Type I, occur at far off-resonance at $\Delta _{p} \sim 11 \gamma _{13}$, similar to the previously reported superflash in a two-level system [1]. The maximum value of the Type I superflash is $I_{s} \sim 3 I_0$, which resembles the superflash in a two-level system. The corresponding phase shift at $\Delta _{p} \sim 11 \gamma _{13}$ was close to $\pi$, which was anticipated in the superflash condition of the two-level system. When $\Delta _{p} \sim \pm 11 \gamma _{31}$, the steady-state value of the transmitted intensity is approximately $0.5 I_{0}$, as indicated by the dashed red line at the bottom of Fig. 3(a), which is consistent with the respective steady-state transmission in Fig. 2.

For nonzero Rabi frequencies, the EIT transparency window emerges near the resonance, $\Delta _p \sim 0$, as shown by the dashed red lines in Fig. 3(b–d). Enhanced steady-state transmission near the resonance is generated by the coupling laser $\Omega _c$. These EIT-enabled cases lead to a new spectral feature in $I_{s}$ which we refer to as Type II peaks over $\Delta _{p}$. They are located near the resonance, showing sharper peaks than Type I as a function of $\Delta _{p}$. Considering only the positive detuning for $\Omega _{c} = 16.8\gamma _{31}$, the Type I peak occurs at a detuning $\Delta _{p} = 15.4\gamma _{31}$, outside the EIT windows, generating a peak of $I_s \sim 3 I_{0}$. The Type II superflash peak occurs at $\Delta _{p} = 4.5\gamma _{31}$ inside the EIT window, and generates almost the same peak height as Type I. For negative detunings or for other Rabi frequencies at $\Omega _{c} = 4.2\gamma _{31}$ or $8.4\gamma _{31}$, the detunings for Type I superflashes are also located outside the EIT window, and the new type II superflash peaks emerge near resonance at the EIT window edges. No such phenomenon was observed in the two-level system shown in Fig. 3(a) (dashed red line). Immediately next to the transparency window, the significantly reduced steady-state transmitted intensity seen in Fig. 2(b), 2(e), and 2(h), can also be observed in the steady-state spectra in Fig. 3(a–c), respectively.

The superflashes in the three-level EIT-enabled system exhibit maximum peaks at phase shifts close to $\phi _{s} = \pm \pi$, regardless of the type, as indicated by the solid blue lines in Fig. 3. Interestingly, we also see local maxima with smaller heights (superflashes) near $\phi _{s} = \pm 3\pi$. We expect to see such phenomena for all other odd multiples, $\phi _{s} = \pm \pi$, $\pm 3\pi$, $\pm 5\pi \ldots$, as the OD increases [3]. In Fig. 4(c-d), more superflashes with smaller heights appear beyond OD $\sim 100$. The OD tunable superflash might be suitable for two-level systems, whereas the Rabi frequency tunability in EIT enables external control even for a fixed OD medium.

 

Fig. 4. Surface plot of the forward-scattered intensity $I_{s}$ as a function of $\Delta _p$ and OD. Each surface plot corresponds to a different value of $\Omega _{c}$, namely, a) $0$, b) $4.2\gamma _{31}$, c) $8.4\gamma _{31}$, and d) $16.8\gamma _{31}$. The color bar denotes the normalized forward-scattered intensity ($I_s/I_0$) generated numerically. The brightest area corresponds to the maximum intensity peaks. The slope of $\Delta _{peak}$ (OD) is positive for the Type I superflash, while it is negative for the new (Type II) superflash. The vertical black dashed lines denote the constant OD at 62.

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For a better understanding of the superflash in the three-level with EIT conditions, we generated surface plots of $I_s$ as functions of OD and $\Delta _{p}$ for given $\Omega _{c}$, as shown in Fig. 4. The brightest [darkest] region indicates the maximum [minimum] values. Figure 4(a) shows a zero-Rabi frequency case ($\Omega _{c} = 0$), which is consistent with the previous result of the two-level system [1]. The bright yellow region diverges from the origin towards the upper-right region of the figure. The straight line in the middle of this yellow region indicates the points of maximum superflash. The line also implies a linear dependence between $\Delta _{p}$ and OD. The black dashed line denotes the $I_s (\Delta _p)$ values for a fixed OD$=62$ corresponding to Fig. 3. The Type I superflashes are essentially equivalent to the superflash that has been extensively discussed in the literature [1].

In contrast, we observe the emergence of new features for non-zero Rabi frequencies ($\Omega _c \neq 0$). Figure 4(b–d) show the surface plots for the three-level EIT system with $\Omega _{c} = 4.2\gamma _{31}$, $8.4\gamma _{31}$, and $16.8\gamma _{31}$, respectively. The features resemble those shown in Fig. 4(a) when the detunings are sufficiently large; however, there are notable fundamental differences. First, we observe the formation of a new streak of bright yellow region near the horizontal axis, which corresponds to the Type II superflash discussed previously. As we increase $\Omega _c$ as shown in Fig. 4(c) and 4(d), these new lines shift upwards accompanied by the widened yellow region of Type II. This feature shows that both $\Delta _p$ and $\Omega _c$ should be considered in obtaining an optimal superflash. Second, we observe that the dependence of the Type I superflash exhibits a nonlinear behavior via upward shifting, as indicated by the bright yellow regions. This nonlinear behavior implies that superflash formation with respect to the phase shift is not exactly the same as the anticipated arguments in the two-level system, which agrees with a recent work [19]. Third, the dependence of Type II superflash as a function of the OD exhibits the opposite behavior - inverse linearity with negative slopes as opposed to the linear behavior with positive slopes observed in Type I. This shows that the Type I and Type II peak detuning behave oppositely as the OD of the system increases. Therefore, by tuning an external control parameter, such as $\Omega _c$, we have a knob to control $\Delta _{peak}$ that renders superflash. The maxima of Type I and Type II superflashes are bound to the OD values.

4. Tunable superflash via Rabi frequency, $\Omega _c$

To investigate the tunability of the superflash by $\Omega _{c}$ in further detail, we obtained the numerical data of the peak detuning as a function of the Rabi frequency, $\Delta _{peak}$ ($\Omega _{c}$), for $I_s$ peak values from the temporal transmission data in Fig. 2. We obtain the numerical data by implementing an algorithm to extract $\Delta _{peak}$ from the $I_{s}$ spectrum for different values of $\Omega _{c}$. We then plot $\Delta _{peak}$ ($\Omega _{c}$) for Type I (solid circles) and Type II (solid triangles), as shown in Fig. 5. The forward-scattered field intensity values are shown in a color bar, as before. The dotted data in the brightest region, which indicate the maximum values near $3 I_0$, demonstrate the quadratic nature of $\Delta _{peak} (\Omega _c)$. The numerical values are compared with the analytic expression, as discussed below.

 

Fig. 5. Surface plot of the forward-scattered intensity $I_s$ as a function $\Delta _{p}$ and $\Omega _{c}$. The color bar denotes the normalized forward-scattered intensity, $I_s/I_0$, generated numerically. The black circles (triangles) denote the numerically-evaluated peak superflash $\Delta _{peak} (\Omega _{c})$ for Type I (Type II). Consequently, the red line (blue dashed line) denotes the quadratically-dependent values obtained analytically from Eq. (8) for Type I (Type II).

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The origin of the maximum $I_{s}$ values can be understood using the coherent phase shift of the forward-scattered field, $\phi _{s}(\Delta _{p})$ [1]. Therefore, the analytic expression of the tunable peak detuning $\Delta _{peak} (\Omega _{c})$ can be obtained by equating to a certain phase-shift condition. Ideally, the maximum peak of the superflash occurs when $\phi _{s}(\bar {\Delta }_{peak})=\pm \pi$. From Eq. (6) in a three-level system, a quartic equation in terms of $\bar {\Delta }_{p}$ can be written as

This equation can be used to test the validity of the phase shift condition by comparing it to the numerical data in Fig. 5. However, we obtained poor agreements for both Types I and II in the ideal phase case. This implies that we may adjust additional phase shifts that deviate from $\phi _{s}(\bar {\Delta }_{peak})=\pm \pi$ while analyzing the EIT-enabled three-level system for the maximum superflash condition.

By introducing the additional phase shift as a free-fitting parameter, we fit the data for both Type I and Type II peaks. Table 1 shows the obtained results of the phase shifts $\phi _{s}$ at $\Delta _{peak}$ for Types I and II, respectively. As anticipated from the ideal case, both Type I and Type II peaks possess absolute values of their phase shifts close to $\pi$, which leads to the magnitude of the forward-scattered field $E_{s}$ being maximally close to the limit of $2 E_{0}$ as in the two-level case [3]. However, as shown in Table 1, the EIT-enabled three-level system undergoes additional phase shifts oppositely in Type I and Type II. The additional phase shift results in approximately $-\pi +0.1\pi$ for Type I whereas it is approximately $+\pi -0.1\pi$ for Type II. The finite OD of our medium is one of the reasons for deviations from the ideal condition [3]. If we equate the phase shift to $\pm \pi + \varphi$ where $\varphi$ is a correction term and $\phi _{s}(\Delta _{peak}) = \pm \pi + \varphi$, we obtain

Table 1. List of $\Delta _{peak}$ and phase shift $\phi _{s} = \pm \pi + \varphi$ for the best fit of the analytic expression of $\Delta _p (\Omega _c)$ with our numerical data for Type I and Type II peaks at different $\Omega _{c}$. The phase shifts for the best fit deviate by about ten percent from the ideal value $\pm \pi$.

View Table

Here, the normalized correction term is $\bar {\varphi } = \varphi /\pi$. Now if we plot Eq. (8) implicitly with $\bar {\Delta }_{p}$ and $\Omega _{c}/\gamma _{31}$ as the y- and x-axes, respectively, good agreement with the numerical data is obtained, as indicated by the solid line (dashed line) for Type I (Type II) in Fig. 5. For the best-fit agreement between the numerical data and analytic results in Eq. (8), we find that the Type I and Type II peaks require $\bar {\varphi } \approx 0.1$, and $\bar {\varphi } \approx -0.1$, respectively. Interestingly, the direction of the phase deviations from $\pm \pi$ for Types I and II are opposite. These originate from the EIT-enabled three-level configuration in Fig. 3. The phase shift $\phi _s (\bar {\Delta }_p)$ in Eq. (6) is antisymmetric (the blue solid lines in Fig. 3), whereas the superflash peaks in $I_s (\bar {\Delta }_p)$ are symmetric (the red solid lines in Fig. 3). Consequently, we observed that the deviation of peak detunings in Type II superflash is to be shifted towards the resonance and to be shifted away for the Type I. It might be associated with the detuning-dependent absorption, as shown in the steady-state transmission $I_{ss}(\bar {\Delta }_p)$ (the red dashed lines in Fig. 3). Therefore, $\bar {\varphi }$ for Type I (outside the EIT windows) and Type II (inside the EIT window) have opposite signs that agree with our observation.

5. Conclusion

We investigate the tunability of the superflash peak detuning $\Delta _{peak}$ by varying the $\Omega _{c}$ associated with the intensity of the EIT coupling laser. The values of $\Delta _{peak}$, which are extracted from the superflash $I_{s}$ spectrum that shows the EIT transparency windows for different values of $\Omega _{c}$. Compared with the superflash discussed in the two-level system (Type I), the EIT feature provides a new class of superflash near the medium resonance (Type II). We confirm the OD-dependent tunability exhibits linear behavior for Type I with a positive slope of $\Delta _{peak}$ (OD) as discussed previously [1], whereas Type II exhibits a negative slope (inverse-linearity) as in the most recent study [19].

We observe quadratic tunability of $\Delta _{peak} (\Omega _c)$ for both Type I and Type II. The analytic expression for $\Delta _{peak} (\Omega _c)$ can be derived from the phase shift $\phi _{s} \sim \pm \pi$. Our numerical data are consistent with the analytic expression when the phase was further shifted by $\varphi = \mp \, 0.1 \pi$ as $\phi _{s}(\Delta _{peak}) = \pm \pi + \varphi$. The deviation $\varphi$ from the ideal phase shift $\phi _{s}= \pm \pi$ has opposite signs for Type I and Type II. Further investigation is needed for such opposite signs and the ten percent deviation that might be associated with the detuning-dependent absorption. The result can also be compared to coherent stacks [6] which have a peak $\sim 3 I_0$ that requires a series of multiple square pulses propagating through a two-level system and $\sim 8 I_0$ in an EIT medium [6,17]. For further investigation, multi-level extension beyond three-level might enables delicate controllability with noble characteristics, although the possible drawback would be adding more experimental complexity. It might also be interesting to trigger additional transient phase differences by switching the coupling laser together with the probe. Our results show that the external Rabi frequency can facilitate the continuous and viable control of the superflash of a given fixed OD medium.