Anti-relaxation coating-induced velocity-dependent population re-distribution in electromagnetically induced transparency

Abstract

In an uncoated vapor cell, transmission spectra obtained for electromagnetically induced transparency (EIT) for D$_2$ line of $^{87}$Rb show asymmetry and (or) absorption in the presence of a magnetic field. In this study, complete conversion from asymmetry/absorption to transmission is found using octadecyltrichlorosilane (OTS) as an anti-relaxation coating for a $\Lambda$ system. The experimental results were interpreted in terms of velocity-dependent population re-distribution in the ground states induced by the coating, eventually resulting in the conversion from absorption to transmission. A simple theoretical model based on density matrix formalism is presented for qualitative interpretation of the results.

1 Introduction

For experiments related to electromagnetically induced transparency (EIT) [1], optical magnetometry [2, 3], slow light [4, 5], optical quantum memory [6, 7], atomic frequency standards [8, 9] and many others [10,11,12], the lifetime of the ground state coherence plays an important role. The decay of ground state coherence in an alkali metal vapor is mainly due to the collisions (both elastic and inelastic) between the atoms and between atoms and the wall of the cell containing the vapor. To minimize the effect of collisions, either a buffer gas (such as N$_2$ (nitrogen)) is added into the cell or the wall is coated with an anti-relaxation material such as paraffin, ethene or octadecyltrichlorosilane (OTS). The addition of the buffer gas increases the time that an atom spends inside the volume where the light is present by slowing down the diffusion process. The collision with the buffer gas preserves the polarization of the atom while changing its velocity, giving rise to what is known as velocity-changing coherence-preserving collisions (VCCPC) [13, 14]. The addition of the anti-relaxation coating on the other hand preserves the ground state coherence over thousands of collisions between atoms and the wall of the cell [15]. The phenomenon of EIT has been studied extensively in both the buffer gas and anti-relaxation-coated cell [16, 17]. In both systems, the narrowing of the EIT peak as compared to the uncoated vapor cell has been observed due to increased ground state coherence time. Moreover, in the case of an anti-relaxation coated cell, a dual structure EIT is observed. Due to atomic motion-induced Ramsey narrowing, a narrow peak resides on top of a broad peak in the dual structure EIT [18, 19].

The application of magnetic field lifts the degeneracy of the hyperfine levels that leads to multiple EIT sub-peaks. However, due to the involvement of a large number of levels due to the lifting of the degeneracy, the spectra show absorption in place of transmission or some competition between them [20] in an uncoated cell. Such a process is system specific and depends upon the polarization of the pump and probe beam in general. In another work as reported in [21], the authors found that a paraffin-coated cell probe beam undergoes absorption in $D_1$ line of $^{87}$Rb in the presence of a magnetic field. The presence of such absorption was discussed in terms of the re-alignment of the population in the ground state brought about by the anti-relaxation coating. Unlike $D_1$ line, at room temperature, the frequency separation between the hyperfine excited levels is smaller than the Doppler width of the system in the case of D$_2$ line of $^{87}$Rb. Hence, the EIT spectrum in D$_2$ line in the presence of a magnetic field using anti-relaxation coatings can have modified lineshapes due to Doppler effects.

In this work, OTS-coated cell is used to study the EIT in D$_2$ line of $^{87}$Rb to study the modified lineshapes. To have a reference, the measurements of the uncoated cell are used for comparisons. In an uncoated cell, with a magnetic field the authors have shown in earlier works [20, 22] that it is possible to resolve an off-resonant process in an uncoated vapor cell under a certain pump–probe configuration. Such off-resonant process was observed as absorption and it was also found that by carefully tuning the power of the beams, absorption can be converted to transmission. The results in this work show that all the absorption peaks present in the uncoated cell convert to the transmission when a coated cell is used. The result is different from what has been observed in the $D_1$ line. The ability to switch resonances from absorption to transmission (vice versa) is an interesting process that can be useful in various experiments [23,24,25]. Such conversion due to the coating in the EIT spectrum has a different physical origin as compared to the one in an uncoated cell where it was due to the power of the probe beam [22]. The study performed in the present work may also have applications related to atomic vapor magnetometry [26, 27] and quantum memory experiments [28].

In the experimental results, four resonances were visible with the application of a magnetic field. The results were interpreted in terms of velocity-dependent re-distribution of the population in the ground hyperfine states. A simple model where such re-distribution of the population was included by defining a velocity-dependent decay of the population from one ground state to the other is also presented to substantiate the experiment. A minimum model approach is used where only four energy levels were included which forms a double-$\Lambda$ type of system to understand the behavior of individual resonance.

The general approach to include the effect of anti-relaxation coating in the density matrix equation is to modify the equation which becomes an integro-differential type of equation. A similar method is also followed in the case of buffer gas-filled vapor cell experiments [13], where the coherence-preserving collisions occur with buffer gas instead of the wall. Such systems are solved by appropriately identifying the collisional kernel, which can be different for different systems. Based on the work by Ref. [29], Bhattarai et al. [30] followed a different approach where a coupled dark and bright zone density matrix equation in the steady-state condition was solved to explain the observation. The Ramsey-like feature in the EIT, which is the signature of relaxation coating, needs a different approach where precise time-dependent study or a study based on atomic flow is required [18, 31]. To explain the observation in the current experiment, a rather simple model was pursued due to the following two reasons: (i) the absence of the Ramsey feature in the experiment, which was certainly because the system under study was the hyperfine system and (ii) the region of interest was the steady state where small fluctuations in the system are averaged out smoothly. To substantiate, the recorded spectra is compared with the theoretical result.

2 Experiment

The schematic of the experimental setup is described in Fig. 1. The two laser sources are external cavity diode (ECD) lasers lasing at 780 nm. Both the lasers were combined using a polarizing beam splitter (PBS), which was then allowed to pass through the experimental cell. The laser power was controlled using a combination of half-wave plate (HWP) and PBS. After passing through the cell, another PBS was used to separate the two beams. The probe laser was detected using a photodetector and recorded in an oscilloscope. Saturated absorption spectroscopy (SAS) was used for both locking the laser and also for calibrating the measured data. The experimental cell was kept inside a chamber where the cell temperature and externally applied magnetic field could be controlled. Two kinds of cells were used in the experiment, one was the uncoated cell (without any buffer) and the other was an anti-relaxation-coated cell with OTS (octadecyltrichlorosilane) as the coating material. Both the experimental cells consisted of Rb atoms having both the isotopes of Rb with a concentration of 78% for $^{87}$Rb isotope.

Fig. 1

Schematic of the experimental setup. a External cavity diode laser (ECDL)-probe laser (Toptica DL); b EDCL laser-pump laser(Teach spin DL); c Digilock 110 laser locking module; d PBS (polarizing beam splitter); e zeroth-order HWP (half wave plate); f temperature controlling chamber; g Rb vapor cell (uncoated and OTS(octadecyltrichlorosilane)-coated cell); h pair of Helmholtz coil; i temperature controller electronics; j DC power supply for generating magnetic field, and the increase of magnetic field with applied DC voltage is linear; k 4 channel oscilloscope; l lens; m photo detector with responsivity of 0.6 V/A at 800 nm; n beam dump. The experimental cell is kept at room temperature, and the stray magnetic field is canceled using mu-metals which cover the entire box (f)

Both the beams were of a diameter of 0.2 cm. The pump power for the entire experiment was kept fixed at 3.5 mW (111.4 mW/cm$^2$), which is well above the saturation intensity for the system (3.5 mW/cm$^2$ for D$_2$ line of Rb). The probe laser was kept in the locked mode in the F$_g$=1 to F$_e$=2 transition and the pump laser was allowed to scan around F$_g$=2 to F$_e$=2 transition. The relevant energy level diagram for the system is shown in Fig. 2. The magnetic field was varied from 0 to 30 G using a Helmholtz coil and the direction was perpendicular to the propagation direction of the lasers. The polarization of the beams was linear and orthogonal to each other. The direction of the magnetic field was such that the probe laser was $\pi$ polarized and the pump laser was $\sigma$ $\pm$ polarized as seen from the atomic frame of reference. Only three velocity selective optical pumping (VSOP) dips instead of five for the pump–probe intensity regime were seen in the spectra (Fig. 3). This was because of the power broadening and saturation effects induced by the strong pump laser [32], which washes away the other two small VSOP peaks in the broad Doppler background. More details of the experimental setup can be found in [33]. Experimental measurements for both cells were performed under the same cell temperature of 23$\,^\circ$C.

Fig. 2

Energy-level diagram showing D${_2}$ line of $^{87}$ Rb, where a weak probe beam was locked in the F$_g=1 \rightarrow$ F$_e=2$ and the pump was scanning around F$_g=2 \rightarrow$ F$_e=2$ transition

3 Results and discussion

Fig. 3

Experimental transmission spectra as a function of pump detuning. a For uncoated vapor cell with fixed pump and probe intensity at 111.4 mW/cm$^2$ and 3.1 mW/cm$^2,$ respectively. b For OTS-coated cell with fixed pump intensity at 111.4 mW/cm$^2$. Probe intensity was changed from 3.1 mW/cm$^2$(curve 1) to 15.9 mW/cm$^2$(curve 2) for the study

Table 1 Values of velocity (v) and pump detuning ($\delta _{c}$) for three different VSOP dips present in the experimental spectra

At first, the transmission profile of the probe beam was recorded at zero magnetic fields for both uncoated and OTS-coated cells (Fig. 3). The value of the probe and pump intensity was $I_{pr}$=3.1 mW/cm$^2$ and $I_{pu}$=111.4 mW/cm$^2,$ respectively. For the case of an uncoated cell (Fig. 3a), three dips in the transmission profile due to velocity selective optical pumping (VSOP) were observed at $\delta _c=+/-157$ MHz and $\delta _c=0$ MHz with an EIT peak at $\delta _c=0$ MHz.$\delta _c$ is the pump detuning defined as $\omega _{22} - \omega _c$, where $\omega _{22}$ is the optical frequency gap between the levels F$_g=2$ and F$_e=2$ with $\omega _c$ as the frequency of the pump beam. At $\delta _c=0$ MHz, the condition of two-photon resonance is satisfied, ensuring the formation of a dark state which leads to enhanced transmission (EIT). The positions of the three VSOPs matched the frequency gap (157 MHz) between the excited levels $F_e=1$ and $F_e=2,$ which was expected, as these states are well within the Doppler width for $^{87}$Rb at room temperature (around 500 MHz). The class of velocities which are responsible for the VSOP are listed in table 1 and their energy-level positions are shown in Fig. 4.

As shown in Fig. 3a an asymmetry around the peak position of EIT was present in the transmission profile due to unequal absorption of the probe beam around $\delta _c=0$ MHz. The asymmetry of the EIT line shape is due to the coupling of the light fields with excited state $F_e=1$. If the frequency gap between the excited levels is smaller than the Doppler width (as for the present case), an asymmetric feature of EIT appears in the transmission profile [34]. For the case of an OTS-coated cell, under identical experimental conditions, it was found that the VSOP dips were substantially quenched as shown in curve (1) of Fig. 3b. At $\delta _c=0$ MHz, an EIT peak with a larger amplitude in comparison to that of an uncoated cell was visible in the spectra. An approximate estimation shows that the amplitude of the EIT for OTS-coated cell is ten times that of an uncoated one. Also, it is evident from curve (1) of Fig. 3b that the asymmetry around the EIT peak position was negligible for the OTS-coated cell.

Fig. 4

Energy-level diagram showing three positions of the scanning pump beam and corresponding velocity class where three VSOP dips in the transmission spectra were observed

The underlying mechanism of VSOP depends upon the process known as spectral hole burning, where the same velocity class of atoms is pumped and probed, leading to enhanced absorption of the probe beam [35]. In a coated cell, a fresh batch of atoms enters the interaction zone where light fields are present. The atom–light interaction zone is usually referred to as the bright zone. In the bright zone, atoms get polarized through their interaction with the light fields. Such polarized atoms come out of the bright zone and spend some time in the region where there is no interaction with the light field (dark zone). In the dark zone, these atoms undergo numerous collisions with the wall and again re-enter the bright zone with its polarization intact [36]. The preservation of the polarization increases the effective lifetime of ground state coherence for the processes such as coherent population trapping (CPT) and EIT, where the atoms are trapped in the dark state [37]. On the other hand, collision with the wall completely re-thermalizes the atoms which then follows the Maxwell–Boltzmann distribution governed by the temperature of the cell wall. Due to the re-thermalization of atoms upon the collisions with the wall, the process of VSOP is limited up to a single pass through the bright zone.

On the other hand, the phenomenon of EIT is related to the ground state coherence, which is preserved over many collisions with the wall. This obvious difference in the time scale is the reason behind the quenching of the VSOP in the spectra. However, this assumption is valid when the intensity of the light fields is such that all the atoms are not optically pumped into the dark state during a single pass through the bright zone. With the increase in the intensity of the light fields, the atoms can be optically pumped into the dark states during a single pass and the system can equilibrate at a faster rate. Curve (2) in Fig. 3b shows the spectra at an increased intensity of the probe beam from 3.1 to 15.9 mW/cm$^2,$ where prominent VSOP dips were visible in the spectra for a coated cell.

Fig. 5

Experimental transmission spectra for OTS-coated cell at different values of the magnetic field. Four distinct EIT peaks were visible at a non-zero magnetic field. The value of probe and pump intensity was kept fixed at 3.1 mW/cm$^2$ and 111.4 mW/cm$^2,$ respectively

Fig. 6

Transmission as a function of pump detuning at B = 30 G for both uncoated (a) and (b) and OTS-coated cell (c) and (d) at two different values of probe intensity. Pump intensity was fixed at 111.4 mW/cm$^2$ for all the cases. Probe intensity was varied from 3.1 mW/cm$^2$ (a) and (c) to 15.9 mW/cm$^2$ (b) and (d)

Next keeping the intensity of the pump and probe beam fixed at 111.4 mW/cm$^2$ and 3.1 mW/cm$^2,$ respectively, a magnetic field up to 30 G was applied perpendicular to the propagation direction of light. The experimental results for increasing magnetic field are shown in Fig. 5. With the application of the magnetic field, a single EIT peak splits into four peaks whose positions are the two-photon resonance position for the various systems formed due to the coupling of light fields with the Zeeman sub-levels. The relevant energy-level diagram and the values of peak positions for the same system are discussed in detail in a previous work by the authors [22].

To compare the response of the system under the magnetic field, the uncoated and coated cell were studied at 30 G at the same intensity as the pump and probe. A change in the sign of resonance was observed for the case of B=30 G, I$_{pr}$=3.1 mW/cm$^2$ and I$_{pu}$=111.4 mW/cm$^2$ for the two cells. The case of an uncoated cell showed four two-photon absorptions in the spectra as shown in Fig. 6a. However, for the same parameters, four EIT peaks were present as and when the coated cell was used (Fig. 6c). The observation of the absorption in place of transmission for an uncoated cell was reported earlier [22] where it was found that the coupling of the light field with the Zeeman states of excited level F$_{e}$=1 was the reason behind such an observation. The completely different observation in the case of OTS-coated cells highlights the fact that the inclusion of the anti-relaxation coating can minimize the effect of the coupling of light fields with the nearby excited states. When the intensity of the probe beam was increased from 3.1 to 15.9 mW/cm$^2,$ all the absorption peaks converted to transmission peaks for the uncoated cell as shown in Fig. 6b. Such conversion from absorption to transmission was understood as the increased rate of probe light scattering brought about by the increase in intensity. For comparison, the probe intensity was increased for the case of the coated cell, which resulted in negligible power broadening as shown in Fig. 6d.

In a hyperfine system such as D$_2$ line of $^{87}$Rb, the thermal motion of atoms in the form of Doppler effects needs to be considered. At a finite temperature, the light fields are Doppler shifted, which results in different non-zero one-photon detuning for different velocity classes of atoms. If there are multiple excited states, then some classes of atoms can satisfy the condition of the two-photon absorption process for one state and the EIT condition for the other. At low values of the light fields, due to negligible coherence among the ground state, the absorption process can dominate the spectra as in the case of the uncoated cell. Such atoms, which lead to the absorption of the probe beam, are not trapped in the dark state. On the other hand, the atoms fulfilling the condition of EIT are trapped in the dark state leading to the transmission of the probe beam [38]. These two velocity classes of atoms can be named on-resonant atoms (fulfilling the condition of EIT) and off-resonant atoms (which enhances the two-photon absorption process).

The inclusion of the anti-relaxation coating results in the re-pumping of both of these classes of atoms into the bright zone numerous times before the polarization information is lost. On the other hand, re-thermalization of the atoms due to the collision with the wall completely mixes these classes (complete mixing of velocity) [39]. Therefore, the atoms trapped in a dark state in a single pass through the bright zone can lead to absorption in the second pass due to a change in the velocity (and vice versa). The off-resonant classes of atoms which are responsible for the two-photon absorption of the probe beam are continuously (during passes through bright zone) subjected to the loss from Zeeman states of F$_g$=1 to F$_g$=2. Hence, multiple passes through the bright zone lead to the re-distribution of the population such that the velocity distribution for the atoms trapped in the dark state sharply peaks for the resonant classes only. Such re-distribution of the population in the ground state ensures that the only common velocity classes for the pump and probe beam are the resonant ones, which explain the absence of absorption for the coated cell.

4 Theoretical model

Following the same approach as in [22], a four-level system, each of which (with two three-level systems) are responsible for the four two-photon resonance peak in the experimental spectra, is taken as a model. For discussion, the levels responsible for peak A in Fig. 6, which are F$_g$=1:$m_F=-1$ (1), F$_g$=2:$m_F=-2$ (2), F$_e$=1:$m_F=-1$(3) and F$_e$=2:$m_F=-1$ (4), are taken. The effect of the anti-relaxation coating is to preserve the polarization of the atoms even after numerous collisions with the wall. In a coated cell, multiple passes through the bright zone create re-distribution of the population in the ground state. This re-distribution occurs because of the loss of the off-resonant atoms from F$_g$=1:$m_F=-1$ to F$_g$=2:$m_F=-2$ due to two-photon absorption, which happens during each passing of such atoms through the bright zone. Although in an uncoated cell, such re-distribution is possible during a single pass, it quickly gets destroyed when the atoms collide with the wall. Therefore, an effective decay rate ($\gamma _a$) of the population from F$_g$=1:$m_F=-1$ to F$_g$=2:$m_F=-2$ was defined to study the steady-state behavior of the system for the case of the coated cell. The rate of decay was introduced in the equation of motions for the ground state populations only (see Appendix for the complete set), i.e., for $\rho _{11}$ and $\rho _{22}$ as

$$\begin{aligned} \dot{\rho _{11}}= & {} \frac{1}{2} i \Omega _ p (c_{13} \rho _{31}+c_{14} \rho _{41}-c_{31}\rho _{13}-c_{41} \rho _{14})\nonumber \\{} & {} + \gamma _p \rho _{22} - (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+\rho _{44}), \end{aligned}$$

(1)

$$\begin{aligned} \dot{\rho _{22}}= & {} \frac{1}{2} i \Omega _ c (c_{23} \rho _{32}+c_{24} \rho _{42}-c_{32}\rho _{23}-c_{42}\rho _{24})\nonumber \\{} & {} - \gamma _p \rho _{22} + (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+\rho _{44}), \end{aligned}$$

(2)

where $\Omega _p$ and $\Omega _c$ are the Rabi frequencies of the probe and pump beams, respectively. $c_{ij}$ are the transition strengths defined by the dipole matrix elements for the transitions $i\rightarrow j$. $\Gamma$ (2$\pi$. 6 MHz for Rb atom) is the rate of spontaneous emission from the excited states and $\gamma _p$ is the depopulation rate due to the collisions with the wall (of the order of a few Hz in case of a coated cell). The introduction of decay rate $\gamma _a$ separates the process of absorption and transmission. It was assumed that the population decay rate due to the absorption of off-resonant classes of atoms is equal to the inverse of the time taken by atoms for a single pass through the bright zone. For an atom traveling in the bright zone with an average speed <v> with beam diameter d, decay rate $\gamma _a$ introduced in the equation was taken to be of the following form:

$$\begin{aligned} \gamma _a = \left\{ \begin{array}{ll} 0, &{} \text {for},\ \text {v} = \text {v} _r \\ \frac{<\text {v}>}{d}, &{} \text {for},\ \text {v} \ne \text {v}_r \end{array} \right. , \end{aligned}$$

(3)

where v is the axial velocity of the atoms, and v$_r$ is the resonant velocity class. There are two such classes $\frac{\delta _p}{k}$ and $\frac{\delta _p - \Delta }{k}$ with respect to the excited states F$_e$=2:$m_F=-1$ and F$_e$=1:$m_F=-1,$ respectively. $\delta _p$ is the detuning in the locked probe beam due to the application of the magnetic field, $\Delta$ is the frequency separation between the excited levels (157 MHz) and k is the wave vector.

Fig. 7

Numerical results showing the transmission for a single two-photon resonance (B = 30 G) for both uncoated (a) and (b) and OTS-coated cell (c) and (d) at two different values of probe Rabi frequency. The parameters were $\Omega _c=3 \Gamma$, $\gamma _p=\gamma _d=0.0001 \Gamma$,$\gamma _a=(120 \pi )^{-1} \Gamma$ with $\Gamma =2\pi .6$ MHz. $\Omega _p=0.1 \Gamma$ for (a) and (c) with, $\Omega _p=1 \Gamma$ for (b) and (d)

The transmission was evaluated as T($\delta$) $\propto$ Exp[-($\rho _{13}(\delta )+\rho _{14}(\delta )$)], where $\rho _{13}(\delta )$ and $\rho _{14}(\delta )$ are the Doppler averaged solutions of the equations (Appendix) evaluated at steady-state condition.

Fig. 8

Comparison between the experimental and numerical result for peak A for both coated and uncoated cells. a Uncoated cell, $\gamma _a=0$. b Coated cell, $\gamma _a=0.1$ MHz. The dotted and solid lines represent the numerical and experimental curves, respectively. The pump and probe intensities were fixed at 111.4 mW/cm$^2$ and 3.1 mW/cm$^2,$ respectively

For completeness, results (steady-state solutions) for both the cases of an uncoated and coated cell are shown in Fig. 7. Throughout the calculation, $\gamma _a$ for off-resonant classes was taken to be equal to 0.1 MHz for a beam diameter of 0.2 cm and average speed of 200 ms$^{-1}$. For uncoated cell, an absorption around two-photon resonance (Fig. 7a) was found for which $\gamma _a$ was not included in the equations. At the same set of parameters used for the uncoated cell, it was found that the addition of the decay rate $\gamma _a$ leads to the appearance of transmission in a coated cell as shown in Fig. 7c. Figure 7b and d shows the result found for uncoated and coated cells, respectively, at a slightly larger value of the probe intensity.

Figure 8 shows the comparison between the experimental result and the numerical one for peak A. Since all of the peaks can be compared by using the same model, the comparison is only included for the case of peak A. The theoretical results are in agreement with the experimental one. In summary, the population re-distribution which is dependent on the velocity now creates a scenario where the populations in F$_g$=1:m$_F$=-1 are all resonant with the excited states, which always leads to the transmission of the beam in the case of a coated cell.

5 Conclusion

D$_2$ line of $^{87}$Rb was investigated using the pump–probe spectroscopy under the EIT condition for an OTS-coated cell. The results were compared with those obtained for an uncoated cell at different intensities of the probe beam. In an uncoated cell, the experimental transmission spectra showed the presence of three VSOP dips with one asymmetric EIT peak. At the same intensity of the pump and probe beam, the VSOPs were quenched in the case of OTS-coated cell and a strong EIT was present with negligible asymmetry. With an increase in the intensity of the probe beam, prominent VSOP dips were observed along with asymmetric EIT for OTS-coated cells. For the case of high intensity, both the spectra for coated and uncoated cells had similar features. The difference in the time scale of the VSOP and EIT phenomenon was the reason behind the differences in the spectra at low intensity of the light fields. The study was further extended by applying a transverse magnetic field up to 30 G. Four EIT peaks were visible in the spectra, whose position matched with the two-photon resonance position for different systems which form due to the coupling of light fields with Zeeman states. A similar study for the case of an uncoated cell resulted in four absorption peaks in place of transmission for the low intensity of the probe beam. The difference in the spectra observed at a non-zero value of the magnetic field was due to the increase in the number of resonant classes of atoms inside the bright zone in the case of a coated cell. The study can find possible applications in experiments related to the optical switch, magnetometry, and quantum memory.

Appendix

A four-level system interacting with two lasers was taken as a model. Energy levels 1 and 2 are dipole forbidden (E$_{1}$ E$_{2}$). A locked weak probe excites the transitions 1$\rightarrow$3 and 1$\rightarrow$4 simultaneously, while a strong pump scans the transitions 2$\rightarrow$3 and 2$\rightarrow$4. The excited states 3 and 4 have a frequency separation of $\Delta$. The density matrix equations for the system were taken to be of the following form:

$$\begin{aligned} \dot{\rho _{11}}= & {} \frac{1}{2} i \Omega _ p (c_{13} \rho _{31}+c_{14} \rho _{41}-c_{31}\rho _{13}-c_{41} \rho _{14})\\{} & {} + \gamma _p \rho _{22} - (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+ \rho _{44}) \\ \dot{\rho _{12}}= & {} - \frac{1}{2} i [\Omega _ c (c_{32} \rho _{13}+c_{42} \rho _{14})-\Omega _p(c_{13}\rho _{32}\\{} & {} +c_{14} \rho _{42})- 2\delta \rho _{12}] - \frac{1}{2} (\gamma _p + \gamma _d) \rho _{12}\\ \dot{\rho _{13}}= & {} - \frac{1}{2} i [\Omega _ p (c_{13} \rho _{11} - c_{13} \rho _{33}- c_{14} \rho _{43})+\Omega _c c_{23}\rho _{12}\\{} & {} - 2(\delta _p- \Delta -k \text {v}) \rho _{13}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{14}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{11} - c_{14} \rho _{44}- c_{13} \rho _{34})+\Omega _c c_{24}\rho _{12}\\{} & {} - 2(\delta _p-k \text {v}) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{22}}= & {} \frac{1}{2} i \Omega _ c (c_{23} \rho _{32}+c_{24} \rho _{42}-c_{32}\rho _{23}-c_{42}\rho _{24})\\{} & {} - \gamma _p \rho _{22} + (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+\rho _{44})\\ \dot{\rho _{23}}= & {} - \frac{1}{2} i [\Omega _ c (c_{23} \rho _{22} - c_{23} \rho _{33}- c_{24} \rho _{43})+\Omega _p c_{13}\rho _{21}\\ {}{} & {} - 2(\delta _p-\Delta - k \text {v}-\delta ) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{24}}= & {} - \frac{1}{2} i [\Omega _ c (c_{24} \rho _{22} - c_{24} \rho _{44}- c_{23} \rho _{34})+\Omega _p c_{14}\rho _{21}\\ {}{} & {} - 2(\delta _p- k \text {v}-\delta ) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{33}}= & {} - \frac{1}{2} i [\Omega _ p (c_{13} \rho _{31} - c_{31} \rho _{13})+\Omega _c (c_{23}\rho _{32}\\ {}{} & {} -c_{32}\rho _{23})] - 2\Gamma \rho _{33}\\ \dot{\rho _{34}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{31} - c_{31} \rho _{14})+\Omega _c (c_{24}\rho _{32}\\ {}{} & {} -c_{32}\rho _{24}+(2 (\delta _p+k \text {v})-\Delta )\rho _{34})]\\ \dot{\rho _{44}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{41} - c_{41} \rho _{14})+\Omega _c (c_{24}\rho _{42}\\ {}{} & {} -c_{42}\rho _{24})] - 2\Gamma \rho _{44}, \end{aligned}$$

with $\rho _{ij}=(\rho _{ji})^*$. In the equations, $\delta _p$=$\omega _{44}-\omega _p$, $\delta$=$\delta _p-\delta _c$, k v is the Doppler shift, and $\gamma _d$ is the dephasing rate due to collisions (taken to be of the order of few Hz). It should be noted that the non-zero detuning in the probe beam with respect to the excited state where the excited states are within the Doppler width of the system is important for the re-distribution discussed in the article. Usually in a three-level system, two-photon absorption processes are resolved when the conditions $\delta _p$ $>>$ $\Gamma$ and $\Omega _c>>\Omega _p$ are satisfied. However, in four-level system, the condition $\delta _p$ $>>$ $\Gamma$ is not a strict one due to the presence of the nearby excited state. It can be argued that if the Doppler shift of light makes some atoms near resonant with respect to the excited state (say 4), the same class of atoms may be far resonant to the other excited state (3). Hence every class of velocity except the ones which satisfy the condition $\Vert \delta _p-k v\Vert \ll \Gamma$ can undergo two-photon absorption. If the frequency separation between the excited states is large as compared to the Doppler width, no such re-distribution can occur (for e.g., in D$_1$ line of $^{87}$Rb).