## Abstract

We present discrimination of the effect of one-photon and two-photon coherences in electromagnetically induced transparency for a three-level ladder-type atomic system. After the optical Bloch equations for a three-level atom, with either cycling or non-cycling transitions, were solved numerically, the solutions were averaged over the velocity distribution and finite transit time. Through this we were able to discriminate one-photon and two-photon coherence parts of the calculated spectra. We also found that the spectra showed peaks as the branching ratio of the intermediate (excited) state increased (decreased). The experimental results of previous reports [H. S. Moon, *et al.*, Opt. Express **16**, 12163 (2008); H. S. Moon and H. R. Noh, J. Phys. B **44**, 055004 (2011)] could well be accounted for by this discrimination of one-photon and two-photon coherences in the transmittance signals for the simplified three-level atomic system.

©2011 Optical Society of America

## 1. Introduction

Atomic coherence plays an important role in various spectroscopies including electromagnetically induced transparency (EIT) [1, 2], electromagnetically induced absorption (EIA) [3], and coherent population trapping (CPT) [4]. Of these, EIT in particular has drawn considerable interest since the discovery of this phenomenon [5] owing to its various potential applications such as precision magnetometers [6], slow light and light storage [7], and quantum information [8]. Although a basic underlying mechanism in EIT is known to be a two-photon atomic coherence resulting from a coherent interaction of the coupling and probe lasers with atoms, the optical pumping, in other words, the effect of one-photon resonance also affects the lineshapes in EIT significantly. The effect of optical pumping on the EIT lineshape was investigated in several papers, especially for the Λ-type atomic system [9–11].

The study of EIT spectra in ladder-type atomic systems has been reported for various atomic species such as Rb [12–17], Cs [18, 19], and Na [20]. From the perspective of trying to discriminate the effects of one-photon and two-photon resonances, to the best of our knowledge, only a few papers exist. Abi-Salloum presented a theoretical study of the interference between two competing pathways in three-level ladder atoms [21]. Recently, Hayashi *et al.* conducted an experimental and theoretical study on the interference between EIT and two-step excitation in three-level ladder systems [20]. Although two previous papers studied EIT in three-level atomic system, accurate discrimination of the effects of one-photon and two-photon resonances in the spectra was not reported.

Recently, studies of the effects of optical pumping and coherence in ladder-type EIT for the transitions 5*S*
_{1/2}−5*P*
_{3/2}−5*D*
_{5/2} [22] and 5*D*
_{3/2} [23] of ^{87}Rb atoms were carried out by the authors of this paper. We provided a satisfactory explanation of the dips and peaks in signals using a simple theoretical model. Based on experimental and theoretical observations in our previous reports, this paper presents a theoretical study of complete discrimination of one-photon and two-photon coherences in the spectra for a three-level ladder-type atomic system. In order to reveal the effect of the branching ratio in the intermediate or excited states, we assumed that the intermediate and excited levels possess other decay channels. We can understand the signals succinctly using the variation of the branching ratio at each level. This paper is organized as follows. Section 2 describes the method for calculating the contribution of the one-photon and two-photon coherences in EIT spectra. In Section 3, we discuss the calculated results for different branching ratios. The final section summarizes the results.

## 2. Theory

Figure 1(a) shows the energy level diagram for the transitions 5*S*
_{1/2}−5*P*
_{3/2}−5*D*
_{5/2,3/2} of ^{87}Rb atoms. Figure 1(b) shows the EIT signals, this figure was previously published in [22, 23], and many places. In Fig. 1(b), we observe three kinds of signals: signal (A) shows the double structure observed in the case of a transition from 5*P*
_{3/2}(*F′* = 3) to 5*D*
_{5/2}(*F″* = 4), which is attributed to the EIT effect [22]. The signal (B) is absorptive for the transition from 5*P*
_{3/2}(*F′* = 2) to 5*D*
_{3/2}(*F″* = 1), which was ascribed to a two-photon absorption signal [23]. The final one (C) is the peak for the other four signals. These transmittance signals were attributed to the effect of a double resonance optical pumping (DROP) [22–25]. In this paper, we accurately interpret the observed signals by means of discrimination of one-photon and two-photon coherences.

As can be seen in Fig. 1(b), there exists a variety of signals in the spectra. This variety mainly results from the branching ratio of the excited or intermediate levels. In the case of signal (A) in Fig. 1(b), the branching ratio of the intermediate state [5*P*
_{3/2}(*F′* = 3)] to the ground state [5*S*
_{1/2}(*F* = 2)] is *b*
_{1} = 1, whereas that of the excited state [5*D*
_{5/2}(*F″* = 4)] to the intermediate state [5*P*
_{3/2}(*F′* = 3)] is *b*
_{2} = 0.74. Although we mentioned that transition 5*S*
_{1/2}(*F* = 2)–5*P*
_{3/2}(*F′* = 3)−5*D*
_{5/2}(*F″* = 4) was cycling, accurately speaking, the excited state has a decay channel to 6*P*
_{3/2}. However, since the branching ratio is quite large and the transition strength is strong, the assumption of a cycling transition is still approximately correct. In the case of signal (B) in Fig. 1(b), the branching ratios are *b*
_{1} ≠ 1 and *b*
_{2} ≠ 1. The branching ratios for the signals (C) in Fig. 1(b) are *b*
_{1} = 1 and *b*
_{2} ≠ 1. The difference between (A) and (C) lies in the magnitude of *b*
_{2}. In the case of (C), *b*
_{2} is much smaller than 0.74. In Fig. 1(b), we can say, roughly, that when *b*
_{1} = 1 (*b* ≠ 1) we have transmittance (absorption) signals. The discussion for these observations will be given later.

In order to study the effect of the branching ratios on the EIT spectra, we employed a simplified three-level ladder-type system, as shown in Fig. 2. In Fig. 2, the excited, intermediate, and ground states are denoted by |*e*〉, |*i*〉, and |*g*〉, respectively. The resonant wavelength between the states |*i*〉 and |*g*〉, which is the probe line, is *λ*
_{1}. In our calculation *λ*
_{1} was chosen as 780.2 nm, which is the resonant wavelength for the 5*S*
_{1/2}–5*P*
_{3/2} transition line in Rb atoms. The probe laser frequency is scanned near this transition line. The coupling laser of the wavelength *λ*
_{2} is fixed at the transition between *|e*〉 and *|i*〉. In our calculations *λ*
_{2} was 775.8 nm, which is the resonant line for 5*P*
_{3/2}−5*D*
_{3/2,5/2}. The decay rates of the excited and the intermediate states are *γ*
_{2}(= 2*π* × 0.97 MHz) and *γ*
_{1}(= 2*π* × 6 MHz) [26], respectively. The branching ratios of the excited and intermediate states are defined to be *b*
_{2} and *b*
_{1}, respectively. Therefore, when *b*
_{1} = *b*
_{2} = 1, the three-level system is cycling, whereas the system becomes open when either *b*
_{1} ≠ 1 or *b*
_{2} ≠ 1.

Using the usual density matrix equation, the optical Bloch equations for each density matrix element can be derived (e.g. Eq. (1) in Ref. [13]). From the optical Bloch equations, the equations for the populations are

_{1}(Ω

_{2}) is the Rabi frequency of the probe (coupling) beam. In the optical Bloch equations,

*p*denotes the population of the state |

_{j}*j*〉 and

*ρ*is the coherence between the states |

_{j j′}*j*〉 and |

*j′*〉 with

*j, j′*=

*e,i,g*. The absorption coefficient of the probe beam is then given by

*δ*

_{p}(=

*ω*

_{1}−

*ω*

_{10}) and

*δ*

_{c}(=

*ω*

_{2}−

*ω*

_{20}) are the detunings of the probe and coupling lasers, respectively, where

*ω*

_{1(2)}= 2

*πc*/

*λ*

_{1(2)}and

*ω*

_{10(20)}are the resonant frequencies between the states |

*i*〉 and |

*g*〉 (|

*e*〉 and |

*i*〉).

*δ*

_{1(2)}is the effective detuning of the probe (pump) beam felt by an atom moving at velocity

*v*, and is given by

*δ*

_{1(2)}=

*δ*

_{p(c)}−

*k*

_{1(2)}

*v*. While we have

*k*

_{1}= 2

*π/λ*

_{1}for both schemes, we assume

*k*

_{2}= +(−)2

*π/λ*

_{2}for the copropagating (counterpropagating) scheme.

Since the absorption coefficient at a given time *t* in Eq. (5) is now given by *α*
_{0}(*δ _{p}* –

*k*

_{1}

*v*,

*δ*–

_{c}*k*

_{2}

*v*,

*t*), it must be averaged over the Maxwell-Boltzmann velocity distribution as follows [27]:

*u*= (2

*k*

_{B}*T/M*)

^{1/2}is the most probable speed (

*T*= temperature of the vapor cell;

*M*= the mass of an atom). Finally, the signals must be averaged over the interaction time while the atoms are crossing the laser beam as follows: $\alpha \hspace{0.17em}=\hspace{0.17em}\left(1/{t}_{\text{av}}\right)\hspace{0.17em}{\int}_{0}^{{t}_{\text{av}}}{\alpha}_{1}\hspace{0.17em}(t)\mathit{\text{dt}}$, where ${t}_{\text{av}}\hspace{0.17em}=\hspace{0.17em}\left(=\left(\sqrt{\pi}/2\right)d/u\right)$ is the average transit time with

*d*being the diameter of the laser beam [28].

Since the transmittance is given approximately by *e*
^{−αl} where *l* is the length of the cell, it is roughly proportional to Im*ρ _{ig}*. In what follows, we calculate and discuss the transmittance in terms of Im

*ρ*. Using the rate equation approximation and solving the steady-state solutions of Eq. (4), Im

_{ig}*ρ*can be expressed in terms of populations (

_{ig}*p*and

_{i}*p*) and coherence (

_{g}*ρ*) as follows:

_{eg}Diagrams of typical pathways of interaction responsible for the effects of one-photon and two-photon coherences are shown in Figs. 3(a) and 3(b), respectively. In Fig. 3(a), the diagrams for Ω_{1},
${\Omega}_{1}^{3}$, and also the higher interactions of Ω_{1} (not shown) represent the background signal of the probe beam. The diagram for
${\Omega}_{1}^{3}{\Omega}_{2}^{2}$ denotes the two-step excitation term, which is the dominant interaction for determining the shape of the signals in one-photon resonance. Figure 3(b) shows two important pathways in the two-photon coherence term. The first one denotes the EIT term originating from the destructive interference between the two pathways from the ground state to the intermediate state, and is proportional to Ω_{1} [13, 14]. In Fig. 3(b), only the interaction for
${\Omega}_{1}{\Omega}_{2}^{2}$ is shown, and other higher interactions of Ω_{2} are not shown. The next one for
${\Omega}_{1}^{3}{\Omega}_{2}^{2}$ represents a part of the two-photon absorption (TPA) term. Again, other higher interactions of Ω_{2} are not shown. It should be noted that the interaction of five photons for the term
${\Omega}_{1}^{3}{\Omega}_{2}^{2}$ occurs via either one-photon or two-photon coherence term. Therefore, this term exists in the one-photon and two-photon coherence parts simultaneously. Thus, the two-photon coherence term in Eq. (6) can be further decomposed into two parts: one is the EIT term (∝ Ω_{1}) and the other is the TPA term (∝
${\Omega}_{1}^{3}$). Since the contribution of the higher interactions of
${\Omega}_{1}^{n}$ (*n* ≥ 5) is much smaller than the EIT term, we define all interactions except EIT (∝ Ω_{1}) as TPA term roughly.

The coherence *ρ _{eg}* on the right-hand side of Eq. (6) can be expressed in terms of the populations from the steady-state solutions of Eqs. (2)–(4). Thus, the second term on the right-hand side of Eq. (6) is given by:

_{0}≡ 2(

*δ*

_{1}+

*δ*

_{2}) +

*iγ*

_{2}, Δ

_{1}≡ 2

*δ*

_{1}+

*iγ*

_{1}, and Δ

_{2}≡ 2

*δ*

_{2}+

*i*(

*γ*

_{1}+

*γ*

_{2}). Therefore, the EIT term, which is proportional to Ω

_{1}can be obtained by setting

*p*=

_{e}*p*= 0 and

_{i}*p*= 1 in Eq. (7). Thus we can obtain the explicit form of the contribution of the EIT effect in Eq. (6) as follows:

_{g}## 3. Calculated Results

We study the validity of the discrimination between the two contributions in Eq. (6) by averaging the solutions of the optical Bloch equations over the various velocities and atomic transit times. Figure 4(a) shows the spectra as a function of *δ _{p}*, while

*δ*is fixed at 0, 250 MHz, 500 MHz, and 750 MHz, and where the lasers are set in a counterpropagating scheme. In Fig. 4, the Rabi frequencies were Ω

_{c}_{1}= 0.2

*γ*

_{1}and Ω

_{2}= 2.5

*γ*

_{2}, and the diameter of the laser beam was 1.5 mm. When the coupling laser is fixed at the resonance frequency, we could find a striking contribution from the two-photon coherence term in the signal. As the frequency of the coupling laser is detuned far from the resonance while at the same time the two-photon resonance condition is maintained, the one-photon resonance part is suppressed and becomes the same as that of the background signal. This is because the one-photon resonance part results from the single resonance between the ground and the intermediate states. In contrast, we found that the signal for the copropagating scheme was mainly composed of the one-photon resonance part, as shown in Fig. 4(b). It is quite reasonable that the effect of two-photon coherence is large for the counterpropagating scheme and small for the copropagating scheme. This is because the large effect of two-photon coherence for the counterpropagating scheme results from the fact that the sub-Doppler resonance condition can be satisfied for quite a broad range of velocity. The results in Fig. 4 verify that the discrimination of the one-photon and two-photon coherence parts in Eq. (6) is correct.

In order to study the phenomena shown in Fig. 1(b) in ladder-type three-level atoms, we calculated the imaginary part of the coherence between the intermediate and the ground states by varying the values of *b*
_{1} and *b*
_{2}. Figure 5(a) [5(b)] shows the signals and their one-photon resonance part [two-photon coherence part] as a function of *δ _{p}* where

*δ*= 0. The opposite case, the scanning of

_{c}*δ*, shows similar behavior to Fig. 5. The Rabi frequencies were Ω

_{c}_{1}= 0.2

*γ*

_{1}and Ω

_{2}= 2.5

*γ*

_{2}, and the diameter of the laser beam was 1.5 mm. In Fig. 5(a), when the transition line for the probe beam is cycling (

*b*

_{1}= 1), and as the branching ratio of the excited state (

*b*

_{2}) decreases (from left to the right at the lowest row), i.e., the leakage increases, the magnitude of the transmittance signal increases. At the same time, the relative contribution of the one-photon resonance effect does not change significantly. In contrast, as shown in the first panel of Fig. 5(a), with

*b*

_{2}= 1, as the branching ratio of the intermediate state into the ground state (

*b*

_{1}) decreases, the transmission peak changes into a transmission dip very quickly. In particular, the contribution of the one-photon resonance part shows abrupt variation.

From Eq. (1), we have the following equation for the populations: *ṗ _{e}* +

*ṗ*+

_{i}*ṗ*= − (1 −

_{g}*b*

_{1})

*γ*

_{1}

*p*− (1 −

_{i}*b*

_{2})

*γ*

_{2}

*p*. Therefore, the decay rate of the leakage of the total population is given by

_{e}*δ*= ∞ and

_{p}*δ*= 0. Although the results with the scanned

_{p}*δ*are shown in Fig. 5, it is more convenient to explain the behavior of the spectrum by using a scanned

_{p}*δ*and a fixed

_{c}*δ*= 0 configuration. Assuming

_{p}*δ*= 0, let us consider the populations at

_{p}*δ*= ∞ and at the atomic transit time. As the detuning approaches

_{c}*δ*= 0,

_{c}*p*decreases, while

_{i}*p*increases due to the increased transition rate. Therefore, as can be seen in Eq. (9), the first term decreases and the second term increases. Thus, at

_{e}*b*

_{2}= 0 and

*b*

_{1}≠ 0, i.e., in the presence of leakage from the intermediate state, the leakage of the population at

*δ*= 0 is weaker than at

_{c}*δ*= ∞. Therefore, the population at

_{c}*δ*= 0 is greater than that at

_{c}*δ*= ∞. Accordingly, the absorption becomes larger, and the transmission becomes smaller. Thus, we observe the dip signals as shown in Fig. 5(a). In addition, at

_{c}*b*

_{2}≠ 0 and

*b*

_{1}= 0, we can do a similar analysis to before. Therefore, we can observe the peak signal. To sum up, the behavior of the one-photon resonance component of the spectra shown in Fig. 5(a) can be approximately explained in terms of the population leakage. The results in Fig. 5 show also that the branching ratio of the intermediate state (

*b*

_{1}) plays an important role in forming the shape of the signal.

The two-photon coherence contribution of the total spectrum is presented in Fig. 5(b). Further discrimination of the two-photon coherence part using EIT and TPA is presented in Fig. 5(c). When the system is cycling (*b*
_{1} = *b*
_{2} = 1), we observe a transmittance signal, which is composed of transmittance (EIT) and an absorptive (TPA) signals. As the leakage of the population increases, the spectrum shows dip signals. As the leakage increases further, the dip signals also increase. Unlike the behavior of the one-photon resonance part, the two-photon component shows dip signals regardless of the branching ratio. Because the strength of the EIT is independent of branching ratio, it remains constant regardless of the branching ratios. In contrast, the dip signal in the TPA part increases more as the leakage increases (branching ratios decrease). This is because the two-photon absorption process is expedited as the decay into states other than the ground state increases. The width of the two-photon resonance signal looks comparable to that of the one-photon resonance signal. This is because of the slight difference in the wavelengths of the probe and the coupling lasers. When *δ _{p}* = 0, two-photon resonance occurs at the frequency of
${\delta}_{c}\hspace{0.17em}=\hspace{0.17em}2\pi ({\lambda}_{1}^{-1}\hspace{0.17em}-\hspace{0.17em}{\lambda}_{2}^{-1})v$ for atoms moving at a velocity of

*v*. Therefore, considering the Maxwell-Boltzmann velocity distribution, we observe a broad two-photon resonance signal.

As can be seen in Fig. 5(a), the two-photon absorption contribution is much weaker than the contribution of the one-photon resonance signal when the intermediate state is almost cycling, i.e. *b*
_{1} ≃ 1. This corresponds to all signals except for (B) in Fig. 1(a). Therefore, we conclude that the peak signals in Fig. 1(a) are mostly composed of the one-photon resonance component. In contrast, in the case of signal (A), because of the strong transition strength, the two-photon coherence component exists as a relatively narrow signal on top of the broad one-photon resonance signal. As explained above, because the two-photon coherence signal is not so much narrower than the one-photon resonance signal, the two-photon resonance component contributed to the broad signal in signal (A) in Fig. 1(a) to a certain extent. If we are to discriminate the two-photon component accurately in signal (A) in Fig. 1(a), it is necessary to accurately calculate the signal by taking into account all the substates of the energy level. When the intermediate state has a decay rate (*b*
_{1} ≠ 1), we have a dip signal. This corresponds to signal (B) in Fig. 1(a). As can be seen in Fig. 5(a), both the one-photon and two-photon coherence components contribute to the overall dip signal.

## 4. Conclusions

In this paper we described a theoretical study of EIT spectra for ladder-type three-level atoms. The signals are decomposed into two-parts: one is the one-photon resonance part and the other is the two-photon coherence part. We showed that the two-photon coherence part exhibited absorption and increased as the leakage to other states increased. For fixed values of the branching ratios, as the coupling laser intensity was increased, the EIT effect surpassed by the two-photon absorption effect in the two-photon resonance part of the signals. In contrast to the two-photon coherence part, the one-photon resonance part is dependent on the branching ratios. As the branching ratio of the intermediate (excited) state increased (decreased), the signals were transformed to peaks from dips. This interesting phenomenon can be understood by comparing the leakage of the populations at *δ _{c}* = ∞ and

*δ*= 0. The total population at

_{c}*δ*= 0 is smaller than that at

_{c}*δ*= 0 when the excited (intermediate) state is not cycling. Therefore, the signal peaked as the branching ratio of the intermediate (excited) states increased (decreased). In order to more accurately explain these phenomena, a more elaborate calculation for real atoms is necessary. This calculation is currently being undertaken.

_{c}## Acknowledgments

This work was supported by the Korea Research Foundation Grant funded by the Korean Government( KRF-2008-313-C00355 and 2009-0073051). Also, this work was supported by the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government(MOST) (No. R01-2007-000-11636-0).

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