## Abstract

We investigated the relationship between two- and three-photon coherence in terms of the transition routes and coupling field intensities in a Doppler-broadened ladder-type atomic system for the 5S_{1/2}–5P_{3/2}–5D_{5/2} transition in ^{87}Rb atoms. Three-photon electromagnetically induced absorption (TPEIA) due to three-photon coherence was observed in the only transition route that exhibited a dominant two-photon coherence effect. We showed that two-photon coherence is a necessary condition for three-photon coherence phenomena. A comparison of the relative magnitudes of the electromagnetically induced transparency and TPEIA as a function of the coupling field intensity revealed that the increase of three-photon coherence was faster than that of two-photon coherence. Considering three-photon coherence in a Doppler-broadened ladder-type three-level atomic system, the relation between two- and three-photon coherence was numerically calculated.

© 2015 Optical Society of America

## 1. Introduction

Atomic coherence, which is generated by the interaction of an atom with coherent electromagnetic fields, has been studied in the field of atomic physics since coherent light sources were first developed. In particular, two-photon coherence phenomena due to the interaction of two coherent fields and a three-level atomic system, such as electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA), are widely regarded as important [1–4]. Two-photon coherence phenomena are understood as a quantum interference effect between two quantum transitions. Recently, quantum interference effects have been intensively studied in artificial atomic systems such as superconducting circuits [5], quantum dots [6], metamaterials [7], optomechanics [8], and nitrogen vacancy centers in diamond [9], because the EIT observed in an artificial atomic system has been proved to be quantum interference.

However, multi-photon coherence effects involving more than two photons, such as three-photon electromagnetically induced absorption (TPEIA), four-wave mixing, and six-wave mixing [10–19], are also very interesting. Pandey reported significant results regarding the role of different types of multi-photon coherence, such as those in three-level Λ-type, four-level N-type, and five-level M-type systems [14]. Ben-Aroya and Eisenstein reported high-contrast TPEIA in an N-type configuration of a three-level Λ-type system interacting with three separate electromagnetic fields and proposed the application of TPEIA to a small atomic clock [15]. Many studies of multi-photon coherence have been performed in degenerate two-level, three-level Λ-type, and four-level N-type atomic systems [10–15]. Basically, those studies considered three-level Λ-type atomic systems.

Comparing ladder-type and Λ-type atomic systems, multi-photon coherence in a ladder-type atomic system has been the subject of relatively little investigation [16–20]. The spectral features of a ladder-type atomic system differ from those of a Λ-type atomic system because the decay channels of a ladder-type atomic system differ from those of a Λ-type atomic system. Understanding the multi-photon coherence phenomena in a ladder-type atomic system is meaningful because of interesting applications in quantum optics [21–23]. Recently, TPEIA in ladder-type atomic systems was experimentally demonstrated and theoretically decomposed into two-photon coherence and three-photon coherence (TPC) components [19]. Whereas the important characteristic of EIT is transmittance at a two-photon resonance, TPEIA due to TPC exhibits absorption at a three-photon resonance [14, 19]. The even- and odd-photon coherence contributed to the transmittance and absorption spectra, respectively. Although EIT and TPEIA in a ladder-type atomic system have been observed and described using a simple atomic model [19], no findings on the relationship between two- and three-photon coherence in ladder-type atomic systems have been reported.

In this paper, we examine the relationship between two- and three-photon coherence in terms of the transition route and coupling intensity in a 5S_{1/2}–5P_{3/2}–5D_{5/2} ladder-type system of ^{87}Rb atoms. The spectral features of TPEIA in each transition between hyperfine states were investigated to confirm the relationship between two- and three-photon coherence. In addition, we measured the relative magnitudes of EIT and TPEIA as a function of the coupling field intensity to discuss the strength of TPC. To illuminate the experimental results for TPEIA and EIT, the relationship between two-photon coherence and TPC was numerically calculated using a Doppler-broadened ladder-type three-level atomic system.

## 2. Experimental setup

Figure 1(a) shows the three-level ladder-type atomic system interacting with a probe field (Ω_{p}) and two coupling fields (Ω_{C1} and Ω_{C2}) in the 5S_{1/2} –5P_{3/2} –5D_{5/2} transition of ^{87}Rb. ${\delta}_{p}$, ${\delta}_{C1}$, and ${\delta}_{C2}$ are the detuning frequencies from the resonances of the probe and the two coupling fields, respectively. The two-photon detuning is ${\delta}_{p}+{\delta}_{C1}$, and the three-photon detuning is ${\delta}_{p}+{\delta}_{C1}-{\delta}_{C2}$, where ${\delta}_{C}={\delta}_{C1}={\delta}_{C2}$. The decay channels and transition probabilities depend on the hyperfine states of the ladder-type atomic system, which consists of a ground state (5S_{1/2}), an intermediate state (5P_{3/2}), and an excited state (5D_{5/2}). The branching ratios of three-level ladder-type atomic systems determine the spectral features due to two-photon coherence.

Our experimental setup, shown in Fig. 1(b), is the same as that used for TPEIA in Doppler-broadened ladder-type atomic systems [19]. Two external cavity diode lasers (one for the Ω_{p} field and the other for Ω_{C1}, and Ω_{C2} fields) were independently operated at wavelengths of 780 nm and 775.8 nm, respectively. The linewidths of the two lasers were estimated to be less than 1 MHz. They were linearly polarized in the perpendicular direction, and the intensities of the probe and coupling lasers were 2.7 μW/mm^{2} and 6.5 mW/mm^{2}, respectively. The probe and coupling fields counter-propagate through an 5cm-long pure Rb atom vapor cell, and the mirror-reflected coupling field roles as the Ω_{C2} field whose power was adjusted by rotating quarter-wave plate. All of three fields completely overlap. The temperature of vapor cell was not controlled in a room-temperature. The effect of the Earth’s magnetic field was minimized by having the vapor cell be housed in three-layers μ-metal chamber. Although the two coupling fields (Ω_{C1} and Ω_{C2}) are counter-propagated like a standing-wave coupling field, TPC does not require spatial intensity modulation of the coupling fields. To measure the atomic coherence effects when the probe or coupling laser frequency is scanned, the probe field was measured using a photocurrent detector.

## 3. Experimental results and discussion

The ladder-type EIT and TPEIA of the 5S_{1/2}–5P_{3/2}–5D_{5/2} transition of ^{87}Rb are the transmittance due to two-photon coherence and absorption due to TPC, respectively, as shown in Figs. 2(a) and 2(b). To investigate the relationship between the EIT and TPEIA in terms of the transition routes, we observed the EIT and TPEIA spectra using both methods, scanning the probe and coupling laser frequencies, respectively. In particular, we focused on the spectral features at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition, considering a simple three-level ladder-type atomic system.

Figure 2(a) shows the narrow ladder-type EIT and TPEIA spectra with the Doppler background, where the probe laser frequency was scanned over ± 150 MHz at the resonance of the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition, and the coupling laser frequency was free running at the 5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition; the intensities of the probe and coupling lasers were 2.7 μW/mm^{2} and 6.5 mW/mm^{2}, respectively. As shown by the black curve in Fig. 2(a), the ladder-type EIT spectrum of this transition has a double structure, with a narrow EIT transmittance due to two-photon coherence and a broad transmittance due to a saturation effect [24, 25]. Interestingly on the EIT spectrum, we could observe the clear two steep side-peaks at the boundary between the EIT and the broad transmittance. These two side-peaks arise from the wavelength mismatch between probe and coupling lasers due to the Doppler effect. To date, it has been observed in several ladder-type atomic systems with large difference of wavelength between probe and coupling lasers [26, 27]. It is a bit difficult to observe the two side-peaks in our system since the two wavelengths 780 nm and 775.8 nm are close. This means that our experimental setup is useful for coherent spectroscopy in a ladder-type atomic system. We obtained the TPEIA spectrum by adding Ω_{C2}, as shown by the red curve in Fig. 2(a). The magnitude of the TPEIA is twice that of the EIT. We will discuss the relative magnitudes of the EIT and TPEIA as a function of the coupling field intensity in detail.

Figure 2(b) shows the narrow ladder-type EIT and TPEIA spectra without the Doppler background, where the coupling laser frequency was scanned around the 5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition, and the probe laser frequency was locked at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition. As mentioned above, the double structure of the ladder-type EIT is clearly apparent. Although the EIT spectrum in Fig. 2(a) is similar to that in Fig. 2(b) except for the Doppler background, the TPEIA spectrum in Fig. 2(b) differs from that in Fig. 2(a). In particular, the TPEIA in the Doppler-broadened atomic medium included the velocity-selective effect needed to satisfy the three-photon resonance condition. Because the probe laser frequency is fixed at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition, the only atom group with zero velocity contributed to the TPEIA spectrum. The spectrum obtained using coupling laser frequency scanning is useful for discussing the relationship between the EIT and TPEIA in detail in terms of the transition routes.

The well-known selection rule between hyperfine states, ΔF = 0, ± 1, determines the spontaneous decay paths. An atomic coherence, i.e., coupled atomic states with coherent fields, is significantly related to the decay rates between hyperfine states. Figure 3 shows the transition routes between hyperfine states and the TPEIA spectra according to the coupling laser detuning. For the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2)–5D_{5/2}(F″ = 1, 2, 3) transition, when the atoms are resonant with the coupling and probe lasers, the population of the 5S_{1/2}(F = 2) state may be depleted by single and double resonance optical pumping. However, the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition may be modeled as a simple three-level system, even though there is a transition channel from the 5D_{5/2} state to the 6P_{3/2} state. As mentioned above, the ladder-type EIT and TPEIA are strongly apparent in this transition.

Figure 3(b) shows the absorption spectra of the probe laser in each section according to the coupling laser detuning, where the probe laser frequency was scanned over the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2, 3) transition. When the coupling laser frequency is resonant with the 5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 3 or 4) transition, we observed TPEIA signals at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2)–5D_{5/2}(F″ = 3) and 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transitions (red and blue, respectively). However, in other two sections (green and black), TPEIA is not observed. In that cases, although the two-photon resonance of the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2) + ${\delta}_{p}$–5D_{5/2}(F″ = 1, 2, 3) and 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) + ${\delta}_{p}$–5D_{5/2}(F″ = 2, 3, 4) transitions is satisfied by the probe laser detuning ${\delta}_{p}$, there is no three-photon resonance, as shown in Figs. 3(a) and 3(b) [19].

Regarding the magnitude of the two TPEIA peaks of F″ = 3 and 4, the TPEIA signal of F″ = 4 is significantly larger than that of F″ = 3 because of the different decay routes. To illustrate the dependence of the TPEIA on a transition, we consider the relationship between two-photon coherence and TPC. A ladder-type EIT due to two-photon coherence is dominant at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition, whereas double resonance optical pumping (DROP) and two-photon absorption (TPA) effects are more dominant than EIT in the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2)–5D_{5/2}(F″ = 3) transition [28]. For a three-photon process to occur, two-photon coherence is necessary because of the interconnection between two-photon and three-photon processes [16].

Figure 4(a) shows the transition routes between hyperfine states obtained by scanning the coupling laser detuning, where the frequency of the probe laser was set to the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2 or 3) transition. In the case of the scanning the frequency of the coupling, there is always three-photon resonance as well as two-photon resonance at the hyperfine states of 5D_{5/2} state, as shown in Fig. 4(a). When the probe laser is resonant with the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2) transition, atoms in the 5S_{1/2}(F = 2) ground state can be optically pumped into the other ground state, 5S_{1/2}(F = 1). The 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2)–5D_{5/2} transition is an open ladder-type atomic system that includes decays into the other ground state, 5S_{1/2}(F = 1). In an open ladder-type atomic system, whether DROP or TPA occurs is determined by the decay rates of the hyperfine states [24]. Figure 4(b) shows the absorption spectra of both EIT and TPEIA configurations as a function of the coupling laser frequency detuning, where the probe laser frequency is resonant with the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2) transition. As shown in Fig. 4(b), although the transmittance signal in the EIT configuration of the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 2)–5D_{5/2}(F″ = 3) transition is weak, the TPEIA peak was clearly observed. However, the very weak two-photon coherence signal of the 5D_{5/2}(F″ = 2) state was transformed to weak TPEIA because of weak TPC.

When the frequency of the probe laser was fixed to the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition, we clearly observed the TPEIA peaks in the 5D_{5/2}(F″ = 3 and 4) state in Fig. 4(c). In contrast to the TPEIA spectra of Fig. 3(b), the signal of this 5D_{5/2}(F″ = 3) state changed from transmittance to absorption and the TPEIA spectra exhibit hyperfine structure. This reason is the velocity-selective effect needed to satisfy with the three-photon resonance condition according to the coupling laser detuning. Although the transmittance signal of the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 3) transition includes the DROP effect, two-photon coherence still occurs in this transition, and the two-photon coherence component contributed to the TPEIA signal. However, the signal in the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 2) transition is transmittance, because the DROP is the dominant effect for this transmittance signal. Comparing the ladder-type EIT and the TPEIA spectra in Figs. 4(b) and 4(c), TPEIA due to TPC was observed in the transition routes in which a dominant two-photon coherence effect appeared. Therefore, we could confirm that the condition for two-photon coherence is a good condition for TPC phenomena.

Atomic coherence depends significantly on the intensities of the interacting coherent fields. Two-photon coherence has been studied under conditions of a weak probe and a strong coupling laser [11–13]. The additional coupling field intensity needed for the three-photon process switches the phenomenon from EIT due to two-photon coherence to TPEIA due to TPC. We investigated the relationship between two-photon and three-photon coherence in terms of the spectral features according to the additional coupling intensity when the probe and coupling laser intensities were constant. Figure 5 shows the absorption spectra of the probe laser according to the additional coupling intensity, where the frequency of the probe laser was fixed at the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition, and that of the coupling laser was scanned over the 5P_{3/2}(F′ = 3)–5D_{5/2} transition. The two lasers were linearly polarized in the perpendicular direction, and the intensities of the probe and coupling lasers were 2.7 μW/mm^{2} and 6.5 mW/mm^{2}, respectively. For the 5D_{5/2}(F″ = 4) hyperfine state, EIT was transformed to TPEIA as the additional coupling laser intensity increased. At an additional coupling laser intensity of about 1.5 mW/mm^{2}, a crossover between EIT and TPEIA appears to occur. If the observed absorption spectrum is decomposed into one-photon, two-photon, and three-photon coherence components, the total spectrum is a linear combination of the three components [19]. This crossover signal can be understood as the sum of the two-photon and three-photon coherence components. Therefore, when the coherent channel for TPC is generated by the interaction with the additional coupling laser, TPC is added to the already generated and fixed two-photon coherence.

Comparing the relative magnitudes of the EIT and TPEIA as a function of the coupling field intensity, we discuss the dependence of the coupling laser intensity on the relative magnitudes of two-photon and three-photon coherence. Figures 6(a) and 6(b) show the ladder-type EIT spectra according to the coupling field intensity (Ω_{C1}) without Ω_{C2}, and the TPEIA spectra according to both coupling field intensities (Ω_{C1} and Ω_{C2}), respectively. As the intensity of Ω_{C1} increased, the narrow EIT component of the 5D_{5/2}(F″ = 4) state increased significantly compared with the other transmittance peaks of the 5D_{5/2}(F″ = 2 and 3) states, as shown in Fig. 6(a). When the intensities of both Ω_{C1} and Ω_{C2} increased simultaneously, the change in the TPEIA in Fig. 6(b) appears similar to that of the EIT. However, the magnitude of the TPEIA in the 5D_{5/2}(F″ = 4) state was approximately twice that of the EIT.

Figure 6(c) shows the relative magnitude of the EIT (blue squares) with that of normalized TPEIA (red circles) for 5D_{5/2}(F″ = 4) and growing slopes of the magnitude with exponential fit. The measured spectrums were averaged 4 times and each measurement was repeated more than 5 times. The errors of the data points in Fig. 6(c) were estimated to be from 0.01 to 0.03 which could be negligible. When the coupling laser power was increased, the magnitudes of both the EIT and TPEIA increased exponentially. In the regime of not strong intensities of coupling, the magnitude of the EIT is approximately proportional to Ω_{C1}^{2}, while the density matrix elements for TPEIA can be composed by terms which are proportional to Ω_{C1}^{4}, Ω_{C1}^{2}, Ω_{C2}^{2} and Ω_{C2}^{4}. Thus, regarding the relative magnitudes of the EIT and TPEIA as a function of the coupling field intensity, the TPEIA due to TPC grows faster than that of the EIT due to two-photon coherence. Therefore, we confirmed that the TPC in the ladder-type atomic system increases more rapidly than the two-photon coherence because of the additional coupling field (Ω_{C2}).

To understand the dynamics of the multi-photon coherence in ladder-type atomic systems, we employed a three-level atomic system considering TPC, as shown in Fig. 7(a). To consider the TPC of a ladder-type three-level atomic system, we assumed a modified three-level atomic system with two intermediate states ($|2\u3009$ and $|2\text{'}\u3009$). This model is the same as that of our previous work [19]. The decay rates (*γ*_{1} and *γ*_{2}) of the intermediate and excited states are 6 MHz and 0.97 MHz, respectively. The branching ratios of the intermediate and excited states are denoted as *b*_{1} and *b*_{2}, respectively. For the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–5D_{5/2}(F″ = 4) transition, the branching ratios are determined to *b*_{1} = 1 and *b*_{2} = 0.75, considering the decay channel of the 6P_{3/2} state.

The numerical results of the density matrix were averaged over the Maxwell–Boltzmann velocity distribution in the Doppler-broadened ladder-type atomic system; Fig. 7(b) shows the resulting calculated EIT spectrum (Ω_{p} = 0.4 MHz, Ω_{C1} = 10 MHz, and Ω_{C2} = 0 MHz) and TPEIA spectrum (Ω_{p} = 0.4 MHz, Ω_{C1} = 10 MHz, and Ω_{C2} = 8 MHz), considering the reflected coupling field Ω_{C2}. The calculated ladder-type EIT spectrum shows the double structure due to the EIT and a saturation effect. However, the spectral feature of TPEIA is narrow absorption with broad transmittance, which is in good agreement with the experimental result in Fig. 2(b). If the two-photon coherence terms in the density matrix elements are not considered, not only EIT but also TPEIA did not appear. Therefore, we confirmed that two-photon coherence is a precondition for TPEIA due to TPC.

We numerically investigated the transformation from EIT to TPEIA according to the additional coupling field Ω_{C2}, which is similar to the experimental results shown in Fig. 5. When the coupling field Ω_{C1} was fixed at 10 MHz, and Ω_{C2} was increased from zero to 8 MHz, the calculated spectra changed from EIT due to two-photon coherence to TPEIA due to TPC, as shown in Fig. 8(a). In addition, Fig. 8(b) shows the calculated TPEIA spectra for various Ω_{C1} and Ω_{C2}. In the experiment, Ω_{C2} was the coupling field Ω_{C1} reflected by a mirror after passing through a Rb cell. The intensity of Ω_{C2} decreased because of reflection by the cell windows and absorption of Rb atoms. When the ratio of Ω_{C2} to Ω_{C1} was set to 0.8, the calculated TPEIA spectra in terms of the Rabi frequency of the coupling fields are in good agreement with the experimental results in Fig. 6(b). Although the simple atomic model for considering TPC differs from the real atomic system with hyperfine structures and Zeeman sublevels, the observed spectra were shown to be in good agreement with the numerical calculations.

## 4. Conclusion

We investigated the relationship between two-photon and three-photon coherence, examining ladder-type EIT and TPEIA spectra in terms of the transition routes between the hyperfine states and the coupling laser intensity for the 5S_{1/2}–5P_{3/2}-5D_{5/2} transition of ^{87}Rb atoms. When we investigated the relationship between the EIT and TPEIA in terms of the transition routes, the TPEIA due to TPC was observed only in the transition routes that generated a dominant two-photon coherence effect. From the observed EIT and TPEIA spectra as a function of the probe or coupling laser frequency detuning, we confirmed that the condition for two-photon coherence was a good condition for TPC phenomena. When a coherent channel for TPC was generated by interaction with the additional coupling laser, the ladder-type EIT was transformed to TPEIA as the additional coupling laser intensity increased. This crossover spectrum between EIT and TPEIA was understood as a linear summation of the two-photon and three-photon coherence components. We also measured the relative magnitudes of the EIT and TPEIA as a function of the coupling field intensity; the increase of TPEIA due to TPC is faster than that of the EIT due to two-photon coherence. From this result, we could estimate the relationship between the two-photon and three-photon coherence and the coupling laser intensity. We believe that our results contribute to an understanding of the relationship between two-photon and three-photon coherence in ladder-type atomic systems.

## Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2012R1A2A1A01006579).

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