## Abstract

We investigate stimulated four-wave mixing (FWM) in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of warm ^{87}Rb atoms. Here, the telecommunication wavelength (1529 nm) of the 5P_{3/2}–4D_{5/2} transition is nearly twice that of the 5S_{1/2}–5P_{3/2} transition (780 nm). The observed FWM signals of the 5P_{3/2}–4D_{5/2} transition indicate that the FWM process is significantly influenced by the two-photon Doppler broadening due to the wavelength difference between both transitions and the double-resonance optical pumping (DROP) effect due to two-step excitation. We elucidate the suppression of the FWM process due to the DROP effect using a simple six-level atomic model.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Nonlinear optical processes are at the heart of photon-pair-source development for photonic quantum information processing and quantum communications [1–18]. In this context, the parametric down-conversion process with nonlinear crystals [1–3] and the four-wave mixing (FWM) process with atomic ensembles have been exploited for the development of high-performance quantum photonic sources [4–18]. In particular, for long-distance quantum communication and long-lived atomic memory, it is essential to generate telecommunication-wavelength photons with narrow bandwidths for interactions with atoms [15–18]. In this regard, photon-pair generation from cold or warm atomic ensembles has been intensively studied [4–18].

In the above context, the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of ^{87}Rb atom is interesting atomic system as it covers a resonance frequency of the telecommunication wavelength (1529 nm). Previous studies on cascade-type atomic systems have reported on various two-photon coherence phenomena resulting from the interaction of an atom with two coherent electromagnetic fields, as in the cases of electromagnetically induced transparency (EIT), double-resonance optical pumping (DROP), and two-photon absorption (TPA) [19–24]. However, it is necessary to consider the Doppler effect of the two-photon resonance in a warm atomic ensemble. Particularly in the case of the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition, we note that the telecommunication wavelength of the 5P_{3/2}–4D_{5/2} transition between the upper excited states is approximately twice that of the 5S_{1/2}–5P_{3/2} transition. Because of the wavelength difference between both transitions in this cascade-type atomic system, the Doppler shift of an atomic velocity group is double that of the upper and lower transitions, and a narrow EIT signal cannot be observed. Meanwhile, the Doppler-free transparency spectrum with the hyperfine structure of the 4D_{5/2} state has been understood as being due to DROP, wherein the population of one ground state is optically pumped into another ground state through an intermediate state after excitation. Although photon-pair generation via spontaneous FWM processes in this cascade-type atomic system has been studied [16–18], the spectral features of enhanced FWM nonlinearity in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition with the two-photon Doppler shift and the DROP have not been analyzed thus far. Furthermore, the FWM signals in this transition have not thus far been reported to the best of our knowledge. It is not easy to understand the physics of the spectral features of FWM of this transition in the Doppler-broadened atomic ensemble. The properties of FWM signal is important for understanding the characteristics of generated photon-pair such as the biphoton spectral and temporal waveforms of the photon pair via the spontaneous FWM process.

In this study, we investigate the Doppler-broadened FWM under DROP in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition in a warm ^{87}Rb atomic ensemble. We examine how the two-photon Doppler shift and DROP affect the spectral features of the FWM signal. To understand the causes underlying the FWM spectral features in this transition, the FWM spectrum is numerically calculated and decomposed into the pure two-photon coherence term dominantly related to the enhanced FWM.

## 2. Experimental setup for FWM light generation from warm ^{87}Rb atomic ensemble

The enhancement of FWM light in a cascade-type atomic system requires two-photon coherence between the two states interacting with the pump and coupling lasers, because FWM light generation is based on stimulated emission via two-photon coherence. The FWM light is induced by the driving laser under the phase-matching condition. Figure 1(a) shows the energy diagram of the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of ^{87}Rb [25]. The pump and driving lasers interact with the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition at a wavelength of 780.2 nm, and the coupling laser interacts with the 5P_{3/2}(F′ = 3)–4D_{5/2}(F″ = 2, 3, 4) transition at a wavelength of 1529.4 nm. Parameters δ_{p}, δ_{d}, and δ_{c} denote the detuning frequencies of the pump, driving, and coupling lasers, respectively. The natural linewidths of the 5P_{3/2} and 4D_{5/2} states are 6.1 MHz and 1.7 MHz, respectively [26,27].

In this transition, the wavelength ratio of the pump and coupling lasers is ∼2. Here, the *two-photon Doppler broadening* (Δ*ω*_{two}) is defined as follows [28]:

*k*

_{p}and

*k*

_{c}denote the wave vectors of the pump and coupling lasers, respectively, and Δ

*v*is the width of the atom velocity distribution. The Δ

*ω*

_{two}value at Δ

*v*= 270 m/s is calculated as ∼170 MHz, which is 100 times the natural linewidth of the 4D

_{5/2}state. Because the FWM process is strongly correlated with the two-photon coherence, only the velocity classes within the spectral width of the two-photon resonance can coherently contribute to FWM enhancement. Therefore, the atoms of the velocity classes contributing to the FWM process in the Doppler-broadened atomic ensemble significantly decrease in number. Although the pump and coupling lasers propagate in opposite directions, as shown in the experimental setup in Fig. 1(b), the two-photon resonant condition in this transition cannot be Doppler-free.

The experimental setup for the generation of FWM light with a warm atomic vapor cell of the cascade-type atomic system is similar to that described in previous studies [29–31]. The warm atomic ensemble is a 12.5-mm-long vapor cell containing the ^{87}Rb isotope, housed in μ-metal chambers for shielding against the earth’s magnetic field. In our study, the temperature of the warm atomic ensemble was maintained at 55°C. For our experiment, we used two external cavity diode laser (ECDL) sources at wavelengths of 780.2 nm and 1529.4 nm, with the pump and driving lasers sourced from a single ECDL. The *e*^{-2} full width value of the pump laser was 1.2 mm and that of the coupling laser was 0.8 mm. In the setup, both lasers were focused on the vapor cell with two lenses with a focal length of 300 mm. The waists of the focused pump and coupling beams were estimated to be 60 μm and 180 μm, respectively. The polarizations of the pump and coupling laser were mutually orthogonal and linear polarized. The vertically polarized pump and horizontally polarized coupling lasers were counter-propagated and spatially overlapped in the atomic vapor cell. We measured the transmittance spectrum (DROP signal) of the pump laser using a photodiode (PD1), and the mutually orthogonal polarized FWM signal with the linearly polarized driving laser by using a polarizing beam splitter (PBS) and a second photodiode (PD2).

FWM light can be generated in the 5P_{3/2}–4D_{5/2} transition between the upper excited states under the phase-matching condition of the energy conservation and wave-vector conservation of the contributing lasers. In our experiment, when the two-photon resonance condition with the counter-propagating pump and coupling lasers is satisfied, the propagating direction of the generated FWM is determined by the tilt angle (*θ _{d}*) of the driving laser. The phase-matching function $\varphi ({} )$ [17] is related to the propagating angle (

*θ*) of the generated FWM as follows:

*k*denotes the wave-vector mismatch of the four fields, i.e., $\Delta k = {k_p} + {k_c} - {k_d} - {k_F}$, with ${k_{p,c,d,F}}$ denoting the wave vectors of the pump, coupling, driving, and FWM fields, respectively, and

*L*denotes the length of the

^{87}Rb vapor cell.

Figure 2 shows the calculated phase-matching function as a function of the FWM propagating angle for *θ _{d}* values ranging from 0 to 2°. In our experiment, the driving laser is tilted at 1° relative to the propagation direction of the pump laser, and the propagating angle (

*θ*) of the generated FWM is calculated as ∼1.4°, with an acceptance angle width of 0.25°. Here, we note that because the wavelength difference between the pump and coupling lasers is large, the acceptance angle width of the generated FWM is narrow. Therefore, we can set up the experiment for FWM signal generation using the experimental configuration shown in Fig. 1, based on the calculated result.

## 3. Experimental results and discussion

In the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of ^{87}Rb, the Doppler shift of the two-photon coherence is significantly larger than the spectral width of the two-photon resonance. The DROP signal is not due to EIT effect, but to optical pumping effect from one ground state through an intermediate state to another [24]. Furthermore, the ratio of the frequency interval in the energy diagram of Fig. 1(a) to that of the DROP peaks with frequency scanning of the pump laser is the same as the ratio of the Doppler shifts of the coupling and pump lasers (*k*_{p}:*k*_{c} = 1:1.96). As mentioned above, the FWM process is strongly correlated with two-photon coherence because of the nonlinear optical process enhancement via two-photon coherence. At this point, we consider two questions: (1) Is it possible to observe the FWM spectrum in this transition under the conditions of two-photon Doppler broadening and DROP? (2) How does the varying Doppler shift of the probe and coupling lasers affect the spectral features of the FWM spectrum?

Figure 3(a) shows the FWM (blue curve) and DROP (gray curve) signals for the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–4D_{5/2}(F^{″} = 4, 3, 2) transition as a function of the detuning frequency (δ_{c}) of the coupling laser. The powers of the coupling, pump, and driving lasers were 52 μW, 80 μW, and 100 μW, respectively. In the figure, the DROP peaks with the hyperfine structure (F^{″} = 4, 3, 2) of the 4D_{5/2} state are observed on two-photon resonance. Interestingly, we clearly observe that the asymmetric FWM spectrum is suppressed in the region of the DROP peaks. The two-photon coherence effect is dominant in the two-photon cycling transition of 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3)–4D_{5/2}(F^{″} = 4). When δ_{c} is varied over the scanning range, the two-photon resonance is Doppler-broadened owing to the presence of atoms of different velocity classes in the Doppler-broadened atomic ensemble, where ${\delta _c} = ({k_c} - {k_p}) \cdot v$. The FWM signal in this transition can be understood as being stimulated via the interaction of the driving laser with the atomic medium under weak residual pure two-photon coherence. Therefore, we can elucidate the FWM spectrum in Fig. 3(a) as a fraction of the Doppler-broadened FWM spectrum with DROP suppression.

Next, we note that the ratio of the Doppler shifts of the coupling and pump lasers is 1.96, which may affect the spectral features of the FWM spectrum. Figure 3(b) shows the FWM spectrum as a function of the detuning frequencies of the pump and driving lasers when the frequency of the coupling laser is fixed at the 5P_{3/2}(F′ = 3)–4D_{5/2}(F^{″} = 4) transition. Compared with the FWM spectrum in Fig. 3(a) under the same experimental parameters, we observe that the FWM spectral shape is very similar; however, the spectral broadening is approximately double. Thus, we can confirm that the Doppler-broadening and different Doppler-shift effects due to the different atomic velocity groups affect the spectral features of the FWM spectrum, as shown in Fig. 3(b).

To explain the spectral features of the FWM signals in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition, we theoretically investigated the FWM process using a simple six-level cascade-type atomic model, as shown in Fig. 4(a). Our six-level atomic model is composed of a ground state ($|1 \rangle$), two degenerate intermediate states ($|2 \rangle$ and $|3 \rangle $), and three excited states ($|4 \rangle$, $|5 \rangle$, and $|6 \rangle$) that account for the hyperfine structure (F^{″} = 4, 3, 2) of the 4D_{5/2} state. The density matrix equation of motion in this case can be described as

*i*and

*j*indicate the $|i \rangle $ and $|j \rangle $ states, respectively. Moreover, ${\rho _{ij}}$ denotes the density-matrix element and ${H_{ij}}$ denotes the effective interaction Hamiltonian, which is composed of the atomic and interaction Hamiltonians. Furthermore, ${\Gamma _{ij}}$ denotes the relaxation term describing the relaxation processes from the $|j \rangle $ state to the $|i \rangle $ state. To consider the DROP effect from the excited 4D

_{5/2}state to another ground state ($|{1^{\prime}} \rangle$), we additionally include the relaxation terms of ${\gamma _j}$ (

*j*= 4, 5, 6) of the three excited hyperfine states, which are dependent on the open degree of the hyperfine states. The branching ratio of each transition depends on the transition routes, as governed by the selection rule. However, we don’t include the $|{1^{\prime}} \rangle$ state in the atomic model for simulation, because the $|{1^{\prime}} \rangle$ state does not interact with lasers. The electric-field amplitude of the generated FWM signal is proportional to the coherence term ${\rho _{34}}$ because of the contribution of the cycling transition. Therefore, the FWM signal is proportional to ${|{{\rho_{34}}} |^2}$. Under the weak field condition (${\Omega _P},{\Omega _d} \ll {\Omega _c}$) in the four-level atomic model considering only the excited state ($|4 \rangle$), ${\rho _{34}}$ can be simplified as [30]

_{p}, Ω

_{c}, and Ω

_{d}, and the corresponding detuning frequencies as ${\delta _p}$, ${\delta _c}$, and ${\delta _d}$, respectively. The decay rates of the intermediate and excited states are 6.1 MHz (Γ

_{12}= Γ

_{13}= 3.05 MHz) and 1.7 MHz (Γ

_{2j}+ Γ

_{3j}+ γ

*= 1.7 MHz), respectively. Because the analytical equation of Eq. (4) is limited to weak-pump-intensity, we numerically investigated the spectral features of the FWM using the density matrix equation of Eq. (3) and incorporating the Maxwell–Boltzmann velocity distribution to consider the case of a Doppler-broadened atomic medium.*

_{j}First, to understand the spectral features of the FWM signal using the simplified atomic model, we consider only the excited hyperfine state $|4 \rangle$, corresponding to the four-level atomic model. The calculated FWM spectrum in the blue curve in Fig. 4(b) has a broad FWM spectral shape due to two-photon Doppler broadening and a central dip relating to FWM suppression due to the DROP effect (red curve in Fig. 4(b)), where Ω_{p}/2π, Ω_{c}/2π, and Ω_{d}/2π are set to 10 MHz, 15 MHz, and 10 MHz, respectively. To elucidate the components contributing to the FWM signal, we decomposed the calculated FWM signal into the cascade-type (red curve) and V-type (pink curve) two-photon components, as shown in Fig. 4(c). From the figure, we can intuit the relation between the FWM signal and both two-photon coherence components, that is, the cascade (${\rho _{14}}$) and V (${\rho _{23}}$)-type two-photon coherence components. From the calculated result in Fig. 4(c), we observe the narrow FWM via the cascade-type two-photon coherence contribution and the broad FWM via the V-type two-photon coherence contribution.

However, the FWM spectrum in Fig. 4(b) calculated with the simple four-level atomic model does not satisfactorily agree with the observed FWM signal in Fig. 3(a). Next, we consider the hyperfine structure (F^{″} = 4, 3, 2) of the 4D_{5/2} state with the six-level atomic model shown in Fig. 4(a). In this case, the hyperfine splittings of the 4D_{5/2} state are applied as in the energy-level diagram in Fig. 1(a). The relaxation terms of γ_{4} (for F^{″} = 4), γ_{5} (for F^{″} = 3), and γ_{6} (for F^{″} = 2) to another ground state are set to 0.25 × 1.7 MHz, 0.5 × 1.7 MHz, and 0.75 × 1.7 MHz, respectively, considering the branching ratio of each transition and the transition routes. As the γ value increases, the DROP effect increases, and the total population of the atomic system decreases. As mentioned above, DROP is the main cause of FWM suppression.

Figure 5(a) shows the FWM spectrum calculated as per the six-level atomic model. Although the simple atomic model used for FWM analysis differs from an actual atomic system with hyperfine structures and Zeeman sublevels, the calculated FWM spectral feature successfully accounts for the asymmetric spectral shape of the experimental result. When we consider the hyperfine states of the excited state, we note that the cause of the asymmetric FWM spectrum is the DROP effect related to the F^{″} = 2 and 3 states of the 4D_{5/2} states.

To analyze the contribution of the two-photon coherence component to the FWM spectrum, the two-photon spectrum, except for the driving laser, is decomposed into cascade-type (red curve) and V-type (pink curve) two-photon components, as shown in Fig. 5(b). The V-type two-photon coherence term (${\rho _{23}}$) can be obtained by neglecting the cascade-type two-photon coherence components (${\rho _{14}}$, ${\rho _{15}}$, and ${\rho _{16}}$) between the ground and excited states. Meanwhile, the cascade-type two-photon coherence term can be calculated by neglecting the V-type two-photon coherence term (${\rho _{23}}$) of both the intermediate $|2 \rangle$ and $|3 \rangle$ states. The spectrum (blue curve) due to both two-photon coherence components contributes to the FWM process. The DROP effect in both the $|5 \rangle$ and $|6 \rangle$ excited states can be described as the relaxation terms of γ_{5} (for F^{″} = 3) and γ_{6} (for F^{″} = 2), respectively, which are related to spontaneous transfer into another ground state. In particular, the DROP effect contributes to the transfer of the population in the 5S_{1∕2} ground state to the 4D_{5∕2} excited state via the 5P_{3∕2} intermediate state [23]. We note that the DROP effect in the F^{″} = 2 and 3 states is dominant relative to the two-photon coherence. Therefore, considering the excited $|5 \rangle$ and $|6 \rangle$ states, we can confirm that the asymmetric FWM spectral shape is due to the FWM suppression resulting from the DROP of the F^{″} = 2 and 3 states.

The two-photon coherence and DROP in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition change according to the powers of the coupling and pump lasers. To understand the spectral dynamics of the FWM signal, we investigated the FWM spectra as a function of the coupling detuning frequency according to the powers of the coupling and pump lasers. Figure 6(a) shows the FWM spectra as a function of the coupling power in the range from 52 μW to 2.0 mW when the pump and driving laser powers are fixed at 80 μW and 100 μW, respectively. As the coupling power increases, the spectral shape of the FWM changes from narrow to broad, and the magnitude of the FWM spectrum increases and finally saturates. Although the pump and driving powers are small, the DROP effect at the two-photon resonance is non-negligible. Except for the suppression region of the FWM owing to the DROP effect, the right tail of the broadened FWM spectrum increases. The broadening of the FWM spectral width and the magnitude of the upsurge of the FWM signal with an increase in the coupling power can be understood as an increase in the AC Stark splitting and two-photon coherence effects due to the strong coupling laser.

Meanwhile, Fig. 6(b) shows the generated FWM signals according to the pump power when the coupling and driving laser powers are fixed at 52 μW and 100 μW, respectively. As the pump power increases from 80 μW to 450 μW, the FWM spectral shape exhibits broadening; however, the magnitude of the FWM signal is saturated at a pump power of 450 μW. An increase in the pump power induces the DROP effect due to two-step excitation, and the right tail of the broadened FWM spectrum does not change significantly. Because the pump laser is resonant on the 5S_{1/2}(F = 2)–5P_{3/2}(F′ = 3) transition, the atoms in the 5S_{1/2}(F = 2) ground state are two-step excited to the 4D_{5/2}(F^{″} = 2, 3, 4), and subsequently, the excited atoms spontaneously decay to the 5S_{1/2}(F = 1) state via the intermediate 5P_{3/2} state. This spontaneous process does not contribute to the FWM process because the generated FWM light is induced by the driving laser via pure two-photon coherence. Therefore, when the pump power is high, FWM signal generation may be suppressed because of optical pumping.

## 4. Conclusion

In conclusion, we experimentally demonstrated the generation of Doppler-broadened FWM light at telecommunication wavelength in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of a warm ^{87}Rb atomic ensemble. The FWM light generation in the 5P_{3/2}–4D_{5/2} transition between the upper excited states was elucidated as the stimulation process by the driving laser via two-photon coherence. We found that the FWM spectral features are significantly influenced by both two-photon Doppler broadening and the DROP effect. We clarified the suppression of the FWM process due to the DROP effect from the results calculated using a simple six-level atomic model. The main cause of the asymmetric FWM spectrum is the suppression of the FWM generation by the DROP effect. We believe that a better understanding of the FWM process in the 5S_{1/2}–5P_{3/2}–4D_{5/2} transition of ^{87}Rb atoms is necessary for the high-performance generation of correlated photon pairs at telecommunication wavelengths for application to long-distance quantum communication.

## Funding

Institute for Information and Communications Technology Promotion (IITP-2021-2020-0-01606); National Research Foundation of Korea (2020M3E4A1080030, 2021R1A2B5B03002377).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper may be available from the corresponding author upon reasonable request.

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