1. Introduction

Quantum optics based on atom-photon interaction is important for long-distance quantum communication and optical quantum-information processing by a linear optics quantum computer [1–3]. In particular, this behavior is essential for fundamental quantum optics and quantum communications that require high-performance generation of correlated photon pairs via spontaneous four-wave mixing (SFWM) processes in various atomic systems [4–14]. To date, many studies have demonstrated photon-pair source generation via SFWM in Λ-type atomic systems with two ground states, because of the long coherence time in such systems [4–12]. Recently, the two-photon transitions of a ladder-type atomic system have attracted interest with regard to applications involving high-performance correlated photon-pairs and low-noise quantum memory [13–18]. Furthermore, it has been studied that the quantum-correlated photon pairs via the spontaneous processes can be understood by the classical measurement via the stimulated processes [19-20].

However, multi-photon interaction with ladder-type atomic systems has also been studied in relation to various two-photon coherence phenomena, such as electromagnetically induced transparency (EIT), two-photon absorption (TPA), and double-resonance optical pumping (DROP) [21–27]. Three-photon coherence occurring in a ladder-type atomic system interacting with three coherent fields has also been studied, because of interesting phenomena such as three-photon electromagnetically induced transparency [28], three-photon electromagnetically induced absorption (TPEIA) [29], and enhanced nonlinear optical processes [30]. Four-wave mixing (FWM) is one of the well-known nonlinear phenomena requiring the interaction of a strong optical field with a nonlinear medium. In addition, the nonlinearity in such atomic media can be enhanced significantly via multi-photon atomic coherence effects, which are generated by the interaction of an atom with coherent electromagnetic fields [31–39]. Therefore, the FWM occurring in a ladder-type atomic system is related to the two- and three-photon coherence phenomena. In particular, the relationship between two-photon coherence and FWM has been reported under EIT and TPA conditions in ladder-type atomic systems [39].

To date, studies on FWM in ladder-type atomic systems have focused on ladder-type two-photon coherence effects, because the FWM signal is very strongly correlated to two-photon coherence. Essentially, it is understood that an FWM light including two-photon coherence is stimulated by a weak seed field from a prepared two-photon coherent atomic ensemble. Understanding the role of the seed field in FWM is very important for quantum optics applications based on the SFWM processes in ladder-type atomic systems. Although the relationship between TPEIA and FWM in the 5S_1/2 – 5P_3/2 – 5D_5/2 transition of ⁸⁷Rb atoms has been investigated recently [40], the role of the seed field in the FWM occurring in a ladder-type atomic system has never been reported.

In this paper, we investigate the stimulated process in a four-level atomic system prepared to achieve ladder-type two-photon coherence for the 5S_1/2 − 5P_3/2 − 5D_5/2 transition of the ⁸⁷Rb atom. To understand the role of the seed field on the generation of FWM in the ladder-type atomic system, we compare FWM spectra for two cases: scanning of the ladder-type two-photon resonance and detuning of the seed-field frequency. The dependence of the spectral features of the FWM signals on the seed-field intensities and frequency detuning is investigated. In addition, the FWM signals are numerically calculated as functions of the seed-field detuning frequency using a Doppler-broadened four-level atomic model.

2. Experimental setup for FWM in ladder-type atomic system

Figure 1(a) shows the generation of an FWM signal induced by a seed field in a ladder-type atomic medium coherently interacting with pump and coupling fields. In our experiment, the 5S_1/2 – 5P_3/2 – 5D_5/2 transition of ⁸⁷Rb was used as the ladder-type atomic system, which consists of a 5S_1/2 ($|1〉$) ground state, 5P_3/2 ($|2〉$ and $|3〉$) first excited state, and 5D_5/2 ($|4〉$) second excited state. The total frequency of the pump and coupling fields satisfied the conditions for two-photon resonance for the 5S_1/2 – 5P_3/2 – 5D_5/2 ($|1〉$–$|2〉$–$|4〉$) transition. To stimulate an FWM signal between the upper excited states, the seed-field frequency was scanned around the 5S_1/2 – 5P_3/2 ($|1〉$ – $|3〉$) transition. Here, δ_p, δ_C, and δ_s are the detuning frequencies of the pump, coupling, and seed fields, respectively. To investigate the role of the seed field in the generation of FWM in the ladder-type atomic system, two-photon detuning between the pump and coupling fields was maintained under the two-photon resonance condition (δ_p + δ_C = 0).

 

Fig. 1 Experimental configuration for FWM generation. (a) Energy-level diagram of 5S_1/2 – 5P_3/2 – 5D_5/2 transition of ⁸⁷Rb atoms. (b) Experimental setup for FWM generation in ladder-type atomic system (PBS: polarizing beam splitter; M: Mirror; B: beam block; PD: photo-detector; M: Mirror).

Download Full Size | PDF

The experimental setup for generation of the FWM field in the ladder-type atomic system is shown in Fig. 1(b). Three external cavity diode lasers (ECDLs) were operated independently at wavelengths of 780.2 (seed and pump lasers) and 775.8 (coupling laser) nm. The pump and coupling lasers were counter-propagated to satisfy the two-photon resonant condition in the Doppler-broadened ladder-type atomic system. The pump and coupling laser powers were 1 and 10 mW, respectively, with 1.2-mm diameters; the polarizations of both lasers were perpendicularly linear. To conduct the FWM experiment under the two-photon resonant condition, we observed the ladder-type EIT spectrum in the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) cycling transition and locked the frequencies of the pump and coupling lasers to the EIT spectrum. A 12.5-mm-long vapor cell containing the ⁸⁷Rb isotope was installed in a three-layer μ-metal chamber, so as to reduce the external magnetic field. The residual magnetic field in the three-layer μ-metal chamber is less than 10 μgauss. To maintain a warm atomic vapor, the vapor cell temperature was stabilized to 80 °C.

The seed laser with 780.2-nm wavelength was aligned with a small tilted angle of 1.3° from the pump laser beam, to maximize the FWM generation. The FWM light from the warm atomic medium was stimulated by the seed laser under the condition of two-photon resonance and detected by a photon detector (PD) in the phase-matched direction. The seed-laser polarization was linearly polarized perpendicular to that of the pump laser. The seed-laser power could be adjusted from 0.1 mW to 1.2 mW, with the same beam diameter of 1.2 mm, to investigate the dependence of the FWM signal on the seed power.

3. Experimental results and discussion

To elucidate the role of the seed laser in the FWM process, the FWM spectrum as a function of the seed-laser detuning frequency from the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) resonance was obtained, as shown in Fig. 2(a), for a seed-laser frequency scanned in the vicinity of the 5S_1/2(F = 2) − 5P_3/2(F′ = 1, 2, 3) transition. The frequencies of the pump and coupling lasers were fixed at the hyperfine transitions of 5S_1/2(F = 2) − 5P_3/2(F′ = 3) and 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4). Here, the seed-laser power was adjusted to 0.6 mW. As shown in Fig. 2(a), an FWM signal was stimulated by the seed laser in the atomic medium under the condition of two-photon resonance. The FWM light generated from the atoms excited by two-photon coherence using the seed laser was emitted in the phase-matching direction. FWM is induced by three-photon coherence which is related to pure two-photon coherence. When the pump- and coupling-laser powers were 1 mW and 10 mW, the maximum of the FWM signal was red-shifted by approximately 20 MHz and the full width at half maximum (FWHM) of the FWM spectrum was measured to be 65 MHz. The appearance of the FWM signal shown in Fig. 2(a), which was obtained via frequency scanning of the seed laser, indicates that FWM phenomena were induced by the seed laser in the atomic medium in the presence of two-photon coherence. The gray curve in Fig. 2 indicates the saturated absorption spectrum (SAS) of the seed laser, which is used to mark the seed-laser optical frequency. In particular, an interesting peculiar characteristic of FWM signal is the dip on the resonance indicated by the gray arrow in Fig. 2(a). Below, we discuss the dip in the FWM spectrum by investigating the FWM spectra according to the detuning conditions of the two-photon resonance and the seed-laser power.

 

Fig. 2 (a) FWM spectrum as function of seed-laser detuning frequency from 5S_1/2(F = 2)−5P_3/2(F′ = 3) resonance of ⁸⁷Rb. (b) FWM spectrum for pump-laser frequency scanning from 5S_1/2(F = 2)−5P_3/2(F′ = 3) resonance of ⁸⁷Rb. The gray curve is the saturated absorption spectrum (SAS) of the frequency-scanning laser.

Download Full Size | PDF

In contrast, Fig. 2(b) shows the FWM spectrum obtained via frequency scanning of the pump laser in the vicinity of the 5S_1/2(F = 2) − 5P_3/2(F′ = 1, 2, 3) transition, where the frequencies of the coupling and seed lasers were fixed at the 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) and 5S_1/2(F = 2) − 5P_3/2(F′ = 3) transitions, respectively. Figure 2(b) presents the resonant FWM spectrum as a function of the pump-laser detuning frequency obtained under the same conditions as the dip in the FWM spectrum in Fig. 2(a). As frequency scanning of the pump laser was employed, the two-photon frequencies of both lasers were detuned from the ladder-type two-photon resonance of the 5S_1/2(F = 2) − 5D_5/2 transition. Because the FWM process has a significant influence on the two-photon coherence, the FWM spectrum of Fig. 2(b) shows the FWM generation as a function of ladder-type two-photon coherence. Here, the FWM spectral shape in the cycling transition has a narrow structure because of the two-photon coherence. In this case, the FWM signal of the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) cycling transition can be significantly enhanced because of the strong two-photon atomic coherence. On the other hand, the weak FWM signal in the 5D_5/2(F″ = 3) state can be understood as being due to weak two-photon coherence; however, note that there is no FWM signal in the 5D_5/2(F″ = 2) state.

Comparing the FWM spectra of Fig. 2(a) and 2(b), the spectral shape of the FWM spectrum of Fig. 2(a), which was obtained through frequency scanning of the seed laser, differs significantly from that of Fig. 2(b), generated through frequency scanning of the pump laser. In particular, comparing the FWHMs of both spectra, the spectral width of Fig. 2(a) is 10 times broader than that of Fig. 2(b). Although the wavelength and propagation direction of the seed laser are similar to those of the pump laser, from both FWM spectra of Fig. 2, it is apparent that the pump and seed lasers play completely different roles in the ladder-type FWM scheme.

Previous studies of ladder-type FWM were performed using a single ECDL to produce both the pump and seed lasers, by splitting the lasers from the ECDL [38–40]. Thus, the optical frequencies of both lasers could not be independently scanned. In our experiment, because the seed-laser optical frequency was independently scanned, we could observe the influence of the seed laser alone on the FWM signals. As mentioned above, for generation of the ladder-type FWM, it is necessary to maintain the two-photon resonance condition (δ_p + δ_C = 0). Various two-photon coherence phenomena have been observed in ladder-type atomic systems, such as EIT, TPA, and DROP, depending on the transition routes and optical detuning frequencies of both lasers [39]. Under our experimental conditions, two-photon coherence phenomena were observed for the pump-laser transmittance spectra obtained for four δ_C values (0, 266, 532, and 798 MHz) and in the absence of the seed laser, as shown in Fig. 3(a). The gray curve in Fig. 3 indicates the SAS of the pump laser, which is used to mark the pump-laser optical frequency. The Doppler background is saturated at near resonance at an optical density (OD) of approximately 4.3. When δ_C is increased, various spectral dynamics are apparent, because of the change of two-photon coherence. EIT, DROP (δ_C = 0, 266, and 532 MHz), and TPA (δ_C = 798 MHz) effects are visible in the transmittance spectra of Fig. 3(a), where the pump- and coupling-laser powers are 1 mW and 10 mW, respectively. Because the pump-laser intensity is stronger than the saturation intensity in this transition, splitting of the EIT signal in the cycling transition is apparent in the Doppler-broadened spectrum. However, beyond the linear absorption spectrum, the TPA spectrum for the hyperfine structure is clearly apparent.

 

Fig. 3 (a) Pump-laser transmittance spectra in accordance with δ_C.(b) FWM spectra for seed-laser frequency scanning in accordance with δ_C ( = −δ_P), where the x-axis is δ_s. (c) FWM spectra with pump-laser frequency scanning in accordance with δ_s, where the x-axis is δ_P. The gray curve is the saturated absorption spectrum (SAS) of the frequency-scanning laser.

Download Full Size | PDF

Next, with the seed laser activated under the two-photon resonance condition, we investigated the FWM spectra as functions of δ_s obtained for the different δ_C (−δ_P) in the Doppler-broadened medium, as shown in Fig. 3(b), where the seed-laser power was 0.6 mW. As δ_C and −δ_P increased, the FWM resonance peak as a function of δ_s was blue-shifted. As shown in the energy-level diagram of Fig. 1(a), the pump and coupling lasers generate the ladder-type two-photon coherence between the 5S_1/2(F = 2) and 5D_5/2(F″ = 4) states, as well as the pump and seed lasers generate the V-type two-photon coherence [39]. The V-type two-photon coherence is due to interaction of the co-propagating pump and seed lasers with the Doppler-broadened medium. Therefore, the role of the seed laser for FWM generation is not simply stimulation of FWM light from ladder-type two-photon coherence, because it is also related to V-type two-photon coherence. In particular, as shown by the arrows of Fig. 3(b), the dip in the FWM spectrum can be understood as resulting from the three-photon atomic coherence effect of the cycling transition due to quantum interference in the Doppler-broadened medium. Under our experimental conditions, the three-photon atomic coherence effect in the Doppler-broadened medium could be achieved when the pump and seed lasers were co-propagating. Because the three-photon resonance condition can be defined by δ_p + δ_C − δ_d = 0, the two-photon resonance conditions for each V-type (δ_P − δ_d = 0) and ladder-type (δ_P + δ_C = 0) system are always satisfied for a wide-ranging Maxwell-Boltzmann distribution [39]. Therefore, the dip in the FWM spectrum means that the three-photon resonance condition is satisfied. In addition, we can interpret the frequency shift of the FWM signal in accordance with the detuning frequencies in terms of the three-photon resonance condition including V-type two-photon coherence.

Figure 3(c) shows the FWM spectra as functions of δ_P for four different δ_s, where the δ_C was fixed around the 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) transition. When the pump-laser frequency was scanned around the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) transition, narrow FWM signals with the hyperfine structure of the 5D_5/2(F″ = 2, 3, 4) state could be observed. Interestingly, despite the change in δ_s, the peaks of the FWM spectra of Fig. 3(c) were not shifted. The result shown in Fig. 3(c) indicates that the ladder-type two-photon resonance condition should be satisfied to generate a ladder-type FWM signal. However, although we did not measure the optical frequencies of the generated FWM light, they should vary for the four different δ_s, because the FWM light is induced by the seed laser under the phase-matching condition. Therefore, we can interpret the FWM process as follows: the seed laser stimulates the FWM light from the ladder-type two-photon coherence generated by the pump and coupling lasers.

FWM generation is highly dependent on the intensities of the lasers interacting with the atoms because of the nonlinear phenomenon in the atomic medium used. We investigated the change in the FWM signal in accordance with changes in the seed-laser power for pump and coupling powers of 1.0 mW and 10 mW, respectively, as shown in Fig. 4. When the seed-laser optical frequency was independently scanned and those of the coupling and pump lasers were fixed on the two-photon resonance, we could easily observe the influence of the seed laser only on the FWM signals. As the seed power was increased from 0.1 to 1.2 mW, the spectral features of the FWM signals changed from narrow to broad. The broadening of the FWM spectral width is due to the power broadening of the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) transition resulting from interaction with the seed laser. However, the FWM magnitude increased up to a seed power of 0.6 mW, but then decreased for higher seed powers. Thus, although the seed laser is required to achieve stimulated FWM, a certain seed-laser condition that yields the optimal FWM signal exists. This decrease can be understood as a decrease of the ladder-type two-photon coherence and a population change for the ground states resulting from the strong seed laser. It is possible to disturb the ladder-type two-photon coherence with a strong seed laser, because the 5S_1/2(F = 2) ground state is common with the ladder-type two-photon transition. Therefore, a weak seed laser is appropriate for stimulation of FWM light from atoms with pure two-photon coherence.

 

Fig. 4 Ladder-type FWM spectra in accordance with seed power.

Download Full Size | PDF

However, a dip in the FWM spectrum due to three-photon resonance was observed for the total range of the seed-laser power shown in Fig. 4. The relative magnitude of the dip in the FWM signal increased as the seed-laser power increased. Further, the dip in the FWM spectrum was clearly observed even at a weak seed power of 0.1 mW. This dip in the FWM means that the FWM light decreased for the three-photon resonance due to the three coherent lights (pump, coupling, and seed lasers) of the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) transition. The FWM spectra are asymmetric because of other hyperfine states except the cycling transition. This dip in the FWM spectrum in the case of three-photon resonance has not been reported previously, because the previous results for ladder-type FWM have shown the FWM spectra obtained for changes in the ladder-type two-photon coherence due to frequency scanning of the coupling or pump lasers [38–40]. In the case of three-photon resonance including V-type two-photon coherence, a significant population change should occur for each state in response to increments in the seed intensity, because the seed laser is resonant on the 5S_1/2(F = 2) ground state. The population change due to a single-photon resonance does not contribute to the FWM process. Although three-photon coherence is related with FWM, the dip in the FWM spectrum appears for three-photon resonance.

To theoretically investigate the role of the seed laser in the FWM process between the upper excited states in ladder-type atomic systems, we studied these phenomena in the simplified four-level atomic system ($|1〉$, $|2〉$, $|3〉$, and $|4〉$) shown in Fig. 1(a) from a theoretical perspective. From the density matrix equations [29], we analytically calculated the coherence component (${\rho }_{34}$) related to the FWM. In the weak-pump-intensity limit, ${\rho }_{34}$ can be expressed as [40]

As mentioned above, under weak pump- and weak seed-intensity conditions, the V-type two-photon coherence ${\rho }_{23}$ can be neglect. However, if the seed power is zero, the ${\rho }_{34}$ of Eq. (1) is zero and the FWM signal cannot be generated. Considering a spontaneous FWM process without a seed laser, the seed light is a weak spontaneously-emitted light from the $|3〉$ state and ${\rho }_{23}$ can be neglect. Therefore, the role of the seed laser for FWM generation should be focused on the stimulation of FWM than the V-type two-photon coherence or the three-photon coherence including the seed laser.

We numerically investigated the change in the FWM signals according to seed power, as shown in Fig. 5. This figure shows ${\left|{\rho }_{34}\right|}^2$ for FWM for different Ω_s, as determined by applying the density matrix equation to the four-level atomic system of Fig. 1(a). The FWM spectra were calculated for the Doppler-broadened atomic medium and the branching ratios of the 5S_1/2(F = 2) – 5P_3/2(F′ = 3) – 5D_5/2(F″ = 4) cycling transition, where Ω_p and Ω_C were set to 1 MHz and 15 MHz, respectively. In the FWM spectra calculated as functions of δ_s, the dip in the FWM is clearly apparent. Note that the calculated spectra of Fig. 5 are not in complete agreement with the observed FWM spectral shapes of Fig. 4; this is because we did not consider the hyperfine structures of the 5P_3/2 and 5D_5/2 states. Furthermore, the energy difference between the hyperfine states (F″ = 2, 3, 4) of the 5D_5/2 state is smaller than the spectral width of the FWM spectrum. The hyperfine states of the 5D_5/2 state may significantly influence the spectral shape of the calculated FWM spectrum. However, although the simple four-level atomic model cannot be considered to exhibit the hyperfine structures and Zeeman sublevels of a real atomic system, we can elucidate the role of the seed laser in the ladder-type FWM process; that is, the seed laser is a stimulator for FWM generation but a disturber due to V-type atomic coherence. Here, the dips in the FWM signal for three-photon resonance were numerically calculated using a Doppler-broadened four-level atomic model.

 

Fig. 5 Numerically calculated FWM spectra in accordance with seed-field Rabi frequency (Ω_s) using Doppler-broadened four-level atomic model.

Download Full Size | PDF

4. Conclusion

We experimentally investigated the stimulated FWM process of the 5S_1/2 − 5P_3/2 − 5D_5/2 transition of the ⁸⁷Rb atom. We measured and compared the FWM spectra of two cases, i.e., for FWM signals obtained via scanning of the seed laser under the ladder-type two-photon resonance condition and for those achieved through scanning of the detuning frequency related to two-photon resonance. Although the seed-laser wavelength and propagation direction were similar to those of the pump laser in our experimental configuration, the two resultant FWM spectra differed significantly from each other. We confirmed that the roles of the pump and seed lasers in the ladder-type FWM scheme are completely different. From our experimental results, it was revealed that the seed laser induces the FWM phenomena from the atomic medium in the presence of two-photon coherence.

When we investigated the FWM spectra obtained via scanning of the seed laser in the Doppler-broadened medium, the frequency shift of the FWM signal in accordance with the detuning frequency was related to the three-photon resonance condition including V-type two-photon coherence. However, in contrast with the FWM spectra obtained via scanning of the pump laser, no shifting of the peaks of the FWM spectra was observed, despite the change in the seed-laser detuning frequency. This is because the ladder-type two-photon resonance condition should be satisfied to generate a ladder-type FWM signal. From the variance of the FWM spectra according to the detuning frequency, we confirmed that the FWM process can be interpreted as follows: the seed laser acts to stimulate the FWM light from the ladder-type two-photon coherent atomic ensemble.

As the seed-field power increased to 0.6 mW, the magnitude of the FWM increased; however, this magnitude then decreased for greater seed-field power. The reason for this subsequent FWM decrease can be understood as disturbance of the ladder-type two-photon coherence by the strong seed laser. Therefore, it was found that a weak seed laser is appropriate for stimulation of the FWM light from atoms with pure two-photon coherence. In particular, an interesting spectral feature is the dip in the FWM spectrum for the three-photon resonance. This dip in the FWM means that the FWM light decreased under the three-photon resonance because of the three coherent lights of the 5S_1/2(F = 2) − 5P_3/2(F′ = 3) − 5D_5/2(F″ = 4) transition. We noted that the dip in the FWM spectrum appears on three-photon resonance as a result of three-photon coherence including the V-type two-photon coherence. Additionally, from theoretical results obtained for the Doppler-broadened four-level atomic model, we performed a theory-based investigation of the role of the seed laser in the FWM process between the upper excited states in the simplified four-level atomic system. In future, we hope that our results are contributed to a better understanding of the properties of photon pairs obtained via a spontaneous FWM process in a ladder-type atomic medium.