## Abstract

We study the spectral features and phase of four-wave mixing (FWM) light according to the relative phase-noise of the optical fields coupled to a double Λ-type atomic system of the 5S_{1/2}–5P_{1/2} transition of ^{87}Rb atoms. We observe that the spectral shape of the FWM spectrum is identical to that of the two-photon absorption (TPA) spectrum due to two-photon coherence and that it is independent of the relative phase-noise of the pump light. From these results, we clarify that the two-photon coherence plays a very important role in the FWM process. Furthermore, we measure the relative linewidth of the FWM signal to the probe and pump lasers by means of a beat interferometer. We confirmed that the phase of the FWM signal is strongly correlated with that of the pump laser under the condition of phase-locked probe and coupling lasers for two-photon coherence.

© 2016 Optical Society of America

## 1. Introduction

Four-wave mixing (FWM) has long been one of the well-known phenomena in nonlinear optics [1,2]. The atomic coherence arising due to the interaction of coherent optical fields with atomic media may be used to enhance nonlinear optical interactions [2]. In this context, an electromagnetically induced transparency (EIT) is a representative quantum interference phenomenon occurring due to atomic coherence, wherein the presence of a coupling field makes the medium transparent to a probe field [3]. Nonlinear susceptibility is enhanced in an EIT medium because the optical properties of the medium are dramatically modified. The resulting enhanced nonlinear susceptibility is useful in frequency-mixing processes. The enhanced FWM nonlinearity with the use of atomic coherence has been exploited in various atomic systems [4–15]. The enhanced FWM processes in a double Λ-type atomic system are very important because of their many applications to atomic spectroscopy, nonlinear optics, and quantum optics [12–21]. Recently, the FWM process in double Λ-type atomic systems has been intensively studied in quantum optics, because of interesting applications such as quantum memory and photon-pair generation that can be realized with the use of long-lived atomic coherence between two ground states in a Λ-type atomic system [16–21].

However, the characteristics of light beams coupled to an atomic medium affect the properties of the generated atomic coherence. In this regard, there are studies of the spectral width of EIT obtained with the use of narrow and noise-broadening lasers [22–25]. The relative noise between the two coupled light beams in EIT strongly influences the spectral features of the EIT [22,23]. However, as is well-known, the FWM process in the double Λ-type atomic system is related to four optical fields, which are composed of one Λ-type EIT configuration with the probe and coupling lasers and the second Λ-type configuration with the pump laser and the generated FWM signal. Although the FWM signal have investigated various properties [26–29], the spectral features and phase of the generated FWM signals in relation with the phase-noise of the pump laser applied to the FWM process have not thus far been investigated.

In this work, we experimentally study the characteristics of the FWM signals generated from the double Λ-type atomic system of the 5S_{1/2}−5P_{1/2} transition of ^{87}Rb atom. We investigate the relation between FWM and the two-photon coherence between two ground states in the double Λ-type atomic system. We examine the dependence of the spectral features of the FWM spectrum on the relative phase-noise of the pump light. Furthermore, the phase of the FWM with respect to that of the phase-unlocked pump laser is investigated by comparing the phases of the probe and pump light beams by means of a beat interferometer, respectively.

## 2. Experimental setup

In our study, we investigated the properties of the FWM signal in the double Λ-type atomic system of the 5S_{1/2}(F = 1 and 2)−5P_{1/2}(F′ = 2) transition of the ^{87}Rb atom, whose schematic is shown in Fig. 1. Our experimental setup consists of the FWM setup and a beat interferometer for measurement of the relative phase-noise of the generated FWM signal, as shown in Fig. 1(a). As shown in the energy-level diagram in Fig. 1(b), three co-propagating lasers are used for the generation of a Doppler-free FWM signal. The three lasers include probe (Ω* _{pr}*) and coupling (Ω

*) lasers for the Λ-type EIT configuration and a pump (Ω*

_{C}*) laser for a second Λ-type configuration with the generated FWM (Ω*

_{P}*) signal. δ*

_{FWM}*and δ*

_{pr}*represent the detuning of the probe and coupling lasers, respectively. The FWM signal is generated in the 5S*

_{C}_{1/2}(F = 1)−5P

_{1/2}(F′ = 2) transition under the phase-matching condition.

The laser system for our experiment is composed of three external cavity diode lasers (ECDLs), as shown in Fig. 1(a). The linewidths of these independently operated ECDLs were estimated to be less than 1 MHz. We locked the relative phase-noises of the “Ω* _{pr}*” and “Ω

*” lasers to the “Ω*

_{P}*” laser using the phase-locking method. For phase-locking among the three lasers, we use optical injection locking for the Ω*

_{C}*laser and electrical phase locking for the Ω*

_{pr}*laser, and the Ω*

_{P}*laser is used as the master laser. The Ω*

_{C}*laser is optically phase-locked to the 6.8-GHz side mode of the modulated master laser via its passage through an electro-optic modulator (EOM) with a 6.8-GHz modulation frequency. The Ω*

_{pr}*laser is electrically phase-locked to an 8.0-GHz RF reference via beating between the master laser and Ω*

_{P}*. In the study, the three lasers were coupled to single-mode optical fibers and delivered to the setup for the FWM experiment.*

_{P}The intensities of the three fields (Ω* _{pr}*, Ω

*, and Ω*

_{C}*) were adjusted independently with the use of a half-wave plate (HWP) and polarized beam splitter (PBS). The Ω*

_{P}*and Ω*

_{pr}*fields co-propagated and completely overlapped in the atomic vapor cell. The Ω*

_{C}*field passed through the main cell at a tilt angle of 1.3°, and the FWM field was generated in the direction determined by the phase-matching condition. The atomic vapor cell for the FWM was 5 cm long and contained*

_{P}^{87}Rb with Ne buffer gas at a pressure of 4 Torr. The cell was housed in μ-metal chambers, which acted as shields against the earth’s magnetic field. In our experiment conditions, we found the best temperature condition of the atomic vapor cell for high-contrast and narrow FWM signal. The temperature of the main cell was maintained at 50 °C. In order to compare the Ω

*output with the transmitted Ω*

_{FWM}*signal, both signals were simultaneously measured by two photocurrent detectors (PD1 and PD2). In particular, to investigate the relative phase-noise of the Ω*

_{pr}*light beam with respect to those of the Ω*

_{FWM}*and Ω*

_{pr}*beams, the beat signals were measured between the Ω*

_{C}*and Ω*

_{FWM}*light beams and between the Ω*

_{pr}*and Ω*

_{FWM}*light beams for the cases of the phase-locked and phase-unlocked Ω*

_{P}*laser. In this work, we don’t directly measure the relative phase value of the Ω*

_{P}*light beam with respect to the phase of the Ω*

_{FWM}*or Ω*

_{pr}*light beams.*

_{P}## 3. Experimental results and discussion

Two-photon coherence is important for FWM processes, because FWM generation is based on stimulated emission via two-photon coherence [10, 30]. In our study, in order to investigate the relation between FWM and two-photon coherence in the double-Λ-type atomic system shown in Fig. 1(b), we simultaneously measured the spectra of the FWM signal and the two-photon coherence. Figure 2 shows the transmittance spectra of Ω* _{pr}* (black curve) and the FWM signal (red curve) as a function of the two-photon detuning δ = δ

*‒ δ*

_{C}*, which were simultaneously acquired by PD1 and PD2. The Ω*

_{pr}*and Ω*

_{C}*laser frequencies were fixed at δ*

_{P}*/2π = 1.2 GHz from the 5S*

_{C}_{1/2}(F = 1)−5P

_{1/2}(F′ = 2) transition and at the 5S

_{1/2}(F = 2)−5P

_{1/2}(F′ = 2) transition, respectively, while that of Ω

*was scanned in the vicinity of δ*

_{pr}*/2π = 1.2 GHz from the 5S*

_{pr}_{1/2}(F = 2)−5P

_{1/2}(F′ = 2) transition. The relative phases of the Ω

*and Ω*

_{pr}*lasers were locked with respect to that of the Ω*

_{P}*laser. The powers of Ω*

_{C}*, Ω*

_{pr}*, and Ω*

_{C}*were set to 2 μW, 1.0 mW, and 1.0 mW, respectively, and the beam diameters of all three beams were 2 mm. The polarizations of Ω*

_{P}*and Ω*

_{pr}*were perpendicularly linear, and Ω*

_{C}*was linearly polarized perpendicular to the Ω*

_{P}*polarization. However, when the Ω*

_{pr}*laser was blocked in the FWM experiment corresponding to Fig. 1(a), in the Λ-type configuration with the Ω*

_{P}*and Ω*

_{pr}*lasers, the two-photon absorption (TPA) spectrum due to two-photon coherence between the two ground states of 5S*

_{C}_{1/2}(F = 1 and 2) was observed at PD1, as indicated by the blue curve in Fig. 2. The reason underlying the observation of TPA instead of EIT is that the frequencies of the Ω

*and Ω*

_{pr}*lasers are detuned far from 1.2 GHz beyond Doppler-broadening (~530 MHz). Under the two-photon resonant condition of the two “far-detuned” lasers from the optical transition, the two-photon coherence effect contributes to non-resonant TPA. The non-resonant TPA based on three-level atomic systems is an absorption phenomenon caused by the two-photon coherence between both ground states, where one-photon detuning lies far from the transition between the ground and intermediate states.*

_{C}Upon comparing the spectral features of the three observed signals (probe, FWM, and TPA), we note that the three spectral shapes with sub-natural width are similar. In particular, the spectral shapes of the normalized FWM and TPA spectra are identical, as can be observed in the inset in Fig. 2. Under the condition of far detuning of 1.2 GHz, the TPA signal arises due to pure two-photon coherence, and subsequently, the FWM signal is strongly correlated with the two-photon coherence. We thus confirmed that the two-photon coherence plays a very important role in the FWM process. In this work, we investigated the spectral width and the phase correlation of the generated FWM light beam with respect to Ω* _{P}* in the FWM process.

In order to investigate the change in the FWM signal according to the relative phase-noise of Ω* _{P}*, we compared the FWM spectra for the cases of both phase-locked and phase-unlocked Ω

*configurations. First, we measured the relative phase-noise of Ω*

_{P}*with that of Ω*

_{P}_{C}. Figure 3(a) shows the spectral density curve of the beat signal between the Ω

*and Ω*

_{P}*lasers at the center frequency of 8.0 GHz. In the case of the phase-unlocked Ω*

_{C}*laser [see the blue curve in Fig. 3(a)], the width of the spectral density curve was measured to be about 1 MHz, corresponding to the linewidth of the ECDL. When Ω*

_{P}*was electrically phase-locked to the 8.0-GHz frequency shift from Ω*

_{P}*, the width of the spectral density curve was limited by the resolution of the RF spectrum analyzer, as indicated by the red curve in Fig. 3(a).*

_{C}Figure 3(b) shows the FWM spectra for both the phase-locked and phase-unlocked Ω* _{P}* cases, wherein the detuning δ

*/2π of Ω*

_{pr}*is scanned at two-photon resonance with Ω*

_{pr}*. Interestingly, the FWM spectrum in the case of the phase-locked Ω*

_{C}*[see the red curve in Fig. 3(b)] is identical to that in the case of the unlocked Ω*

_{P}*laser [see the blue curve in Fig. 3(b)]. The spectral width of the FWM spectrum is less than 0.1 MHz, which is narrower than the relative phase-noise of the unlocked Ω*

_{P}*. This result indicates that the FWM signal is independent of the phase-noise of Ω*

_{P}*. This is counter-intuitive because FWM generation is based on stimulated emission by the Ω*

_{P}*laser via two-photon coherence. However, to understand this counter-intuitive result, we first remark that the FWM spectra in Fig. 3(b) were obtained by scanning the frequency of Ω*

_{P}*. As shown in Fig. 2, because the FWM signal is strongly correlated with the two-photon coherence, the spectral shape of the FWM signal is determined by that of the two-photon resonance scanning the frequency of Ω*

_{pr}*. Therefore, the result in Fig. 3(b) can be understood as FWM due to two-photon coherence as a function of the two-photon detuning δ/2π between Ω*

_{pr}*and Ω*

_{pr}*in the Λ-type configuration corresponding to Fig. 1(b). Thus, we confirmed that the spectral feature of FWM spectrum is strongly correlated with the pure two-photon coherence but uncorrelated with the relative phase of Ω*

_{C}*.*

_{P}As mentioned previously, the role of the Ω* _{P}* laser is to induce FWM signals from the atomic system in the presence of two-photon coherence. Thus, we can assume that the phase of Ω

*influences that of the Ω*

_{P}*light beam. To investigate the phase relation of Ω*

_{FWM}*, we compared the phase of Ω*

_{FWM}*to those of Ω*

_{FWM}*and Ω*

_{P}*, respectively, using the beat interference method. Figure 4(a) shows the spectral density of the beat signal between Ω*

_{pr}*and Ω*

_{pr}*in the case of the phase-unlocked Ω*

_{FWM}*laser, where the frequencies of the three lasers (Ω*

_{P}*, Ω*

_{pr}*, and Ω*

_{C}*) are fixed to their respective values corresponding to Fig. 1(b). The offset frequency of the spectral density is 8.0 GHz, corresponding to the frequency difference between Ω*

_{P}*and Ω*

_{pr}*. In Fig. 4(a), the relative phase-noise between Ω*

_{FWM}*and Ω*

_{pr}*was measured to be about 1 MHz, corresponding to the linewidth of the independent ECDL in Fig. 3(a). From this result, we conclude that the phase of Ω*

_{FWM}*is independent of that of Ω*

_{FWM}*. In comparison with the spectral width of the FWM spectrum in Fig. 3(b), the result in Fig. 4(a) appears to be inconsistent because of the significant difference between the spectral widths.*

_{pr}Meanwhile, Fig. 4(b) shows the spectral density curve of the beat signal between Ω* _{P}* and Ω

*. In contrast to Fig. 4(a), it is interesting to note that the relative phase between Ω*

_{FWM}*and Ω*

_{P}*is correlated despite the use of the phase-unlocked Ω*

_{FWM}*. The result indicates that the phases of both Ω*

_{P}*and Ω*

_{P}*are correlated regardless of the phase relation of Ω*

_{FWM}*. In the case of the phase-locked Ω*

_{P}*, we observed narrow spectral densities of both the beat signals between Ω*

_{P}*and Ω*

_{pr}*and between Ω*

_{FWM}*and Ω*

_{P}*.*

_{FWM}From Figs. 4(a) and (b), the phase relation of Ω* _{FWM}* can be understood as the atomic coherence in a double Λ-type atomic system. The phase relation between the Ω

*and Ω*

_{pr}*lasers for the generation of two-photon coherence in one Λ-configuration is dependent on the phase relation of Ω*

_{C}*and Ω*

_{P}*for the generation of FWM of the other Λ-type configuration, because Ω*

_{FWM}*induces Ω*

_{P}*from the atomic coherence between the two ground states. Therefore, we confirmed that the phase of Ω*

_{FWM}*is correlated with that of Ω*

_{FWM}*, but uncorrelated with that of Ω*

_{P}*.*

_{pr}In order to further understand the interesting results that the spectral width of Ω_{FWM} does not depend on the spectral width of Ω* _{P}* and that the relative phase of Ω

*is correlated with that of Ω*

_{FWM}*regardless of the phase-noise of Ω*

_{P}*, we theoretically investigated the FWM in a double Λ-type atomic system, whose schematic is shown in Fig. 5(a). The four-level atomic model in Fig. 5(a) is composed of two ground states (|1> and |2>) and two excited states (|3> and |4>). δ*

_{P}

_{pr}_{,}

_{C}_{,}

*and γ*

_{P}*represent the detuning and phase-noise bandwidths of the probe, coupling, and pump lasers, respectively. The density matrix equation of motion can be expressed as*

_{pr, C, P}where the subscript indices *i* and *j* indicate the | *i* > and | *j* > states, respectively. Further, *ρ _{ij}* denotes a density-matrix element and

*H*the effective interaction Hamiltonian, which is composed of the atomic and interaction Hamiltonians. The two-photon coherence in one Λ-type configuration is

_{ij}*ρ*

_{12}and the electric-field amplitude of the generated FWM signal is directly related to the coherence

*ρ*

_{14}. It is convenient to transform into a co-rotating frame to eliminate the fast rotation. We can transform the density-matrix elements (

*ρ*) into the rotating frame of a slowly varying density operator (

_{ij}*σ*). Under weak probe and far detuning conditions, we can deal with experimental atomic system as a simple four-level system [31,32]. In the four-level atomic model in Fig. 5(a), we calculated the steady-state coherences

_{ij}*σ*

_{12}and

*σ*

_{14}analytically, according to

Here, Δ_{12} = Γ_{12} + γ_{12} ‒ *i*(δ* _{C}* ‒ δ

*) where γ*

_{pr}_{12}denotes the relative phase-noise between the Ω

*and Ω*

_{pr}*lasers, Δ*

_{C}_{13}= Γ

_{13}+ γ

*‒*

_{C}*i*δ

*where γ*

_{C}*denotes the phase-noise bandwidth of the Ω*

_{C}*laser, and Δ*

_{C}_{14}= Γ

_{14}+ γ

_{14}‒

*i*(δ

*‒ δ*

_{C}*+ δ*

_{pr}*) where γ*

_{P}_{14}denotes the relative phase-noise among the Ω

*, Ω*

_{pr}*, and Ω*

_{C}*lasers. Considering the broadening effect of the 4-Torr Ne buffer gas, the decay rates are set to Γ*

_{P}_{13}/2π = Γ

_{23}/2π = 23 MHz, Γ

_{14}/2π = Γ

_{24}/2π = 23 MHz, and Γ

_{12}/2π = 0.1 MHz [34]. Although the phase-noise bandwidths γ

_{pr}_{,}

_{C}_{,}

*/2π of the Ω*

_{P}*, Ω*

_{pr}*, and Ω*

_{C}*lasers are set to 1 MHz considering the linewidth of the ECLDs, the relative phase-noise γ*

_{P}_{12}is zero because of phase-locking between Ω

*and Ω*

_{pr}*. In particular, to consider both the phase-locked and phase-unlocked Ω*

_{C}*cases, the relative phase-noise γ*

_{P}_{14}is set equal to the phase-noise bandwidth γ

*under the condition that γ*

_{P}_{12}= 0, and set to zero or 2π × 1 MHz for the phase-locked or phase-unlocked Ω

*cases, respectively.*

_{P}Figure 5(b) shows the calculated Im(*σ*_{23}) values for TPA and |*σ*_{14}|^{2} for FWM in the four-level atomic model corresponding to Fig. 5(a). Here, the relative phase-noise of the Ω* _{pr}*, Ω

*, and Ω*

_{P}*lasers is zero, and the frequencies of Ω*

_{C}*and Ω*

_{C}*are fixed at δ*

_{P}*/2π = 1.2 GHz and resonance, as in Fig. 2(b), while the detuning δ*

_{C}*/2π of Ω*

_{pr}*is scanned in the vicinity of δ*

_{pr}*/2π = 1.2 GHz. The Rabi frequency of Ω*

_{pr}*, Ω*

_{pr}*, and Ω*

_{C}*were set to 0.02 MHz, 1.0 MHz, and 1.0 MHz, respectively. From the figure, we note that the normalized Im(*

_{P}*σ*

_{23}) and |

*σ*

_{14}|

^{2}spectra are identical. In addition, the calculated spectra are in good agreement with the spectral shape of the experimental FWM and TPA spectra shown in Fig. 2. From the calculated results, we confirmed that

*σ*

_{14}is proportional to the two-photon coherence

*σ*

_{12}, generated from one Λ-type configuration with the phase-locked Ω

*and Ω*

_{pr}*lasers. The Ω*

_{C}*absorption signal is proportional to Im(*

_{pr}*σ*

_{23}) and the TPA is due to the pure two-photon coherence

*σ*

_{12}[35]. Upon comparing the normalized Im(

*σ*

_{23}) for TPA and |

*σ*

_{14}|

^{2}for FWM signals in Fig. 5(b), we can identify both the calculated signals. The calculation results considering Maxwell-Boltzmann velocity distribution in Fig. 5(b) are in good agreement with the experimental FWM and TPA spectra shown in the inset of Fig. 2. Therefore, we can clearly conclude that the spectral shape of the FWM signal is identical with that of TPA.

The experimental results shown in Fig. 3 were observed under the condition of the coupling detuning δ* _{C}*/2π = 1.2 GHz, the two-photon detuning δ/2π = 0, and Γ

_{13}+ γ

*<< δ*

_{C}*, where Γ*

_{C}_{13}/2π and γ

*/2π are equal to 23 MHz and 1 MHz, respectively. The coherence*

_{C}*σ*

_{14}of Eq. (3) can be simply written as

*= Γ*

_{P}_{24}+ γ

*‒*

_{P}*i*δ

*. When the optical frequency of the pump laser is fixed at δ*

_{P}*= 0, the coherence*

_{P}*σ*

_{14}is just proportional to the two-photon coherence

*σ*

_{12}. To confirm that the spectral shape of the FWM signal is independent of the relative phase noise of the Ω

*, we calculated the |*

_{P}*σ*

_{14}|

^{2}for FWM signals according to the relative phase noise γ

*of Ω*

_{P}*, as shown in Fig. 6. When the γ*

_{P}*/2π increased from zero to 100 MHz, we could see that the spectral shape of the FWM signal maintains sub-natural linewidth. The calculated FWM spectrum in the case of the γ*

_{P}*/2π = 0 MHz (red curve) is identical to that in the case of the γ*

_{P}*/2π = 1 MHz (blue curve), as can be observed in the inset in Fig. 6. The calculation result is in good agreement with the observed FWM spectra shown in Fig. 3(b). Therefore, the spectral width of the FWM field has narrow linewidth regardless of phase noise of Ω*

_{P}*with the others.*

_{P}Finally, we discuss the phase relation of the FWM signal as shown in Fig. 4. To consider the phase relation of Ω* _{FWM}*, the rapid time coherence

*ρ*

_{14}can be rewritten as

*ω*and

_{FWM}*ϕ*(

_{FWM}*t*) are the optical frequency and phase of the generated FWM signal, where

*ω*= (

_{FWM}*ω*‒

_{C}*ω*) +

_{pr}*ω*and

_{P}*ϕ*(

_{FWM}*t*) = [

*ϕ*(

_{C}*t*) ‒

*ϕ*(

_{pr}*t*)] +

*ϕ*(

_{P}*t*), respectively.

*ϕ*(

_{pr,}_{C,}_{P}*t*) denote the random phase fluctuation of the Ω

*, Ω*

_{pr}*, and Ω*

_{C}*lasers. Under our experimental conditions of the phase-locked Ω*

_{P}*and Ω*

_{pr}*lasers [*

_{C}*ϕ*(

_{C}*t*) ‒

*ϕ*(

_{pr}*t*) = 0], as can be inferred from Eq. (5), the phase of the FWM signal is identical to that of the pump laser [

*ϕ*(

_{FWM}*t*) =

*ϕ*(

_{P}*t*)]. Therefore, we can conclude that the phase of Ω

*is strongly correlated with that of Ω*

_{FWM}*regardless of the relative phase of Ω*

_{P}*with respect to those of Ω*

_{P}*or Ω*

_{C}*.*

_{pr}## 4. Conclusion

We investigated the properties of FWM signals generated from a double Λ-type atomic system of the 5S_{1/2}(F = 1 and 2)−5P_{1/2}(F′ = 2) transition of ^{87}Rb atoms under the condition of narrow two-photon resonance between the two ground states coupled by phase-locked probe and coupling lasers. Firstly, via comparing the spectral features of the normalized FWM and TPA due to the pure two-photon coherence, we experimentally demonstrated that the FWM signal is proportional to two-photon coherence, which plays a very important role in the FWM process. Secondly, we observed the spectral width and the phase-noise of FWM light signals according to the relative phase-noise of the pump laser with respect to those of the phase-locked probe and coupling lasers. Interestingly, in both the phase-locked and phase-unlocked pump-laser cases_{,} the spectral features of both FWM signals as a function of the detuning of the probe laser were identical. From both these FWM spectra, we confirmed that the spectral features of the FWM spectrum are correlated with the pure two-photon coherence but uncorrelated with the relative phase of the pump laser. In addition, we investigated the phase correlation of the generated FWM signal with respect to those of the probe and pump lasers using the beat interferometer method. To compare the phase of the FWM signal with those of the pump and probe lasers, we measured the spectral densities of the beat signals between the FWM light signals and the probe or pump lasers. The phase of FWM is correlated with the phase of the pump light but independent of the phase-noises of the probe and coupling light beams. Therefore, we confirmed that the phase of the FWM light beam is strongly correlated with that of the pump laser, but uncorrelated with that of the probe laser. Using the double-Λ-type four-level atomic model, we could theoretically analyze the properties of the generated FWM signal, which are significantly related to the two-photon coherence. The spectral features of the FWM light signals do not depend upon the phase-noise of the pump laser, but the phase of the FWM signal is strongly correlated with that of the pump laser regardless of the phase-noise of the pump laser. We believe that our results can contribute to a better understanding of the properties of photon-pairs obtained via a spontaneous four-wave mixing process in a double-Λ-type atomic system and that of the retrieval light signal in quantum memories based on atomic coherence in a Λ-type atomic system.

## Funding

KIST Institutional Program (2E26681); National Research Foundation of Korea (2015R1A2A1A05001819).

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