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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 5S1/2–5P1/2 transition of 87Rb 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 5S1/2−5P1/2 transition of 87Rb 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 5S1/2(F = 1 and 2)−5P1/2(F′ = 2) transition of the 87Rb 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 (Ωpr) and coupling (ΩC) lasers for the Λ-type EIT configuration and a pump (ΩP) laser for a second Λ-type configuration with the generated FWM (ΩFWM) signal. δpr and δC represent the detuning of the probe and coupling lasers, respectively. The FWM signal is generated in the 5S1/2(F = 1)−5P1/2(F′ = 2) transition under the phase-matching condition.
Fig. 1 (a) Schematic of experimental setup for examining the characteristics of four-wave mixing (FWM) in the double-Λ-type atomic system of 87Rb with the use of 795-nm external cavity diode lasers (ECDLs) for probe (Ωpr), coupling (ΩC), and pump (ΩP) lasers (PD: photo-current detector; PBS: polarization beam splitter; BS: beam splitter; OI: optical isolator; HWP: half-wave plate; M: Mirror). (b) Energy-level diagram of the 5S1/2(F = 1 and 2)−5P1/2(F′ = 2) transition of 87Rb atoms.
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 “Ωpr” and “ΩP” lasers to the “ΩC” laser using the phase-locking method. For phase-locking among the three lasers, we use optical injection locking for the Ωpr laser and electrical phase locking for the ΩP laser, and the ΩC laser is used as the master laser. The Ωpr 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 ΩP laser is electrically phase-locked to an 8.0-GHz RF reference via beating between the master laser and ΩP. In the study, the three lasers were coupled to single-mode optical fibers and delivered to the setup for the FWM experiment.
The intensities of the three fields (Ωpr, ΩC, and ΩP) were adjusted independently with the use of a half-wave plate (HWP) and polarized beam splitter (PBS). The Ωpr and ΩC fields co-propagated and completely overlapped in the atomic vapor cell. The ΩP 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 87Rb 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 ΩFWM output with the transmitted Ωpr signal, both signals were simultaneously measured by two photocurrent detectors (PD1 and PD2). In particular, to investigate the relative phase-noise of the ΩFWM light beam with respect to those of the Ωpr and ΩC beams, the beat signals were measured between the ΩFWM and Ωpr light beams and between the ΩFWM and ΩP light beams for the cases of the phase-locked and phase-unlocked ΩP laser. In this work, we don’t directly measure the relative phase value of the ΩFWM light beam with respect to the phase of the Ωpr or ΩP light beams.
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 Ωpr (black curve) and the FWM signal (red curve) as a function of the two-photon detuning δ = δC ‒ δpr, which were simultaneously acquired by PD1 and PD2. The ΩC and ΩP laser frequencies were fixed at δC/2π = 1.2 GHz from the 5S1/2(F = 1)−5P1/2(F′ = 2) transition and at the 5S1/2(F = 2)−5P1/2(F′ = 2) transition, respectively, while that of Ωpr was scanned in the vicinity of δpr/2π = 1.2 GHz from the 5S1/2(F = 2)−5P1/2(F′ = 2) transition. The relative phases of the Ωpr and ΩP lasers were locked with respect to that of the ΩC laser. The powers of Ωpr, ΩC, and ΩP 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 Ωpr and ΩC were perpendicularly linear, and ΩP was linearly polarized perpendicular to the Ωpr polarization. However, when the ΩP laser was blocked in the FWM experiment corresponding to Fig. 1(a), in the Λ-type configuration with the Ωpr and ΩC lasers, the two-photon absorption (TPA) spectrum due to two-photon coherence between the two ground states of 5S1/2(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 Ωpr and ΩC 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.
Fig. 2 Transmittances of probe laser (black), two-photon absorption (TPA, blue), and four-wave mixing (FWM, red) spectra as functions of the detuning δpr/2π of the Ωpr laser, where TPA is due to the pure two-photon coherence by the phase-locked Ωp and ΩC lasers. The inset shows the normalized FWM and TPA spectra.
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 ΩP in the FWM process.
In order to investigate the change in the FWM signal according to the relative phase-noise of ΩP, we compared the FWM spectra for the cases of both phase-locked and phase-unlocked ΩP configurations. First, we measured the relative phase-noise of ΩP with that of ΩC. Figure 3(a) shows the spectral density curve of the beat signal between the ΩP and ΩC lasers at the center frequency of 8.0 GHz. In the case of the phase-unlocked ΩP 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 ΩP was electrically phase-locked to the 8.0-GHz frequency shift from ΩC, 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).
Fig. 3 (a) Relative linewidth measurements in cases of phase-unlocked ΩP (blue curve) and phase-locked ΩP (red curve); spectral density curves of beat signal between the ΩP and ΩC lasers at the center frequency of 8.0 GHz. (b) Comparison of four-wave mixing (FWM) spectra obtained with the phase-unlocked (blue curve) and phase-locked (red curve) ΩP laser.
Figure 3(b) shows the FWM spectra for both the phase-locked and phase-unlocked ΩP cases, wherein the detuning δpr/2π of Ωpr is scanned at two-photon resonance with ΩC. Interestingly, the FWM spectrum in the case of the phase-locked ΩP [see the red curve in Fig. 3(b)] is identical to that in the case of the unlocked ΩP 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 ΩP. This result indicates that the FWM signal is independent of the phase-noise of ΩP. This is counter-intuitive because FWM generation is based on stimulated emission by the ΩP 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 Ωpr. 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 Ωpr. 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 Ωpr and ΩC 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 ΩP.
As mentioned previously, the role of the ΩP 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 ΩP influences that of the ΩFWM light beam. To investigate the phase relation of ΩFWM, we compared the phase of ΩFWM to those of ΩP and Ωpr, respectively, using the beat interference method. Figure 4(a) shows the spectral density of the beat signal between Ωpr and ΩFWM in the case of the phase-unlocked ΩP laser, where the frequencies of the three lasers (Ωpr, ΩC, and ΩP) 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 Ωpr and ΩFWM. In Fig. 4(a), the relative phase-noise between Ωpr and ΩFWM 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 ΩFWM is independent of that of Ωpr. 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.
Fig. 4 Relative phase-noise measurement of ΩFWM with respect to those of ΩP or Ωpr for phase-unlocked ΩP: Spectral density of the beat signal between (a) Ωpr and ΩFWM and (b) ΩP and ΩFWM.
Meanwhile, Fig. 4(b) shows the spectral density curve of the beat signal between ΩP and ΩFWM. In contrast to Fig. 4(a), it is interesting to note that the relative phase between ΩP and ΩFWM is correlated despite the use of the phase-unlocked ΩP. The result indicates that the phases of both ΩP and ΩFWM are correlated regardless of the phase relation of ΩP. In the case of the phase-locked ΩP, we observed narrow spectral densities of both the beat signals between Ωpr and ΩFWM and between ΩP and ΩFWM.
From Figs. 4(a) and (b), the phase relation of ΩFWM can be understood as the atomic coherence in a double Λ-type atomic system. The phase relation between the Ωpr and ΩC lasers for the generation of two-photon coherence in one Λ-configuration is dependent on the phase relation of ΩP and ΩFWM for the generation of FWM of the other Λ-type configuration, because ΩP induces ΩFWM from the atomic coherence between the two ground states. Therefore, we confirmed that the phase of ΩFWM is correlated with that of ΩP, but uncorrelated with that of Ωpr.
In order to further understand the interesting results that the spectral width of ΩFWM does not depend on the spectral width of ΩP and that the relative phase of ΩFWM is correlated with that of ΩP regardless of the phase-noise of ΩP, 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>). δpr, C, P and γpr, C, 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
Fig. 5 (a) Double Λ-type four-level atomic model for four-wave mixing (FWM) generation. (b) Numerically calculated Im(σ23) for two-photon absorption (TPA, black curve) and |σ14|2 for FWM (red curve) spectra in four-level atomic model.
where the subscript indices i and j indicate the | i > and | j > states, respectively. Further, ρij denotes a density-matrix element and Hij the effective interaction Hamiltonian, which is composed of the atomic and interaction Hamiltonians. The two-photon coherence in one Λ-type configuration is ρ12 and the electric-field amplitude of the generated FWM signal is directly related to the coherence ρ14. It is convenient to transform into a co-rotating frame to eliminate the fast rotation. We can transform the density-matrix elements (ρij) into the rotating frame of a slowly varying density operator (σij). 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 σ12 and σ14 analytically, according to
andusing a diagrammatic method [33].Here, Δ12 = Γ12 + γ12 ‒ i(δC ‒ δpr) where γ12 denotes the relative phase-noise between the Ωpr and ΩC lasers, Δ13 = Γ13 + γC ‒ iδC where γC denotes the phase-noise bandwidth of the ΩC laser, and Δ14 = Γ14 + γ14 ‒ i(δC ‒ δpr + δP) where γ14 denotes the relative phase-noise among the Ωpr, ΩC, and ΩP lasers. Considering the broadening effect of the 4-Torr Ne buffer gas, the decay rates are set to Γ13/2π = Γ23/2π = 23 MHz, Γ14/2π = Γ24/2π = 23 MHz, and Γ12/2π = 0.1 MHz [34]. Although the phase-noise bandwidths γpr, C, P/2π of the Ωpr, ΩC, and ΩP lasers are set to 1 MHz considering the linewidth of the ECLDs, the relative phase-noise γ12 is zero because of phase-locking between Ωpr and ΩC. In particular, to consider both the phase-locked and phase-unlocked ΩP cases, the relative phase-noise γ14 is set equal to the phase-noise bandwidth γP under the condition that γ12 = 0, and set to zero or 2π × 1 MHz for the phase-locked or phase-unlocked ΩP cases, respectively.
Figure 5(b) shows the calculated Im(σ23) values for TPA and |σ14|2 for FWM in the four-level atomic model corresponding to Fig. 5(a). Here, the relative phase-noise of the Ωpr, ΩP, and ΩC lasers is zero, and the frequencies of ΩC and ΩP are fixed at δC/2π = 1.2 GHz and resonance, as in Fig. 2(b), while the detuning δpr/2π of Ωpr is scanned in the vicinity of δpr/2π = 1.2 GHz. The Rabi frequency of Ωpr, ΩC, and ΩP were set to 0.02 MHz, 1.0 MHz, and 1.0 MHz, respectively. From the figure, we note that the normalized Im(σ23) and |σ14|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 σ14 is proportional to the two-photon coherence σ12, generated from one Λ-type configuration with the phase-locked Ωpr and ΩC lasers. The Ωpr absorption signal is proportional to Im(σ23) and the TPA is due to the pure two-photon coherence σ12 [35]. Upon comparing the normalized Im(σ23) for TPA and |σ14|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 δC/2π = 1.2 GHz, the two-photon detuning δ/2π = 0, and Γ13 + γC << δC, where Γ13/2π and γC/2π are equal to 23 MHz and 1 MHz, respectively. The coherence σ14 of Eq. (3) can be simply written as
where ΔP = Γ24 + γP ‒ iδP. When the optical frequency of the pump laser is fixed at δP = 0, the coherence σ14 is just proportional to the two-photon coherence σ12. To confirm that the spectral shape of the FWM signal is independent of the relative phase noise of the ΩP, we calculated the |σ14|2 for FWM signals according to the relative phase noise γP of ΩP, as shown in Fig. 6. When the γP/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 γP/2π = 0 MHz (red curve) is identical to that in the case of the γP/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 ΩP with the others.Finally, we discuss the phase relation of the FWM signal as shown in Fig. 4. To consider the phase relation of ΩFWM, the rapid time coherence ρ14 can be rewritten as
Here, ωFWM and ϕFWM(t) are the optical frequency and phase of the generated FWM signal, where ωFWM = (ωC ‒ ωpr) + ωP and ϕFWM(t) = [ϕC(t) ‒ ϕpr(t)] + ϕP(t), respectively. ϕpr,C,P(t) denote the random phase fluctuation of the Ωpr, ΩC, and ΩP lasers. Under our experimental conditions of the phase-locked Ωpr and ΩC lasers [ϕC(t) ‒ ϕpr(t) = 0], as can be inferred from Eq. (5), the phase of the FWM signal is identical to that of the pump laser [ϕFWM(t) = ϕP(t)]. Therefore, we can conclude that the phase of ΩFWM is strongly correlated with that of ΩP regardless of the relative phase of ΩP with respect to those of ΩC or Ωpr.4. Conclusion
We investigated the properties of FWM signals generated from a double Λ-type atomic system of the 5S1/2(F = 1 and 2)−5P1/2(F′ = 2) transition of 87Rb 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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