1. Introduction

Recently, four-wave mixing (FWM) and six-wave mixing (SWM) processes based on third- and fifth-order, nonlinear processes in EIT media have also attracted worldwide attention in recent years [1–3], in which a strong coupling beam renders a resonant, opaque medium nearly transparency while enhancing the nonlinearity. Our group has reported the coexistence of these two nonlinear processes due to double EIT windows [4,5]. The nonlinear process plays important role in entanglement generation [6,7] and cascade-nonlinear optical process [8]. Such nonlinear process can be used to produce narrow band (MHz) and ultra-long coherence time (μs) two-mode and three-mode entanglement source.

The generation of non-classical light has both fundamental scientific significance and potential applications in quantum information and quantum metrology [9–14]. Several interesting works were performed, such as entangled images [6], the slow light [15], and the delay of Einstein-Podolsky-Rosen (EPR) entanglement [16]. The stable, efficient and reliable quantum entangled state is a major technical challenge. Generally, techniques for obtaining entanglement source are based on either broad band parametric down-conversion in solid-state crystals or narrow band spontaneous parametric four-wave mixing (SP-FWM) process in atomic vapors. Biphotons generations from spontaneous parametric down-conversion in nonlinear crystals have very wide bandwidth (THz) and ultra-short coherence time (ps) [17,18]. Moreover, a lot of interesting experiments have been carried out involving two cascaded FWM processes [16]. Paul Lett’s group has also experimentally realized a low-noise amplification of a continuous variable quantum state [19]. Additionally, the cascaded spontaneous parametric down conversion has also been successfully used to generated quantum-correlated photon triplets [20] and three-photon energy-time entanglement [21]. These results have been successfully implemented to develop a quantum imaging technique in nonlinear crystal [22]. The intensity-difference squeezing (IDS) has important applications in quantum metrology and gravitational wave detection along with potential applications in quantum information [23]. However, higher order multi-wave mixing in atomic ensemble are the special candidates to obtain high order controlled nonlinearity for strong correlation, narrow-band IDS [6].

In this paper, we propose a scheme to investigate the eight-wave mixing parametrical amplification. And we report the generation of correlated light beams from the PA-FWM, PA-SWM, PA-EWM processes, respectively, in the same energy level from a hot ⁸⁵Rb atomic ensemble. These nonlinear optical processes are controlled by adjustable dressing fields. Such controllable properties can find potential applications in all-optical communication and high dimensional stereoscopic imaging.

2. Experimental setup

Firstly, let us focus on Fig. 1(b), which shows a five-level atomic system. The 5S_1/2, F = 2 (|0>), 5S_1/2, F = 3 (|1>), 5P_3/2 (|2>), 5D_5/2 (|3>), 5P_1/2 (|4>) are five relevant energy levels in ⁸⁵Rb system. A strong pump beam E₁ (frequency ω₁, wave vector k₁, Rabi frequency G₁, vertical polarization, wavelength 780nm) tuned to the D₂ line transition (780nm), and the weak beam E₂ (ω₂, k₂, G₂, horizontal polarization, 780nm) as a probe. When the frequency detuning of E₁ is tuned far away from the resonance, the SP-FWM process will occur in the “double-Λ” configuration, generating the Stokes field E_S and anti-Stokes field E_aS (satisfying the phase match conditions (PMCs) k_S^F = 2k₁–k_aS^F and k_aS^F = 2k₁–k_S^F in Fig. 1(a2), respectively). E₃ (ω₃, k₃, G₃, vertical polarization, 776nm) and E₄ (ω₄, k₄, G₄, vertical polarization, 795nm) are the parallel cascade dressing fileds. The spatial beams alignment shows in Fig. 1(a1). In our experiment, we use one Ti:sapphire laser with 500 mW output power to generate pump beam (E₁) and one external cavity diode laser (ECDL) with 0.2mW output power to generate probe beam (E₂). All beams (E₁, E₂, E₃ and E₄) are focused at the center of the rubidium cell by optical lenses, where E₂ propagates in E₁ direction with an angle of 0.26°, E₄ propagates in E₁ direction while E₃ counter-propagates with E₁. The two SP-SWM and one SP-EWM processes will occur in this system by satisfying the PMCs k^S1_S + k^S1_aS = 2k₁ + k₃–k₃, k^S2_S + k^S2_aS = 2k₁ + k₄–k₄ and k_S^E + k_aS^E = 2k₁ + k₃–k₃ + k₄–k₄ in Fig. 1(a2). The output signals are taken to be the sum of several optical parametric amplification (OPA) processes, and output probe (FWM, SWM1, SWM2 and EWM) is detected by BHD. Also, the conjugate signal (FWM, SWM1, SWM2 and EWM) is coupling with output probe signal and received by the other BHD.

 

Fig. 1 (a1) Spatial beams alignment of the PA-FWM, PA-SWM and PA-EWM processes; I: isolator: WP; λ/2 wave plate: PBS: polarization beam splitter; M: mirror; BHD: balanced homodyne detector. (a2) Phase-matching geometrical diagram. (b) Energy-level diagram for the inverted-Y configuration in ⁸⁵Rb vapor. (c1) The dressing filed E₃ splits |2> state to |G_{2 ±} >. (c2) the dressing filed E₄ splits |1> (|0>) state to |G_{1 ±} > (|G_{0 ±} >).

Download Full Size | PDF

3. Basic theory

3.1 Third-order density matrix elements of SP-FWM

The generated E_S and E_aS signals in SP-FWM can be described by the third-order density matrix elements:

3.2 Generation process of PA-EWM

In order to investigate PA-EWM, firstly we take dressing effects of E₃ and E₄ into SP-FWM concurrently and the third-order density matrix element of double dressed SP-FWM can be written as:

When the E₂ is injected into the Stokes port of dressed SP-FWM (SP-FWM + SP-SWM1 + SP-SWM2 + SP-EWM) process in probe channel, it can amplify the seeded signal in an appropriate condition. Therefore, the photon numbers of the output Stokes and anti-Stokes fields of OPA can be described as:

That means the cascade dressing fields (E₃, E₄) can adjust the third-order density matrix elements ${{\rho }^{‴}}_{\text{21}\text{(}S)}^{(3)}$and ${{\rho }^{‴}}_{\text{20}\text{(}aS)}^{(3)}$ [Eqs. (7)-(10)], which can affect the nonlinear gain g_D [Eqs. (11) and (12)] of the PA-FWM. Besides, the correlation and squeezing result from g_D, which can be well controlled by the cascade dressing effect [24].

Therefore, the intensity of output signals are taken to be the sum of several interactional OPA processes, we investigate the correlation between PA-FWM, PA-FWM and PA-EWM by considering the suppression and enhancement of dressing effect. We set Δφ₁ = φ₁–φ₃, Δφ₂ = φ₁–φ₅, Δφ₃ = φ₁–φ₇, Δφ₄ = φ₃–φ₅, Δφ₅ = φ₃–φ₇ and Δφ₆ = φ₅–φ₇ as relative phases between these OPA processes. Therefore, with E₃ and E₄, when they both satisfies the enhanced conditions (Δ₃ + λ₃₁₊ = 0, Δ_S–λ_41- = 0), the relative phases (Δφ₁, Δφ₂, Δφ₃, Δφ₄, Δφ₅, Δφ₆) are all infinitely close to 0. Equation (9) of the pure enhancement output probe can be modified as:

Also, when the double dressing effects are E₃ enhanced and E₄ suppressed (Δ₃ + λ₃₁₊ = 0, Δ_S–Δ₄ = 0), or E₃ suppressed and E₄ enhanced (Δ_S + Δ₃ = 0, Δ_S–λ_41- = 0), the output probe [Eq. (9)] of partial enhancement and suppression could be explained as $\left|{\rho }_S^{(3)}\right|+\left|{\rho }_S^{(5)}\right|-\left|{{\rho }^{\prime }}_S^{(5)}\right|-\left|{\rho }_S^{(7)}\right|$and $\left|{\rho }_S^{(3)}\right|-\left|{\rho }_S^{(5)}\right|+\left|{{\rho }^{\prime }}_S^{(5)}\right|-\left|{\rho }_S^{(7)}\right|$, respectively. Besides, by adjusting the relative phases between these OPA processes, we could obtain four other partial enhancement and suppression cases ($\left|{\rho }_S^{(3)}\right|+\left|{\rho }_S^{(5)}\right|+\left|{{\rho }^{\prime }}_S^{(5)}\right|-\left|{\rho }_S^{(7)}\right|$, $\left|{\rho }_S^{(3)}\right|+\left|{\rho }_S^{(5)}\right|-\left|{{\rho }^{\prime }}_S^{(5)}\right|+\left|{\rho }_S^{(7)}\right|$, $\left|{\rho }_S^{(3)}\right|-\left|{\rho }_S^{(5)}\right|+\left|{{\rho }^{\prime }}_S^{(5)}\right|+\left|{\rho }_S^{(7)}\right|$ and $\left|{\rho }_S^{(3)}\right|-\left|{\rho }_S^{(5)}\right|-\left|{{\rho }^{\prime }}_S^{(5)}\right|+\left|{\rho }_S^{(7)}\right|$. When the E₃ and E₄ both suppressed (Δ_S + Δ₃ = 0, Δ_S–Δ₄ = 0), Eq. (9) could be explained as $\left|{\rho }_S^{(3)}\right|-\left|{\rho }_S^{(5)}\right|-\left|{{\rho }^{\prime }}_S^{(5)}\right|-\left|{\rho }_S^{(7)}\right|$ (pure suppression). Same with output probe signal, the conjugate signal also can be explained as $\left|{\rho }_{aS}^{(3)}\right|\pm \left|{\rho }_{aS}^{(5)}\right|\pm \left|{{\rho }^{\prime }}_{aS}^{(5)}\right|\pm \left|{\rho }_{aS}^{(7)}\right|$. According to the above-described correlation, the output photon numbers of probe and conjugate channel in double dressed PA-FWM process can be expressed as:

4. Experiment results

Firstly, we focus on the single dressed PA-FWM to investigate the constructive and destructive effects of by adjusting the phases between PA-FWM and PA-SWM. In the experiment, we scan the probe detuning Δ_p over 36.0 × 2π GHz. By shining dressing beam E₃ and changing its detuning, we observe the intensity variation of Stokes and anti-Stokes signal in probe and conjugate channels in Fig. 2 with large diameter of pump beam. When probe beam E₂ scans between 5P_1/2, F = 2 (|2>) and 5S_1/2, F = 3 (|1>) in Fig. 1(b), the output probe signal is injected into Stokes in Fig. 1(a), and it will generate gain peak ⁸⁵Rb, F = 3 Stokes signal in probe channel [Fig. 2(a)] and its corresponding ⁸⁵Rb, F = 2 anti-Stokes signal in conjugate channel [Fig. 2(c)] with phase match condition (PMC) k_S^F = 2k₁–k_aS^F [Fig. 1(a2)]_. Analogously, when E₂ scans between 5P_1/2, F = 2 (|2>) and 5S_1/2, F = 2 (|0>), it will generate gain peak anti-Stokes signal in probe channel [Fig. 2(b)] and its corresponding Stokes signal in conjugate channel [Fig. 2(d)].

 

Fig. 2 Measured intensities of signal with dressing field E₃ at different detuning Δ₃ in probe and corresponding conjugate channel. (a) With the pump beam E₁ of 780.2345nm, the intensity evolutions of Stokes in probe channel by increasing Δ_3. (b) Anti-Stokes signal in probe channel. (c) Anti-Stokes signal in conjugate channel corresponding to (a). (d) Stokes signal conjugate channel corresponding to (b). From bottom to top, the detuning of E₃ is changed from 0 to 0.7GHz. (a1), (b1), (c1) and (d1) represent the intensity of signal obtained under the condition when E₃ is off.

Download Full Size | PDF

It can be observed from Fig. 2 that the gain peaks splits into two or more peaks. The Stokes signal and anti-Stokes signal raises at the same probe detuning, while sometimes they have a deviation, which stems from the phase mismatch in frequency δ. Both Stokes and anti-Stokes signals have three extremas, denoted as δ_S-, δ_S0, δ_S+ and δ_aS-, δ_aS0, δ_aS+, where δ_aS + δ_S0 = δ_aS0 = 0 and δ_aS+ + δ_S- = δ_aS- + δ_S+ = 3GHz. With large diameter of pump beam, the Stokes signal and anti-Stokes signal in both probe and conjugate channel will split into two smaller peaks. This is caused by δ_S0≠0 (δ_aS0≠0), so that the Stokes (anti-Stokes) signal has higher probability to appear at δ_S- (δ_aS-) and δ_S+ (δ_aS+). The splitting is similar with the so-called Autler-Townes Splitting.

Next, we focus on the intensity variation of the gain peaks. The signal intensity of single dressed PA-FWM in probe channel [Eqs. (11) and (12)] can simply treat as ${\left|{\rho }_{21(p)}^{(3)}\pm {\rho }_{21(p)}^{(5)}\right|}^2$ and ${\left|{\rho }_{20(p)}^{(3)}\pm {\rho }_{20(p)}^{(5)}\right|}^2$, respectively. The signal intensity in conjugate channel could be described as ${\left|{\rho }_{20(c)}^{(3)}\pm {\rho }_{20(c)}^{(5)}\right|}^2$ and ${\left|{\rho }_{21(c)}^{(3)}\pm {\rho }_{21(c)}^{(5)}\right|}^2$. Here, due to the influence of large diameter of E₁, both phase shift factor e^iφ and δ of E₁ should be considered in the third-order matrix density. The injection of E₃ beam could cause strong dressing effect to the PA-FWM signals. Thus, the Stokes and anti-Stokes signals can be expected to get enhanced or suppressed due to the satisfaction of different enhancement and suppression conditions caused by the changing dressing field E₃. The dressed state diagram Fig. 1 (c1) shows that E₃ splits energy level |2> to two states |G₂ ± > and the increasing of the dressing detuning could gradually lead to the satisfaction of the suppression condition (Δ_S + Δ₃ = 0) to Stokes signal in probe field, which is caused by resonance of Stokes and E₃ field. Thus, the phase between the Stokes of FWM and SWM changes to Δφ₁ = φ₁–φ₃ = π in Fig. 2(a4). So the cross term is negative and the intensity of Stokes signal could be expressed as ${\left(\left|{\rho }_{21(p)}^{(3)}\right|-\left|{\rho }_{21(p)}^{(5)}\right|\right)}^2$ (the boson-creation operator of probe channel ${\widehat{a}}^{+}{}_4-{\widehat{a}}^{+}{}_6$). The corresponding anti-Stokes signal in Fig. 2(c2) in conjugate channel first increases due to the satisfaction of the enhancement condition (Δ_aS + λ₃₊ = 0). Corresponding to the above-described, the phase between the anti-Stokes of FWM and SWM will change from $\Delta {{\phi }^{\prime }}_1={\phi }_2-{\phi }_4=0$ and the intensity of anti-Stokes signal could be expressed as ${\left(\left|{\rho }_{21(c)}^{(3)}\right|+\left|{\rho }_{21(c)}^{(5)}\right|\right)}^2$ (the boson-annihilation operator of conjugate channel ${\widehat{b}}_4\text{+}{\widehat{b}}_6$). As shown in Fig. 2(b), the anti-Stokes signal is significantly reduced to the minimum value due to meet the suppression condition Δ_aS + Δ₃ = 0. So the intensity of gain peaks could be expressed as ${\left(\left|{\rho }_{21(p)}^{(3)}\right|-\left|{\rho }_{21(p)}^{(5)}\right|\right)}^2$. Similar to Fig. 2(c), the Stokes signal in conjugate channel increases due to the satisfaction of the enhancement condition (Δ₃ + λ₃₊ = 0) in Fig. 2(d), then decrease due to the deviation from the enhancement conditions.

Physically, the two pairs of generated Stokes and anti-Stokes signals are correlated in frequency and space, so the measured anti-Stokes signal and Stokes signal in conjugate channel should behave similar to the corresponding signal in probe channel and their enhancement and suppression conditions should be the same. However, the first pair of signals in Figs. 2(a) and 2(c) and the second ones in Figs. 2(b) and 2(d) are generated at different times and the powers of E₃ are 3mW and 10mW, respectively. Due to the dressing power increased, the energy |2> splitting becomes wider, and the originally enhanced condition of E_aS becomes suppressed, and the suppressed condition of E_S becomes enhanced.

In the following, we observe another single dressed PA-FWM for different physical principles with E₃. There are two single dressing methods to control the Stokes and anti-Stokes signals at the same time. One method is mentioned above, the other one can achieve the same effect though splitting energy level |1> and |0> with dressing field E₄. We continue to analyze the enhancement and suppression phenomena with PMC (k_S^S2 + k_aS^S2 = 2k₁ + k₄–k₄) by varying detuning of E₄ from –1.30 to 1.56GHz. As the energy level diagram Fig. 1(c3) showed, E₄ field dressed between |1> and |0> state and split these two states into |G₁ ± > and |G₀ ± >, the Stokes signal is enhanced if E₄ field detuning satisfies enhancement condition Δ_S–λ₄₀₊ + 3GHz = 0 (${\lambda }_{\text{40}\pm }\text{=}{\Delta }_{\text{4}}\pm \sqrt{{\Delta }_{\text{4}}^2\pm 4{\left|G_{\text{4}}\right|}^2})/2$). With the varying of E₄ detuning, the Stokes signal gradually meets the enhancement condition in probe channel, and the gain peak reaches the maximum value Fig. 3(a4) with the phase between the Stokes of FWM and SWM2 changing to ∆φ₂ = φ₁–φ₅ = 0 and its intensity can be described as $\left|{\rho }_{22(F=3)}^{(2)}\right|+{\left(\left|{\rho }_{21(p)}^{(3)}\right|+\left|{{\rho }^{\prime }}_{21(p)}^{(5)}\right|\right)}^2$ (probe channel ${\widehat{a}}^{+}{}_4\text{+}{{\widehat{a}}^{\prime }}_6^{+}$) and the intensity of the PA-SWM could be expressed as the difference between (a4) and (a1), which is about 0.4μW. The dip on the baseline in the Fig. 3(a) represents the E₄ field enhanced second-order fluorescence signal due to the term of ${\rho }_{22(F=3)}^{(2)}$. At the same time, the corresponding anti-Stokes signal is suppressed and satisfied suppression condition Δ_aS–Δ₄₀ = 0 in the conjugate channel [Fig. 3 (c)] and the gain peak reaches the minimum value [Fig. 3(c4)] with the phase between the anti-Stokes of FWM and SWM changing to $\Delta {{\phi }^{\prime }}_2={\phi }_2-{\phi }_6=\pi$ and its intensity could be described as $\left|{\rho }_{22(F=2)}^{(2)}\right|+{\left(\left|{\rho }_{20(c)}^{(3)}\right|-\left|{{\rho }^{\prime }}_{20(c)}^{(5)}\right|\right)}^2$ (conjugate channel ${\widehat{b}}_4-{{\widehat{b}}^{\prime }}_6$). The anti-Stokes signal [Fig. 3(b4)] get enhanced and reaches its maximum when it satisfies the enhancement condition ∆_aS–λ_41-–3GHz = 0, the cross term is positive and the intensity could be expressed as$\left|{\rho }_{22(F=2)}^{(2)}\right|+{\left(\left|{\rho }_{20(c)}^{(3)}\right|+\left|{{\rho }^{\prime }}_{20(c)}^{(5)}\right|\right)}^2$ and the intensity difference between (b4) and (b1) is about 0.6μW. In Fig. 3(d), it can be observed that Stokes signal in conjugate channel first decrease due to the satisfaction of the suppression condition (Δ_S–Δ₄₁ = 0) and then increases due to the deviation from the suppression conditions. The intensity difference between (d4) and (d1), which is 0.15μW for Stokes signal. Similar to Fig. 2, the first pair of signals in Figs. 3(a) and 3(c) and the second ones in Figs. 3(b) and 3(d), the dressing powers of E₄ are 10.5mW and 3.5mW, respectively. Due to the dressing power decrease, the splitting width of the energy |0> becomes narrower and the originally suppressed condition of E_aS becomes enhanced. In general, the splitting of the energy |1> is narrowed so that the Stokes signal deviate from the original enhancement condition, and the |G₀₊> level of the energy |0> splitting suppresses the Stokes signal at the same time. This is the reason for the opposite of the two pairs of signal changes.

 

Fig. 3 Measured intensities of signal with dressing field E₄ at different detuning Δ₄ in probe and corresponding conjugate channel. (a) With the pump beam E₁ of 780.2345nm, the intensity evolutions of Stokes in probe channel by increasing Δ_4. (b) Anti-Stokes signal in probe channel. (c) Anti-Stokes signal in conjugate channel corresponding to (a). (d) Stokes signal conjugate channel corresponding to (b). From bottom to top, the detuning of E₄ is changed from –1.30 to 1.56GHz. (a1), (b1), (c1) and (d1) represent the intensity of signal obtained under the condition when E₄ is off.

Download Full Size | PDF

Finally, we concentrate on the double dressed PA-FWM to investigate the constructive and destructive effects of by aligning the phases of PA-FWM, PA-SWM and PA-EWM. Learning from the curve in Figs. 2 and 3, we can select the enhancement or suppression points of E₃ and E₄ to be combined. In Fig. 4, we analyze four different typical phenomena of the double dressed PA-FWM (with PMC k_S^E + k_aS^E = 2k₁ + k₃–k₃ + k₄–k₄). When the phases among FWM, SWM1, SWM2 and EWM change from 0 to π, the double dressed PA-FWM could satisfy the pure enhancement, partial enhancement and suppression, or pure suppression condition, respectively.

 

Fig. 4 Measured Stokes signal of ⁸⁵Rb F = 3 (a) and anti-Stokes signal of ⁸⁵Rb F = 2 (b) in the probe channel. (c) (d) are same as (b) (a), respectively, but in in conjugate channel. (a1), (b1), (c1) and (d1) are the gain peaks with no dressing field; (a2), (b2), (c2) and (d2) with E₃ on; (a3), (b3), (c3) and (d3) with E₄ on; (a4), (b4), (c4) and (d4) are the gain peaks with both E₃ and E₄ on.

Download Full Size | PDF

When the dressing field E₃ is coupled in the system with E₄ blocked, we can find the intensity is reduced (compare (a2) to (a1)), because the suppression condition (Δ_S + Δ₃ = 0) of the Stokes signal in probe channel is satisfied in Fig. 4(a). Then, the enhancement condition (Δ_S–λ_41- = 0) is satisfied with E₃ blocked and the intensity of SWM2 is higher than the one of FWM obviously (compare (a3) to (a1)). Lastly, the double dressed (E₃ E₄) enhancement effect for nonlinear gain has been observed at Δ_S + Δ₃ = 0, Δ_S–λ_41- = 0. At the same time, it reveals that the enhancement effect of single dressed E₄ is stronger than the suppression effect of E₃ (compare (a4) to (a1)). In this case, it is because the relative phases among FWM, SWM1, SWM2 and EWM, which is Δφ₁ = Δφ₃ = Δφ₄ = Δφ₆ = π, Δφ₂ = Δφ₅ = 0, so the intensity (2 suppression and 1 enhancement) of Stokes signal could be described as $\left|{\rho }_{22(F=3)}^{(2)}\right|+{\left(\left|{\rho }_{21(p)}^{(3)}\right|-\left|{\rho }_{21(p)}^{(5)}\right|+\left|{{\rho }^{\prime }}_{21(p)}^{(5)}\right|-\left|{\rho }_{21(p)}^{(7)}\right|\right)}^2$ according to the Eq. (17) and the output photon numbers could be described as $〈N_{aDD}〉=〈{\widehat{a}}_D^{+}{\widehat{a}}_D〉=〈{\widehat{a}}_4^{+}{\widehat{a}}_4〉-〈{\widehat{a}}_6^{+}{\widehat{a}}_6〉+〈{{\widehat{a}}^{\prime }}_6^{+}{{\widehat{a}}^{\prime }}_6〉-〈{\widehat{a}}_8^{+}{\widehat{a}}_8〉$ [Eq. (14)].

Then, in Fig. 4(b), for the anti-Stokes signal of ⁸⁵Rb F = 2 in probe channel, the single dressed signal show destructive effects of SWM1 and SWM2 [compare (b2) (b3) to (b1)]. So the suppression conditions (Δ_aS + Δ₃ = 0 and Δ_aS–Δ₄ = 0) are satisfied, respectively. Then, the double dressed (E₃, E₄) suppression effect for nonlinear gain has been observed at Δ_aS + Δ₃ = 0, Δ_aS–Δ₄ = 0 [compare (b4) to (b3)]. This is due to the relative phases Δφ₁ = Δφ₂ = Δφ₃ = π, Δφ₄ = Δφ₅ = Δφ₆ = 0, so the intensity (pure suppression) of anti-Stokes signal can be expressed as $\left|{\rho }_{22(F=2)}^{(2)}\right|+{\left(\left|{\rho }_{21(p)}^{(3)}\right|-\left|{\rho }_{21(p)}^{(5)}\right|-\left|{{\rho }^{\prime }}_{21(p)}^{(5)}\right|-\left|{\rho }_{21(p)}^{(7)}\right|\right)}^2$with photon numbers $〈N_{bDD}〉=〈{\widehat{b}}_D^{+}{\widehat{b}}_D〉=〈{\widehat{b}}_4^{+}{\widehat{b}}_4〉-〈{\widehat{b}}_6^{+}{\widehat{b}}_6〉-〈{{\widehat{b}}^{\prime }}_6^{+}{{\widehat{b}}^{\prime }}_6〉-〈{\widehat{b}}_8^{+}{\widehat{b}}_8〉$ [Eq. (15)]. Similarly, for the Stokes signal of ⁸⁵Rb F = 3 in conjugate channel in Fig. 4(c), the single dressed signal is contrary to Fig. 4(b). The double dressed (E₃, E₄) enhancement effect for nonlinear gain has been observed at Δ₃ + λ₃₁₊ = 0, Δ_S–λ_41- = 0, and it is due to the relative phases Δφ₁ = Δφ₂ = Δφ₃ = Δφ₄ = Δφ₅ = Δφ₆ = 0, so the pure enhancement intensity of Stokes signal could be described as $\left|{\rho }_{22(F=2)}^{(2)}\right|+{\left(\left|{\rho }_{21(p)}^{(3)}\right|+\left|{\rho }_{21(p)}^{(5)}\right|+\left|{{\rho }^{\prime }}_{21(p)}^{(5)}\right|+\left|{\rho }_{21(p)}^{(7)}\right|\right)}^2$. For the anti-Stokes signal in Fig. 4(d), the single dressed signals are contrary to Fig. 4(a). Then, the suppression effect of double dressed (E₃, E₄) for nonlinear gain has been observed at Δ₃ + λ₃₁₊ = 0, Δ_S–Δ₄ = 0, and it is due to the relative phases φ₂ = Δφ₃ = Δφ₄ = Δφ₅ = π, Δφ₁ = Δφ₆ = 0, so the intensity (1 suppression and 1 enhancement) of Stokes signal could be described as$\left|{\rho }_{22(F=2)}^{(2)}\right|+{\left(\left|{\rho }_{21(p)}^{(3)}\right|+\left|{\rho }_{21(p)}^{(5)}\right|-\left|{{\rho }^{\prime }}_{21(p)}^{(5)}\right|-\left|{\rho }_{21(p)}^{(7)}\right|\right)}^2$. Besides, it reveals that the suppression effect of single dressed E₄ is stronger than the enhancement effect of E₃ [compare (d4) to (d1)]. There are common features in the above four cases. Dressing field plays the same role regardless of single or double dressed PA-FWM. Besides, though comparing the dressing effect, we can find out the leading one.

5. Conclusion

In summary, we have observed PA-SWM (PA-EWM) process by single (double) dressed PA-FWM. Besides, we propose a new method to describe the single dressed PA-FWM as the superposition of pure PA-FWM and pure PA-SWM while the double dressed PA-FWM as the superposition of one pure PA-FWM, two different PA-SWMs and one PA-EWM. We investigate the constructive and destructive effects by adjusting the phases of PA-FWM, PA-SWM and PA-EWM. Moreover, there are two single dressing methods to control the Stokes and anti-Stokes signals. One method is that dressing field E₃ splits energy level |2>, the other one is that E₄ splits |1> and |0>. Such properties of superposition can find potential applications in all-optical communication and high dimensional stereoscopic imaging.