1 Introduction

Electromagnetically induced transparency (EIT) attracted a lot of attention from all over the world in past two decades [1, 2], because it can greatly reduce the linear absorption of the generated optical field and enhance the refractive index. The interesting phenomenon was initially observed in a Λ-type three-level atomic system [1], and then in some other type of multi-level atomic systems [3]. Many studies on EIT indicate that this quantum interference effect has potential applications in several fields, such as ultraslow light [4], light storage [5–7], and quantum memory [8, 9], to name a few.

Such EIT is also useful for investigating multi-wave-mixing (MWM) processes since the generated signals can transmit through medium with little absorption under the EIT conditions. Enhanced four-wave-mixing (FWM) processes have been experimentally demonstrated [10]. FWM processes or six-wave-mixing (SWM) processes in semiconductor-doped glasses [11], liquid medium [12], and typical closed-cycle multi-level atomic systems [10] were experimentally observed, and they can coexist in a four-level atomic system due to atomic coherence [10]. We also reported the interplay among coexisting of FWM and SWM signals in a five-level atomic system of 85Rb (the open-cycled atomic system) [13]. The occurrence of EIT in the closed four-level atomic system is mainly because of atomic coherence effect, while in an open five-level atomic system it is because of population grating effect [14]. As far as to our knowledge, the link between the two effects has not been investigated ever before.

In this paper, we first report a new type EIT as well as a new type electromagnetically induced absorption (EIA) [15], which result from a completely different mechanism in comparison with standard EIT and EIA in previous literatures. We find those EIT and EIA can still appear in a V-type multi-level atomic system, even though the coupling beams propagate along the opposite directions of the probe beam, which do not satisfy the standard two-photon Doppler-free configuration. In order to explain the new type EIT and EIA, we raise up a new theory which agrees with our experimental results. Secondly, we investigate the competition and switching of atomic coherence and population grating effects, which exist in a closed four-level atomic system and an open five-level atomic system, respectively. The organization of this paper is that we briefly introduce the theoretical model and the experimental setup in Sect. 2, the analysis and discussions on the experimental results are displayed in Sect. 3, and in Sect. 4, we conclude the paper.

2 Experimental scheme and basic theory

The experimental demonstration of coexisting MWM processes was carried out in atomic vapors of 85Rb with a temperature of 60 °C and without buffer gas. The two relevant experimental systems are shown in Fig. 1b, c. In the first experiment, five energy levels from 85Rb atoms are involved in the experimental schemes. Five laser beams with the typical diameter 0.2 mm were arranged as indicated in Fig. 1a. The probe laser beam E P (E 1) (frequency ω 1, wave vector k 1, and Rabi frequency G 1) connecting |0〉 [5S 1/2(F = 3)] and |1〉 (5P 3/2) with a wavelength 780.24 nm comes from an external cavity diode laser (ECDL) and is horizontally polarized. The vertical polarized laser beams E 2 (ω 2, k 2, and G 2) and E′2 (ω 2, k′2, and G′2) with the same wavelength 776.15 nm connecting |1〉 (5P 3/2) and |4〉 (5D 3/2) are split from another ECDL. Another two vertical polarized laser beams E 3 (ω 3, k 3, and G 3) and E′3 (ω 3, k′3, and G′3) with the same wavelength 794.975 nm connecting |2〉 [5S 1/2(F = 2)] and |3〉 (5P 1/2) are split from a cw Ti:sapphire laser. As Fig. 1b indicated, |0〉, |1〉, and |4〉 form a ladder-type three-level atomic system. As Fig. 1a indicated, E 2, E′2 and E 3, E′3 propagate through the atomic cell with a small angle (about 0.3°) between them. Pump beams E 3 and E′3 together with E 1 give rise to the broaden FWM signal E F2 which is outside EIT windows and the FWM signal E F3 which is inside a new type EIT window, respectively. The generated signals E F2 and E F3 satisfy the same phase-matching condition k F2,3 = k 1 + k 3 − k′3. If E 3 and E′3 are added, two SWM signals E S1 and E S2 respectively satisfying k S1 = k 1 + k′2 − k′2 + k 3 − k′3 (E 1, E′2, E 3, E′3 turned on and E 2 blocked) and k S2 = k 1 + k 2 − k 2 + k 3 − k′3 (E 1, E 2, E 3, E′3 turned on and E′2 blocked) will be generated. The FWM and SWM signals with horizontal polarization all propagate in the opposite direction of E 1 and are detected by an avalanche photodiode detector. The transmitted probe beam is detected by a silicon photodiode.

Fig. 1
figure 1

a Square box pattern beams geometry used in the experiment based on the phase-matching condition. b–e Five-level open loop, four-level closed-loop, four-level open-loop, and three-level closed-loop V-type atomic systems of 85Rb, respectively. f Spatial beams geometry used to generate the coexisting FWM and SWM signals in new type EIT window. g The second auxiliary arrangement of the incident laser beams of the standard V-type EIT configurations

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In next experiment, with the shift of the frequency detuning of pump beam, the four-level system from 85Rb atoms is also involved in the experiment (Fig. 1c), in which pump beams E 3 and E′3 with the same wavelength 794.982 nm turn to connect |0〉[5S 1/2(F = 3)] and |3〉(5P 1/2). The ladder-type atomic system still exists, but E′2 is always blocked in this experiment. The energy levels |1〉, |3〉 together with |0〉 turn to form a V-type three-level atomic system. Only FWM E F2, E F3 and SWM E S2 exist under such experimental condition. On the other hand, with the scanning of the probe beam frequency detuning, ground state for E 1 changes from |0〉 to |2〉; therefore, we attain another five-level system and four-level system. Here, we can get FWM E F4, E F5 and SWM E S3, E S4 with same phase-matching conditions as FWM E F2, E F3 and SWM E S1, E S2. Moreover, when the new type EIT is discussed, we apply an auxiliary beam configuration, with which the incident beams E 1 propagates in the same direction with E 3 and E′3 (Fig. 1g), and the standard EIT can form. This reference probe beam is split from another ECDL and detected by another silicon photodiode.

2.1 Standard EIT and EIA

In the standard two-photon Doppler-free configurations of the V-type three-level (Fig. 1e) and four-level subsystems (Fig. 1d), E 1 propagates in the same direction with E 3 and E′3 (Fig. 1g). According to density matrix dynamic equations

$$\partial \rho_{10} /\partial t = - d_{1} \rho_{10} + i(G_{1} \rho_{00} - G_{1} \rho_{11} ),$$
$$\partial \rho_{13} /\partial t = - [i(\varDelta_{1} - \varDelta_{3} ) + \varGamma_{13} ]\rho_{13} + i(G_{1} \rho_{03} - G_{3}^{*} \rho_{10} ),$$

(in which the density matrix element ρ ij represents the particle population at the state |i〉 if i = j and the atomic coherence between |i〉 and |j〉 if i ≠ j; G i  = E i μ i /ħ is Rabi frequency with transition dipole moment μ i ; d 1 = iΔ 1 + Γ10, where Δ i  = Ω i  − ω i is the detuning; Γ ij is spontaneous decay rate from level |i〉 if i = j and the transverse relaxation rate between states |i〉 and |j〉 if i ≠ j) we obtain the density matrix element

$$\rho_{10} = \frac{{iG_{1} }}{{d_{1} + |G_{3} |^{2} /[i(\varDelta_{1} - \varDelta_{3} ) + \varGamma_{13} ]}}.$$
(1)

The imaginary part of ρ 10 determines the linear absorption of the probe. Therefore, an EIT can be created when the suppression condition Δ 1 − Δ 3 = 0 is satisfied and an EIA when enhancement condition Δ 1 − Δ 3 = |G 3|2/Δ 1 is satisfied.

2.2 New type EIT and EIA in V-type three-level

When beams E 3 and E′3 propagate in the opposite direction of E 1, the EIT regions determined by Eq. (1) will be broadened by the Doppler effect intensively. However, there still exists EIT-like window in a V-type three-level system (Fig. 1f, e), when not only the variations of coherence terms, but also the variations of the populations are considered. This theory can be explained as a new type EIT or EIA window. In such case, the density matrix equations are

$$\dot{\rho }_{10} = - d_{1} \rho_{10} + iG_{1} (\rho_{00} - \rho_{11} ),$$
(2)
$$\dot{\rho }_{00} = \varGamma_{33} \rho_{33} + iG_{3}^{*} \rho_{30} - iG_{3} \rho_{03} ,$$
(3)
$$\dot{\rho }_{30} = - (i\varDelta_{3} + \varGamma_{30} )\rho_{30} + iG_{3} (\rho_{00} - \rho_{33} ),$$
(4)
$$\rho_{00} + \rho_{33} = 1, \, \rho_{30} = \rho_{03}^{*} .$$
(5)

Here, we only consider the pumping effect of pump beams, and so \(\rho_{11}^{(0)} = 0\) and ρ 00 + ρ 33 = 1 can be satisfied. Since the probe transmission signal is considered, we use the non-diagonal matrix element ρ 10 to form Eq. (2). Additionally, Eqs. (3) and (4) are derived from the saturated absorption and burning hole effect. Moreover, under the steady state and weak probe field approximation, we obtain ρ 10 = i(G 1/d 1)ρ 00 = iG 1(d 2 + d 3)/(d 1(d 2 + 2d 3)), where \(d_{2} = \varGamma {}_{33}(\varGamma_{30}^{2} + \varDelta_{3}^{2} )\) and \(d_{3} = 2\varGamma_{30} G_{3}^{2}\). The atomic density with velocity ν is \(N(v) = Ne^{{ - v^{2} /u^{2} }} /(u\sqrt \pi )\) in the Maxwellian distribution, where N is the total number of atoms of 85Rb, and u represents the most probable atomic velocity, depending on the temperature of the cell, the atomic mass of particles, and the gas constant. The absorption coefficient is defined as α = ω 1 n 0 χ″/c, where χ″ is the imaginary part of the susceptibility χ related to the polarization intensity via P = ε 0 Eχ = Nμρ 10, and so we can obtain α∝Im[Nμρ 10/ε 0 E]. When considering the Doppler effect shown as Fig. 1f, the density matrix element ρ 10 is modified into ρ 10 = iN(ν)(G 1/d′1)[1 − d 3/(d′2 + 2d 3)], where d′1 = i(Δ 1 − kν) + Γ10, \(d^{\prime}_{2} = \varGamma {}_{33}[\varGamma_{30}^{2} + (\varDelta_{3} + kv)^{2} ]\) and assuming k 1 = k 3 = k. Since the atomic homogeneous broadening is much weaker than the Doppler broadening and the homogeneous broadening being saturated by the strong pumping, the Lorentz factor with homogeneous broadened effect can be deemed as a δ function, i.e., \({{\varGamma_{10} } \mathord{\left/ {\vphantom {{\varGamma_{10} } {\left[ {\pi \left( {(\varDelta_{1} - kv)^{2} + \varGamma_{10}^{2} } \right)} \right]}}} \right. \kern-0pt} {\left[ {\pi \left( {(\varDelta_{1} - kv)^{2} + \varGamma_{10}^{2} } \right)} \right]}} = \delta (\varDelta_{1} - kv)\). Therefore, α can be integrated over ν as \(\alpha (\varDelta_{1} ) \approx (N\sqrt \pi /u )e^{{ - \varDelta_{1}^{2} /(k^{2} u^{2} )}} \left[ {1 - {{d_{3} } \mathord{\left/ {\vphantom {{d_{3} } {\left( {\varGamma {}_{33}(\varDelta_{1} + \varDelta_{3} )^{2} } \right)}}} \right. \kern-0pt} {\left( {\varGamma {}_{33}(\varDelta_{1} + \varDelta_{3} )^{2} } \right)}}} \right]\), which means the absorption is reduced at Δ 1 + Δ 3 = 0. The result indicates that the two-photon Doppler-free condition is satisfied, i.e., the EIT window appears when suppression condition Δ 1 + Δ 3 = 0 is satisfied and EIA appears when also considering the dressing effect of E 3, and then, enhancement condition Δ 1 + Δ 3 = |G 3|2/Δ 1 will be satisfied. By analogy, we can obtain the density matrix element ρ 10 as

$$\rho_{10} (v) = \frac{{iG_{1} N(v)}}{{d^{\prime}_{1} }}\left[ {1 - \frac{{d_{3} }}{{\varGamma {}_{33}(\varDelta_{1} + \varDelta_{3} )^{2} }}} \right].$$
(6)

2.3 New type EIT and EIA in four-level system

On the other hand, in the four-level system (Fig. 1d) when beams E 3 and E′3 still propagate in the opposite direction of E 1, the satisfied density matrix equations are as follows

$$\dot{\rho }_{00} = \gamma_{20} \rho_{22} - \gamma_{20} \rho_{00}$$
(7)
$$\dot{\rho }_{22} = \gamma_{32} \rho_{33} + \gamma_{20} \rho_{00} - \gamma_{20} \rho_{22} + iG_{3}^{ * } \rho_{32} - iG_{3} \rho_{23}$$
(8)
$$\dot{\rho }_{10} = - d_{1} \rho_{10} + iG_{1} (\rho_{00} - \rho_{11} )$$
(9)
$$\dot{\rho }_{32} = - (i\varDelta_{3} + \varGamma_{32} )\rho_{32} + iG_{3} (\rho_{22} - \rho_{33} )$$
(10)
$$\rho_{00} + \rho_{22} + \rho_{33} = 1,\quad \rho_{23} = \rho_{32}^{*}$$
(11)

where γ 32 denote the spontaneous decay rates from level |3〉 to level |2〉, and γ 20 designates the population damping rate of levels |2〉 and |0〉. By the similar way in Sect. 2, the relevant non-diagonal density matrix element ρ 10 can be obtained as

$$\rho_{10} (v) = \frac{{iG_{1} N(v)}}{{2d^{\prime}_{1} }}\left[ {1 - \frac{{d_{5} }}{{\gamma_{32} (\varDelta_{1} + \varDelta_{3} )^{2} + 3d_{5} }}} \right],$$
(12)

with \(d_{5} = G_{3}^{2} \varGamma_{32}\).

2.4 Coexisting MWM

2.4.1 FWM in Δ 1 + Δ 2 = 0 EIT window

More specifically, when E 1 connects the transition 5S 1/2(F = 3) − 5P 3/2, and E 2 and E′2 connect the transition 5P 3/2 − 5D 3/2, the perturbation chain for the |0〉 − |1〉 − |4〉 FWM process E F1 is \(\rho_{00}^{(0)} \mathop{\longrightarrow}\limits{{\omega_{1} }}\rho_{10}^{(1)} \mathop{\longrightarrow}\limits{{\omega_{2} }}\rho_{40}^{(2)} \mathop{\longrightarrow}\limits{{ - \omega_{2} }}\rho_{10}^{(3)} (F1)\). This chain shows a closed-loop atomic coherence; thus, the third-order nonlinear density matrix element for the E F1 is

$$\rho_{F1} = \frac{{G_{F1} \exp (i{\mathbf{k}}_{F1} \cdot {\mathbf{r}})}}{{d_{6} (d_{1} + |G_{2} |^{2} /d_{6} )^{2} }},$$
(13)

where Δ 2 = Ω4 − ω 2 with resonate frequency Ω4 between the levels |1〉 and |4〉, d 6 = Γ40 + i(Δ 1 + Δ 2) and \(G_{F1} = - iG_{1} G_{2} (G^{\prime}_{2} )^{*}\).

2.4.2 FWM outside EIT window

A broadened FWM process E F2 from V-type subsystem (|0〉 − |1〉 − |3〉) is outside the EIT window. When E 3 and E′3 connect the transition 5S 1/2(F = 3) − 5P 1/2 and E 1 connects the transition 5S 1/2(F = 3) − 5P 3/2, the perturbation chain for this FWM process is \(\rho_{00}^{(0)} \mathop{\longrightarrow}\limits{{\omega_{3} }}\rho_{30}^{(1)} \mathop{\longrightarrow}\limits{{ - \omega_{3} }}\rho_{00}^{(2)} \mathop{\longrightarrow}\limits{{\omega_{1} }}\rho_{10}^{(3)}\) (F2) and a broadened FWM signal is generated by the Doppler effect, with the nonlinear density matrix element being

$$\rho_{{F2^{\prime}}} = \frac{{G_{F3} N(v)\exp (i{\mathbf{k}}_{F2} \cdot {\mathbf{r}})}}{{d^{\prime}_{1} (i\varDelta_{3} + \varGamma_{30} )\varGamma_{00} }},$$
(14)

with G F3 = −iG 1 G 3(G′3)* and Δ′3 = Δ 3 + kv. When E 3 and E′3 turn to connect 5S 1/2(F = 2) − 5P 1/2, we get

$$\rho_{{F2^{\prime\prime}}} = \frac{{G_{F3} N(v)\exp (i{\mathbf{k}}_{F2} \cdot {\mathbf{r}})}}{{d^{\prime}_{1} [i(\varDelta_{3}^{{\prime }} \varDelta_{3}^{{\prime }} + \varDelta^{\prime}) + \varGamma_{30} ]\varGamma_{00} }},$$
(15)

where Δ′ is the detuning constant and here it equals the energy-level frequency difference between |0〉 and |2〉. Finally, we get the total third-order nonlinear density matrix element ρ F2 = ρ F2′ + ρ F2″.

2.4.3 FWM in Δ 1 + Δ 3 = 0 new type EIT and EIA window

Since MWM process can be enhanced significantly in the EIT window, the new type EIT window also can be used to investigate such phenomenon. When E 3 and E′3 connect the transition 5S 1/2(F = 2) − 5P 1/2 and E 1 connects 5S 1/2(F = 3) − 5P 3/2, the interference between E 3 and E′3 induces a static population grating in the medium, which diffracts E 1 and gives rise to the FWM signal E F3′ with frequency ω 1. The corresponding perturbation chain is \(\rho_{00}^{(0)} \mathop{\longrightarrow}\limits{{\omega_{1} }}\rho_{10}^{(1)} \cdots \cdots > \rho_{12} \mathop{\longrightarrow}\limits{{ - \omega_{3} }}\rho_{13}^{(2)} \mathop{\longrightarrow}\limits{{\omega_{3} }}\rho_{12}^{(3)} \cdots \cdots > \rho_{10} (F3')\). The sub-chain \(\rho_{12}^{{}} \mathop{\longrightarrow}\limits{{ - \omega_{3} }}\rho_{13}^{(2)} \mathop{\longrightarrow}\limits{{\omega_{3} }}\rho_{12}^{(3)}\) represents the construction process of the static gratings, while \(\rho_{00}^{(0)} \mathop{\longrightarrow}\limits{{\omega_{1} }}\rho_{10}^{(1)}\) represents the diffraction process. The sub-chain \(\rho_{10}^{(1)} \cdots > \rho_{12}\) and \(\rho_{12}^{(3)} \cdots > \rho_{10}\) represent the transfer of atomic coherence and population grating between |0〉 and |2〉, which is induced by atomic collision. The nonlinear polarization responsible for such FWM is P F3′ = Q F3′ E 1/(d 1Γ00). Here, Q F3′ are the order parameters of the static gratings. An equation ∂Q F3′/∂t + (Γ20 − iΔ 3)Q F3′ = E 3 E′ *3 is satisfied for the chain (F3′), and we have

$$\rho_{{F3^{\prime}}} = \frac{{A_{F1} G_{F3} N\left( v \right)\exp [i{\mathbf{k}}_{F3} \cdot {\mathbf{r}}]}}{{2(d^{\prime}_{1} )^{2} [\varGamma_{13} + i(\varDelta_{1} + \varDelta_{3} + \varDelta^{\prime})]}},$$
(16)

where \(A_{F1} = 1 - d_{5} /[\gamma_{32} (\varDelta_{1} + \varDelta_{3} + \varDelta^{\prime})^{2} + 3d_{5} ]\). However, when E 3 and E′3 are turned to construct a closed-loop atomic coherence, the FWM process E F3″ from the V-type three-level subsystem |1〉 − |0〉 − |3〉 will be generated via the perturbation chain \(\rho_{00}^{(0)} \mathop{\longrightarrow}\limits{{\omega_{1} }}\rho_{10}^{(1)} \mathop{\longrightarrow}\limits{{ - \omega_{3} }}\rho_{13}^{(2)} \mathop{\longrightarrow}\limits{{\omega_{3} }}\rho_{10}^{(3)} (F3^{\prime\prime})\). The density matrix element related to such FWM process is

$$\rho_{{F3^{\prime\prime}}} = \frac{{A_{F2} G_{F3} N(v)\exp [i{\mathbf{k}}_{F3} \cdot {\mathbf{r}}]}}{{(d^{\prime}_{1} )^{2} [\varGamma_{13} + i(\varDelta_{1} + \varDelta_{3} )]}},$$
(17)

where \(A_{F2} = 1 - {{d_{3} } \mathord{\left/ {\vphantom {{d_{3} } {\left( {\varGamma {}_{33}(\varDelta_{3} + \varDelta_{1} )^{2} } \right)}}} \right. \kern-0pt} {\left( {\varGamma {}_{33}(\varDelta_{3} + \varDelta_{1} )^{2} } \right)}}\). Finally, we can obtain the total third-order nonlinear density matrix element as ρ F3 = ρ F3′ + ρ F3″. In addition, E 3 and E′3 cannot be considered as the dressing field and have no dressing effect since they are participating in constructing the static population gratings.

2.4.4 SWM in Δ 1 + Δ 2 = 0 EIT and EIA window

Under the condition E 3 and E′3 connect the transition between 5S 1/2(F = 3) − 5P 1/2 and E 1 connects the transition between 5S 1/2(F = 3) − 5P 3/2, the interference between E 3 and E′3 induces a static population grating in the medium, which diffracts E 1 and gives rise to the |0〉 − |1〉 − |4〉 − |3〉 SWM signals E S1′ and E S2′ with same frequency ω 1. Similarly, we have

$$\rho_{{S1^{\prime},2^{\prime}}} \propto \frac{{A_{F2} G_{S1,2} N(v)\exp (i{\mathbf{k}}_{S1,2} \cdot {\mathbf{r}})}}{{(d^{\prime}_{1} + |G_{2} |^{2} /d_{6} )^{2} d_{6} \varGamma_{00} (\varGamma_{30} + i\varDelta_{3}^{{\prime }} )}},$$
(18)

where G S1 = iG 1 G 3(G′3)* G′2(G′2)*, G S2 = iG 1 G 3(G′3)* G 2(G 2)*, and they satisfy the phase-match condition k S1 = k 1 + k′2 − k′2 + k 3 − k′3 (for S1) or k S2 = k 1 + k 2 − k 2 + k 3 − k′3 (for S2), respectively.

When E 3 and E′3 are turned to connect 5S 1/2(F = 2) − 5P 1/2, the density matrix element related to SWM signals E S1″ and E S2″ can be attained as

$$\rho_{{S1^{\prime\prime},2^{\prime\prime}}} \propto \frac{{A_{F1} G_{S1,2} N(v)\exp (i{\mathbf{k}}_{S1,2} \cdot {\mathbf{r}})}}{{2(d^{\prime}_{1} + |G_{2} |^{2} /d_{6} )^{2} d_{6} \varGamma_{00} [\varGamma_{30} + i(\varDelta^{\prime}_{3} + \varDelta^{\prime})]}}$$
(19)

Finally, we have the total fifth-order nonlinear density matrix element ρ S1,2 = ρ S1′,2′ + ρ S1″,2″.

When E 1 connects 5S 1/2(F = 2) − 5P 3/2, we can also obtain similar density matrix elements ρ F4 (ρ F5) for FWM signals E F4 (E F5) and ρ S3 (ρ S4) for SWM signals E S3 (E S4), respectively.

3 Experimental results

3.1 Different wave-mixing processes in open-cycled system

First, we conduct our experiment in the open-cycled atomic system to demonstrate different wave-mixing processes with different coupling laser beams blocked. When E 3 and E′3 beams are off, the coupling beams E 2 (a power of 28 mW) and E′2 (10 mW) between |1〉 and |4〉 together with the weak probe beam E 1 (7 mW) between |0〉 and |1〉 can generate a pure FWM signal E F1 (with k F1 = k 1 + k 2 − k′2) at frequency ω 1, which falls into ladder-type EIT window P (lower curve) and is very efficient as shown in Fig. 2a labeled as F1. This EIT satisfies the two-photon resonant condition at Δ 1 + Δ 2 = 0 according to the dressing term |G 2|2/d 6 in the corresponding density matrix ρ 10 = iG 1/(d 1 + |G 2|2/d 6). Next, when E′3 (with a power of 30 mW) is on (E 3 blocked) between |2〉 and |3〉, its optical pumping effect from |2〉 and |0〉 can enhance the FWM process E F1. The resulting FWM signal is in the direction of k D1 and labeled as D1 in Fig. 2b. Similarly, when E 3 (50 mW) is on (E′3 blocked), the system can also generate an enhanced FWM signal k D2 (in Fig. 2c labeled as D2), which gets very high since E 3 is much stronger than E′3. Here, the wave vectors of the unenhanced and enhanced signals are k F1 = k D1 = k D2. These three FWM signals (F1, D1, and D2) fall in the same Δ 1 + Δ 2 = 0 EIT window caused by the term d 6 in Eq. (13).

Fig. 2
figure 2

a Measured pure E F1 signal (the upper curve) and probe transmission (the bottom curve with standard ladder-type EIT window) versus the probe detuning Δ 1. b Measured a enhanced E F1 signal versus Δ 1 with E′3 on. c Measured another enhanced E F1 signal versus Δ 1 with E 3 on. d Measured pure E F2 signal. e Measured dressed E F2 and SWM signals E S1 versus Δ 1 with E′2 on. f Measured dressed E F2 and SWM signals E S2 versus Δ 1 with E 2 on. g Measured total MWM signals with E 1, E 2, E′2, E 3, E′3 on

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Without E 2 and E′2, as shown in Fig. 1b, the strong pump beams E 3 and E′3 (80 mW) together with E 1 generate a pure FWM signal E F2 outside the EIT window (Fig. 2d labeled as F2). Furthermore, we consider the case with either E′2 or E 2 further on. When E′2 is turned on (E 2 blocked), E′2 will dress FWM signal E F2 to generate a dressed FWM signal E D3 with the direction of k D3 (labeled as D3) and there will generate a SWM signal E S1 (labeled as S1) simultaneously (Fig. 2e) in ladder-type EIT window P. In the same way, when E′2 instead of E 2 is blocked, a dressed FWM signal E D4 with k D4 (labeled as D4) and a SWM signal (labeled as S2) can coexist and propagate in the same direction (Fig. 2f). Moreover, the wave vectors fulfill k F2 = k D3 = k D4. SWM signals S1 and S2 fall into the same EIT window P, but dressed FWM signals D3 and D4 are outside EIT windows.

Now, with all five laser beams (E 1, E 2, E′2, E 3, and E′3) on, especially with E′2 ≪ E 2, both FWM and SWM processes can coexist and propagate in the same direction, which can be shown by the fact that the total generated MWM signals shown in Fig. 2g (labeled as M) are larger than the pure FWM signal F1 (in Fig. 2a), SWM signal S1 (in Fig. 2e), or SWM signal S2 (in Fig. 2f). We can easily control the ratio of SWM/FWM intensities by adjusting the power of E′2. When E′2 has the similar intensity as E 2, only a very small percentage of SWM signal will exist since the efficient FWM process dominates. As the intensity of E′2 decreases, the FWM process gets suppressed, as the percentage of SWM increases, until the FWM process is completely turned off when the power of E′2 → 0.

3.2 Switching and competition between atomic coherence and population grating

Next, when we investigate the switching and competition between open-cycled atomic system and closed-cycled atomic system, E′2 is always off. Figure 3 presents the moving of EIT (EIA) and MWM signals when the frequency detuning of E 3 and E′3 increases from the top to bottom under two different beam configurations. Under the beam configuration shown in Fig. 1f, Fig. 3a1 shows the group of probe transmission signals in F = 3 and F = 2 sides, which are obtained near Δ 3 = 0 and Δ 3 = −3.03 GHz with E 1 connected |0〉 − |1〉 and |2〉 − |1〉, respectively. The new type EIT satisfying Δ 1 + Δ 3 = 0 and new type EIA satisfying Δ 1 + Δ 3 = |G 3|2/Δ 1 move to negative direction along Δ 1-axis as Δ 3 increasing, shown by oblique dot lines in the experimental results in Fig. 3a1, b1. Oblique dot lines in Fig. 3a2, b2 indicate that the corresponding FWM signal E F3 and E F5 also move when Δ 3 changes. The moving of two types of FWM signal E F3 in F = 3 side in Fig. 3a2 can be predicted by the term Γ13 + i(Δ 1 + Δ 3) in Eq. (17) when E 1 and E 3 and E′3 connect |0〉 − |1〉 and |0〉 − |3〉, while by Γ13 + i(Δ 1 + Δ 3 + Δ′) in Eq. (16) when E 1 and E 3 and E′3 connect |0〉 − |1〉 and |2〉 − |3〉. This phenomenon indicates the primary cause of E F3 switches from atomic coherence to population grating. FWM signal E F2 is outside the Δ 1 + Δ 3 = 0 EIT window and has the same phase-matching condition as E F3. E F2 is clearly Doppler broadened by the optical pumping effect due to Γ10 + iΔ 1 in Eq. (14). Due to the strong optical pumping effect, the peaks of E F2 overwhelm FWM signals E F3 when pump beams connect 5S 1/2(F = 2) − 5P 1/2 and probe beam simultaneously connects 5S 1/2(F = 3) − 5P 3/2 (F = 3 side in Fig. 3a2). Similarly, E F4 overwhelms E F5 in F = 2 side of Fig. 3a2. SWM signal E S2 and E S4 fall into the EIT satisfying Δ 1 + Δ 2 = 0 and EIA satisfying Δ 1 + Δ 2 = |G 2|2/Δ 1 shown by vertical dot lines in Fig. 3a1, b1. Since the detunings of coupling beams are fixed, the peaks of E S2 and E S4 are also fixed in Fig. 3a2, b2.

Fig. 3
figure 3

a1 New type of EIT in probe transmission signal versus Δ 1 with increasing Δ 3 from top to bottom. a2 FWM and SWM signals E F3 (E F5) and E S2 (E S4) in F = 3 (F = 2) side corresponding to a1. a3 EIT windows under standard Doppler-free condition versus Δ 1 with increasing Δ 3 from top to bottom. b1–b3 E 3 and E′3 are close to F = 2 side, which are different from (a1–a3) at F = 3 side, respectively

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Figure 3a3, b3 shows the EIT and EIA under the standard configuration in which E 1 and E 3, E′3 propagate in the same direction as shown in Fig. 1g. We notice that the two peaks of those standard EIT and EIA (satisfying Δ 1 − Δ 3 = 0 and Δ 1 − Δ 3 = |G 3|2/Δ 1, respectively) move in opposite direction along the Δ 1-axis to new type EIT and EIA, and they have the similar wave profile at the same time. It is worth to say that unlike standard EIT depending on both atomic system configuration and the propagating direction of related beams [15], the new type EIT only depends on the propagating direction. Two contra-propagating and two co-propagating beams in such new type EIT correspond to Δ 1 + Δ 3 = 0 in Fig. 1f and Δ 1 − Δ 3 = 0 in Fig. 1g, respectively.

Electromagnetically induced grating (EIG) was extensively studied [16–18]. Many experimentalists used the configuration of two contra-propagating beams [19]. In this work, two co-propagating beams with a small angle induce atomic coherence grating (satisfying standard EIT and EIA condition) or optical population grating (satisfying new type EIT and EIA condition) that deflects the probe, generating multi-wave-mixing signals. We can switch between coherence and population grating by changing the frequency detuning of pump the beams.

In the following, we concentrate on the frequency dependence of E 3 and E′3, as shown in Fig. 4, where the experimental condition is same as in Fig. 3. Figure 4a shows the dependence of FWM signal E F3 intensity versus Δ 3 at Δ 1 = 0. When Δ 1 = Δ 3 = 0, the intensity of E F3 reaches its right peak corresponding to the maximum of the atomic coherence effect, which is induced by i(Δ 1 + Δ 3) + Γ13 in Eq. (17). Meanwhile, the influence of population grating effect approaches minimum because E 3 and E′3 and E 1 connect the transition |0〉 − |3〉 and |0〉 − |1〉, respectively. When Δ 3 is turned to −3.03 GHz (i.e., connecting |2〉 − |3〉) but Δ 1 still satisfies Δ 1 = 0 (i.e., connecting |0〉 − |1〉), the population grating effect, induced by the pump beams E 3 and E′3, attains its maximum and the atomic coherence effect reaches the minimum, which results in the left peak in E F3, and can be explained by sub-terms i(Δ 1 + Δ 3 + Δ′) + Γ13 in Eq. (16). Analogously, as shown in Fig. 4b, another FWM signal E F5 reaches its left peak due to atomic coherence at Δ 1 = Δ 3 = −3.03 GHz and reaches its right peak due to population grating effect when Δ 1 = −3.03 GHz but Δ 3 = 0. We can conclude that it is the switching and competition between the atomic coherence and population grating effects that produce the double peaks as shown in Fig. 4a, b; however, the population grating effect is stronger than atomic coherence effect, which can be easily seen from the fact that the peak induced by population grating is higher than that induced by atomic coherence. When E 1 connects the transition |0〉 − |1〉, Fig. 4c shows SWM E S2 intensity in Δ 1 + Δ 2 = 0 EIT window satisfying two-photon Doppler-free condition, and it reaches maximal values at Δ 3 = 0 and Δ 3 = −3.03 GHz due to the sub-terms iΔ′3 + Γ30 and i(Δ′3 + Δ′) + Γ30 in Eqs. (18) and (19), respectively. Figure 4d shows the SWM E S3 with the similar physical process when E 1 connects the transition |2〉 − |1〉, and it also arrives two maximal values with similar physical schemes to E S2. Figure 4e, f show the frequency dependence of FWM E F2 and E F4 on Δ 3. They are outside the EIT window and are enhanced by optical pumping to maximal values with Doppler broadened effect, which can be explained by sub-terms iΔ′3 + Γ30 and i(Δ′3 + Δ′) + Γ30 in Eqs. (14) and (15), respectively. Figure 4g–l are the corresponding theoretical plots of MWM versus Δ 3, calculated by ρ F3, ρ F5, ρ S2, ρ S3, ρ F2, and ρ F4, respectively, which are matched quite well with the experimental results shown in Fig. 4a–f. Such theoretical calculation confirms the existence of competition and switching between the atomic coherence and population grating effects.

Fig. 4
figure 4

a, b Intensities of E F3 at Δ 1 = 0 and E F5 at Δ 1 = −3.03 GHz versus Δ 3, respectively. c, d Intensities of E S2 at Δ 1 = 0 and E S4 at Δ 1 = −3.03 GHz versus Δ 3, respectively. e, f Intensities of E F2 at Δ 1 = 0 and E F4 at Δ 1 = −3.03 GHz outside the EIT window versus Δ 3 at Δ 1 = 0, respectively. g–l Theoretical results correspond to a–f, respectively

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3.3 Power dependence induced by atomic coherence or population grating

Finally, the power dependence of MWM signals on various incident beams is studied, as shown in Fig. 5. The squares represent the relative intensity of MWM signals under the same conditions as in Fig. 3 and 4 except that E 1 is resonated with 5S 1/2 (F = 3) − 5P 3/2, and the frequency of E 3 and E′3 is fixed. Figure 5a1 shows the power dependence of E F3 on probe beam E 1 when E 3 is fixed around the transition between 5S 1/2 (F = 3) − 5P 1/2. We can clearly see that this FWM signal increases with the increasing of P 1. When E 3 and E′3 are close to the transition between 5S 1/2 (F = 2) − 5P 1/2, Fig. 5a2 shows the similar changes with the power of E 1 with Fig. 5a1, where the probe transmission signal will behave the new type EIT or EIA, and so the intensity of FWM signal E F3 increases with the power of E 1 increasing. However, when E 3 and E′3 are between 5S 1/2 (F = 3) − 5P 1/2 and 5S 1/2 (F = 2) − 5P 1/2, the results dependent on E 1 are shown in Fig. 5a3, which is similar to Fig. 5a1, a2. Figure 5b1–b3 shows the dependence of E F3 on the power of E 2, where the frequencies of E 3 and E′3 are under the same conditions with Fig. 5a1–a3. Since E 2 is not involved in producing FWM signal E F3, the power of beam E 2 has little relation with E F3, which results in that the intensity of E F3 is almost unchanged shown in Fig. 5b1–b3. Figure 5c shows the dependence on P 3, in which the intensities of FWM signals have the similar trending profile with case of the power dependence of E 1, since E 1, E 3, and E′3 together generate the population grating and the atomic coherence effects to generate FWM signal E F3.

Fig. 5
figure 5

a–c The intensity of FWM signal E F3 versus P 1, P 2, and P 3, respectively. (1)–(3) E 3 and E′3 is closed to the hyperfine ground state 5S 1/2 (F = 2), 5S 1/2 (F = 3), and between them, respectively. d–f Intensity of SWM signal E S2 in the same situation as a–c

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In consideration of the power dependence of SWM E S2 on k 1, k 2, and k 3, we take the similar analysis method used in the case of FWM, and the results are shown in Fig. 5d–f, respectively. We can see the power dependence on E 1 and E 3 of SWM has the analogues trend with that of FWM, i.e., the intensity of E S2 is almost proportional to the powers of E 1 and E 3. However, the power dependence of E S2 on E 2 is also proportional to P 2 (in Fig. 5c), which is completely different with the case of FWM signal (in Fig. 5b). The crucial reason lies in that E 2 participates in the generation of SWM signal E S2, as given in the phase-matching condition k S2 = k 1 + k 2 − k 2 + k 3 − k′3.

4 Conclusion

In conclusion, we have studied the coexistence and switching effects of MWM in four- and five-level atomic systems both theoretically and experimentally. Those MWM signals fall into the new type EIT window which is induced by electromagnetically induced population grating effect. In the competition between atomic coherence and population grating effect, the atomic coherence effect will play a main role when Δ 1 = Δ 3 = 0 or Δ 1 = Δ 3 = −3.03 GHz. However, the population grating effect will play a main role when the difference between Δ 1 and Δ 3 is 3.03 GHz. This competition leads to the double-peak effect of MWM and EIT if the detuning of the pump beams is scanned. Our research may contribute to the development of quantum information processing and MWM photon-pair generations.