Spatial-Multiplexed Four-Channel Optical Amplification via Multiple Four-Wave Mixing in a Double-Λ Atomic System

1. Introduction

Optical amplification and spatial multiplexing technologies play a crucial role in overcoming the limits of information transmission capacity and enhancing information processing speed [1,2,3]. These technologies have wide-ranging applications in quantum communication and photonic quantum networks. A major challenge lies in achieving multi-channel and high-efficiency optical signal amplification while preserving the quantum properties of high-dimensional information encoding and signal transmission [4]. Orbital angular momentum (OAM), which corresponds to orthogonal spatial modes with different topological charges, theoretically offers infinite-dimensional information encoding resources. OAM-based spatial multiplexing technology can construct high-dimensional Hilbert spaces, providing a critical physical foundation for high-capacity quantum communication, parallel quantum computing, and complex quantum state simulation and manipulation [5,6,7,8]. However, traditional optical amplification technologies face challenges such as gain imbalance and mode crosstalk when simultaneously amplifying multiple channels and preserving OAM modes [9,10]. These limitations constrain their applicability to complex quantum systems requiring high-dimensional and high-fidelity operation.

Four-Wave Mixing (FWM)‌ is a parametric process originating from the third-order nonlinear polarization of media, exhibiting unique characteristics including low noise, phase sensitivity, tunable bandwidth, and the capacity to generate quantum-correlated photon pairs [11,12,13]. As such, it has become a core nonlinear technology for achieving near-noiseless optical amplification [14,15], all-optical wavelength conversion [16,17], all-optical quantum gate operations [18,19], and deterministic quantum light sources [20,21]. However, traditional FWM systems driven by a single pump field face fundamental limitations in channel number scalability and OAM mode integration, which fall short of meeting the evolving demands of photonic quantum information technology for higher-dimensional and more complex architectures [22,23]. In recent years, significant advancements have been achieved in FWM utilizing alkali metal atoms (such as rubidium and cesium). The incorporation of electromagnetically induced transparency (EIT) enables the formation of highly coherent dark states near atomic resonance transition frequencies while simultaneously enhancing nonlinear optical responses. This breakthrough has opened new avenues for developing high-efficiency and low-noise FWM processes [24,25,26,27]. The applications of FWM have expanded from initial single-channel optical amplification [28,29,30] and correlated photon pair generation to the production of multiple correlated light beams through cascaded or parallel FWM configurations [31,32]. Nevertheless, most research efforts have predominantly focused on intensity or frequency multiplexing approaches [33,34], with comparatively limited attention devoted to the cooperative amplification and precise spatial mode manipulation of light fields.

As a prototypical multi-level system, the double-Λ atomic energy level configuration demonstrates pronounced nonlinear optical phenomena and exceptional controllability under the coherent interaction of multiple light fields, thereby serving as an ideal platform for realizing efficient multi-channel FWM processes [35,36,37]. By introducing coupling fields and control fields at distinct transition energy levels, the atomic system enables the construction of dual or multiple EIT windows. This not only effectively suppresses the linear absorption of probe light across multi-frequency channels but also concurrently enhances the third-order nonlinear susceptibility in the vicinity of multiple resonant frequency points, thereby facilitating the realization of multi-channel optical amplification based on FWM [38,39]. Moreover, the system opens a new avenue for the parallel processing and coherent manipulation of multi-dimensional photon quantum states, including those carrying OAM [40,41,42].

In this study, we propose and implement a four-channel optical amplification transmission scheme utilizing multiple FWM in a double-Λ cesium atomic system. Under the action of a single pump field, the nonreciprocal amplification (NRA) of co-propagating dual FWM fields is achieved by accounting for the Doppler effect of thermal atoms. Subsequently, by introducing a counter-propagating collinear pump field, a spatially multiplexed multi-FWM process is established, enabling bidirectional four-channel FWM signal amplification. Furthermore, we use spiral phase plates (SPP) to modulate the signal and pump beams to high-order Laguerre-Gaussian (LG) beams carrying different OAM quantum numbers, then experimentally and theoretically investigate the transfer characteristics of OAM among various FWM fields driven by the double counter-propagating pump fields. This scheme not only surpasses the channel number limitations of conventional FWM techniques but also significantly enhances spatial multiplexing capacity through the integration of OAM degrees of freedom. It thereby provides a novel approach for constructing high-capacity, multi-dimensional optical quantum information processing systems.

2. Experimental Setup and Results Analysis

The energy level diagrams of the atoms are shown in Figure 1 (a). A double-Λ atomic system includes two ground states |1⟩ and |2⟩ and one excited state |3⟩. This system can be realized in the D1 line of the ¹³³Cs atoms with |6S_1/2, F=4⟩, |6S_1/2, F=3⟩ and |6P_1/2, Fʹ=4⟩ acting as states |1⟩, |2⟩ and |3⟩, respectively. The frequency difference between the two lower levels is δ≈9.192 GHz. The two transitions |1⟩→|2⟩ and |1⟩→|3⟩ are simultaneously driven by two strong counter-propagating pump fields (forward pump field PF and backward pump field PB with same frequency ω_P) with different single-photon detunings Δ=ω_P –ω₃₁ and Δ-δ. The transition |3⟩ to |2⟩ is also coupled by a weak signal light S with frequency ω_S and detuning Δ_S=ω_S –ω₃₂. The multiple FWM can be satisfied when Δ_S =Δ, that is, while S is being amplified, three new FWM fields are simultaneously generated, which denoted as C, S′ and C′.

The schematic diagram of our experimental setup is shown in Figure 1 (b). We use two continuous frequency-tuned diode lasers, each shaped to fundamental mode Gaussian beam (TEM₀₀) by optical fibers, as the strong pump beam and the weak signal beam, respectively. The strong pump beam is divided into two parts, PF and PB, which are reflected to the L=25 mm-long Cs cell with vertical polarization through two Glan-Taylor prisms (GT1 and GT2, with the extinction ratio of 10⁵:1) in a collinear opposite direction. The weak signal beam S is reflected by a beam splitter (TS, with reflectivity 1%), and is then passed through the Cs cell with horizontal polarization to overlap with the PF at a small angle θ (θ≈0.23°). The e⁻² beam widths of PF, PB and S at the center of the cell are approximately 0.69 mm, 0.68 mm, and 0.49 mm, respectively. The cell temperature is stabilized at T_cell=100 °C. The forward FWM fields S and C are detected by PD1 and PD2, and the backward FWM field S′ and C′ are detected by PD3 and PD4, respectively.

Figure 2 shows the normalized transmission spectra under different pumping conditions. When only the forward PF and S are injected into the Cs cell, the forward FWM process occurs due to the phase-matching relation is achieved in this forward Doppler-free configuration, which satisfies the condition of nonreciprocity amplification [43,44,45]. At the two-photon resonance (e.g., Δ_S =Δ≈1.1 GHz), S is amplified, and a newly forward conjugate beam C is created in the same direction of its symmetry with the PF field, as shown by the red and blue curves in Figure 2 (a). At this time, no backward beams can be detected by PD3 and PD4, as shown by the light blue and green lines in inset of Figure 2 (a). Furthermore, when both PF and PB are present, multiple FWM processes occur simultaneously (see Figure 1). Along the opposite directions of the S and C, two new FWM beams denoted S′ and C′ are generated at the two-photon resonance, as shown the light blue and green curves in Figure 2 (b). The optical gain G_i of the FWM process is defined as the ratio of the output power P_i to the input signal power P_S0 [46], i.e., G_i = P_i/P_S0 (i=S, C, S′ and C′). By comparing Figure 2(a) and 2(b), it can be seen that on the basis of forward FWM (only PF injected), when the PB is introduced, the gain of the forward FWM fields G_S increases from 1.5 to 3, while G_C increases from 1.2 to 3.9. At this time, the gain of new generate backward FWM fields G_S′ ≈2.3, and G_C′ ≈1.8.

Actually, the four-channel optical amplification of Figure 2(b) is the result of the mutual superposition and enhancement of multiple FWM processes which include four interactions processes between light and atoms, as shown in Figure 3. The first FWM process (named FWM_f) is induced under the action of forward PF and S fields [see Figure 3(a1)]: The atom initially in the ground state |1⟩ first absorbs a forward pump photon PF and is stimulated to the excited state |3⟩, and then emits a forward signal photon S and jumps to another ground state |2⟩. Immediately after, the atom absorbs a forward pump photon PF and is stimulated to the excited state |3⟩, and finally emits a forward conjugate photon C, and finally jumps back to the ground state |1⟩. FWM_f can be described as$\left|1\right.〉\underset{A}{\overset{\mathrm{PF}}{\to }}\left|3\right.〉\underset{E}{\overset{\mathrm{S}}{\to }}\left|2\right.〉\underset{A}{\overset{\mathrm{PF}}{\to }}\left|3\right.〉\underset{E}{\overset{\mathrm{C}}{\to }}\left|1\right.〉$ (A and E represent the absorption and emission of the atom, respectively), with energy conservation relation 2ω_P=ω_S+ω_C and phase matching relation 2k_PF=k_S+k_C, see Figure 2 (b1). When considering the backward PB, the forward amplified S and generated C can respectively serve as seed beam, interacting with the two counter-propagating pump fields (PF and PB) to generate two additional FWM processes, named as FWM_S and FWM_C, as shown in Figure 3 (a2) and (a3). The interaction process of FWM_S can be expressed as $\left|1\right.〉\underset{A}{\overset{\mathrm{PF}}{\to }}\left|3\right.〉\underset{E}{\overset{\mathrm{S}}{\to }}\left|2\right.〉\underset{A}{\overset{\mathrm{PB}}{\to }}\left|3\right.〉\underset{E}{\overset{ \mathrm{S} \prime }{\to }}\left|1\right.〉$with 2ω_P=ω_S+ω_S′ and k_PF+k_PB=k_S+k_S′ [see Figure 3 (b2)], and that of FWM_C can be expressed as$\left|2\right.〉\underset{A}{\overset{\mathrm{PF}}{\to }}\left|3\right.〉\underset{E}{\overset{\mathrm{C}}{\to }}\left|1\right.〉\underset{A}{\overset{\mathrm{PB}}{\to }}\left|3\right.〉\underset{E}{\overset{ \mathrm{C} \prime }{\to }}\left|2\right.〉$with 2ω_P=ω_C+ω_C′ and k_PF+k_PB=k_C+k_C′ [see Figure 3 (b3)]. Finally, the newly generated S′(C′) and PB contribute the backward FWM (named as FWM_b) expressed as $\left|2\right.〉\underset{A}{\overset{\mathrm{PB}}{\to }}\left|3\right.〉\underset{E}{\overset{ \mathrm{S} \prime }{\to }}\left|1\right.〉\underset{A}{\overset{\mathrm{PB}}{\to }}\left|3\right.〉\underset{E}{\overset{ \mathrm{C} \prime }{\to }}\left|2\right.〉$with 2ω_P=ω_S′+ω_C′ and 2k_PB=k_S′+k_C′ , as shown in Figure 3 (a4) and (b4).

If the fundamental Gaussian optical fields injected into the Cs vapor cell are replaced by vortex beams carrying lℏ OAM (l is the OAM quantum number), the topological charge can also be transferred into the generated FWM beams through multiple phase-matched FWM interaction which in accordance with the principle of OAM conservation [9,10]. That is, the rules should be 2l_PF=l_S+l_C for FWM_f, l_PF+l_PB =l_S+l_S′ for FWM_S, l_PF+l_PB =l_C+l_C′ for FWM_C and 2l_PB=l_S′+l_C′ for FWM_b. Furthermore, to validate this inference, we employed spiral phase plate (SPP, which includes one vortex phase plate and two quarter-wave plate‌‌s) to modulate PF, PB, as well as S into Laguerre-Gaussian (LG) beams [see Figure 1 (b)], respectively, and observed the output pattern of the four amplified FWM fields by a camera.

When the injected S is modulated to LG₀₁ mode with l_S=1, and the two counter-propagating pump beams PF and PB are both in the TEM₀₀ mode (l_PF=l_PB=0), the four output FWM fields in both forward and backward directions exhibit LG₀₁ modes, while the modes of PF and PB remain unchanged, as shown the top in Figure 4 (a) and (b). Using the tilted lens method [47,48]，the OAM quantum numbers of the forward amplified FWM fields are: l_S=1 and l_C=-1, see the bottom in Figure 4 (a), and those of the backward generated FWM fields are l_S′=-1 and l_C′=1, see the bottom in Figure 4(b). Obviously, the experimental results demonstrate that all the four FWM processes adhere to the angular momentum conservation principle.

Actually, in each complete four-wave mixing process, while atoms absorb two pump photons and emit a pair of conjugate photons, OAM is simultaneously transferred from the pump photons to the amplified photons, in which the topological charge number transferred to each four-wave mixing light field depends on the original topological charge number of the injected signal light. Taking l_PF=1, l_PB=0 as an example, when the pattern of the injected S is TEM₀₀, it is found that the OAM quantum number of the amplified S contains unchanged with l_S=0, and that of the generated C is modulated to l_C=2 according to angular momentum conservation in FWM_f, as shown in Figure 5 (a). At the same time, the backward generated S′ (C′) is changed to LG₀₁ with l_S′=1 (l_C′=-1) by the process of FWM_S (FWM_C), as shown in Figure 5(b). As for the FWM_b, the angular momentum conservation is also satisfied. Similarly, for the case of l_PF=0, l_PB=1, the forward S and C are the same OAM quantum number with l_S= l_C=0, and those of the backward S′ and C′ are the same with l_S′=l_C′=1, as shown in Figure 5(c) and (d).

3. Theoretical Simulation of OAM Transfer

In order to quantitatively describe the dynamic behavior of OAM conversion in the multiple FWM processes, here we give the theoretical simulation using semiclassical approach. See Figure 1(a), under the condition of two counter-propagating pump fields, the Hamiltonian of the system can be shown as:

$$\begin{array}{l}\stackrel{\wedge }{H}=\hslash {\omega }_1\left|1\right.〉\left.〈1\right|+\hslash {\omega }_2\left|2\right.〉\left.〈2\right|+\hslash {\omega }_3\left|3\right.〉\left.〈3\right|\\ -\hslash \left[{\Omega }_{\mathrm{PF}}{e}^{-i{\omega }_{\mathrm{P}}t}+{\Omega }_{\mathrm{PB}}{e}^{-i{\omega }_{\mathrm{P}}t}+{\Omega }_{\mathrm{C}}{e}^{-i{\omega }_{\mathrm{C}}t}+{\Omega }_{ \mathrm{S} \prime }{e}^{-i{\omega }_{{\mathrm{S}}_{\prime }}t}\right]\left|3\right.〉\left.〈1\right|\\ -\hslash \left[{\Omega }_{\mathrm{PF}}{e}^{-i{\omega }_{\mathrm{P}}t}+{\Omega }_{\mathrm{PB}}{e}^{-i{\omega }_{\mathrm{P}}t}+{\Omega }_{\mathrm{S}}{e}^{-i{\omega }_{\mathrm{S}}t}+{\Omega }_{ \mathrm{C} \prime }{e}^{-i{\omega }_{ \mathrm{C} \prime }t}\right]\left|3\right.〉\left.〈2\right|+h.c.\end{array},$$

(1)

where ћ is the reduced Planck constant, Ω_i=μE_i/ ћ (i=PF, PB, S, C, S′, C′) denotes the Rabi frequency of each field in multiple FWMs, and h.c. represents the Hermitian conjugate term. By inserting the Hamiltonian from Equation (1) into the Liouville–von Neumann equation (see Appendix A (A1)), and using the rotating wave approximation (see Appendix A. (A2a)-(A2d)), we obtain the zero-order coupled equations (see Appendix A. (A3a)-(A3i)) and the first-order coupled equations (see Appendix A. (A4a)-(A4g)) for the density matrix elements. we then derive the expressions for the atomic coherence terms ${{\sigma }^{\prime }}_{31}$,${{\sigma }^{″}}_{31}$,${{\sigma }^{\prime }}_{32}$and${{\sigma }^{″}}_{32}$ under the first-order approximation of the four output optical fields. A_S(S′), A_C(C′), B_S(S′) and B_C(C′) are the parameters and their expressions see Appendix B.

$${{\sigma }^{\prime }}_{31}=A_{ \mathrm{S} \prime }{\Omega }_{ \mathrm{S} \prime }+B{}_{ \mathrm{S} \prime }\Omega {\mathrm{S}}^{\ast }+B_{ \mathrm{S} \prime }{{\Omega }_{ \mathrm{C} \prime }}^{\ast }$$

(2a)

$${{\sigma }^{″}}_{31}=A_{\mathrm{C}}{\Omega }_{\mathrm{C}}+B{}_{\mathrm{C}}\Omega {\mathrm{S}}^{\ast }+B_{\mathrm{C}}{{\Omega }_{ \mathrm{C} \prime }}^{\ast }$$

(2b)

$${{\sigma }^{\prime }}_{32}=A_{\mathrm{S}}{\Omega }_{\mathrm{S}}+B{}_{\mathrm{S}}\Omega {\mathrm{S}\prime }^{\ast }+B_{\mathrm{S}}{{\Omega }_{\mathrm{C}}}^{\ast }$$

(2c)

$${{\sigma }^{″}}_{32}=A_{ \mathrm{C} \prime }{\Omega }_{ \mathrm{C} \prime }+B{}_{ \mathrm{C} \prime }\Omega {\mathrm{S}\prime }^{\ast }+B_{ \mathrm{C} \prime }{{\Omega }_{\mathrm{C}}}^{\ast }$$

(2d)

Combining with the coupled Maxwell-Bloch equations describing the propagation of optical fields in the medium [49], the evolution equations for the four output optical fields along the propagation direction z of the medium are derived as:

$${\displaystyle \frac{\partial {\Omega }_{\mathrm{S}}}{\partial z}}=i\eta {{\sigma }^{\prime }}_{32}$$

(3a)

$${\displaystyle \frac{\partial {\Omega }_{\mathrm{C}}}{\partial z}}=i\eta {{\sigma }^{″}}_{31}$$

(3b)

$$-{\displaystyle \frac{\partial {\Omega }_{ \mathrm{S} \prime }}{\partial z}}=i\eta {{\sigma }^{\prime }}_{31}$$

(3c)

$$-{\displaystyle \frac{\partial {\Omega }_{ \mathrm{C} \prime }}{\partial z}}=i\eta {{\sigma }^{″}}_{32}$$

(3d)

where η=3N₀λГ/4π is the coupling constant (N₀ is atomic number density, λ is wavelength, and Г is attenuation coefficient). When numerically solving the above evolution equations, the initial conditions are set as: Ω_S(z=0)=Ω_S0, and Ω_C (z=0)=0, Ω_S′(C′) (z=L)=0

To compare with the three different combinations of input OAM topological charges verified experimentally (the experimental configurations are shown in Figure 4 and Figure 5), we conducted corresponding theoretical simulations and obtained the intensity and phase patterns of the four output optical fields. First, for the first topological charge configuration: l_S=1 and l_PF=l_PB=0, Figure 6 (a1)-(a4) show that the intensity distributions of all four output fields exhibit typical annular structures with a phase singularity (dark core) at the center of the ring, indicating they are all vortex beams. The core criterion for the topological charge of OAM can be characterized by the azimuthal evolution of the helical phase distribution: a phase variation of 2π corresponds to a topological charge l=1 when rotating counterclockwise around the optical axis for a full cycle, whereas a phase variation of -2π corresponds to l=-1 when rotating clockwise for a full cycle. As shown in Figure 6 (b1)-(b4), the topological charges of the output S and C’ satisfy: l_S=l_C′=1, and the topological charges of the output fields S’ and C satisfy: l_C=l_S′=-1.

Similarly, considering the second topological charge configuration (l_PF=1, l_S=l_PB =0) and the third topological charge configuration (l_PB=1, l_S=l_PF=0), the results are shown in Figure 7 and Figure 8, respectively. By combining the helical characteristics of the phase distributions and the phase evolution rule around the optical axis, the OAM topological charges of the four output fields under the two configurations are obtained as follows: for the second case: l_S=0, l_C=2, l_S′=1 and l_C′=-1; for the third case: l_S=l_C=0, l_S′=l_C′=1.

The above three theoretical simulations are agree well with the experimental results, and demonstrate that in our multiple FWM processes, whether it’s FWM_f or FWM_S, when an atom absorbs a pump photon and emits a signal photon S, it always transfers the same topological charge as the initial injected signal photon. Then, in accordance with the law of angular momentum conservation, the topological charge of the newly generated conjugate photon C or S′ is determined. Furthermore, the topological charge of C′ can also be determined by FWM_C or FWM_b. Based on our experimental scheme, while achieving four-fold spatial multiplexing optical amplification, the OAM of each newly generated FWM signal can be effectively controlled by modulating the topological charge of the two counter-propagating pump fields.

4. Conclusions

In this research, a scheme of four-channel optical amplification is experimentally demonstrated based on the multiple-stimulated FWM processes in hot Cs atoms. With the action of a single pump field, the necessary condition for the generation of dual-channel conjugate NRA signals is the establishment of FWM in the same direction and break in the opposite direction. However, a pair of optical amplified signals can be generated in the opposite direction by introducing a counter-propagating pump field. Based on this, when the signal and pump lights are modulated into higher-order LG beams carrying OAM to participate in the FWM process, the transfer and conservation of OAM among various amplified FWM fields are verified, which achieves OAM transfer in higher dimensions. These results pave the way for realizing complex structured light non-reciprocal devices by modulating OAM. Since optical non-reciprocal devices such as optical isolators, circulators, and phase shifters are fundamental units of all-optical communication, this conclusion holds potential applications for achieving high-capacity optical communication and high-dimensional signal processing.