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Abstract
The nonlinear parametric process is of great significance for achieving high-quality coherent optical signals and quantum correlated photons. With the development of classical and quantum information processing, the study of the properties of parametric processes is evolving in complex scenarios of multimode, which is limited in conventional nonlinear media due to strict phase matching, e.g. nonlinear crystals. Here we study the dressing-energy-level-cascaded four-wave mixing process to generate multimode optical parametric signals. Via cascading double-Λ type configuration of 85Rb D1 line, the non-degenerate energy-level-cascaded FWM is constructed to generate multimode self-parametric amplification. Moreover, with the dressing effects based on atomic coherence, the spatial and frequency multimode characteristics of energy-level-cascaded FWM parametric amplification, i.e., the modes number and pattern, are actively modulated by the pump fields detuning. Also, the spatial modes from the coupling of two coexisting spontaneous parametric FWMs can be controlled to reach tremendous scalability via the atomic coherence and Kerr non-linearity. The atomic coherence effects and unique phase-matching symmetry nature allow flexible modulation of the multimode property of the generated parametric signals within a nonlinear device, which paves a way for multimode classical and quantum information processing.
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1. Introduction
Nonlinear parametric process, including high-order harmonics parametric amplification [1,2], optical parametric oscillators (OPO) [3,4], spontaneous parametric down-conversion (SPDC) [5,6], four-wave mixing (FWM) [7,8], etc, is one of the most important nonlinear optical processes playing a pivotal role for realizing various classical and quantum tasks. In particular, the nonlinear parametric process is essential for generating quantum light sources, which is widely applied for quantum information processing and quantum metrology, e.g., quantum teleportation [9], quantum computing [10,11], quantum precision measurement [12–15], etc. With the development of quantum information technology, larger information capacity with multimode is very demanded in applications of the real world. The multimode parametric processes, which can be used to realize large-scale quantum states and entanglement, provide the foundation for realizing quantum information processing in complex scenarios [16–18], like quantum communication networks [19–21], scalable quantum computing [22–24] and quantum multi-parameter estimation [25–28], etc.
Hitherto, based on the nonlinear parametric processes, there are many ways to generate non-classical quantum light sources, such as bi-photon sources from spontaneous parametric down-conversion (SPDC) in $\chi ^2$ nonlinear crystals [29], spontaneous four-wave mixing (FWM) process in fibers [30,31], and squeezed light from nonlinear crystal based on OPO [3,4,32]. However, strict phase-matching conditions in such nonlinear media, e.g., birefringence phase-matching, limit it to reaching the multimode in the parametric processes. According to so, the generation of multimode quantum correlated signals relies on the superposition and cascading of bipartite nonlinear processes [24,33–35], and the modulation of the multimode properties needs more quantum sources and additional post-operation of linear optics. This leads to a lack of scalability and flexibility, restricting the further increase in the number of modes and information capacity. Besides, recently, multipartite entanglement and clusters states have been realized based on multimode OPOs, such as with frequency comb [25], sideband modes [36], temporal modes [22,23], and spatial modes [35]. But still, the modulation of the multimode properties needs additional post-operation or projective measurements set.
Another promising candidate for realizing multimode parametric processes is the FWM with atoms [19,37–39]. Atomic systems exhibit unique advantages, such as naturally isotropic symmetry and strong nonlinear susceptibility. The FWM of atoms process exhibits multimode properties due to the relaxed phase-matching condition and naturally isotropic symmetry of atoms [38]. The strong nonlinear susceptibility allows the FWM process to escape the cavity limitation. Many theories and experiments have been proposed based on the FWM processes of atoms to generate multimode entangled light. The intensity-difference squeezing of the signal and the idlers beams can be efficiently generated via the FWM process of rubidium atoms [40,41]. Dual-pumped FWM processes were also introduced to obtain intensity squeezing of multiple signal outputs [42]. The multi-spatial quantum squeezing was observed from the FWM of rubidium atoms [43]. Also, researchers achieved multiple modes through cascaded FWM processes of two rubidium vapors [44,45] and using orbital angular momentum (OAM) [20,37,46]. Moreover, the nonlinear susceptibility can be directly regulated with atomic coherence, which enables flexible and active modulation of multimode and coherent channels [47–50]. The atomic coherence modulation of FWMs has been applied to realize hyper entanglement and energy-time entanglement [51,52].
Multimode properties of nonlinear parametric amplification are important for extending the information capacity in both classical and quantum information applications. In this paper, we focus on employing atomic level cascading combined with atomic coherence-based dressing effects to realize frequency and spatial multimode parametric amplification, via non-degenerate dual pumped FWM processes within an atomic vapor. The dressing-energy-level-cascading FWM, shown in Fig. 1, can directly generate multiple signals and allow active modulation of the multimode properties within a nonlinear device. Figure 1(a) represents the usual parametric amplification process based on third-order nonlinear four-wave mixing. When a weak signal/idler and strong pump fields are injected with a tiny angle, the parametric process can achieve parametric amplification (PA-FWM) to generate the amplified signal and the idler beams. Under specific conditions, quantum correlations exist between the two produced beams. In the frequency domain, a third-order gain peak signal appears, as shown in Fig. 1(a) frequency mode, while in the spatial domain, due to the isotropic nature of the rubidium atomic system, the signal spots and the signal rings correspond to the condition with and without seeding, respectively. Moreover, Fig. 1(b) shows the one-step generation of four signal outputs within a single rubidium vapor, by employing an energy level cascaded parametric amplification (ECPA) process with dual pumps. Note that it is distinguished from the multiple atomic vapors cascading scheme [44]. In the energy-level cascading FWM scheme, all the cascading processes and the multimode modulations are within an atomic vapor, without introducing transmitting losses or more nonlinear devices when the modes number is increased. Figure 1(c) shows the active modulation of the ECPA process by dressing effects based on atomic coherence, which can simultaneously generate multiple coherent channels of FWM processes and thus expands the modes number in the frequency and spatial domain.
Fig. 1. Schemes of multimode parametric amplification process with frequency and spatial multimode. (a) The two-partite parametric amplification process scheme in the frequency and spatial domain. (b) The scheme of four-partite energy cascaded parametric amplification process in the frequency and spatial domain. (c) The scheme of the four-partite with frequency and spatial multimode via dressing energy cascaded parametric amplification process (in our self dressing system, the dressing fields are the same as generating dual pump field).
In this paper, multimode nonlinear parametric amplification from the energy-level-cascaded FWM of dressed atoms is investigated. Via cascading double-$\Lambda$ type configuration of $^{85}$Rb D$_1$ line, the non-degenerate energy-level-cascaded FWM structure is constructed to generate quadripartite self-parametric amplification signals. We observe the formation of spots via the quadripartite energy-level-cascaded FWM process in the spatial domain. Moreover, the modulation of the quadripartite signal modes is implemented from unimodal to multimodal verse pump detuning in both frequency and spatial domains. In particular, the spatial modes from the coupling of the two-coexisting spontaneous FWMs can be controlled to reach tremendous scalability via the atomic coherence and Kerr non-linearity [53], providing a viable path to an all-optical controlled quantum platform.
2. Theoretical model
The interaction Hamiltonian of the energy-level-cascaded FWM process, as seen in Fig. 2, can be expressed as
Fig. 2. (a) The energy-level diagrams of spontaneous parametric processes, i.e., two single-pumped FWMs with $\boldsymbol {E}_1$ ($\boldsymbol {E}_3$), and energy-level cascaded FWM. (b) The phase-matching diagrams of ECPA process and the phase-mismatching diagrams of ECPA process. (c) The spatial diagram of the energy-level-cascaded process. (d) The simulation of second-order fluorescence signals with dressing effect based on atomic coherence. (e) The dispersion of parametric signals $|\chi _{Si}^{(3)'}|^{2}$ ($i=1,2,3,4$) versus the frequency detuning $\delta$ ($\delta =\Delta _1-\Delta _{S1}$) obtained from ECPA process. (f) The simulation of spontaneous halos with dressing effect in the probe channel (Stokes channel).
Using FWM$_1$ as an example, $\chi _{S1}^{(3)}$ (Stokes signal) can be described by the perturbation chain [54] $\rho _{22}^{(0)}\overset {\omega _1 }{\rightarrow } \rho _{32}^{(1)}\overset {\omega _{S2} }{\rightarrow }\rho _{12}^{(2)}\overset {\omega _1 }{\rightarrow }\rho _{32(S1)}^{(3)}$, $\chi _{S2}^{(3)}$ (anti-Stokes signal) can be described by the perturbation chain $\rho _{11}^{(0)}\overset {\omega _1 }{\rightarrow } \rho _{31}^{(1)}\overset {\omega _{S1} }{\rightarrow }\rho _{21}^{(2)}\overset {\omega _1 }{\rightarrow }\rho _{31(S2)}^{(3)}$, where $\omega _{i}$ (i=1, S1, S2) represent the frequencies of pump $\boldsymbol {E}_1$, signal Stokes ($\boldsymbol {E}_{S1}$) and anti-Stokes ($\boldsymbol {E}_{S2}$) under ideal phase-matching conditions, shown in Fig. 2(b), respectively. The density matrix element can be expanded as $\rho _{mn}=\rho _{mn}^{(0)}+\rho _{mn}^{(1)}+\rho _{mn}^{(2)}+\dots +\rho _{mn}^{(k)}+\dots$, where the subscript $mn$ of $\rho _{mn}^{(k)}$ is its position in the density matrix and the superscript $k$ represents the order of its perturbation. In our case, $m$ (or $n$) indicates the corresponding energy levels, i.e., $m$ ($n$)=1, 2, 3 represent the 5S$_{1/2}$ F=2, 5S$_{1/2}$ F=3 and 5P$_{1/2}$ energy levels, respectively. The energy-level structure is shown in Fig. 2(a).
FWM$_i$ ($i=2,3,4$) have similar forms, the detailed deductions and expressions of theoretical models are not described here. The third-order non-linearity of $\boldsymbol {E}_{S1}$ and $\boldsymbol {E}_{S2}$ signals can be expressed as
where $A=N\mu _{31}^2\mu _{32}^2 /(\hbar ^3 \varepsilon _0)$, $D_1=\Gamma _{32}+i\Delta _2$, $D_2=\Gamma _{12}+i\Delta _2-i\Delta _{S2}$ and $D_3=\Gamma _{32}+i\Delta _2-i\Delta _{S2}+i\Delta _1$, $D_4=\Gamma _{31}+i\Delta _1$, $D_5=\Gamma _{21}+i\Delta _1-i\Delta _{S1}$ and $D_6=\Gamma _{31}+i\Delta _1-i\Delta _{S1}+i\Delta _2$. $\Delta _{S1}$ and $\Delta _{S2}$ are the detuning of the $\boldsymbol {E}_{S1}$ and $\boldsymbol {E}_{S2}$ signals, $\Delta _1$ and $\Delta _2$ are the detuning of the $\boldsymbol {E}_i$ ($i=1,1'$) fields, which represents the frequency difference between the resonant transition frequency $\Omega _i$ and the frequency $\omega _i$ of $\boldsymbol {E}_i$ ($i=1,1'$) ($\Delta _i=\Omega _i-\omega _i(i=1,1')$), $G_i=\mu _{ij}\boldsymbol {E}_i/\hbar$ is the Rabi frequency of $\boldsymbol {E}_i$ and $\Gamma _{ij}=(\Gamma _i+\Gamma _j)/2$ is the decoherence rate between $|i\rangle$ and $|j\rangle$. Considering the atomic coherence effect of dressing field $\boldsymbol {E}_3$ [55,56], the intensities of generating field $\boldsymbol {E}_{S1}$ and $\boldsymbol {E}_{S2}$ can be described by the $\chi _{S1}^{(3)'}$ and $\chi _{S2}^{(3)'}$ using the perturbation chain where $D_1'=\Gamma _{32}+i\Delta _2+|G_3|^2/d_1$, $d_1=\Gamma _{12}+i\Delta _2-i\Delta _{1}$, $D_4'=\Gamma _{31}+i\Delta _2+|G_3|^2/d_3$, $d_3=\Gamma _{11}+i\Delta _2-i\Delta _{S1}$. Considering the atomic coherence effect of dressing fields $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, the intensities of generating fields $\boldsymbol {E}_{S1}$ and $\boldsymbol {E}_{S2}$ can be described by the $\chi _{S1}^{(3)''}$ and $\chi _{S2}^{(3)''}$ using the perturbation chainThe main goal of the work is to study the multimode parametric amplification via dressing-controlled cascading FWM processes. However, besides the third-order nonlinear process, other atomic coherence effects co-exist in the generation of multimode signals, such as the first-order dressing susceptibility, e.g., electromagnetic induced absorption (EIA) [57] and the second-order perturbation process, e.g., fluorescence,
Considering the existence of the dressing effect, the realistic frequencies of them are expressed as $\omega _{S1}+\delta$ and $\omega _{S2}-\delta$, where $\delta =\Delta _1-\Delta _{S1}$ and $-\delta =\Delta _2-\Delta _{S2}$ and there will be multiple coherent channels in our system. The coherent channels for the signal $\boldsymbol {E}_{S1}$ is obtained by solving the equation by bringing $\delta$ and $-\delta$ into the equations $\chi _{S1}^{(3)'}$ above ($C_1=\Delta _1$, $C_2= (\Delta _1-\sqrt {\Delta _1^2+4\Gamma _{12}\Gamma _{32}+4G_1^2}/2)$, $C_3=(\Delta _1+\sqrt {\Delta _1^2+4\Gamma _{12}\Gamma _{32}+4G_1^2}/2)$) and $C_1$, $C_2$, $C_3$ are the values of $\delta$ at the resonance position. The coherent channels for the signal $\boldsymbol {E}_{S2}$ is obtained by solving the equation by bringing $\delta$ and $-\delta$ into the equations $\chi _{S2}^{(3)'}$ above ($C_1=-\Delta _1$, $C_2= (-\Delta _1+\sqrt {\Delta _1^2+4\Gamma _{12}\Gamma _{32}+4G_1^2}/2)$, $C_3= (-\Delta _1-\sqrt {\Delta _1^2+4\Gamma _{12}\Gamma _{32}+4G_1^2}/2)$).
The simulations of $|\chi _{Si}^{(3)'}|^{2}$ versus $\delta$ in four channels shown in Fig. 2(e). The number of spatial modes in Stokes channel is simulated considering the effects of $\boldsymbol {E}_{S1}$ dressing field and phase-mismatching condition (shown in Fig. 2(b)) shown in Fig. 2(f).
where $\delta _i=\Delta _i-\Delta _{Si}$ ($i=1,1',3,3$) is the detuning, $\omega _i$ is the laser frequency and $n_j$ is the refractive index. The equation establishes the link between $\Delta \boldsymbol {K}$ and $\delta$, the coherent channels in the frequency domain (Fig. 2(e)) and the coherent annulus in the spatial domain (Fig. 2(f)). The phase-mismatching and dressing effects cause the multimode properties of the signals, and the relationship of $\delta$ and $\Delta \boldsymbol {K}$ illustrates the uniformity of the multimode in both frequency and spatial domains. Besides, the Kerr non-linearity ($n_2=Re(\chi ^{(3)})/(\varepsilon _0cn_1)$) [58,59] can exacerbate the multimodal nature of the spatial domain, where $\chi ^{(3)}$ is the third-order nonlinear susceptibility. $\rho _{S1}^{(3)}$ ($\rho _{S1}^{(3)}=\chi ^{(3)}/(N\mu _{31}^2\mu _{32}^2 /\hbar ^3 \varepsilon _0))$) denotes the third-order non-linearity density matrix element expressed by $\rho _{22}^{(0)}\overset {G_{S1}^*}{\longrightarrow }\rho _{32}^{(1)}\overset {G_{S1}}{\longrightarrow }\rho _{22}^{(2)}\overset {G_{S1}^*}{\longrightarrow }\rho _{32}^{(3)}$.The total number of patterns is equal to the number of light spots multiplied by the number of patterns contained in each light spot:
where $n_{\text {dressing }}$ denotes the atomic coherent dressing effect’s light spot splitting. The number of spatial modes contained in each light spot is represented by $N_{\text {spatial }}$, which is as follows [60]: where the FWM phase-matching area $(\text {Area}_\text {pm})$ can be calculated from the solid angle $\Delta \Omega =\pi \Delta \Phi ^{2}$ , where $\Delta \Phi$ is the maximum interior angle of the signal. Based on the dielectric characteristics of the thermal atomic ensemble, hundreds of modes are verified in one spot area [38,43]. Note that the modes number, $N_{spatial}$, is based on an FWM channel, and thus modes number can be further enhanced in our scheme under the impact of atomic coherence assisted multiple co-existing coherence FWM channels. Note that, distinguished from using the usual nonlinear crystals, here we realize multimode parametric amplification via directly modulating the nonlinear susceptibility of the atomic media, where the waveforms of the nonlinear signals outputs are actively controlled via the atomic coherence. Also, compared to a recent work [42], instead of controlling the number of modes by modulating the phase-matching angle, our system directly controls the atomic coherence through dressing modulation via coupling the pump filed to the atomic energy levels.3. Results
3.1 Experimental setup
The schematic diagram of the experimental setup and the detection are shown in Fig. 3(a). The rubidium atom serves as the main source of third-order nonlinear ($\chi ^{(3)}$) interactions as well as parametric amplification, the corresponding energy-level structure is shown in Fig. 2(a). In cascaded double-$\Lambda$ system, two hyperfine ground state energy levels ($5s_{1/2}$ [F=2 ($|1\rangle$), F=3 ($|2\rangle$)]) and one excited state energy level ($5p_{1/2}$ [F=2 ($|3\rangle$)]) are encapsulated. When only pump $\boldsymbol {E}_1$ (frequency $\omega _1$, wave vector $\boldsymbol {k}_1$, Rabi frequency $G_1$, vertical polarization) existing (without the probe field), the region of spontaneous parametric process forms an annulus around the pump $\boldsymbol {E}_1$ due to the axial symmetry of atomic scheme, shown in Fig. 3(c). Adding the pump $\boldsymbol {E}_3$ (frequency $\omega _3$, wave vector $\boldsymbol {k}_3$, Rabi frequency $G_3$, vertical polarization) to the system (with random angle), two separate spontaneous FWM processes occur, producing two annulus, shown in Fig. 3(d), due to the weak cascaded phase-matching condition. Adjusting the optimal angle of pump fields $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, two separated spontaneous FWM processes couple to a self-parametric amplification energy-level-cascaded FWM process (the energy-level shown in Fig. 2(a)), satisfying the phase-matching condition ($2\boldsymbol {k}_1+2\boldsymbol {k}_3=\boldsymbol {k}_{S1}+\boldsymbol {k}_{S2}+\boldsymbol {k}_{S3}+\boldsymbol {k}_{S4}$), the symmetry of the annular region of the spontaneous process is broken, self-localized to four spots, shown in Fig. 3(e). Injecting the probe filed $\boldsymbol {E}_2$ (frequency $\omega _2$, wave vector $\boldsymbol {k}_2$, Rabi frequency $G_2$, horizontal polarization), an energy-level-cascaded FWM parametric amplification process occurs and the signals including the probe, conjugate, cascaded I and cascaded II. are amplified, shown in Fig. 3(f).
Fig. 3. The schematic diagram for generating energy-level-cascaded FWM signals and measurement devices. (a) The experimental setup. CCD, charge-coupled device; PBS, polarizing beam splitter; HR, high reflector; PD, Photo-detector; AOM, acoustic-optical modulator; Green, pump beam $\boldsymbol {E}_1$; Blue, pump beam $\boldsymbol {E}_3$; Red, probe beam $\boldsymbol {E}_2$ ($\boldsymbol {E}_{S1}$); Black, conjugate beam $\boldsymbol {E}_{S2}$; Purple, cascaded I beam $\boldsymbol {E}_{S3}$; Yellow, cascaded II beam $\boldsymbol {E}_{S4}$. (b) The diagram is obtained by PDs in the frequency domain when the probe field is generated by a separate laser with a wide scanning range (defining the injecting frequency of the probe field). Here, $\Delta _1$=−1.5 GHz, $P_1$=222 mW, $P_{S2}$=31 $\mu$W, $P_3$=57 mW. $\textrm {I}$, $\textrm {II}$, $\textrm {III}$ and $\textrm {IV}$ signals are generated when the detuning of the probe field is $\omega _{s1}$ -$\omega _1$=−3.8 GHz. In practice, $\Delta _1$ is fixed at −1 GHz while $\Delta _{S1}$ is scanning. (c)-(f) The diagrams are obtained by CCD in the spatial domain. (c) The spontaneous parametric halo via pump field $\boldsymbol {E}_1$. (d) Two spontaneous parametric halos via pump fields $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$. (e) Four spontaneous spots via pump fields $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$. (f) Four energy-level-cascaded FWM spots via probe field $\boldsymbol {E}_2$, pump fields $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ and the spatial location of the $\textrm {I}$, $\textrm {II}$, $\textrm {III}$ and $\textrm {IV}$ signals.
In the signal detection section, the pump $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ are split by the half-wave plate and PBS combination, based on polarization characteristics. The signals, generated by the energy-level-cascaded FWM process, can choose two paths via a foldable high reflector, the first path for frequency domain signals will be detected by four PDs, and the other path for their spatial images are detected by a CCD camera. The frequency of the probe $\boldsymbol {E}_2$ and pump $\boldsymbol {E}_3$ are shifted approximately 3 GHz and −0.8 GHz by two acoustic-optical modulators to satisfy the phase-matching and energy conservation conditions, respectively. Figure 3(b) is obtained by scanning the probe field $\boldsymbol {E}_2$ from another laser with a large periodic scanning range to determine the phase matching condition of the probe field. As shown in the Fig. 3(b), the signals appear at $+/-$3 GHz within one scanning cycle, and the symmetry between the left and right halves of the signals exhibits fine stability in our system. $\textrm {I}$ (probe), $\textrm {II}$ (conjugate), $\textrm {III}$ (cascaded $\textrm {I}$) and $\textrm {IV}$ (cascaded $\textrm {II}$) signals in Fig. 3(b) correspond to $\textrm {I}$, $\textrm {II}$, $\textrm {III}$ and $\textrm {IV}$ spots in Fig. 3(e).
When coupling the probe, conjugate, cascade I and cascade II signals via the $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ pump fields, the process can be generated efficiently within a certain detuning $\Delta _1$, considering the existing of the dressing effect based on atomic coherence, the dressing fields will split the energy levels, allowing phase-matching conditions more flexible. The signals in four channels evolve from single mode to multiple modes, indicating that the interior dressing effect of the two pump fields can make bright and dark states evolution’s [61] via modulating the pump fields’ detuning.
3.2 Multimode Experimental Results
The evolving multimode signals are observed by CCD and PDs in spatial and frequency domains. The experimental results are shown in Fig. 4. The pump light $\boldsymbol {E}_1$ detuning $\Delta _1$ is defined as the frequency of the $^{85}$Rb F=2 resonance absorption pit minus the $\boldsymbol {E}_1$ laser frequency ($\Delta _1=\Omega -\omega _1$). The image’s horizontal coordinate ranges from −1.2 GHz to −0.9 GHz with a 0.05 GHz interval from left to right.
Fig. 4. Frequency and spatial multimode parametric signals. (a)-(d) $\textrm {I}$- $\textrm {IV}$ Signals of the four outputs in the frequency domain via changing the pump fields detuning ($\Delta _1$ and $\Delta _3$) and scanning the probe detuning ($\Delta _{S1}$). Here, $P_1$ = 133 mW, $P_{S2}$ = 197 $\mu$W, $P_3$ = 77.8 mW, $T_{Rb}$ = 130$^{\circ }$C. (e) Upper part: The evolving profiles of the experimental signals in the conjugate channel (signals shown in Fig. 4(b)). Lower part: The three theoretical simulations of the signals based on Eqs. (2)–4. (f) The single dressing energy-level diagram of ECPA process for generating the multimode signals of Fig. 4(a)-(d). (g)-(l) The far-field images of ECPA process in the spatial domain via the changing of the pump detuning ($\Delta _1$ and $\Delta _3$). In this experiment, $\Delta _3$ simultaneously changes with $\Delta _1$. The upper insets show four self-localized spots without injecting the probe, and the bottom insets show the detailed spots of the amplified probe.
In the frequency domain, the fluorescence ($\chi ^{(2)}$) [60] and gain signals ($\chi ^{(3)}$) appear in all output channels $\boldsymbol {E}_{S1}$, $\boldsymbol {E}_{S2}$, $\boldsymbol {E}_{S3}$ and $\boldsymbol {E}_{S4}$) and the EIA just appear in the $\boldsymbol {E}_{S1}$ channel. The probe in Fig. 4(a) shows the evolution of the first-order EIA signals, the second-order fluorescence signals and the third-order gain signals ($\chi ^{(1)}+\chi ^{(2)}+\chi ^{(3)}$) superposition. The other three channels show the evolution of the second-order fluorescence signals and the third-order gain signals ($\chi ^{(2)}+\chi ^{(3)}$) superposition, as seen Fig. 4(b-d). Considering the first-order EIA (bright state) in probe channel (shown in Eq. (5)), when the pump detuning $\Delta _1$ increases, the intensity of EIA signal ($|\chi ^{(1)}|^2$) is declining. This is because the frequency of the probe field approaches the $^{85}$Rb F=3 resonance absorption position to satisfy the phase-matching condition with the pump fields. Proximity to the resonance position enhances the absorption of the probe signal by the atoms. Both atomic absorption of laser light and EIA are first-order dressing signals in competition (classical and quantum feature). The fluorescence signals increase when the pump detuning $\Delta _1$ increases (shown in Eq. (6)). The fluorescence signal (spontaneous radiation) is enhanced when the pump frequency approaches the $^{85}$Rb F=2 resonance position. Considering the dressing effects, there are four main cases for the fluorescence, i.e., the bright state, half-bright half-dark state, half-dark half-bright state and dark state, shown in Fig. 2(d).
To analyze multimode properties, the conjugate signals’ mode evolution in the frequency domain is divided into four regions with different dressing effects, as seen in Fig. 4(e). The regions from ① to ③ correspond to the simulations of no dressing, single dressing and double dressing processes, respectively, and the region ④ corresponds to the double dressing with fluorescence. When the pump detuning $\Delta _1$ increases, the intensity of gain peaks is rising. The dressing effects of the pump $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ split the signal into multi-peaks. Our simulation, with Eqs. (2)–4, agrees well with the experimental multi-peak results, shown in Fig. 4(e), which also proves that our model is well scalable for generating multimode signals.
In Fig. 4(a)-(d), all four output channels have similar multimode properties of dressing effects. The dressing energy level is shown in Fig. 4(f). In the case of single dressing, the energy level $|G_3\rangle$ is dressed by the pump field $\boldsymbol {E}_1$ and splitted into two new energy-levels $|G_{3+}\rangle$ and $|G_{3-}\rangle$. Its theoretical expression is shown in Eq. (3) and the simulation is shown in the middle of Fig. 4(e). In the case of double dressing, the energy level $|G_{3-}\rangle$ ($|G_{3+}\rangle$) can be further separated with the influence of the dressing effect of the pump filed $\boldsymbol {E}_3$. Its theoretical expression is shown in Eq. (4) and the simulation is shown in the right of Fig. 4(e). Some small peaks appear in the region $\textrm {IV}$, this might be the superimposition of third-order gain signals and second-order fluorescence signals (bright state). Note that to show the multimode properties with the multi-peaks of the spectra, we did not take the optimal gains of the nonlinear processes. These gains can be optimized by adjusting the system parameters, such as the dual pump angle or the pump’s powers, as shown in Fig. 7. Also, experimentally, the optimized gain of the energy-cascading FWM process can reach 10, shown in Fig. 3(b).
In the spatial domain, the main diagrams of Fig. 4(g)-(l) show the facula of the energy-level-cascaded FWM parametric amplification process with the probe field $\boldsymbol {E}_2$, the upper insets of Fig. 4(g)-(l) show the self-parametric amplification signal spots and the bottom insets of Fig. 4(g)-(l) show the probe signal spots. Similar to the frequency domain, the strong dressing fields based on atomic coherence split the energy levels, allowing more spatial modes to appear. When increasing the detuning of the pump fields $\Delta _1$, the pump $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ gradually approach the resonance position, and the spontaneous process produces spots of light that are constantly getting strong indicting the superposition of multimode spots. Injecting the probe field, the multimode spots generated by the dressing effects can be observed by the non-Gaussian spots. Combined with the analysis of the frequency multimode section, we can infer that spatial and frequency multimode evolve simultaneously with each other, which also matches with the model shown in Eq. (7).
The gain (G) and the signal-to-noise ratio (SNR) of the parametric FWM processes are defined as $G=(I_s-I_{\textrm {noise}}+I_0)/I_0$ and $SNR=10log(I_s/I_{\textrm {noise}})$, where $I_s$ is the intensity of the measured signal, $I_0$ is the intensity of the original probe signal, as well $I_{\textrm {noise}}$ is the corresponding background noise. The gain of the four outputs ($\boldsymbol {E}_{S1},\boldsymbol {E}_{S2},\boldsymbol {E}_{S3}$ and $\boldsymbol {E}_{S4}$) is shown with the black solid lines in Fig. 5(a)-(d). Note that for cascading FWM processes, here the gain has a vector form [62], including the gains of the four outputs. In the far-detuned region, the gains rise as the frequency of the laser approaches the atomic resonance position, which fits well with the dressing theory model in Eqs. (2)–4 ($G\propto |\chi ^{(3)}|^2$). Since the absorption of the signal is not corrected in the calculation of G, the real gain should be a bit greater than those in Fig. 5. The SNR of the four channels is shown as red dashed lines in Fig. 5(a)-(d). Interestingly, we find that there is a trade-off between the gains and the SNRs. For instance, in Fig. 5, the gain and the SNR curves cross and turn down when the laser approaches to the resonant position. This is due to the fact that as the coupling of the laser to the energy level is enhanced, a stronger second-order fluorescence signal appears, leading to a rise in the background noise. Also, the dressing effect with the near-resonant pumps can induce a suppression in gain, which is from the destructive superposition of the third-order nonlinear gain and higher order effects [63]. The optimal signal is obtained when $\Delta _1$ is around −1.1 GHz when both the gain and the SNR are simultaneously in a superior position. Besides, in our scheme, the lambda-type electromagnetic induced transparency (EIT) configuration, as seen in Fig. 2(a), within the four wave mixing processes protects the gain signals in the absorption pits from being absorbed by rubidium atoms. The EIT effect is related to the first-order dressing susceptibility model in Eq. (5).
Fig. 5. The gain and signal-to-noise ratio (SNR) of ECPA process. (a)-(d) The gain (black solid line) and signal-to-noise ratio (SNR) (red dashed line) of $\textrm {I}$-$\textrm {IV}$ signals in the frequency domain via the changing of the pump fields detuning ($\Delta _1$ and $\Delta _3$).
3.3 Discussions and conclusions
Atomic coherence can enhance and control the nonlinear effects, some of which are usually quite weak or can not co-exist with the parametric processes within conventional nonlinear crystals. In this part, the atomic Kerr effect is further modulated to control the spatial properties of the multimode parametric signals. The pump light $\boldsymbol {E}_1$ detuning $\Delta _1$ is defined as the frequency of the $^{85}$Rb F=3 resonance absorption pit minus the $\boldsymbol {E}_1$ laser frequency ($\Delta _1=\Omega -\omega _1$). The images’ horizontal coordinate ranges from −0.45 GHz to 1.05 GHz with a 0.15 GHz interval from left to right. The interaction between the pumps and the $^{85}$Rb F=3 energy level is employed for further achieving a large-scale spatial multimode. The light spot of the multimode parametric signals evolution is shown in Fig. 6. In Fig. 6(a)-(i), the main diagrams correspond to the spots signals of both the pumps $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, and the upper and bottom insets are the spots signals of the pumps $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, respectively. In Fig. 6(m)-(o), the main diagrams are the parametric amplification spots signals with the pump field $\boldsymbol {E}_1$ and the probe field $\boldsymbol {E}_2$, while the insets are the spots signals of the pump $\boldsymbol {E}_1$ without seeding.
Fig. 6. Spontaneous parametric spots in the spatial domain with the atomic coherence and Kerr non-linearity. (a)-(l) represent the spots from the interaction of the pumps $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ via the changing of the pumps detuning ($\Delta _1$ and $\Delta _3$). The upper and the bottom insets show the spots with only injecting the pumps $\boldsymbol {E}_1$ or $\boldsymbol {E}_3$, respectively. Here, $P_1$ = 150 mW, $P_{S2}$ = 150 $\mu$W, $P_3$ = 80 mW, $T_{Rb}$ = 85$^{\circ }$C. (m)-(o) Show the spots injecting the probe field via the changing of the pump ($\Delta _1$) detuning, where the insets are the corresponding spots without injecting the probe field.
Let us first observe the spots evolution of the pump $\boldsymbol {E}_1$, shown in the upper insets of Fig. 6. When the pump $\boldsymbol {E}_1$ at wavelength away from the resonance position, the spot is a stable, bright and concentrated circular spot. When the pump approaches the resonance position, the spot is unstable, resulting in many scattering spots, and the pump is expanding to large areas as shown in Fig. 6(a)-(h). There are two reasons for the light field expansion. Firstly, when the detuning $\Delta _1$ approaches the resonance position, the wave vector angle increases ($2\boldsymbol {k}_1=\boldsymbol {k}_s+\boldsymbol {k}_{as}$) due to the larger wavelength. The variation of the Stokes and anti-Stokes signals with wavelength is natural, due to energy conservation. Secondly, the third-order Kerr non-linearity cause the appearance of spatial instabilities and expansion as the wavelength changes. The generated Stokes and anti-Stokes signals are split into separated spots by the Kerr non-linearity [53]. Also, as the pump $\boldsymbol {E}_3$ is obtained by shifting the pump $\boldsymbol {E}_1$, the corresponding evolutionary rules are thus consistent with the pump $\boldsymbol {E}_1$. The changes of the pump detuning destroy the spatial symmetry, and together with the influence of the Kerr non-linearity, lead to the evolution from the ring to multi-spots. Figure 6(m)-(o) shows the images of the pump $\boldsymbol {E}_1$ with the probe field $\boldsymbol {E}_2$ injecting. The insets show the images of the signal pump $\boldsymbol {E}_1$ without the probe field. Multiple areas are amplified when injecting the probe field, proving that the resulting Stokes and anti-Stokes were split into spatial multimode.
Then we analyse the evolution of the spots of double pumps. Such spots can be seen as a linear superposition of the pump spots of $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, when the pump field detuning is far away from the resonance position, due to the weak interaction between the laser and the rubidium media. As the pump field detuning ($\Delta _1$ and $\Delta _3$) increases, close the resonance position, the pumps $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$ simultaneously interact with the atomic energy level, shown in Fig. 2(a). There is a nonlinear interaction between the spots generated by the involved spontaneous parametric processes of the pumps $\boldsymbol {E}_1$ and $\boldsymbol {E}_3$, and a self-amplification between the generated parametric signals occurs, which exacerbates the spatial instability and further splits the spatial modes, as shown in Fig. 6(e)-(i). Another interesting phenomenon is the restriction in the lateral direction of the light spot. The spots show significant laterally constrained features when the wavelength is far from the resonance position, as shown in Fig. 6(a)-(c). This phenomenon might be induced by the combined effects of the Kerr non-linearity and the gain of the four-wave mixing [64]. When the wavelength is close to the resonance position, the lateral constrain is released, and the Kerr effect become dominant for controlling the spots.
Besides, the nonlinear gains can be modulated and optimized by adjusting the system parameters, such as the phase-matching angle or the pump’s powers. The gains of the signals are investigated on changing the angle of the dual pumps and the power of the pump field. Figure 7(a)-(d) shows the gains and shapes of the signals when the angle between the dual pumps is modulated, as shown in Fig. 7(e). The gains of the four signals vary with the phase-matching angle, as shown in Fig. 7(a)-(d). The gain can reach a maximum of 30, corresponding to the dark green signal in the graph in Fig. 7(c), and the gain is greater than one except when the signal is within a dark state (shown as a pit-shape spectrum), as shown in Fig. 7(a). Here, the system parameters are set with the temperature of the atoms, $T_{Rb}$=130$^{\circ }$C, the detuning of the laser frequency, $\Delta _1$=−1.05 GHz, and the powers of the pumps and the seed, $P_{1}$=69 mW, $P_{S2}$=29 $\mu$W, $P_{3}$=41.6 mW. Figure 7(f)-(i) shows the four signals gains and shapes when the power of the pump E$_1$ is changed. The gain of the signal increases as the power of the pump $\boldsymbol {E}_1$ field rises. Here, the system parameters are set with the temperature of the atoms, $T_{Rb}$=125$^{\circ }$C, the detuning of the laser frequency, $\Delta _1$=−1.45 GHz, and the powers of the pump and the seed, $P_{S2}$=64.4 $\mu$W, $P_{3}$=27.2 mW. Note that the multimode characteristics of the signals are still kept while the gain is increased by optimizing the experimental parameters. Therefore, due to the atomic coherence and the unique phase-matching symmetry nature, in fact, multimode properties and high gains can co-exist in our system, which is usually not possible for conventional nonlinear birefringent crystals.
Fig. 7. The gains profiles of the energy level cascaded FWM parametric amplification signals. (a)-(d) The four signals outputs in the frequency domain via changing the angle of the two pump fields ($\theta$). (e) Diagram of the change of the angle between the dual pumps. Here, $T_{Rb}$=130$^{\circ }$C, $\Delta _{1}$=−1.05 GHz, $P_{1}$=69 mW, $P_{S2}$=29 $\mu$W, $P_{3}$=41.6 mW. (f)-(i) The four signals outputs in the frequency domain via changing the power of the pump field $P_{1}$. Here, $T_{Rb}$=125 $^{\circ }$C, $\Delta _{1}$=−1.45 GHz, $P_{S2}$=64.4 $\mu$W, $P_{3}$=27.2 mW.
In conclusion, we experimentally study the generation process of multimode parametric signals based on non-degenerate energy-level-cascaded FWM. The formation of the multimode spots in the spatial domain is generated via an atomic vapor. Then with the dressing effect based on atomic coherence, our scheme allows the modulation of the multimode characteristics of the multimode signals integrated within the generation process by controlling the pump field detuning. Our work provides a compact approach to classical and quantum information processing. In particular, the interaction between two spontaneous parametric signals enables multiple parametric amplification processes, greatly increasing the number of modes in the spatial domain, which can be directly applied for constructing multimode optical switches [65] and generating multiple entanglement and quantum squeezing sources [66,67].
Funding
National Key Research and Development Program of China (2017YFA0303700, 2018YFA0307500); National Natural Science Foundation of China (11904279, 12074303, 12074306, 12174302, 61975159, 62022066); Key Scientific and Technological Innovation Team of Shaanxi Province (2021TD-56).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. Pereira, M. Xiao, H. Kimble, and J. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38(9), 4931–4934 (1988). [CrossRef]
2. C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, “Strong low-frequency quantum correlations from a four-wave-mixing amplifier,” Phys. Rev. A 78(4), 043816 (2008). [CrossRef]
3. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986). [CrossRef]
4. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555–2557 (1987). [CrossRef]
5. S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18(18), 732–734 (1967). [CrossRef]
6. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25(2), 84–87 (1970). [CrossRef]
7. S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64(10), 1107–1110 (1990). [CrossRef]
8. Y. qing Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21(14), 1064–1066 (1996). [CrossRef]
9. J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, L. Shengkai, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488(7410), 185–188 (2012). [CrossRef]
10. Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, “Generation of high-fidelity four-photon cluster state and quantum-domain demonstration of one-way quantum computing,” Phys. Rev. Lett. 100(21), 210501 (2008). [CrossRef]
11. K. Chen, C.-M. Li, Q. Zhang, Y.-A. Chen, A. Goebel, S. Chen, A. Mair, and J.-W. Pan, “Experimental realization of one-way quantum computing with two-photon four-qubit cluster states,” Phys. Rev. Lett. 99(12), 120503 (2007). [CrossRef]
12. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981). [CrossRef]
13. T. Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7(12), 962–965 (2011). [CrossRef]
14. R. C. Pooser and B. Lawrie, “Ultrasensitive measurement of microcantilever displacement below the shot-noise limit,” Optica 2(5), 393–399 (2015). [CrossRef]
15. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006). [CrossRef]
16. P. Kok, W. Munro, K. Nemoto, T. Ralph, J. Dowling, and G. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(2), 797 (2007). [CrossRef]
17. C. Fabre and N. Treps, “Modes and states in quantum optics,” Rev. Mod. Phys. 92(3), 035005 (2020). [CrossRef]
18. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). [CrossRef]
19. Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91(1), 013843 (2015). [CrossRef]
20. W. Wang, K. Zhang, and J. Jing, “Large-scale quantum network over 66 orbital angular momentum optical modes,” Phys. Rev. Lett. 125(14), 140501 (2020). [CrossRef]
21. S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3(1), 1026 (2012). [CrossRef]
22. M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, “Deterministic generation of a two-dimensional cluster state,” Science 366(6463), 369–372 (2019). [CrossRef]
23. W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. Alexander, S. Takeda, J.-i. Yoshikawa, N. Menicucci, H. Yonezawa, and A. Furusawa, “Generation of time-domain-multiplexed two-dimensional cluster state,” Science 366(6463), 373–376 (2019). [CrossRef]
24. X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4(1), 2828 (2013). [CrossRef]
25. Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps, “Multimode entanglement in reconfigurable graph states using optical frequency combs,” Nat. Commun. 8(1), 15645 (2017). [CrossRef]
26. T. J. Proctor, P. A. Knott, and J. A. Dunningham, “Multiparameter estimation in networked quantum sensors,” Phys. Rev. Lett. 120(8), 080501 (2018). [CrossRef]
27. R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, “Multiparameter gaussian quantum metrology,” Phys. Rev. A 98(1), 012114 (2018). [CrossRef]
28. X. Guo, C. Breum, J. Borregaard, S. Izumi, M. Larsen, T. Gehring, M. Christandl, J. Neergaard-Nielsen, and U. Andersen, “Distributed quantum sensing in a continuous-variable entangled network,” Nat. Phys. 16(3), 281–284 (2020). [CrossRef]
29. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995). [CrossRef]
30. Y.-H. Li, W.-T. Fang, Z.-Y. Zhou, S.-L. Liu, S.-K. Liu, Z.-H. Xu, C. Yang, Y. Li, L.-X. Xu, G.-C. Guo, and B.-S. Shi, “Quantum frequency conversion for multiplexed entangled states generated from micro-ring silicon chip,” Opt. Express 26(22), 28429–28440 (2018). [CrossRef]
31. W.-T. Fang, Y.-H. Li, Z. Zhou, L. Xu, G.-C. Guo, and B.-S. Shi, “On-chip generation of time-and wavelength-division multiplexed multiple time-bin entanglement,” Opt. Express 26(10), 12912 (2018). [CrossRef]
32. M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maître, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67(2), 023808 (2003). [CrossRef]
33. J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zeilinger entanglement,” Nature 403(6769), 515–519 (2000). [CrossRef]
34. Y. Huang, B. Liu, L. Peng, Y.-H. Li, L. Li, C.-F. Li, and G.-C. Guo, “Experimental generation of an eight-photon greenberger–horne–zeilinger state,” Nat. Commun. 2(1), 546 (2011). [CrossRef]
35. R. Ukai, N. Iwata, Y. Shimokawa, S. Armstrong, A. Politi, J.-I. Yoshikawa, P. Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106(24), 240504 (2011). [CrossRef]
36. M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112(12), 120505 (2014). [CrossRef]
37. V. Boyer, A. M. Marino, and P. D. Lett, “Generation of spatially broadband twin beams for quantum imaging,” Phys. Rev. Lett. 100(14), 143601 (2008). [CrossRef]
38. N. V. Corzo, A. M. Marino, K. M. Jones, and P. D. Lett, “Noiseless optical amplifier operating on hundreds of spatial modes,” Phys. Rev. Lett. 109(4), 043602 (2012). [CrossRef]
39. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef]
40. M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A 88(3), 033845 (2013). [CrossRef]
41. Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, “Double-Λ microscopic model for entangled light generation by four-wave mixing,” Phys. Rev. A 82(3), 033819 (2010). [CrossRef]
42. S. Liu, H. Wang, and J. Jing, “Two-beam pumped cascaded four-wave-mixing process for producing multiple-beam quantum correlation,” Phys. Rev. A 97(4), 043846 (2018). [CrossRef]
43. C. S. Embrey, M. T. Turnbull, P. G. Petrov, and V. Boyer, “Observation of localized multi-spatial-mode quadrature squeezing,” Phys. Rev. X 5(3), 031004 (2015). [CrossRef]
44. Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113(2), 023602 (2014). [CrossRef]
45. A. Marino, R. Pooser, V. Boyer, and P. Lett, “Tunable delay of einstein–podolsky–rosen entanglement,” Nature 457(7231), 859–862 (2009). [CrossRef]
46. S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020). [CrossRef]
47. X. Li, J. Wu, S. Xiong, M. Chen, H. Yan, Z. Wang, and Y. Zhang, “Spatial and frequency multimode in the dressing parametric amplified multiwave mixing process,” Photonics Res. 7(12), 1454–1460 (2019). [CrossRef]
48. Y. Zhang, A. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99(12), 123603 (2007). [CrossRef]
49. M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, “Resonant enhancement of parametric processes via radiative interference and induced coherence,” Phys. Rev. Lett. 81(13), 2675–2678 (1998). [CrossRef]
50. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]
51. Y. Mei, Y. Zhou, S. Zhang, J. Li, K. Liao, H. Yan, S.-L. Zhu, and S. Du, “Einstein-podolsky-rosen energy-time entanglement of narrow-band biphotons,” Phys. Rev. Lett. 124(1), 010509 (2020). [CrossRef]
52. H. Yan, S. Zhang, J. Chen, M. Loy, G. Wong, and S. Du, “Generation of narrow-band hyperentangled nondegenerate paired photons,” Phys. Rev. Lett. 106(3), 033601 (2011). [CrossRef]
53. Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A 80(5), 055804 (2009). [CrossRef]
54. Z. Nie, H. Zheng, P. Li, Y. Yang, Y. Zhang, and M. Xiao, “Interacting multiwave mixing in a five-level atomic system,” Phys. Rev. A 77(6), 063829 (2008). [CrossRef]
55. M. Yan, E. G. Rickey, and Y. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64(1), 013412 (2001). [CrossRef]
56. M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, “Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence,” Phys. Rev. Lett. 82(9), 1847–1850 (1999). [CrossRef]
57. J. Dimitrijević, D. Arsenović, and B. M. Jelenković, “Intensity dependence narrowing of electromagnetically induced absorption in a doppler-broadened medium,” Phys. Rev. A 76(1), 013836 (2007). [CrossRef]
58. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef]
59. R. S. Bennink, V. Wong, A. M. Marino, D. L. Aronstein, R. W. Boyd, C. R. Stroud, S. Lukishova, and D. J. Gauthier, “Honeycomb pattern formation by laser-beam filamentation in atomic sodium vapor,” Phys. Rev. Lett. 88(11), 113901 (2002). [CrossRef]
60. F. Devaux and E. Lantz, “Spatial and temporal properties of parametric fluorescence around degeneracy in a type i lbo crystal,” Eur. Phys. J. D 8(1), 117–124 (2000). [CrossRef]
61. U. Khadka, Y. Zhang, and M. Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81(2), 023830 (2010). [CrossRef]
62. Y. Cai, L. Hao, D. Zhang, Y. Liu, B. Luo, Z. Zheng, F. Li, and Y. Zhang, “Multimode entanglement generation with dual-pumped four-wave-mixing of rubidium atoms,” Opt. Express 28(17), 25278–25292 (2020). [CrossRef]
63. Y. Liu, K. Li, W. Li, S. Li, Y. Cai, and Y. Zhang, “Dressing-controlled quantum steering in energy-level cascaded parametric amplified four-wave mixing processes,” Adv. Quantum Technol. 3(8), 2000029 (2020). [CrossRef]
64. C. E. Whittaker, B. Dzurnak, O. A. Egorov, G. Buonaiuto, P. M. Walker, E. Cancellieri, D. M. Whittaker, E. Clarke, S. S. Gavrilov, M. S. Skolnick, and D. N. Krizhanovskii, “Polariton pattern formation and photon statistics of the associated emission,” Phys. Rev. X 7(3), 031033 (2017). [CrossRef]
65. A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308(5722), 672–674 (2005). [CrossRef]
66. Y. Liu, J. Wei, M. Niu, Y. Lin, Z. Chen, J. Yan, B. Luo, F. Li, Y. Cai, and Y. Zhang, “Atomic-coherence-assisted multipartite entanglement generation with dressing-energy-level-cascaded four-wave mixing,” Phys. Rev. A 106(4), 043709 (2022). [CrossRef]
67. J. Li, J. Zeng, F. Li, Y. Zhang, and Y. Cai, “Optimal bright multimode quantum squeezing via multi-seeding energy-level cascaded four-wave mixing,” Opt. Express 30(22), 39762–39774 (2022). [CrossRef]
