Evolution of linear and nonlinear optical responses in single- and double-dressing quadphoton correlations

Yuan Feng; Zhaoyang Zhang; Irfan Ahmed; Irfan Ahmed; Sifan Li; Yuliang Liu; Yin Cai; Yanpeng Zhang

doi:10.1364/OE.395780

1. Introduction

Over the past 50 years, non-classical light has become a powerful tool for exploring fundamental quantum optics theory. Quantum correlations and entanglements among multiparticles have great significance in terms of efficient quantum information transmission. There has been significant research on correlated and entangled multiple-photon resources [1–6]. Furthermore, strong dressing effect may enhance the efficiency of six-wave mixing (SWM) and eight-wave mixing (EWM) [7,8] and also the generation efficiency of tirphotons and quadphotons [9,10]. Additionally, for quantum transmission research, controllable wave function systems are of great significance. For single photons, quantum waveforms can be manipulated using external modulations [11,12]. Subsequently, Du et al. demonstrated a technique for shaping the temporal quantum waveforms of narrowband biphotons both theoretically and experimentally, and generated a cold atomic ensemble via four-wave mixing by periodically modulating two input classical lasers [13,14]. They also proposed the use of spatial light modulation to shape biphoton waveforms [15]. Recently, dressing laser beams have been used to alternate both the linear and nonlinear susceptibilities of the hot rubidium vapor, resulting in the modification of biphoton temporal correlation functions [16].

In this study, we observed the competition and coexistence of linear and nonlinear optical responses through an EWM process in ⁸⁵Rb atomic vapor. Nonlinear susceptibility plays a major role in determining the wave packets of quadphotons when the effective coupling Rabi frequencies and linewidths are significantly smaller than the phase-matching and electromagnetically induced transparency (EIT) bandwidths. In the presence of multimodal quadphoton beating or destructive interference, the wave function of the quadphoton exhibits periodic Rabi oscillation. When the effective coupling Rabi frequency and linewidths are significantly larger than both the phase-matching and EIT bandwidths, linear susceptibility governs the shape of quadphoton wave packets in the group delay regime. When the phase-matching bandwidth is smaller than the EIT bandwidth, the wave function exhibits a rectangular profile. Otherwise, it exhibits a rectangular attenuation profile. These results have many potential applications in long-distance and long coherent time quantum communication, quantum information processing, and quantum storage.

2. Experimental setup and basic theory

A simplified experimental setup and atomic energy-level diagrams are shown in Figs. 1(a) and 1(b), respectively. Here, a ⁸⁵Rb atomic vapor medium is used to generate a quadphoton. A thin and long cylindrical volume with a length of L confines the rubidium vapor. Initially, the four-level atoms are prepared at ground level |1 > . All four beams are then coupled and focused onto the center of the Rb atomic medium by optical lenses. Two beams with wavelengths of 795 nm act as weak pump beams E₁ (frequency ω₁, wave vector k₁, detuning Δ₁, and Rabi frequency G₁) and E₃ (ω₃, k₃, Δ₃ and G₃). The detuning, which is expressed as Δ_i=Ω_i−ω_i, is defined as the difference between the resonant transition frequency Ω_i and the laser frequency ω_i(E_i). The pump beams E₁ and E₃ have the same frequency ω and detuning parameter Δ. However, there is a small angle between the wave vectors k₁ and k₃. The atomic transitions |1>→|3 > and |2>→|3 > are pumped by E₁ and E₃, respectively. Additionally, there are two coupling beams E₂ (ω₂, k₂, Δ₂, G₂, 780 nm) and E₄ (ω₄, k₄, Δ₄, G₄, 780 nm). The strong coupling beam E₂ counter-propagates with E₁ near the resonant frequency of the atomic transition |2>→|4 > . Additionally, the coupling beam E₄ propagates along E₁ and is applied to the atomic transition |2>→|4 > with a detuning parameter Δ₄. The EWM process occurs spontaneously and satisfies the phase-matching condition k₁ + k₂ + k₃ + k₄= k_S₁ + k_S₂ + k_S₃ + k_S₄. Therefore, correlated quadphotons (E_S1, E_S2, E_S3, and E_S4) represented by the four small blue circles in Fig. 1(c) can be generated and detected by four single-photon counting modules (SPCMs). Let ω_S_i(i = 1,2,3,4) and δ_i(i = 1,2,3,4) represent the frequency and detuning of the generated photons, respectively. The entangled correlated quadphotons satisfy the conservation of total energy (i.e., ω₁ + ω₂ + ω₃ + ω₄= ω_S₁ + ω_S₂ + ω_S₃ + ω_S₄, and δ₁ + δ₂ + δ₃ + δ₄= 0). According to the energy conservation condition, coherent channels represented by the large orange area in Fig. 1(c) are interrelated and interfere with each other. Here, polarization effects, Doppler broadening, and quantum Langevin noise are ignored.

Fig. 1. (a) Spatial beam alignment for the generation of quadphotons in the EWM process. All incident beams are coupled and focused onto the Rb atomic vapor by a reflective mirror (R) and polarization beam splitter (PBS). Generated photons are detected by four single-photon counting modules (SPCMs) and the distances between the SPCMs and the center of the vapor cell are the same in all directions; (D1-D4: SPCM1-SPCM4). (b) Mechanisms of quadphoton generation given a four-level configuration in ⁸⁵Rb vapor. In a case with two pumps (ω₁, ω₃) and two coupling beams (ω₂, ω₄), quadphotons (ω_S1, ω_S2, ω_S3, and ω_S4) are generated spontaneously by the EWM process in the low-gain regime. (c) Energy conservation of a multimodal quadphoton.

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According to perturbation theory, the Hamiltonian interaction describes the EWM process and provides a clear picture of the quadphoton generation mechanism [9]. Here, the quadphoton state is expressed as

(1)$$\begin{aligned} |\psi \rangle &= \int {d{\omega _{S1}}d{\omega _{S2}}d{\omega _{S3}}d{\omega _{S4}}\kappa } \Phi (\frac{{\Delta kL}}{2})\hat{a}_{S1}^{\dagger} \hat{a}_{S2}^\dagger \hat{a}_{S3}^{\dagger} \hat{a}_{S4}^{\dagger} \delta ({\Delta \omega } )|0 \rangle \\ &= \int {d{\omega _{S1}}d{\omega _{S2}}d{\omega _{S3}}d{\omega _{S4}}\kappa } \Phi (\frac{{\Delta kL}}{2})\hat{a}_{S1}^\dagger \hat{a}_{S2}^{\dagger} \hat{a}_{S3}^\dagger \hat{a}_{S4}^{\dagger} |0 \rangle \end{aligned}, $$

where L is the length of the ⁸⁵Rb medium, $\kappa ={-} i\sqrt {{\varpi _{S1}}{\varpi _{S2}}{\varpi _{S3}}{\varpi _{S4}}/{c^4}} {\chi ^{(7)}}({\omega _{S1}},{\omega _{S2}},{\omega _{S3}},{\omega _{S4}}){E_1}{E_2}{E_3}{E_4}$ is a nonlinear parametric coupling coefficient related to the nonlinear susceptibility, $\Phi (\Delta kL/2) = \sin c(\Delta kL/2){e^{ - i({{k_{S1}} + {k_{S2}} + {k_{S3}} + {k_{S4}}} )L/2}}$is a longitudinal detuning function related to the linear susceptibility, and Δk = k_S₁ + k_S₂ + k_S₃ + k_S₄ − (k₁ + k₂ + k₃ + k₄) = δ₂ / v_S₂ + δ₃ / v_S₃ + δ₄ / v_S₄ is the phase mismatch along the z axis, v_Si (i = 2, 3, 4) is the group velocity of the entangled quadphotons. If Δk = 0, then the phase-matching condition is satisfied appropriately. Here, we ignore the imaginary parts of Δk. The quadphoton amplitude can be obtained as follows [9]:

(2)$$B({{\tau_{S1}},{\tau_{S2}},{\tau_{S3}},{\tau_{S4}}} )= {W_1}\int {d{\omega _{S1}}d{\omega _{S2}}d{\omega _{S3}}d{\omega _{S4}}\kappa ({{\omega_i}} )} \Phi (\Delta kL){e^{ - i\left( {\sum {{\omega_{Si}}{\tau_{Si}}} } \right)}},$$

where $\sum {{\omega _{Si}}{\tau _{Si}} = } {\omega _{S1}}{\tau _{S1}} + {\omega _{S2}}{\tau _{S3}} + {\omega _{S3}}{\tau _{S3}} + {\omega _{S4}}{\tau _{S4}}$ and W₁ is a constant that absorbs all constants and slowly varying terms. According to Eq. (2), the pattern of the quadphoton amplitude is determined by both the nonlinear parametric coupling coefficient $\kappa $ and longitudinal detuning function Φ. The averaged coincidence counting rate is defined as

(3)$${R_{cc}} = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_0^T {d{t_{S1}}} d{t_{S2}}d{t_{S3}}d{t_{S4}}{M_i}({t_i}){|{B({{\tau_{S1}},{\tau_{S2}},{\tau_{S3}},{\tau_{S4}}} )} |^2},$$

where M_i (t_i) = M₁ (t_S₂ – t_S₁) M₂ (t_S₃ – t_S₁) M₃ (t_S₄ – t_S₁) are the coincidence window functions within the time bin width t_cc and we consider M_i = 1 for | t_i –t_S₁ | < t_cc and M_i = 0, otherwise.

According to the internal dressing effect of the field E₂, the seventh-order nonlinear susceptibility can be expressed as follows [9,17–20]:

(4)$$\begin{aligned} \chi _{S2}^{(7)}& = {N_0}/[({\Gamma _{31}} + i{\Delta _3})({\Gamma _{21}} + i{\delta _3})({\Gamma _{41}} + i{\delta _3}\textrm{ + }i{\Delta _4})({\Gamma _{11}}\textrm{ + }i{\delta _\textrm{3}}\textrm{ + }i{\delta _\textrm{4}}\textrm{ + }d)\\ &\times ({\Gamma _{31}}\textrm{ + }i{\delta _\textrm{3}}\textrm{ + }i{\delta _\textrm{4}}\textrm{ + }i{\Delta _1})({\Gamma _{41}} - i{\delta _2} + i{\Delta _2})({\Gamma _{21}} - i{\delta _2} + \frac{{G_2^2}}{{{\Gamma _{41}} - i{\delta _2} + i{\Delta _2}}})], \end{aligned}$$

where ${N_0} = 2N{\mu _{13}}{\mu _{32}}{\mu _{24}}{\mu _{41}}{\mu _{13}}{\mu _{32}}{\mu _{24}}{\mu _{41}}/{\varepsilon _0}{\hbar ^7}$ is a constant, µ_ij denotes the electric dipole matrix elements, Γ_ij= (Γ_i+ Γ_j) / 2 is the dephasing rate between |i > and |j>, δ_i (i = 1, 2, 3, 4) = ω_si − ϖ_si represents the photon detuning of the generated quadphotons, ϖ_si (i = 2, 3, 4) is the central frequency of the generated photons, and G₂ is the dressing field corresponding to E₂. When there is a single dressing field, the term d in Eq. (4) is equal to zero. However, when there is a double-dressing field, another strong dressing field E₄ also plays an important role and the term d is expressed as $d = G_4^2/({\Gamma _{31}}\textrm{ + }i{\delta _3}\textrm{ + }i{\delta _4}\textrm{ + }i{\Delta _1})$, where G₄ is the dressing field generated by E₄.

The linear susceptibilities of the generated quadphotons (E_S1, E_S2, E_S3, and E_S4) with a single-dressing field are defined as follows:

(5)$$\begin{aligned} {\chi _{S1}} & = \frac{{{N_0}\mu _{23}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{11}} - {{|{{G_2}} |}^2}/{d_{12}}}},\\ {\chi _{S2}} &= \frac{{{N_0}\mu _{24}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{21}} - {{|{{G_2}} |}^2}/{d_{22}}}},\\ {\chi _{S3}} &= \frac{{{N_0}\mu _{14}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{31}} - {{|{{G_4}} |}^2}/{d_{32}}}},\\ {\chi _{S4}} & = \frac{{{N_0}\mu _{14}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{41}} - {{|{{G_4}} |}^2}/{d_{42}}}}, \end{aligned}$$

where d₁₁= δ₁ + Δ₁ − iΓ₃₂, d₁₂= δ₁ + Δ₁ – Δ₃ − iΓ₃₄, d₂₁= δ₂ + Δ₂ + Δ₂ − iΓ₄₂, d₂₂= δ₂ + Δ₂ − iΓ₂₂, d₃₁= δ₃ + Δ₄ − iΓ₃₂, d₃₂= δ₃ − iΓ₃₄, d₃₁= δ₄ − iΓ₄₁, d₄₂= δ₄ + Δ₄ − iΓ₂₁, and Δ_i= Ω_i − ω_i is the detuning parameter.

Similarly, the linear susceptibilities of the generated quadphoton with double-dressing (E₂ and E₄) are defined as follows:

(6)$$\begin{aligned} \chi _{S1}^{\prime} & = \frac{{{N_0}\mu _{23}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{11}} - {{|{{G_2}} |}^2}/{d_{12}} - {{|{{G_4}} |}^2}/{d_{13}}}},\\ \chi _{S2}^{\prime} & = \frac{{{N_0}\mu _{24}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{21}} - {{|{{G_2}} |}^2}/{d_{22}} - {{|{{G_4}} |}^2}/{d_{23}}}},\\ \chi _{_{S3}}^{\prime} & = \frac{{{N_0}\mu _{14}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{31}} - {{|{{G_4}} |}^2}/{d_{32}} - {{|{{G_2}} |}^2}/{d_{33}}}},\\ \chi _{S4}^{\prime} & = \frac{{{N_0}\mu _{14}^2}}{{{\varepsilon _0}\hbar }}\frac{{ - 1}}{{{d_{41}} - {{|{{G_4}} |}^2}/{d_{42}} - {{|{{G_2}} |}^2}/{d_{43}}}}, \end{aligned}$$

where d₁₃=δ₁ + Δ₁ – Δ₄ − iΓ₃₄, d₂₃=δ₂ + Δ₂ + Δ₄ − iΓ₂₂, d₃₃=δ₃ – Δ₃ − iΓ₃₄, and d₄₃=δ₄ + Δ₃ − iΓ₂₁. According to v_si= c / {1 + ω_mnd(Re[χ_Si(ω)] / dω) / 2} [21], the group velocity v_si (i = 2, 3, 4) of an entangled quadphoton with single-dressing can be expressed as follows:

(7)$$\begin{aligned} {v_{S2}} & = \frac{c}{{1 + \frac{{{\lambda _{24}}{\omega _{42}}{\Gamma _{42}}O{D_2}}}{{4\pi L}}(\frac{{{\Gamma _{\textrm{22}}}^2 - {{|{{G_\textrm{2}}} |}^2}}}{{{{({{|{{G_\textrm{2}}} |}^2} + {\Gamma _{\textrm{22}}}{\Gamma _{\textrm{42}}})}^2}}})}},\\ {v_{S\textrm{3}}}& = \frac{c}{{1 + \frac{{{\lambda _{41}}\; {\omega _{\textrm{41}}}{\Gamma _{\textrm{32}}}O{D_3}}}{{4\pi L}}(\frac{{{\Gamma _{\textrm{31}}}^2 - {{|{{G_4}} |}^2}}}{{{{({{|{{G_4}} |}^2} + {\Gamma _{\textrm{31}}}{\Gamma _{\textrm{32}}})}^2}}}\textrm{)}}},\\ {v_{S\textrm{4}}} & = \frac{c}{{1 + \frac{{{\lambda _{41}}{\omega _{\textrm{41}}}{\Gamma _{41}}O{D_4}}}{{4\pi L}}{\{ }\frac{{{\Gamma _{\textrm{21}}}^2 - {{|{{G_4}} |}^2}}}{{{{({{|{{G_\textrm{4}}} |}^2} + {\Gamma _{\textrm{21}}}{\Gamma _{\textrm{41}}})}^2}}}{\} }}}, \end{aligned}$$

where c is the speed of light, ω_mn (m,n = 1, 2, 3, 4) is the resonant transition frequency between two energy levels when generating photons, OD_i is the optical depth, OD₂= Nσ₂₄L, σ₂₄= 2π|µ₂₄|² / (ɛ₀ћλ₂₄Γ₂₄), OD₃= Nσ₂₃L, σ₂₃= 2π|µ₂₃|² / (ɛ₀ћλ₂₃Γ₂₃), OD₄= Nσ₁₄L, σ₁₄= 2π|µ₁₄|² / (ɛ₀ ћλ₁₄Γ₁₄). Similarly, we can also obtain the group velocity v_si (i = 2, 3, 4) of an entangled quadphoton with double-dressing. The expressions for v_si are defined as follows:

(8)$$\begin{aligned} {v_{S2}} & = \frac{c}{{1 + \frac{{{\lambda _{42}}{\omega _{42}}{\Gamma _{42}}O{D_2}}}{{4\pi L({{{|{{G_2}} |}^2} + {{|{{G_4}} |}^2}} )}}}},\\ {v_{S3}} & = \frac{c}{{1 + \frac{{{\lambda _{14}}{\omega _{14}}O{D_3}{{\Gamma }_{23}}}}{{4\pi L\left( {{{\left| {{G_2}} \right|}^2} + {{\left| {{G_4}} \right|}^2}} \right)}}}},\\ {v_{S4}} &= \frac{c}{{1 + \frac{{{\lambda _{24}}{\omega _{\textrm{24}}}O{D_4}{\Gamma _{14}}}}{{4\pi L({{{|{{G_2}} |}^2} + {{|{{G_4}} |}^2}} )}}}}. \end{aligned}$$

The bandwidth and time determined by the group delay velocity (v_si) are referred to as the phase-matching bandwidth (Δω_gi) and group delay time (τ_gi), respectively. The phase-matching bandwidth is one of the key factors controlling the profile of quadphoton coincidence counting. The expressions for Δω_gi and τ_gi are defined as follows:

(9)$$\Delta {\omega _{gi}} = 2\pi {v_{si}}/L,$$

(10)$${\tau _{gi}} = L/{v_{si}}.$$

Additionally, Δω_tr is referred to as the EIT bandwidth. The specific expressions for Δω_tr2 and Δω_tr4 are defined as follows:

(11)$$\begin{aligned} \Delta {\omega _{tr2}} & = |{G_2^2} |/{\gamma _{24}}\sqrt {O{D_2}},\\ \Delta {\omega _{tr4}} & = |{G_4^2} |/{\gamma _{14}}\sqrt {O{D_4}}. \end{aligned}$$

According to Eq. (2), the properties of the quadphoton amplitude are dependent on both the nonlinear parametric coupling coefficient $\kappa$ and longitudinal detuning function Φ. Although the effective coupling Rabi frequency Ω_e and linewidth Γ_e are far smaller than the phase-matching bandwidth Δω_g and EIT bandwidth Δω_tr, the phase mismatch Δk is approximately equal to zero. In this case, the nonlinear susceptibility $\chi _{S2}^{(7)}$ plays a significant role in determining the spectral width of the quadphoton. Additionally, the effective coupling Rabi frequency Ω_e can result in multimodal EWM processes, thereby generating multimodal quadphotons. Based on the beating and destructive interference generated by multimodal quadphotons, the corresponding wave function exhibits the form of damped Rabi oscillation. Conversely, when Δω_g and Δω_tr are far smaller than Γ_e and Ω_e, the nonlinear parametric coupling coefficient $\kappa$ is constant and the linear susceptibility ${\chi _{Si}}$(i = 1, 2, 3, 4) determines the spectral width of the quadphotons. In this case, when the phase-matching bandwidth is smaller than the EIT bandwidth (Δω_g < Δω_tr), the wave function of the quadphoton exhibits a rectangular profile and rectangular attenuation profile.

3. Counting measurement

In this section, we investigate the competing and coexisting of single- and double-dressing linear and nonlinear optical responses in quadphoton correlation. Three different cases are discussed below.

3.1 Group delay regime

As suggested by Balić et al. [3] and Kolchin [22], and demonstrated by Du et al. [23], when the linewidth Γ_e is greater than the phase-matching bandwidth Δω_g, the quadphoton coincidence counting rate lies within the group delay regime. In this case, the quadphoton bandwidth depends on the linear susceptibility. The EIT profile controls the wave function through a transparency window and slow light.

When the phase-matching bandwidth Δω_g is smaller than the EIT bandwidth Δω_tr, quadphoton wave packets appear as rectangular functions. In this case, we only consider the real part of Δk. By calculating Eq. (2), the amplitude functions of generated photons with single- and double-dressing fields can be expressed uniformly as follows:

(12)$$B{}_1({\tau _2},{\tau _3},{\tau _4}) = {W_{_2}}\Pi ({\tau _\textrm{2}};0,\frac{L}{{{v_{S\textrm{2}}}}})\Pi ({\tau _3};0,\frac{L}{{{v_{S3}}}})\Pi ({\tau _4};0,\frac{L}{{{v_{S4}}}}),$$

where W₂ is a constant, τ_i= t_Si − t_S₁ is the coherence delay time between S_i and S₁, and the rectangular function п ranges from τ = 0 to L / v_Si, suggesting that the S_i photon is always delayed by the slow light effect. The group delay time τ_gi= L / v_Si determines the quadphoton correlation time. The difference between single- and double-dressing fields lies in the expressions for the group velocity v_Si, which can be predicted using Eqs. (7) and (8).

For a single-dressing field E₂, the numerical simulation described by Eq. (12) is presented in Fig. 2(a). The axes τ₂, τ₃, and τ₄ represent the coherence times of coincidence counting between photons E_S2, E_S3, E_S4, and E_S1. Here, the value of OD is 300, P₂ represents the power of dressing field G₂ and it is equal to 10 mW. The linewidths Γ_ei and EIT bandwidths Δω_tri are greater than the phase-matching bandwidths Δω_gi in all directions. Therefore, the quadphoton wave packets exhibit rectangular profiles in all three directions. From Fig. 2(a), we can obtain sectional views of four-dimensional photon counting measurements, as shown in Figs. 2(b) to 2(d). In Fig. 2(b), the sectional view represents the τ₂ and τ₃ axes. The coherence time in the direction of τ₂ is approximately 150 ns. In the direction of τ₃, it is approximately 108 ns. In Fig. 2(c), the sectional view represents the τ₂ and τ₄ axes. The coherence time in the direction of τ₄ is approximately 150 ns. In Fig. 2(d), the sectional view represents the τ₃ and τ₄ axes.

Fig. 2. Simulated diagram of the quadphoton coincidence counting rate in the group delay regime with the single-dressing effect and seventh-order nonlinear susceptibility |χ_S2⁽⁷⁾|. All photons exhibit rectangular profiles (functions). (a) Quadphoton coincidence counts measurement triggered by a photon E_S1. The axes τ₂, τ₃, and τ₄ represent the coherence times of coincidence counting between photons E_S2, E_S3, E_S4, and E_S1, respectively. (b) A cross-sectional view of Fig. 2(a) along axes τ₂ and τ₃. (c) Same as (b), but along axes τ₂ and τ₄. (d) Same as (b), but along axes τ₃ and τ₄. Here, OD = 300 and P₂= 10 mW.

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Similar to Fig. 2, we present a simulated diagram of the quadphoton coincidence counting rate with double-dressing effect in Fig. 3. During our simulations, we ensured the consistency of parameters between Figs. 2 and 3. According to Fig. 3(a), in the directions of τ₂, τ₃, and τ₄, the coherence times are 75 ns, 54 ns, and 75 ns, respectively. In our previous description, the double-dressing field is actually cascade double-dressing. In fact, we can also implement another kind of double-dressing case as shown in Fig. 3(b), namely nested double-dressing. According to Fig. 3(b), in the directions of τ₂, τ₃, and τ₄, the coherence times are 195 ns, 140 ns, and 195 ns, respectively. It is obvious that the coherence time of each direction in Fig. 3(b) is much longer than that in Fig. 3(a). In view of the fact that these two cases are similar except for the length of the coherence time, in the following discussion, we will choose cascaded double-dressing as an example to compare with the single-dressing case. We refer to cascade double-dressing as double -dressing by default.

Fig. 3. Similar to Fig. 2, this figure presents the quadphoton coincidence counting rate in the group delay regime with the different double-dressing effect with |χ_S2⁽⁷⁾|. The parameters used here are the same as those used in Fig. 2 and P₄ = 10 mW. (a) quadphoton coincidence counting rate in the group delay regime with cascade double-dressing effect. (b) quadphoton coincidence counting rate in the group delay regime with nested double-dressing effect.

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Furthermore, when the EIT bandwidth Δω_tr is smaller than the phase-matching bandwidth Δω_g, quadphoton wave packets exhibit rectangular attenuation profiles. In this case, by extending Eq. (2), the amplitude functions for generated quadphotons with single- and double-dressing fields can be expressed uniformly as follows:

(13)$$\begin{array}{l} {B_2}({\tau _2},{\tau _3},{\tau _4})\\ = {W_3}\Pi ({\tau _2};0,\frac{L}{{{v_{S1}}}})\Pi ({\tau _3};0,\frac{L}{{{v_{S3}}}})\Pi ({\tau _4};0,\frac{L}{{{v_{S4}}}}){e^{ - i({{\varpi_{S2}}{\tau_2} + {\varpi_{S3}}{\tau_3} + {\varpi_{S4}}{\tau_4}} )}}, \end{array}$$

where W₃ is a constant. The difference between the single- and cascade double-dressing fields lies in the value of v_Si, as indicated by Eqs. (7) and (8). In this case, the quadphoton coincidence counting rate exhibits a rectangular attenuation profile. A more detailed discussion is presented in the following subsection.

3.2 Coexistence mechanism for different delay times

With different delay times, the optical properties of quadphoton amplitude [Eq. (2)] are determined by both nonlinear and linear optical responses. In such cases, the quadphoton coincidence counting rate is subject to a coexistence mechanism between the Rabi oscillation and group delay regime. In other words, a portion of the photons are in the Rabi oscillation regime (Γ_e << Δω_g), while others lie in the group delay regime (Γ_e >> Δω_g).

To avoid repetition, we will only consider the condition with a single-dressing field as a representative example. In this case, a photon S₂ is generated in the Rabi oscillation regime and other photons are generated in the group delay regime. The coincidence counts of the quadphotons can be rewritten as

(14)$$\begin{aligned} Rc{c_1} &= {W_4}\xi ({\tau _\textrm{3}}\textrm{,}{\tau _\textrm{4}}){\psi _D}({\tau _\textrm{2}})\\ \xi ({\tau _\textrm{3}}\textrm{,}{\tau _\textrm{4}}) &= {\Pi ^2}({\tau _3};0,\frac{L}{{{v_{S3}}}}){\Pi ^2}({\tau _4};0,\frac{L}{{{v_{S4}}}})\\ {\psi _D}({\tau _\textrm{2}}) &= {e^{ - 2{\Gamma _e}{\tau _\textrm{2}}}}[{R_1} + {R_2}\cos ({\textrm{2}{\Omega _e}{\tau_2}} )\textrm{ + }{R_3}\cos ({({\Omega _e} + {\Delta _1}){\tau_2}} )\\ & + {R_4}\cos ({({\Delta _1} - {\Omega _e}){\tau_2}} )] \end{aligned},$$

where W₄, R₁, R₂, R₃, and R₄ are all constants. The physics of the cosine function in Eq. (14) decides the period of the Rabi oscillation. Additionally, ${\Omega _e} = {\Delta _2} \pm \sqrt {{\Gamma _{41}}{\Gamma _{21}} + G_2^2} $ and ${\Gamma _e} ={\mp} ({\Gamma _{41}} + {\Gamma _{21}} + {\Gamma _{21}}{\Delta _2})/(2\sqrt {G_2^2 + {\Gamma _{41}}{\Gamma _{21}}} )$. Similar to P₂, P₄ represents the power of dressing field G₄. The simulated diagram of the quadphoton coincidence counting rate in Fig. 4(a) was calculated with OD = 800 and P₂= P₄= 20 mW. Sectional views of the three-dimensional photon counting measurements in Fig. 4(a) are presented in Figs. 4(a₁) to 4(a₃). Under these conditions, in the directions of τ₃ and τ₄, the linewidth Γ_e is much larger than the phase-matching bandwidth Δω_g, meaning the linear optical response plays a dominant role. Therefore, the competitive relationship between Δω_g and Δω_tr determines the shape of the wave packets. Because Δω_tr > Δω_g, the directions of τ₃ and τ₄ generate rectangular functions, as shown in Fig. 4(a). The coherence times in the directions of τ₃ and τ₄ are 160 ns and 230 ns, respectively. However, in the direction of τ₂, Δω_g is much larger than Γ_e, meaning the nonlinear optical response plays a dominant role. Therefore, we can obtain τ₂ is under the Rabi oscillation regime. Based on the beating or destructive interference of multimodal quad-photons, the photon coincidence counts in the direction of τ₂ exhibit Rabi oscillations with three periods of 2π / 2Ω_e, 2π / (Ω_e + Δ₁), and 2π / (Δ₁ − Ω_e).

A double-dressing field with the same parameters discussed above is presented in Fig. 4(b). The coherence times in the directions of τ₃ and τ₄ are 80 ns and 115 ns, respectively. Based on Eqs. (7) to (10), we can obtain the following expression via comparisons:

(15)$$\frac{{{\tau _{gs}}}}{{{\tau _{gd}}}} = \frac{{\Delta {\omega _{gs}}}}{{\Delta {\omega _{gd}}}} = \frac{{{v_{ss}}}}{{{v_{sd}}}} \propto \frac{{G_2^2\textrm{ + }G_4^2}}{{G_2^2}},$$

where τ_gs, Δω_gs, and v_ss represent the group delay time, phase-matching bandwidth, and group velocity, respectively, for the case of single-dressing. τ_gd, Δω_gd, and v_sd represent the group delay time, phase-matching bandwidth, and group velocity, respectively, for the case of double-dressing. Based on Eq. (15), we can obtain that τ_gs is greater than τ_gd. In Figs. 4(a) and 4(b), the power of both G₂ and G₄ is set to 20 mW. For the representative example of the τ₃ axis, comparison results are presented in Fig. 4(c). Curves A and B correspond to the two-dimensional cross sections in the cases of double- and single-dressing, respectively. The coherence times of τ₃ are 80 ns and 160 ns for curves A and B, respectively. The width of τ_gs3 is twice that of τ_gd3.

Fig. 4. Simulated diagram of the quadphoton coincidence counting rate with the coexistence of Rabi oscillation and group delay based on the dressing effect. a) The dressing effect is a single-dressing effect. τ₂ lies in the Rabi oscillation regime, but τ₃ and τ₄ lie in the group delay regime. (a₁–a₃) Cross-sectional views of Fig. 4(a). (b) Similar to (a), but with a double-dressing effect. (c) Curves A and B correspond to the two-dimensional cross-sections along the τ₃ axis in the cases with double- and single-dressing, respectively. Here, OD = 800 and P₂= P₄= 20 mW.

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Similar to Fig. 4, although Δω_tr is smaller than Δω_g, the corresponding direction exhibits a rectangular attenuation function. Simulation results for single- and double-dressing fields are presented in Figs. 5(a) and 5(b), respectively. In the directions of τ₃ and τ₄, the linear optical response still dominates. Because Δω_tr is smaller than Δω_g, the wave packets of τ₃ and τ₄ exhibit rectangular attenuation functions. In the direction of τ₂, Γ_e is still greater than Δω_g and the nonlinear optical response plays a dominant role. Similar to Fig. 4(c), two-dimensional cross-sectional comparison results in the direction of τ₃ are presented in Fig. 5(c). The coherence times of τ₃ are 80 ns and 160 ns for curves A and B, respectively. However, the width of τ_gs3 is still twice that of τ_gd3.

Fig. 5. Similar to Fig. 4, but in the directions of τ₃ and τ₄, the wave packets both appear with rectangular attenuation profiles. The parameters used here are the same as those used in Fig. 4.

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Similar to Fig. 4(a), but one direction still lies in the group delay regime and the others lie in the Rabi oscillation regime. When the directions of τ₃ and τ₄ lie in the Rabi oscillation regime, the coincidence counting rate of the quadphotons can be rewritten as

(16)$$\begin{aligned} Rc{c_2}& = \\ &= |\int {d{\delta _4}} \int {d{\delta _{_3}}} \int {d{\delta _2}} \chi _{S2}^{(7 )}\Phi ({\delta _2}){e^{ - i{\delta _2}{\tau _2}}}{e^{ - i{\delta _{_3}}{\tau _3}}}{e^{ - i{\delta _{_4}}{\tau _4}}}{|^2}\\ &= {W_5}\{ R_\textrm{1}^{\prime}\textrm{ + }R_2^{\prime}{e^{i{\tau _\textrm{4}}{\Delta _1}}} + R_3^{\prime}{e^{i{\tau _3}{\Delta _4}}} + R_4^{\prime}{e^{ - i{\tau _4}{\Delta _4}}} + R_5^{\prime}{e^{i{\Delta _1}({\tau _3} + {\tau _4})}}\\ &+ R_6^{\prime}{e^{i{\tau _3}{\Delta _4}}}{e^{ - i{\tau _4}({\Delta _4} - {\Delta _1})}} + R_7^{\prime}{e^{ - i{\tau _4}{\Delta _4}}}{e^{i{\tau _4}({\Delta _4} - {\Delta _1})}}\} \times {\Pi ^2}({\tau _2};0,\frac{L}{{{v_{S2}}}}) \end{aligned}, $$

where W₅ and R’_i (i = 1, 2, 3, 4, 5, 6, 7) are constants. The calculated quadphoton coincidence counting rate based on Eq. (16) is presented in Fig. 6(a) with OD = 500, P₂= 10 mW, and P₄= 15 mW. Sectional views of the three-dimensional photon counting measurements in Fig. 6(a) are presented in Figs. 6(a₁) to 6(a₃). Under these conditions, the linewidth is much smaller than the phase-matching bandwidth in the directions of τ₃ and τ₄, indicating the dominant role of the nonlinear optical response. However, the linewidth Γ_e₂ is far greater than Δω_g2 in the direction of τ₂. Additionally, Δω_tr2 is greater than Δω_g2, resulting in a rectangular profile for the photon coincidence counting rate with a starting position of 0 ns and width of 230 ns. From τ_3,4= 0 ns to τ_3,4= 300 ns, the coincidence count exhibits a clear Rabi oscillation waveform in both directions. According to Euler's formula, the coefficients of τ_i in exponential terms represent the corresponding periods. Therefore, based on the beating or destructive interference of coherent channels, we can obtain two periods in the direction of τ₃ with specific values of 2π / Δ₁ and 2π / Δ₄. The photon coincidence counts in the direction of τ₄ exhibit coherent Rabi oscillations with three periods of 2π / Δ₁, 2π / Δ₄, and 2π / (Δ₄ – Δ₁).

Fig. 6. Similar to Fig. 4, but τ₂ is exhibits a rectangular function, and τ₃ and τ₄, exhibit a Rabi oscillation mechanism. Here, OD = 500, P₂= 10 mW, and P₄= 15 mW.

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Figure 6(b) is similar to Fig. 6(a), except that it represents a double-dressing field. In the directions of τ₃ and τ₄, the photon coincidence counts exhibit periodic Rabi oscillations. The coincidence count in the direction of τ₂ exhibits a rectangular profile. To avoid repetition, we will not describe these factors in detail. Under these conditions, comparison results for the τ₂ axis are presented in Fig. 6(c). Curves A and B correspond to the two-dimensional cross sections in the cases of double- and single-dressing, respectively. The coherence times τ₃ are 90 and 230 ns for curves A and B, respectively. Unlike the parameter values in Fig. 4, because the values of P₂ and P₄ in Fig. 6 are no longer equal, τ_gs2 is no longer twice as large as τ_gd2, as suggested by Eq. (15). However, τ_gs2 is still larger than τ_gd2.

In Fig. 7, similar to Fig. 6, Δω_tr2 is smaller than Δω_g2, resulting in a rectangular attenuation profile for the photon coincidence counts in the direction of τ₂. The wave packets exhibit a clear Rabi oscillation waveform in the directions of both τ₃ and τ₄. The parameters used here are the same as those used in Fig. 6. Figures 7(a) and 7(b) present the quadphoton coincidence counts under the coexistence mechanism with single- and double-dressing fields, respectively. Here, we compare the coherence times based on the example of the τ₂ axis. Comparison results are presented in Fig. 7(c). The comparison results are similar to those in Fig. 6(c) and will not be described in detail here.

Fig. 7. Similar to Fig. 6, but τ₂ exhibits a rectangular attenuation function. The parameters used here are the same as those used in Fig. 6.

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3.3 Transition regime between linear and nonlinear optical responses

When the phase-matching bandwidth Δω_g and Rabi frequency Ω_e are approximately equal, a transition regime between linear and nonlinear optical responses is suggested by the quadphoton wave packets. The quadphoton coincidence counting rate with a single-dressing field can be expressed as follows:

(17)$$\begin{aligned} Rc{c_3}& = {W_6}{\Pi ^2}({\tau _2};0,\frac{L}{{{v_{S2}}}}) \times \{ R_\textrm{1}^{{\prime\prime}}\textrm{ + }R_2^{{\prime\prime}}{e^{i{\tau _\textrm{4}}{\Delta _1}}} + R_3^{{\prime\prime}}{e^{i{\tau _3}{\Delta _4}}} + R_4^{{\prime\prime}}{e^{ - i{\tau _4}{\Delta _4}}} + R_5^{{\prime\prime}}{e^{i{\Delta _1}({\tau _3} + {\tau _4})}}\\ &+ R_6^{{\prime\prime}}{e^{i{\tau _3}{\Delta _4}}}{e^{ - i{\tau _4}({\Delta _4} - {\Delta _1})}} + R_7^{{\prime\prime}}{e^{ - i{\tau _4}{\Delta _4}}}{e^{i{\tau _4}({\Delta _4} - {\Delta _1})}}\} \otimes \Pi ({\tau _\textrm{4}};0,\frac{L}{{{v_{S\textrm{4}}}}}) \otimes \Pi ({\tau _\textrm{4}};0,\frac{L}{{{v_{S\textrm{4}}}}}) \end{aligned},$$

where W₆ and R_i (i = 1, 2, 3, 4, 5, 6, 7) are constants. The operation symbol “U” indicates convolution. Equation (17) can be simplified as follows:

(18)$$\begin{aligned} Rc{c_3}& = {W_6}{\Pi ^2}({\tau _2};0,\frac{L}{{{v_{S2}}}}) \times \{ R_\textrm{1}^{{\prime\prime}}\textrm{ + }R_2^{{\prime\prime}}({e^{i{\tau _\textrm{4}}{\Delta _1}}} + {e^{i({\tau _\textrm{4}} - L/{v_{s4}}){\Delta _1}}})\\ &+ R_3^{{\prime\prime}}({e^{i{\tau _3}{\Delta _4}}} + ({e^{i({\tau _3} - L/{v_{s3}}){\Delta _4}}})) + R_4^{{\prime\prime}}({e^{ - i{\tau _4}{\Delta _4}}} + {e^{i({\tau _\textrm{4}} - L/{v_{s4}}){\Delta _4}}})\\ &+ R_5^{{\prime\prime}}({e^{i{\Delta _1}({\tau _3} + {\tau _4})}} + {e^{i{\Delta _1}{\tau _3} + i{\Delta _1}({\tau _4} - L/{v_{s4}})}} + {e^{i{\Delta _1}{\tau _4} + i{\Delta _1}({\tau _3} - L/{v_{s3}})}}\\ &+ {e^{i{\Delta _1}({\tau _3} - L/{v_{s3}}) + i{\Delta _1}({\tau _4} - L/{v_{s4}})}}) + R_6^{{\prime\prime}}({e^{i{\tau _3}{\Delta _4}}}{e^{ - i{\tau _4}({\Delta _4} - {\Delta _1})}}\\ &+ {e^{i{\tau _3}{\Delta _4}}}{e^{ - i({\tau _\textrm{4}} - L/{v_{s4}})({\Delta _4} - {\Delta _1})}} + {e^{i({\tau _3} - L/{v_{s3}}){\Delta _4}}}{e^{ - i{\tau _4}({\Delta _4} - {\Delta _1})}}\\ &+ {e^{i({\tau _3} - L/{v_{s3}}){\Delta _4}}}{e^{ - i({\tau _\textrm{4}} - L/{v_{s4}})({\Delta _4} - {\Delta _1})}}) + R_7^{{\prime\prime}}({e^{ - i{\tau _4}{\Delta _4}}}{e^{i{\tau _4}({\Delta _4} - {\Delta _1})}}\\ &+ {e^{ - i({\tau _4} - L/{v_{s4}}){\Delta _4}}}{e^{i({\tau _4} - L/{v_{s4}})({\Delta _4} - {\Delta _1})}})\} \end{aligned}.$$

A simulated diagram of the quadphoton coincidence counting rate with a single-dressing effect is presented in Fig. 8(a). Here, the value of OD is 200, and P₂ and P₄ are both 10 mW. Under these conditions, Δω_g2 is much smaller than Γ_e2, and the direction of τ₂ lies in the group delay regime. Additionally, Δω_g2 is smaller than Δω_tr2, so wave packets exhibit a rectangular profile. The coherence time is 40 ns. Δω_g3 and Δω_g4 are approximately equal to Ω_e3 and Ω_e4, respectively. Therefore, these directions both lie in the transition regime. In other words, in these two directions, the wave packets exhibit a mixture of a rectangular profile and Rabi oscillation. In the direction of τ₃, there are two periods of 2π / Δ₁ and 2π / Δ₄. In the direction of τ₄, the periods of the photon coincidence counts are 2π / Δ₁, 2π / Δ₄, and 2π / (Δ₄– Δ₁).

Fig. 8. Simulated diagram of the quadphoton coincidence counting rates in the transition regime with a (a) single-dressing effect and (b) double-dressing effect. The direction of τ₂ exhibits a rectangular function. However, the other two directions lie in the transition regime, where OD = 200 and P₂= P₄= 10 mW.

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To analyze the transition regime in greater detail, we consider the direction of τ₃ as an example. We can control the proportion of linear and nonlinear optical responses by adjusting the value of the dressing field and OD. In the following analysis, we adjust these two parameters separately to observe the resulting change trends in the wave packets.

According to our previous analysis, dressing fields affect both linear and nonlinear optical effects. By changing the power of the dressing field, we can induce the evolution process of the transition regime, as shown in Fig. 9. Here, the value of OD is 200. Figure 9(a_i) represents a single-dressing field. Figures 9(a₁) to 9(a₆) correspond to powers of P₂= 1, 5, 10, 15, 20, and 30 mW. As shown in Fig. 9(a₁), the coincidence counts of the two-dimensional cross sections lie in the transition regime and the linear optical response plays a dominant role. Under these conditions, the total coherence time τ_c is the sum of the linear coherence time τ_g and nonlinear coherence time τ_e (τ_c= τ_g + τ_e). One can observe a very clear rectangular profile with a width of approximately 100 ns and a τ_c value greater than 200 ns. According to Eq. (10), τ_g∝OD/G₂² and τ_e∝G₂. Therefore, we obtain the following expression: τ_g/τ_e∝OD/G₂³. Combined with the results of Fig. 9(a_i), as the power of G₂ increases gradually, the effect of the linear optical response is diminished and τ_g decreases. Additionally, the effect of the nonlinear optical response is enhanced and τ_e increases. Until Fig. 9(a₆), the wave packets exhibit rough Rabi oscillation and τ_c is approximately 120 ns. As τ_g decreases with an increasing value of τ_e, the total coherence time τ_c decreases. It can be predicted that the effect of the nonlinear optical response will be far stronger than that of the linear optical response when G₂ increases significantly. Under these conditions, the waveform will take on the form of pure Rabi oscillation.

Fig. 9. Evolution process between linear and nonlinear optical responses in the direction of τ₂ while varying the power of the dressing field. Part (a_i) represents a single-dressing field. Parts (a₁ to a₆) correspond to powers of P₂= 1, 5, 10, 15, 20, and 30 mW, respectively. Part (b_i) represents a double-dressing field. The changing trend of P₂ is the same as that above, but P₄= 10 mW. Additionally, OD = 200.

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Similarly, Fig. 9(b_i) represents a double-dressing field. The changing trend of P₂ is the same as that above, but P₄ = 10 mW. Under these conditions, the trend of the wave packets is similar to that with a single-dressin The changing trend of P₂ is the same as that above, but P₄= 10 mW.g field. The main difference compared to the previous cause lies in the coherence time. We obtained the following expression: τ_g /τ_e∝OD/ (G₄³+G₄G₂²). As the power of the dressing field increases gradually, the role of the nonlinear optical response becomes more significant. Based on the double-dressing effect, the total coherence time τ_c is shorter than that in the case with single-dressing.

Similar to the effect of varying the dressing field, the transition regime between linear and nonlinear optical responses can also be controlled by varying OD. However, OD only affects the linear optical response. In other words, the coherence time τ_e of the nonlinear optical response is constant. Figure 10 illustrates how the wave packets of two-dimensional cross sections change with increases in OD. Figure 10(a) represents a single-dressing field. Here, the power of P₂ is 10 mW. Figures 10(a₁) to 10(a₆) correspond to the values of OD = 10, 100, 200, 300, 400, and 500, respectively. In Fig. 10(a₁), the wave packets exhibit rough Rabi oscillation. As OD increases, τ_g also increases. Therefore, the proportion of τ_g in the total coherence time τ_c increases. Under these conditions, the linear effect becomes stronger, but there is no variation in the nonlinear effect. Figure 10(a₆) presents a clear rectangular function with little oscillation. The experimental parameters are the same as those in Fig. 4(a), except that OD = 800. In Fig. 4(a), the linear optical response is dominant in the direction of τ₃. This can be regarded as an extreme case in which OD increases significantly, causing the linear optical responses to grow much larger than the nonlinear optical responses. Under these conditions, the wave packets exhibit a pure rectangular profile.

Fig. 10. Similar to Fig. 9, but the evolution process is controlled by varying the value of OD. Parts (a_i) represents a single-dressing field. Parts (a₁ to a₆) correspond to the values of OD = 10, 100, 200, 300, 400, and 500, respectively. Part (b_i) represents a double-dressing field. The change trend of OD is the same as that described above. Here, both P₂ and P₄ are set to 20 mW.

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Figure 10(b_i) is similar to Fig. 10(a_i), except that the dressing fields contain G₂ and G₄, which both have a power of 20 mW. Under these conditions, the trend of the two-dimensional cross sections’ wave packets changes in the same manner as that above, but the coherence time is shorter than that in Fig. 10(a). Figure 4(b) can be considered as an extreme case in which OD increases significantly. Under these conditions, the wave packets exhibit a pure rectangular function. By combining Figs. 9 and 10, we can reach the following conclusions. Based on the double-dressing effect, the total coherence time τ_c is significantly smaller than that in the case with single-dressing. Here, the cascaded scheme is similar to the parallel-cascade scheme [17]. This enhances the overall effect of the dressing field, as well as the group velocity, and the total coherence time is reduced. However, if the cascaded scheme takes on the form of a nested cascade, the situation is reversed. The cascading scheme should be selected according to specific application requirements.

4. Conclusion

In this study, we observed the competition and coexistence of linear and nonlinear optical responses in single- and double-dressing quadphoton correlations. The quadphoton coincidence counting rate exhibits a rectangular function when the linear optical response plays a dominant role. The quadphoton coincidence counting rate exhibits Rabi oscillation when strong dressing effects enhance seventh-order nonlinear susceptibility. By increasing or decreasing the temperature, it is possible to increase or decrease OD, thereby enhancing or diminishing the linearity of quadphotons. Additionally, the power of the dressing field is closely related to the nonlinearity of quadphotons. By varying the power of the dressing field and OD, we simulated the evolution process from a dominant linear optical response to a dominant nonlinear response. We also proved that the double-dressing effect reduces coherence time compared to the single-dressing effect. This theoretical research further provides physical insights into the experimental research of the linear and nonlinear optical response competition. These results also have great significance for the widespread development of long distance quantum communications and quantum storage.

Funding

National Key Research and Development Program of China (2017YFA0303700, 2018YFA0307500); National Natural Science Foundation of China (11604256, 11804267, 11904279, 61605154, 61975159).

Disclosures

The authors declare no conflicts of interest.

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Evolution of linear and nonlinear optical responses in single- and double-dressing quadphoton correlations

Abstract

1. Introduction

2. Experimental setup and basic theory

3. Counting measurement

3.1 Group delay regime

3.2 Coexistence mechanism for different delay times

3.3 Transition regime between linear and nonlinear optical responses

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Equations (18)

Optics Express