Efficient frequency conversions at low optical powers due to the advent of quantum information science has become important, which has been receiving a lot of attention in recent years.^[1–5] At low optical powers, the nonlinear coefficients in most optical materials are usually very small. Traditionally, large optical fields^[6] or equivalent long nonlinear medium^[7,8] are required to obtain the strong nonlinear response. In the past two decades, it was addressed that the nonlinear conversion can be greatly enhanced using the quantum coherence in atomic medium.^[9,10] There are several ways to prepare atomic coherence. One is to use electromagnetically induced transparency (EIT)^[9] for the generation of atomic coherence. Jain et al.^[11] and Merriam et al.^[12] achieved high frequency conversion efficiencies with the help of an atomic coherence prepared via EIT. Another approach is to prepare an atomic spin wave before the Raman conversion process; the atomic spin wave acts as a seed to the Raman amplification process for enhanced Raman conversion.^[13–15] We demonstrated a high Raman conversion of 40% with a lower pump field intensity of 0.1 W/cm².^[13] Nonlinear conversion efficiency can be enhanced with coherent medium prepared by counter-propagating fields and efficient intrinsic feedback.^[16–19] Zibrov et al.^[17] observed a 4% conversion efficiency with laser power of 300 μW and a spot size of 0.3 mm. However, these schemes need other fields to prepare the atomic spin waves and can only operate in pulse mode.

Recently, we have experimentally demonstrated an efficient Raman conversion scheme with coherent feedback.^[20] The experimental setup is simple and the same with usual spontaneous Raman scattering except for a flat mirror. The conversion efficiency of the scheme is as high as 50% for the Stokes field and 30% for the anti-Stokes field with pump field power as low as a few hundreds of a microwatt in both pulsed and continuous wave (CW) modes. By beating two converted fields generated from a common Raman pump field, we observed a narrow line width of 10 kHz, which is determined by the decoherence time of the atomic spin wave in the medium. The mechanism for the efficient conversion is the constructive interference due to the coherent feedback. It relies on the creation of the atomic coherence between the two lower states and the phase correlation between the atomic coherence and converted field in Raman scattering.^[21] Recently, we have also proposed a theoretical scheme to enhance the Raman scattering using quantum correlation, termed as correlation-enhanced Raman scattering (CERS).^[21] In this scheme, a pump field leads to spontaneous emission of the Stokes field, accompanied with the generation of atomic spin waves. Then this Stokes field used as a seeded signal, with the pump field together, is subsequently input into an atomic ensemble to generate a second Stokes field. Due to the correlation of the seeding Stokes field and atomic spin waves a CERS occurs, which can partly explain the experiment of high efficient Raman conversion, but a system theoretical model is required. In this paper, we present a theoretical model to describe this Raman process, and it can be explained by a cascaded double-CERS. In this model on the one hand the Raman process is driven by the quasi-standing-wave pump fields, and on the other hand the process is repeated and the intensities of Stokes fields are coherent enhancement due to the stronger new initial conditions, finally the high efficient conversion is obtained.

The remaining part of this paper is organized in the following way. In Section 2 the general model involving spatial propagation and light–atom coupling in Raman scattering driven by the standing wave is given. In Section 3 we explain our results with the above theory. We draw our conclusions in Section 4.

In this section, we present a model to describe the process of an efficient Raman conversion scheme with coherent feedback. The schematic diagram of the process considered here is shown in Fig. 1(a). We consider a collection of identical Λ-type hot atoms interacting with laser fields as shown in Fig. 1(b). Two counter-propagating strong pump fields P₁ and P₂ simultaneously couple the ground state |1〉 to the excited state |3〉 with one-photon detunings Δ, respectively. Two counter-propagating weak Stokes scattering fields S₁ and S₂ will be generated.

Fig. 1.

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	Fig. 1. (a) The schematic diagram of the experiment. (b) The schematic diagram of Λ-type three-level atoms. P1, P2: pump fields; S1, S2: Stokes fields; Δ: detuning; M: mirror.

In the electric-dipole and rotating-wave approximations, the Hamiltonian of N identical atoms is given (ħ = 1) as follows:^[22,23]

We define continuum atomic operators by summing over the individual atoms in a small volume V, and introduce slowly varying atomic operators , , . Then the equations of motion for the atomic operators in the Heisenberg picture are

Using the slowly varying envelope approximation, we obtain the following propagation equations for the quantum field operators:

In this section, we use the Laplace transform technique to solve Eq. (16). Using the moving frame t′ = t − z/c, z′ = z, then equation (16) can be rewritten as

For analyzing conveniently, we assume the write fields intensity [Ω_P1(t′) = Ω_P1θ (t′)] being constant and real, after being switched on at t′ = 0. Then the coupled equations (19) and (20) are written as

In order to explain the process of efficient Raman conversion scheme,^[20] we calculate the intensities of Stokes field S₁ and S₂. Different initial conditions lead to different output intensities, where the process is described in Fig. 2. The pump field P₁ leads to spontaneous emission of the Stokes field S₁, accompanying with the generation of atomic spin waves S_a1 shown in Fig. 2(b). The atomic spin wave S_a1 stays in the cell, the S₁ and P₁ fields propagate out together and both are reflected back to the atomic medium by a flat mirror M (see Fig. 2(c)). If only P₂ is present, the subsequent Stokes field S₂ in the backward direction will be stimulated by the reflected S₁ and enhanced by the previously produced atomic spin wave at the same time, which was studied by us and we call CERS.^[21] But in Fig. 2(d) the difference is that the initial correlation condition simultaneously leads to two enhanced Stokes fields S₁ and S₂ generation. This process is double CERS, which is also obtained from Eq. (16). Figures 2(a)–2(d) form the first cycle and figures 2(e)–2(f) form the second cycle. Here the cycles are repeated constantly in a period, and the intensity of Stokes field S₂ is coherent enhancement until it is saturated. Therefore, we term as the cascade CERS. Next we present the expression of the intensities under different conditions.

Fig. 2.

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Fig. 2. The schematic diagram of cascade correlation-enhanced Raman scattering. The output intensities of Stokes fields are dependent on the initial conditions. (a) Experimental setup. The experimental setup is simple and the same with usual spontaneous Raman scattering in additional to a flat mirror. (b) The pump field P1 leads to spontaneous emission of the Stokes field S1, accompanied with the generation of atomic spin wave Sa1. (c) The atomic spin wave Sa1 stays in the cell, the S1 and P1 fields propagate out together and both are reflected back to the atomic medium by a flat mirror M. (d) Two correlation-enhanced Raman scattering Stokes fields S1 and S2. Panels (a)–(d) form the first cycle and panels (e)–(f) form the second cycle. The cycles are repeated constantly in a period, and the intensity of Stokes field S2 is coherent enhancement until it is saturated. The dashed curves (green) denote the correlation between the Stokes field and atomic spin wave.

In the first cycle, for Stokes field S₁ no initial Stokes field is externally incident on the ensemble and no initial spin wave is written into the ensemble (see Fig. 2(a)), then the initial conditions are given by

Subsequently, the atomic spin wave S_a stays in the cell, and the S₁ and P₁ fields propagate out together and both are reflected back to the atomic medium by a flat mirror M, where the reflected P₁ is denoted as P₂. As shown in Fig. 2(c), the Stokes field S₁ with its correlated atomic spin wave Ŝ_a is employed as initial seeding, and the atomic emsemble is pumped by the standing-wave formed by P₁ and P₂. Then two new enhanced Stokes fields S₁ and S₂ are generated as shown in Fig. 2(d). For Stokes field S₂, the initial conditions are given by

Next, we will numerically calculate the intensities of , I_S1, and I_S2 using Gaussian-shaped write fields. Assume the write fields Ω_P1 and Ω_P2 have the following Gaussian shapes,

Fig. 3.

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	Fig. 3. The intensities of Stokes fields versus the dimensionless time in the first cycle. The parameters are as follows: Δ = 1.2 GHz, γ21 = 10 kHz, γ31 = 2π × 5.746 MHz, , ΩP10 = ΩP20, T1 = T2 = T.

In conclusion, we have given a theoretical model, termed as the cascade CERS, to describe the efficient Raman process. In the first cycle it is correlation-enhanced Raman scattering, i.e., injecting a seeded light field which is correlated with the initially prepared atomic spin excitation and driven by the quasi-standing-wave pump fields. The cycles are repeated constantly in a period, and the intensities of Stokes fields are coherent enhancement, then high efficient Raman conversion is achieved. Such a cascade correlation-induced enhanced Raman scattering process may find applications in a diversity of technological areas such as optical detection, metrology, imaging, precision spectroscopy, and so on.