Spatial Manipulation via Four-Wave Mixing in Five-Level Cold Atoms

In a recent publication [S. Gan, Laser Phys. 31, 055401 (2021)], a scheme for controlling the vortex four-wave mixing (FWM) in a five-level atomic system has been put forward. Based on this work, we propose a new scheme for the spatial manipulation via four-wave mixing in a five-level atomic system when the radial index is considered. It is found that the phase and intensity of the FWM field can be spatially manipulated. More importantly, we show the superposition modes created by the interference between the FWM field and a same-frequency Gaussian beam, which can also be controlled via the corresponding parameters. Our research is helpful to understand and manipulate optical vortices and can be widely used in quantum computation and communication.


INTRODUCTION
The Laguerre-Gaussian (LG) light carrying orbital angular momentum (OAM) [1] has attracted a lot of interest due to its unique amplitude and phase structures. In particular, the four-wave mixing (FWM) based on OAM light has emerged as a hot topic in recent years. For instance, Qiu et al. proposed a scheme to demonstrate the manipulation of space-dependent four-wave mixing (FWM) in a four-level atomic system [2]. By adjusting the detuning of the control field, one can effectively control the FWM output field. Yu et al. described a theoretical investigation of a FWM scheme in a six-level atomic system driven by a field with OAM and making use of two electromagnetically induced transparency (EIT) control fields [3]. The obtained results allow one to control the helical phase of the output FWM field by varying the intensities of the two EIT control fields as well as the detuning of the probe field. Quite recently, Wang et al. have also proposed some schemes to control vortex FWM carrying OAM in different nanostructures [4][5][6].
In this paper, we investigated the spatial manipulation via four-wave mixing in a five-level atomic system. Quite recently, we have proposed a scheme for modulating the spatial vortex FWM in a five-level atomic system [7]. However, different from this previous study, the major features of applying our considered scheme are as follows. First, the main difference between our scheme and the one in [7] is that we have shaped the LG field as a double-ring LG mode with the radial index p = 1 while in [7] the radial index is 0. This scheme has many advantages for controlling the FWM in comparison with the publication [7]. For example, the double-ring LG mode provides two FWM channels, and in different channels the spatial variation of the FWM are different. Second, in this scheme more physical parameters (e.g., the radial index p = 1) can be manipulated and hence one can select suitable radial index p to explore singularity characteristics of helical phase wavefront in nonlinear processes. Third, we display the superposition modes created by the interference between the FWM field and a samefrequency Gaussian beam, which show a more flexible intensity control or phase control for the superposition modes.

THEORETICAL MODEL AND EQUATIONS
We consider an atomic system as shown in Figure 1. A probe field with Rabi frequency Ω p Ω p0 e −(r/R p ) 2 π/(2 ln 2) exp[−t 2 /(2 ln 2/π 2 )] (Ω p0 and R p are the amplitude and transverse radius with t being the time) is applied to the transition |2〉 ↔|0〉. A control field with Rabi frequency Ω c drives the transition |2〉 ↔|1〉. A pump field with Rabi frequency Ω 1 drives the transition |3〉 ↔|2〉, while a LG field with Rabi frequency Ω v drives the transition |4〉 ↔|3〉. Here Ω v is defined as where Ω(r) )e −r 2 /ω 2 0 , Ω v0 is the initial Rabi frequency, r is the radius and the beam waist is ω 0 . ϕ is the azimuthal angle and L |l| p is the Laguerre polynomial. The radial index and azimuthal index are defined by p and l, respectively.
Making use of the Schrödinger equation, the dynamical equations for the atomic probability amplitudes A j (j = 1-4) in the interaction picture are given by [8].
where Δ p , Δ c , Δ 3 and Δ 4 are the detunings of the fields. The δ k k m − ( k p + k 1 + k v ) is phase mis-matching condition, and k i (i p, 1, v, m) are the wave vectors of the corresponding fields. The γ n (n = 1-4) are decay rates.

RESULTS AND DISCUSSION
In this section, we aim to study the effects of the different parameters on phase and intensity of the FWM field Ω m (z, ω). In order to clearly show the spatial-dependent mechanism, the superposition modes created by the interference between the FWM field and a same-frequency Gaussian beam are also provided.
In Figure 2, we plot the phase and the normalized intensity patterns of the FWM field versus the (x, y) for different probe detuning Δ p . As we expected, both the phase (Figures 2A1-A7) and intensity ( Figures 2B1-B7) patterns are divided into two parts. From this figure, one can find that, upon increasing Δ p from 1 MHz to ±5 MHz, the value of intensity decrease progressively while the phase twist is suppressed. The key factor is that the azimuthal dependent absorption and dispersion properties of the FWM field are modulated, which lead to the corresponding results. Moreover, the real ( Figures 2C1-C7) part and imaginary ( Figures 2D1-D7) part of dispersion relation K(ω) are shown in Figure 2, which have given the physical reason for the phase and the intensity patterns of the FWM field being changed in Figure 2.
In Figure 3, we present the phase and the normalized intensity patterns of the FWM field for different detuning Δ 3 . By direct comparison in Figures 2, 3, we obtain that the situation in Figure 3 is nearly the same as the Figure 2. Such results indicate that the OAM phase is transferred to the FWM field and is modulated via detuning Δ 3 . Here, we also provide the real part and imaginary part of dispersion relation in Figures 3C1-C7 and Figures 3D1-D7. By increasing the detuning Δ 3 , both the imaginary part and real part are changing. So we can see the varying phase and intensity of the FWM field.
Next, to obtain a better understanding of the vortex modulation, we show the superposition patterns of the FWM field and a same-frequency Gaussian beam in Figure 4. As illustrated in Figure 4, the superposition patterns are extremely rotated for different detunings Δ p and Δ 3 . The reason for this is that, due to the equiphase surface of Gaussian beam is a plane, azimuthally phase difference between the FWM field and Gaussian beam is very sensitive to detunings (Δ p , Δ 3 ), which reflects the different superposition patterns. The findings in Figure 4 imply that the OAM phase is indeed transferred to the FWM field and has a spatial dependency, which is originated from the spatial-sensitive absorption and dispersion properties induced by the three fields in the present system.
We note that, very recently, some theoretical schemes for controlling the space-dependent FWM in atoms [2,3] or in semiconductor quantum wells [4][5][6] have been proposed. Comparing with those schemes, the major features of our proposal are the following. First, we have utilized the EIT induced by a additional control beam Ω c . The EIT scheme has