1. Introduction

The study of optical vortices has been one of the hot topics in the field of optics in the last two decades because of its potential applications in large-capacity optical communication [1], rotational speed measurement [2], optical capture [3], space exploration [4] and quantum information process [5,6]. Note that an optical vortex exhibits a helical phase wavefront (represented by the exponential factor $e^{il\theta }$) and a donut-shaped intensity pattern. A phase singularity can occur at the central dark spot. Some common optical vortices include the Laguerre-Gaussian (LG) beam [7,8], Bessel Gaussian beam [9] and perfect vortex beam [10], etc, which carry orbital angular momentum (OAM) of $l\hbar$ per photon with the topological charge (TC) $l$. Since then, a host of breakthroughs have been made to explore the generation and detection of the vortex beams [11–18]. Meanwhile, the manipulation and exchange of the vortex beams have also been studied in optical metasurfaces [19], cylindrical-lens mode converters [20] and cross phase [21].

On the other hand, based on electromagnetically induced transparency (EIT) [22], spatially dependent light-matter interaction induced by optical vortices has been widely explored in atomic vapors [23–48], quantum dots [49,50], quantum wells [51,52], molecular magnets [53], rare earth-doped crystal [54], metal-coated dielectric nanosphere [55] and graphene [56]. Meanwhile, numerous interesting quantum optical phenomena have been found, such as light induced torque [30], storage and retrieval of light beams carrying OAM [31,35], vortex-induced spatial absorption [23,25], spatially dependent electromagnetically induced transparency (EIT) [26], spatially structured Kerr nonlinearity [36], ultraprecise Rydberg atomic localization [27], OAM transfer [24,28,37,43,49], vortex wave mixing [32–34,38,39,48,50–53,56], spatially dependent hyper-Raman scattering [29], spatially dependent optical bistability [40] and electromagnetically induced grating [41,42].

Recently, much attention has been focused on the improvement of the conversion efficiency of optical vortices [28,37,46–48,52]. For example, in the nonclosed loop atomic systems, the transfer of optical vortices between the two weak fields can be achieved by the coherent superposition of the ground states [28] and noise induced coherence [37]. Unfortunately, the maximal conversion efficiency of optical vortices cannot exceed $25{\% }$. Wang et al. proposed a scheme for high-efficiency vortex four-wave mixing (FWM) in asymmetric semiconductor quantum wells and found that the maximal conversion efficiency reaches approximately $50{\% }$ [52]. Another scheme for high-efficiency exchange of optical vortices has been demonstrated via three-wave mixing process in the Autler-Townes splitting regime of a three-level atomic system, where the conversion efficiency is observed to reach $65{\% }$ [46]; By using the rather complicated spatial modulations of between control fields, Hamedi et al. realized the complete energy conversion (i.e., nearly $100{\% }$ conversion efficiency) of optical vortices [47]; Recently, nearly complete transfer of optical vortices has also been achieved via using backward FWM in a dense atomic medium [48]. Thus, it remains us of one question: Can we achieve the high-efficiency transfer of optical vortices via a simple and feasible method?

Inelastic two-wave mixing (ITWM) usually refers to the use of a pair of laser fields and a three-state medium to generate weak atomic coherence, followed by a strong laser field to "retrieve the stored light" from the resulting atomic coherence [57]. This process can also be described as destructive interference between one-photon resonant couplings [58]. This quantum destructive interference in ITWM has now been successfully observed experimentally [57], and may have potential applications in biomagnetic imaging to precision measurement of the magnetic properties of subatomic particles [59]. Thus, can ITWM provide a good platform for achieving high-efficiency transfer of optical vortices?

In this paper, we investigate the high-efficiency transfer of optical vortices in an inverted-Y four-level atomic system via ITWM process. The atoms are initially prepared in a coherent superposition of two ground states. Differing from previous studies, the distinguishing features of this scheme are given as follows: First and foremost, the off-resonance control field allows us to improve the transfer efficiency of optical vortices in a non-closed-loop atomic system. It is found that the maximal vortex conversion efficiency between the probe and signal fields can approach $91{\% }$, which is greatly improved compared with previous schemes [28,37,46,47,52]. Second, the high-efficiency vortex transfer originates from the broken of the destructive interference between two one-photon excitation pathways, which is different from the conversion mechanism in Ref. [47]. Third, we investigate the behaviors of the composite vortex formed by the incident vortex probe and signal fields and focus on the influence of the control field on the composite vortex. The obtained results illustrate that the intensity and phase patterns of the composite vortex are sensitive to the intensity of control field. Our results provide possible potential applications in OAM-based quantum information processing and optical communication.

2. Models and equations

As illustrated in Fig. 1, we consider an inverted-Y four-level atomic system with two ground states ($|g_1\rangle$ and $|g_2\rangle$) and two excited states ($|e_1\rangle$ and $|e_2\rangle$), which can be realized in cold $^{87}$Rb atomic ensemble with $|g_1\rangle$=$|5S_{1/2}, F=1, m_F=-1\rangle$, $|g_2\rangle$=$|5S_{1/2}, F=2,m_F=1\rangle$, $|e_1\rangle$=$|5P_{3/2}, F=1, m_F=0\rangle$ and $|e_2\rangle$=$|5D_{5/2}, F=1, m_F=0\rangle$ [29,60,61]. In the proposed system, the transition $|e_2\rangle \leftrightarrow |e_1\rangle$ is driven by a strong control field $\Omega _c$ with central frequency $\omega _c$. Meanwhile, the transitions $|e_1\rangle \leftrightarrow |g_1\rangle$ and $|e_1\rangle \leftrightarrow |g_2\rangle$ are coupling by a weak probe field $\Omega _1$ and a weak signal field $\Omega _2$, respectively, thereby forming the ITWM process $|g_1\rangle \leftrightarrow |e_1\rangle \leftrightarrow |g_2\rangle$ [57–59]. According to the transition selection rules [62], the control field is $\pi$-polarization ($\pi$), while the probe and signal fields are left circularly polarized ($\sigma ^+$) and right circularly polarized ($\sigma ^-$), respectively. In our proposal, we only consider the condition that both the probe and signal fields are resonant with the corresponding transitions. Therefore, under the electric-dipole and rotating-wave approximations, the interaction Hamiltonian of the inverted-Y four-level atomic system can be written as ($\hbar =1$)

 

Fig. 1. (a) Diagram of the inverted-Y four-level atomic system interacting with three applied fields ($\Omega _c$, $\Omega _1$, $\Omega _2$). (b) Simple block diagram of atomic sample with three optical fields.

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Under the frame of density matrix, the equations of motion for density-matrix elements can be obtained as

At the initial stage, the inverted-Y atomic system is prepared in a superposition of two lower levels, namely, $|g_1\rangle$ to state $|g_2\rangle$. Then, the wave function at this initial time is given by

In the limit of the weak probe and signal fields, i.e., $|{{\Omega }_{1}}|,|{{\Omega }_{2}}|\ll {{\gamma }_{{{e}_{1}}{{g}_{1}}}},{{\gamma }_{{{e}_{1}}{{g}_{2}}}},{{\gamma }_{{{e}_{2}}{{g}_{1}}}},{{\gamma }_{{{e}_{2}}{{g}_{2}}}}$, the depletions of the ground states $|g_1\rangle$ and $|g_2\rangle$ from the initial values are negligible. Then, to the first order one has $\rho _{e_1e_1}=\rho _{e_2e_2}=\rho _{e_2e_1}=0, \rho _{g_1g_1}=|C_{g_1}|^2, \rho _{g_2g_2}=|C_{g_2}|^2, \rho _{g_2g_1}=C_{g_2}C_{g_1}^*$. Therefore, Eqs. (2)–(6) can be simplified as:

By assuming ${{\gamma }_{{{e}_{1}}{{g}_{1}}}}={{\gamma }_{{{e}_{1}}{{g}_{2}}}}={{\gamma }_{{{e}_{1}}}}, {{\gamma }_{{{e}_{2}}{{g}_{1}}}}={{\gamma }_{{{e}_{2}}{{g}_{2}}}}={{\gamma }_{{{e}_{2}}}}$ and solving Eqs. (8)–(11), we can obtain the steady-state solution of ${{\rho }_{e_1g_1}}$ and ${{\rho }_{e_1g_2}}$ as

As shown in Fig. 1(b), we consider an ITWM process in a cold $^{87}$Rb atomic medium with the geometrical length of $L$. In this arrangement, both the probe field ${\Omega }_{1}$ and the signal field ${\Omega }_{2}$ propagate through the atomic medium in the $+z$ direction. $z=0$ and $z=L$ correspond to the input and output ports of the probe and signal fields. Specifically, $\Omega _1(z=0)=\Omega _1(0)$ and $\Omega _2(z=0)=\Omega _2(0)$ represent the input intensities of the probe and signal fields, while $\Omega _1(z=L)=\Omega _1(L)$ and $\Omega _2(z=L)=\Omega _2(L)$ are the output intensities of the probe and signal fields. Thus, ITWM only takes place inside the atomic medium (i.e., $0\leq z\leq L$). Under the slowly varying envelope approximation, the propagation equations of the two weak laser fields $\Omega _1$ and $\Omega _2$ can be expressed as [66,67]

We assume that both the probe and signal fields are the long laser pulses, so that $(1/c){\partial {{\Omega }_{1(2)}}}/{\partial t}=0$. For simplicity, we assume ${\alpha }_{1}={\alpha }_{2}={\alpha }$ and substitute Eqs. (12) and (13) into Eqs. (14) and (15), the propagation equations of the probe and signal fields can be simplified to

Thus, assuming the initial conditions ${{\Omega }_{1}}(z=0)= \Omega _1(0)$ and ${{\Omega }_{2}}(z=0)=0$, the expressions of the probe and signal fields at the position $z$ inside the atomic medium can be written as

3. Results and discussions

Here, a unique mode is given by Laguerre-Gaussian (LG) mode and its Rabi frequency can be given as [34,52]

3.1 Transfer of optical vortices

In this section, we focus on exploring the exchange of optical vortices via ITWM process. From Eq. (19), it is found the Rabi frequency of the generated signal field is proportional to the Rabi frequency of the incident probe field. When the incident probe field $\Omega _1(0)$ is a LG mode described by Eq. (20), the generated signal field $\Omega _2(z)\sim e^{il_1\theta }$ has the same vorticity as the incident probe field $\Omega _1(0)$. Next, we will study in detail the transfer efficiency of optical vortices from the incident probe field to the generated signal field.

In Fig. 2, we plot the dimensionless intensities of the two weak laser fields $|\Omega _{1(2)}(z)|^2/|\Omega _{1}(0)|^2$ as a function of the dimensionless distance $z/L$ inside the atomic medium for two cases: (a) without and (b) with an auxiliary control field. When the control field does not exist, i.e., $\Omega _{c}=0$ MHz, it can be seen from Fig. 2(a) that the incident probe field decreases monotonically as the propagation distance $z$ increases, while the generated signal field increases monotonically. Accordingly, part of the energy of the incident probe field is transferred to the generated signal field. However, after a critical propagation distance (i.e., $z=0.12L$), the probe and signal fields arrive at the same intensity and remain unchanged. As expected [28], the conversion efficiency of optical vortices defined by $\eta =|\Omega _{2}(L)|^2/|\Omega _{1}(0)|^2$ only arrives at $25{\% }$. When an off-resonance control field exists, i.e., $\Omega _{c}=45$MHz and $\Delta _{c}=15$MHz, it can be seen that the incident probe field monotonically decreases and the generated signal field monotonically increases in the whole region (i.e., $0\leq z \leq L$)[see Fig. 2(b)]. In this situation, most of the energy of the incident field is unidirectionally transferred to the signal field and the conversion efficiency of optical vortices can be improved to $76.7{\% }$. Therefore, the off-resonance control field plays an important role in enhancing the transfer efficiency of optical vortices.

 

Fig. 2. The dimensionless intensity of fields $|\Omega _{1(2)}(z)|^2/|\Omega _{1}(0)|^2$ versus the dimensionless distance $z/L$: (a) $\Omega _{c}=0$ MHz, (b) $\Omega _{c}=45$ MHz, $\Delta _{c}=15$ MHz. Other parameters are: $\alpha =150$, $C_{g_1}=C_{g_2}=1/\sqrt {2}$, $\phi =0$, $\Omega _{1_0}=0.1$ MHz, $\gamma _{e_1}=5.6$ MHz and $\gamma _{e_2}=0.76$ MHz.

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The above high-efficiency transfer of optical vortices originates from the broken of the destructive interference between two one-photon excitation pathways [57]. To better digest the physical mechanism, the focus is directed to the case of $C_{g_1}=C_{g_2}=1/\sqrt {2}$. Then Eqs. (18) and (19) can be rewritten as

To inspect the effect of the control field in detail, we first investigate the influence of the detuning $\Delta _c$ of the control field on the transfer of optical vortices. We plot the evolution of the dimensionless intensities $|\Omega _{1}(z)|^2/|\Omega _{1}(0)|^2$ and $|\Omega _{2}(z)|^2/|\Omega _{1}(0)|^2$ for different values of the control detuning $\Delta _c$ in Figs. 3(a) and 3(b), respectively. When the control field is resonant with the transition $|e_2\rangle \leftrightarrow |e_1\rangle$, $\kappa =\alpha \gamma _{e_1}\gamma _{e_2}/(2|\Omega _c|^2+2\gamma _{e_1}\gamma _{e_2})$ and $n=0$. Compared with the case in Fig. 2(a), the additional absorption $\kappa$ greatly decreases, which leads to the incident probe field slightly decreases and the generated signal field slightly increases with the increase of $z$ from $0$ to $L$ [see the red-solid lines in Figs. 3(a) and 3(b)]. Accordingly, the conversion efficiency of optical vortices is less than $1{\% }$. When the control detuning $\Delta _c$ increases to $7.5$MHz and $15$MHz, the coupling length $L_c$ decreases to $2.03L$ and $L$. In this situation, the incident probe field monotonically decreases and the signal field monotonically increases as $z$ increases from $0$ to $L$ [see the blue-dashed and purple-dotted lines in Figs. 3(a) and 3(b)]. In addition, the increase of $\Delta _c$ from $0$MHz to $15$MHz would suppress the output of the incident probe field and enhance the output of the generated signal field. Under the condition of $\Delta _c=22.5$MHz, we have $L_c\approx 0.68L$, which means that the optimal conversion position of optical vortices occurs at $z=0.68L$ inside the atomic medium [see the green dashed-dotted lines in Figs. 3(a) and 3(b)]. With the two laser fields propagating deeper through the atomic medium, the energy is retransferred from the generated signal field to the incident probe field. Thus, these is an optimal value for the detuning of the control field for achieving the high-efficiency transfer of optical vortices.

 

Fig. 3. The dimensionless intensity of fields (a) $|\Omega _{1}(z)|^2/|\Omega _{1}(0)|^2$ and (b) $|\Omega _{2}(z)|^2/|\Omega _{1}(0)|^2$ versus the dimensionless distance $z/L$ for different detuning $\Delta _c$. Other parameters are the same as in Fig. 2(b).

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We then explore the influence of the intensity $\Omega _{c}$ of the control field on the transfer of optical vortices. The variation of the dimensionless intensities $|\Omega _{1}(z)|^2/|\Omega _{1}(0)|^2$ and $|\Omega _{2}(z)|^2/|\Omega _{1}(0)|^2$ for different $\Omega _c$ are plotted in Figs. 4(a) and 4(b), respectively. As $\Omega _c$ increases from $33$ MHz to $45$ MHz, the position of the optimal vortex transfer shifts from $z=0.55L$ to $z=L$ [see the red-solid, blue-dashed and purple-dotted lines in Figs. 4(a) and 4(b)]. It means that the energy retransferred from the signal field to the probe field decreases and the conversion efficiency $\eta$ of optical vortices increases. As $\Omega _c$ increases to $51$ MHz, $L_c\approx 1.30L$. In comparison with the case of $\Omega _c=45$ MHz, the output probe field increases, while the output signal field decreases[see the green dashed-dotted lines in Figs. 4(a) and 4(b)]. Therefore, there also is an optimal intensity of the control field to improve the conversion efficiency of optical vortices.

 

Fig. 4. The dimensionless intensity of fields (a) $|\Omega _{1}(z)|^2/|\Omega _{1}(0)|^2$ and (b) $|\Omega _{2}(z)|^2/|\Omega _{1}(0)|^2$ versus the dimensionless distance $z/L$ for different control fields $\Omega _c$. Other parameters are the same as in Fig. 2.

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To give a direct insight into the coherent transfer of optical vortices, we derive the expressions of the output probe and signal fields at $z=L$ as

By setting $n=\pi$, we can derive the optimal transfer conditions for $(\Delta _c, \Omega _c)$ and $(\Delta _c, \alpha )$ as

We present the variation of the dimensionless intensity of the output signal field $|\Omega _{2}(L)|^2/|\Omega _{1}(0)|^2$ versus $(\Delta _c, \Omega _c)$ and $(\Delta _c, \alpha )$ in Figs. 5(a) and 5(b), respectively. The white dashed lines highlight the optimal relations between $\Delta _c$ and $\Omega _c$ ($\Delta _c$ and $\alpha$) for the highest transfer of optical vortices according to Eq. (28). We can see that the white dashed lines perfectly agree with the numerical results. As shown in Fig. 5(a), the synchronous increase of the detuning and intensity of the control field would enhance the OAM transfer from the probe field to the signal field while maintaining the optimal condition of $n=\pi$. Especially, the maximal intensity of the output signal field occurs at $\Omega _c=106.3$MHz and $\Delta _c=80.0$MHz, where the transfer efficiency of optical vortices arrives at $85{\% }$. As shown in Fig. 5(b), under the conditions of $\Omega _c=140$MHz and $n=\pi$, the variations of the control detuning and OD have very little impact on the output of the signal field. Accordingly, the transfer efficiency of optical vortices is kept at $91{\% }$.

 

Fig. 5. The dimensionless intensity of the output signal field $|\Omega _{2}(L)|^2/|\Omega _{1}(0)|^2$ as a function of (a) ($\Delta _c$, $\Omega _c$) and (b) ($\Delta _c$, $\alpha$). The white dashed lines represent the optimal conditions for the transfer of optical vortices according to Eq. (18). Other parameters are the same as in Fig. 2 expect for $\Omega _c=140$MHz in (b).

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Next, in order to compare the intensity and phase profiles between the incident vortex probe field and the output signal field, we plot the intensity and phase profiles of input probe field and output signal field in Fig. 6. It is not difficult to find that the intensity patterns of the output signal field and the input probe field are the same, and the TC and radial index of the phase of the output signal field are consistent with that of the input probe field. This indicates that the OAM information in the probe field is effectively transferred to the generated signal field. Interestingly, the helical phase of the output signal field is rotated by an angle $\pi /l_1$ in the x-y plane with respect to the helical phase of the input probe field. In order to understand the reasons for this phenomenon, we rewrite the Eq. (27) as

From Eq. (29), it is not difficult to find an additional phase shift $\pi /l_1$ in the output signal field ${{\Omega }_{2}}(L)$.

 

Fig. 6. The intensity (a), (c) and phase (b), (d) profiles of the input probe field and output signal field. Other parameters are the same as in Fig. 5(b) expect for $l_1=2$, $\alpha =400$, $\Delta _c=55.2$ MHz.

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To include the effect of phase mismatch ($\Delta \overset {\scriptscriptstyle \rightharpoonup }{k}={{\overset {\scriptscriptstyle \rightharpoonup }{k}}_{1}}-{{\overset {\scriptscriptstyle \rightharpoonup }{k}}_{2}}\neq 0$) in the ITWM process, we add an additional term. i.e., $i\Delta k \Omega _2$ into Eq. (17), which can be rewritten as [37,39]

Using the boundary conditions ${{\Omega }_{1}}(z=0)= \Omega _1(0)$ and ${{\Omega }_{2}}(z=0)=0$, the solution for can be obtained by solving Eqs. (30) and (16):

 

Fig. 7. The dimensionless intensity of fields $|\Omega _{2}(L)|^2/|\Omega _{1}(0)|^2$ versus the phase mismatch $\Delta k L$. The inset shows the phase pattern of the output signal field when the phase mismatch is $\Delta k L=2$, and other parameters are the same as in Fig. 6.

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In the above discussions, we focus on investigating the OAM transfer from a probe field to a signal field via the ITWM process. As a matter of fact, we can extend our model by considering the propagation of $M$-component light beams through an ($M+2$)-level atomic medium with $M$ ground states and two excited states shown in Fig. 8(a). In this case, the $M$-component light beams can be expressed by Eqs. (44) and (45). Here, we consider an inelastic three-wave mixing in the ($3+2$)-level atomic medium, where the OAM state can be transferred from the incident probe field $\Omega _1(0)$ to the two generated signal fields $\Omega _2(z)$ and $\Omega _3(z)$. When we select $(C_{g_1}, C_{g_2}, C_{g_3})=(1/\sqrt {2}, 1/\sqrt {3}, 1/\sqrt {6})$ and the other parameters are the same as in Fig. 2(b), the coupling length $L_c$ between the three weak fields still satisfies the condition of $L_c=L$. That is to say, the two output signal fields $\Omega _2$ and $\Omega _3$ arrive at their maxima at the output $z=L$, where the probe field $\Omega _1$ reach its minimum. Accordingly, the vortex conversion efficiency from the probe field $\Omega _1(0)$ to the signal field $\Omega _2(L)$ ($\Omega _3(L)$) is 51.1% (25.5%), which is greatly enhanced compared with the case in Ref. [28].

 

Fig. 8. (a) Diagram of the ($M+2$)-state atomic medium with $M$ ground states and $2$ excited states. (b) The dimensionless intensity of fields $|\Omega _{m}(z)|^2/|\Omega _{1}(0)|^2$ versus the dimensionless distance $z/L$ for $(C_{g_1}, C_{g_2}, C_{g_3})=(1/\sqrt {2}, 1/\sqrt {3}, 1/\sqrt {6})$. Other parameters are the same as in Fig. 2 expect for $\Delta _c=0$ MHz and $\Omega _{2_0}=0.1$ MHz.

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3.2 Composite vortices

In this section, we also consider the inverted-Y four-level atomic system, which is initially prepared in a superposition of the two ground states. In the following, we explore the characteristics of composite vortices where the vortex probe field ${{\Omega }_{1}}(z=0)={\Omega }_{1}(0)={\Omega }_{1}(r, \theta )$ and the vortex signal field ${{\Omega }_{2}}(z=0)={\Omega }_{2}(0)={\Omega }_{2}(r, \theta )$ are incident on the atomic medium. Solving Eqs. (16) and (17) with new boundary conditions, the analytical expressions of the output probe and signal fields at $z=L$ can be written as

Then Eq. (33) can be rewritten as

 

Fig. 9. The weights $U_1$, $U_2$ of the incident probe and signal fields as a function of the control field ${{\Omega }_{c}}$. Other parameters are the same as in Fig. 2 expect for $\Delta _c=0$ MHz and $\Omega _{2_0}=0.1$ MHz.

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According to Eq. (34), a composite vortex beam can be generated by the collinear superposition of two incident vortex beams. The intensity and phase profiles of the output signal field ${{\Omega }_{2}}(L)$ are displayed in Figs. 10 and 11, respectively, for ${\Omega }_{c}=0$ MHz, $20$ MHz and $70$ MHz. Different combinations of the LG modes for the incident probe and signal fields are considered, i.e., $(l_1, l_2)=(1,-1), (2,-1)$, and $(3,-1)$ with $p_1=p_2=1$. The intensity of the incident signal field is chosen as $\Omega _{2_0}=0.1$MHz and the other parameters are the same as in Fig. 2. Obviously, the intensity and phase patterns of the composite vortex field are sensitive to the control field ${\Omega }_{c}$. In the case of ${\Omega }_{c}=0$ MHz, the weights of the input signal and probe fields are equal, i.e., $U_1= U_2$. In this case, the complete interference between the two LG modes produces multiple dark spots in the overlapped regions of the inner and outer rings and the number of the dark spots along the ring zone is $|l_1-l_2|$ [see the left column in Fig. 10]. In addition, the same number of phase singularities occur at the positions of dark spots except for the case of $l_1=-l_2$ [see the left column in Fig. 11]. In this situation, the output signal field is a non-composite vortex for $l_1=-l_2$ and composite vortex for $l_1\neq -l_2$. As ${\Omega }_{c}$ increases to $20$ MHz, $U_1< U_2$, the input vortex signal field dominates in the interference of two optical vortices. It is found that these local dark spots would expand outward and the positions of phase singularities are away from the central point [see the middle column in Figs. 10 and 11]. When ${\Omega }_{c}=70$ MHz, $U_1\ll U_2$, the weight of the incident probe field is so small that the effect of the input vortex probe field can be neglected. Consequently, the intensity and phase patterns of the output signal field at $z=L$ are the similar to those of the input signal field $\Omega _2(0)$ [see the right column in Figs. 10 and 11] Therefore, we can effectively manipulate the composite vortex beam via adjusting the control field ${\Omega }_{c}$.

 

Fig. 10. The intensity patterns of the output signal field ${{\Omega }_{2}}(z=L)$ as a function of $x$ and $y$ for different TCs of the input probe field, i.e., $l_1=1,2,3$ and for different control fields ${{\Omega }_{c}}=0$ MHz, $20.0$ MHz, and $70.0$ MHz. $\Omega _{2_0}=0.1$MHz, $p_2=1$, $l_2=-1$ and the other parameters are the same as in Fig. 7.

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Fig. 11. The corresponding phase patterns of the output signal field $\Omega _2(z=L)$ as a function of $x$ and $y$. The parameters used here are the same as in Fig. 8.

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Finally, the possible experimental setup of the proposed vortex ITWM is shown in Fig. 12. The weak probe and signal laser beams passes through two quarter-wave plates ($\textbf {Q1}$ and $\textbf {Q2}$), two half-wave plates ($\textbf {H1}$ and $\textbf {H2}$), a polarization beam splitter ($\textbf {P1}$) and the atomic medium along the $z$ direction, while the strong control beam passes through a quarter-wave plate ($\textbf {Q3}$), a half-wave plate ($\textbf {H3}$) and the atomic medium along the direction of $\theta \approx 0.3^{\circ }$. Therefore, The control beam can be separated from the probe and signal fields. In section 3.1, only the probe beam $\Omega _1(0)$ is incident and transformed into OAM light beam through the vortex phase plate ($\textbf {V1}$). In this case, the vortex signal beam $\Omega _2(L)$ can generated via the vortex ITWM process. In section 3.2, the incident probe $\Omega _1(0)$ and signal $\Omega _2(0)$ beams are transformed into vortex beams via the vortex phase plates $\textbf {V1}$ and $\textbf {V2}$, respectively. In this situation, a composite vortex beam $\Omega _2(L)$ can be generated. In the above two cases, the charge coupled device ($\textbf {CCD}$) camera can be used to detecte the output signal beam.

 

Fig. 12. Schematic view of the experimental setup. H: half-wave plate, Q: quarter-wave plate, V: vortex phase plate, PBS: polarization beam splitter, CCD: charge coupled device

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4. Conclusions

In conclusion, we have theoretically investigated the coherent transfer of optical vortices in an inverted-Y four-level atomic system. With the ITWM process, the OAM transfer between different frequencies can be achieved. It should be noted that the temperature of atomic medium needs to be controlled at a few tens of $\mu$K to effectively eliminate Doppler broadening effects. If not, then, while ITWM processes can still be realized in thermal atoms [57], the Doppler broadening will greatly affect the transfer of optical vortices.

Based on the above analysis, it is found that the matched coupling length $L_c=L$ is the optimal condition for the high-efficiency transfer of optical vortices in our scheme. We have given a suitable physical interpretation for the nearly complete OAM exchange via the broken of the destructive interference between two one-photon excitation pathways. Furthermore, we have extend our model to the inelastic multi-wave mixing process. It has been found that the initial population distribution of multiple ground states would influence the exchange efficiency of optical vortices. Finally, both the vortex probe and signal fields initially act on the atomic medium and generate a composite vortex described by the output signal field. It can be seen that the intensity and phase profiles of the composite vortex beam can be controlled via adjusting the intensity of the control field. Our results provide possible potential applications in OAM-based quantum information processing and optical communication.

Appendix: derivations of the expressions of $M$-component light pulses

In this Appendix we extend our model to a coherently prepared ($M+2$)-level atomic medium with $M$ ground states and two excited states [see Fig. 8(a)]. In the following, we derive the expressions of $M$-component light pulses propagating inside the atomic medium.

Under the electric-dipole and rotating-wave approximations, the interaction Hamiltonian of the ($M+2$)-level atomic medium can be written as ($\hbar =1$)

(35)$${{H}_{I}}={{\Delta }_{c}}\left| e_2 \right\rangle \left\langle e_2 \right| -(\sum_{m=1}^{M}{{\Omega }_{m}}\left| e_1 \right\rangle \left\langle g_m \right|+{{\Omega }_{c}}\left| e_2 \right\rangle \left\langle e_1 \right|+H.c.).$$

In our proposal, all atoms are initially prepared in the superposition of the ground states $|g_1\rangle$, $|g_2\rangle$,…, $|g_M\rangle$, which can be expressed as

(36)$$|\Psi(0)\rangle=\sum_{m=1}^{M} C_{g_m}(0)|g_m\rangle.$$

Thus, the corresponding density-matrix elements can be written as:

(37)$${{{\dot{\rho }}}_{e_1g_\zeta}}={-}i\gamma_{e_1g_\zeta}{{\rho }_{e_1g_\zeta}}+i{{\Omega }_{\zeta}}\rho_{g_\zeta g_\zeta}+i\sum_{m=1;m\neq\zeta}^{M}{{\Omega }_{m}}\rho_{g_m g_\zeta}+i{{\Omega }_{c}^*}{{\rho }_{e_2g_\zeta}},$$

(38)$${{{\dot{\rho }}}_{e_2g_\zeta}}={-}(i\Delta_c+\gamma_{e_2g_\zeta}){{\rho }_{e_2g_\zeta}}+i{{\Omega }_{c}}{{\rho }_{e_1g_\zeta}},$$

with the Rabi frequency of laser pulses $\Omega _\zeta (\zeta =1,2,\ldots,M)$. Assuming the weak atom-light interaction $|\Omega _\zeta |\ll \gamma _{{{e}_{1}}{{g}_{\zeta }}},\gamma _{{{e}_{2}}{{g}_{\zeta }}}$ and $\gamma _{{{e}_{1}}{{g}_{\zeta }}}=\gamma _{{e}_{1}}, \gamma _{{{e}_{2}}{{g}_{\zeta }}}=\gamma _{{e}_{2}}$, then, the steady-state solution of ${{\rho }_{e_1g_\zeta }}$ can be obtained as

(39)$${{\rho }_{e_1g_\zeta}}={-}\frac{{{\Omega }_{\zeta}}|C_{g_\zeta}|^2+\sum_{m=1;m\neq\zeta}^{M}{{\Omega }_{m}}C_{g_m}C_{g_\zeta}^*}{A},$$

with $A=i[|\Omega _c|^2+\gamma _{e_1}(i\Delta _c+\gamma _{e_2})]/(i\Delta _c+\gamma _{e_2})$. Under the slowly varying envelope approximation, the propagation equations of $M$-component laser pulses $\Omega _1$, $\Omega _2$, $\Omega _3$,…, $\Omega _M$ can be expressed as

(40)$$\frac{\partial {{\Omega }_{1}}}{\partial z}={-}i\displaystyle\frac{\alpha\gamma_{e_1}}{2LA} {{\rho }_{e_1g_1}},$$

(41)$$\frac{\partial {{\Omega }_{2}}}{\partial z}={-}i\displaystyle\frac{\alpha\gamma_{e_1}}{2LA}{{\rho }_{e_1g_2}},$$

(42)$$\frac{\partial {{\Omega }_{3}}}{\partial z}={-}i\displaystyle\frac{\alpha\gamma_{e_1}}{2LA}{{\rho }_{e_1g_3}},$$

(43)$$\begin{aligned}& \qquad \vdots \\ &\frac{\partial {{\Omega }_{M}}}{\partial z}={-}i\displaystyle\frac{\alpha\gamma_{e_1}}{2LA}{{\rho }_{e_1g_M}}.\end{aligned}$$

Using the initial condition ${{\Omega }_{1}}(z=0)= \Omega _1(0)$, and ${{\Omega }_{2}}(z=0)={{\Omega }_{3}}(z=0)=\cdots ={{\Omega }_{M}}(z=0)=0$, we can obtain the expressions of $M$-component laser pulses at the position $z$ inside the atomic medium as

(44)$${{\Omega }_{1}}(z)=\Omega_1(0)({{| {{C}_{g_1}}|}^{2}}{e^{{-}iD z/L}}+\sum_{m=2}^{M}{{| {{C}_{g_m}} |}^{2}}),$$

(45)$${{\Omega}_{2}}(z)={-}\Omega_1(0){C_{g_1}}{C_{g_2}^{*}}(1-{e^{{-}iD z/L}}),$$

(46)$${{\Omega}_{3}}(z)={-}\Omega_1(0){C_{g_1}}{C_{g_3}^{*}}(1-{e^{{-}iD z/L}}),$$

(47)$$\begin{aligned}& \qquad \vdots \\ &{{\Omega}_{M}}(z)={-}\Omega_1(0){C_{g_1}}{C_{g_M}^{*}}(1-{e^{{-}iD z/L}})\end{aligned}$$

with $D=\alpha \gamma _{e_1}/(2A)$.

Equations (45)–(47) imply that the generated $M-1$ signal fields $\Omega _2$, $\Omega _3$,…, $\Omega _M$ have the OAM state as the incident vortex probe field $\Omega _1(0)$ described by Eq. (20). For simplicity, we explore the vortex exchange between three laser pulses in a ($3+2$)-level atomic medium. We plot the dimensionless intensities of the three laser fields $\Omega _{1}$, $\Omega _{2}$ and $\Omega _{3}$ versus the dimensionless distance $z/L$ for different combinations of ($C_{g_1}$, $C_{g_2}$, $C_{g_3}$) in Fig. 13. In the case of ($C_{g_1}$, $C_{g_2}$, $C_{g_3}$)=($1/\sqrt {2}$, $1/\sqrt {6}$, $1/\sqrt {3}$), the conversion efficiencies for the two output signal fields $\Omega _2$ and $\Omega _3$ arrive at $25.5{\% }$ and $51.1{\% }$(actually just the opposite to the case shown in Fig. 8(b)), respectively. When $C_{g_1}$ remains unchanged and $C_{g_2}=C_{g_3}=1/2$, as shown in Fig. 13(b), the two genrated signal fields $\Omega _{2}$ and $\Omega _{3}$ have the same evolutionary track and the transfer efficiency of optical vortices is $\eta =38.4{\% }$. According to the above discussions, the output intensities of the generated signal fields are determined by the initial distribution of population of the ground states. These results offer us an effective parameter to control the conversion efficiency of optical vortices in an inelastic multi-wave mixing process.

 

Fig. 13. The dimensionless intensity of fields $|\Omega _{m}(z)|^2/|\Omega _{1}(0)|^2$ versus the dimensionless distance $z/L$ for (a) $(C_{g_1}, C_{g_2}, C_{g_3})=(1/\sqrt {2}, 1/\sqrt {6}, 1/\sqrt {3})$ and (b) $(C_{g_1}, C_{g_2}, C_{g_3})=(1/\sqrt {2}, 1/2, 1/2)$. Other parameters are the same as in Fig. 6(b).

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Funding

National Natural Science Foundation of China (12075036, 12104067, 12375008); Innovation Research Groups of Hubei Natural Science Foundation of China (2023AFA025); Science and Technology Research Project of Education Department of Hubei Province (Q20211314).

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.

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