1. Introduction

Electromagnetically induced transparency (EIT) and related effects in atomic ensembles have been widely investigated for decades [1,2]. The inherent quantum interference effect in EIT can make a resonant, optically thick medium transparent, establishing a coherent platform to realize all-optical control of photons. For example, by adiabatically switching the strong control light on and off in EIT media, a weak probe field can travel slowly almost without dissipation. Further, it can be completely converted into flipped atomic spins (dark states) for long-term storage. Finally, the stored light can be retrieved back to electromagnetic mode [3–5], offering great opportunities toward EIT-based protocols for quantum memories of photons [6–8]. To further improve the processing speed and efficiency, optical images and vortices have been employed in the EIT systems [9–15]. The research efforts have mostly concentrated on the delay, storage, and modulation of spatial optical signals, which may find an array of classical and quantum applications, including image processing [9,11–13], optical sensing [10], all-optical switching and routing [14,15], and spatially multiplexed quantum communication [16–18]. All these studies could help promote the all-optical parallel processing capability for high-dimensional information, thus enabling EIT-based multichannel classical and quantum network architectures.

Optical vector beam, as a single nonseparable light field with the spin and orbital angular momentum (OAM) degrees of freedom, has attracted significant research interest in optics [19,20]. Due to the peculiar polarization and spatial distributions, vector beam has permeated a broad range of applications in classical optics, such as optical imaging [21–24] and trapping [25–27], metrology [28–31], and high-dimensional optical coding [32–34]. Mathematically, because the expression of a vector beam has the same structure with a two-qubit entangled state, a vector beam is also referred to as a classically entangled state of light [35–37], which can be described in the framework of the higher-order or even hybrid-order Poincaré sphere [38,39]. This fact makes vector beam an excellent paradigm of quantum-classical connection, enabling the simulation of diverse quantum processes using classical light [40–42]. Moreover, quantum entanglement has also been realized for the vector beam as carriers with both the spatial and polarization information [43–45], which may have some interesting applications in alignment-free quantum key distribution [46–48].

In recent years, the marriage between vector beams and EIT media has been actively studied, where the vector beam provides a valuable candidate for many appealing EIT-related phenomena. For instance, a spatially dependent EIT has been observed by the interaction between rubidium atoms and vector beams with an azimuthally varying polarization and phase structure [49,50]. Moreover, even at the single-photon level, vector beams have been successfully stored and retrieved in the EIT systems [51–53]. Because both the spin and OAM degrees of freedom can be reversibly mapped onto the collective spin-wave excitation at the same time, a quantum memory based on spatial-polarization-patterned beams can be implemented [51], which may hold great promise for advanced quantum technologies, such as multiple-degree-of-freedom register, networks, and communication. In addition to the EIT effect, the interaction between vector beams and atomic ensembles has also been the subject of intense investigations [54], giving rise to a variety of potential applications, such as atomic spin analyzer [55], polarization rotation [56,57], nonlinear propagation and frequency conversion [58–61], and atomic magnetometer [62,63].

In this article, we explore an EIT-based all-optical scheme to control the spin-orbit nonseparable states of vector beams to traverse the entire higher-order Poincaré sphere (HOPS) with high tunability. In the system, we employ a four-level tripod configuration with cold atoms and consider both the real and imaginary parts of the linear susceptibility. Due to rotational invariance, the paraxial evolution of the vector beam can be described by a generalized Pauli-like equation with a complex pseudo-spin-orbit coupling (PSOC) potential, partially analogous to the spin precession of an electron. By adjusting the EIT parameters to manipulate the PSOC, both the amplitude and phase factors of the two Laguerre-Gaussian (LG) modes in the vector beam can be flexibly tailored. As a result, the spin-orbit nonseparable state of the vector beam can be mapped on the surface of a HOPS, where the spatial orientation and ellipticity of the transmitted polarization can be well controlled. Our finding may potentially improve the EIT-based applications of multiplexed all-optical communication and quantum information processing with spatial and polarization degrees of freedom.

2. Theoretical model

The four-level tripod EIT system and possible experimental setup are shown in Fig. 1. The weak probe field is a vector beam, i.e., a superposition of two LG modes overlapping in space. In the probe field, one mode (P1) has the right circular polarization $\sigma ^{-}$ (i.e., the $| R \rangle$ state) and the OAM index $+\ell$, the other (P2) has the left circular polarization $\sigma ^{+}$ (i.e., the $| L \rangle$ state) and the OAM index $-\ell$, where $\ell$ is a positive integer. In the four-level atomic medium in Fig. 1(a), the P1 (P2) mode interacts with the transition $| 1 \rangle \leftrightarrow | 4 \rangle$ ($| 2 \rangle \leftrightarrow | 4 \rangle$) with the single-photon detuning $\Delta _{1}=\omega _{1}-\omega _{41}$ ($\Delta _{2}=\omega _{2}-\omega _{42}$), where $\omega _{1}=\omega _{2}=\omega _{p}$ because the two LG modes have the same angular frequency $\omega _{p}$. The strong control light drives the transition $| 3 \rangle \leftrightarrow | 4 \rangle$ with the single-photon detuning $\Delta _{c}=\omega _{c}-\omega _{43}$.

 

Fig. 1. (a) Four-level tripod EIT system, where $| 1 \rangle$, $| 2 \rangle$, and $| 3 \rangle$ are the ground states and $| 4 \rangle$ is the excited state. The incident probe field is a vector beam composed of two LG modes (P1 and P2) with opposite OAM indices ($\pm \ell$) and circular polarizations ($\sigma ^{\mp }$), but the same angular frequency $\omega _{p}$. For simplicity, the decay and decoherence rates in Eqs. (1) and (2) are also shown. (b) Possible setup for the all-optical manipulation of a spin-orbit nonseparable state of light in the EIT system, where the weak probe field is a vector beam with space-variant polarization (e.g., a vortex vector beam) and the strong control field provides uniform illumination. All the beams are well-collimated and copropagate through the medium. The thickness of the atomic cloud is $d$. A polarization-sensitive imaging system is used to detect the spatial and polarization distributions of the transmitted vector beam. The hyperfine filter is used to filter out the control field (see Sec. 4 for details).

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Therefore, the light fields and atoms form a standard tripod EIT configuration. Based on the optical pumping technique, we assume that the initial population is equally prepared in the $| 1 \rangle$ and $| 2 \rangle$ levels, i.e., $\rho _{11} \approx \rho _{22} \approx 1/2$, and the population of the other two levels vanishes. Under the electric dipole and rotating wave approximations, one can solve the density-matrix master equations. Considering the steady-state solutions, we can see that the linear susceptibilities for the P1 and P2 modes in the probe field can be described by [64,65]

For simplicity, we assume the dipole moments $\mu _{14}=\mu _{24}=\mu _{p}$, the population decay rates $\gamma _{41}=\gamma _{42}=\gamma$, and the decoherence rates $\gamma _{31}=\gamma _{32}=\tilde {\gamma }$ in the atoms with $\gamma \gg \tilde {\gamma }$. Also, we assume that the control field is always resonant with the $| 3 \rangle \leftrightarrow | 4 \rangle$ transition (i.e., $\Delta _{c}=0$), leading to $\Delta _{1(2)}=\delta _{1(2)}$. Therefore, under the near-resonance EIT conditions (i.e., $\Omega _{c}^{2} \gg \{ \gamma \tilde {\gamma }, \gamma \Delta _{1(2)} \}$), Eqs. (1) and (2) can be rewritten as

Formally, one can always define $\Delta _{1}= + \ell \Omega _{1}$ and $\Delta _{2}= - \ell \Omega _{2}$, where $\Omega _{1}$ and $\Omega _{2}$ are two frequency parameters for the P1 and P2 modes. Therefore, Eqs. (3) and (4) can be rewritten as

Because the vector beam is a paraxial light field, the P1 and P2 modes satisfy the paraxial propagation equations given by [66]

Because the system governed by Eq. (8) obeys rotational invariance, the spinor for the vector beam can be rewritten as $\Psi =(\mathcal {A}_{+} e^{+ i \ell \phi },\ \mathcal {A}_{-} e^{- i \ell \phi })^{T}$ where $\mathcal {A}_{\pm }$ represent the radial evolution and $\phi$ is the azimuthal angle. Furthermore, by performing a unitary transformation $\Psi = U \Phi$ with $U= \left ( \begin {smallmatrix} 1 & 0 \\ 0 & e^{ -2 i \ell \phi } \end {smallmatrix} \right )$, we can achieve the spin-orbit spinor $\Phi = e^{ i \ell \phi } (\mathcal {A}_{+}| +_{z} \rangle + \mathcal {A}_{-}| -_{z} \rangle )$ with $| \pm _{z} \rangle$ being the pseudo-spin-up and -down states [67–70]. Note that the concept of “pseudo-spin” comes from the two-component vector probe field containing two LG modes, which is conceptually inspired by the “pseudo-spin” of electrons in graphene, referring to the two sublattice (spatial) degrees of freedom. We would like to emphasize that the “pseudo-spin” is not the intrinsic spin (polarization) of the photon.

Thus, Eq. (8) can be transferred to a generalized Pauli-like equation with a complex PSOC in a two-dimensional central potential [71]

3. Numerical results

Because EIT systems offer high flexibility for photon manipulation, we will discuss two typical cases for controlling the vector beam as follows.

3.1 Antisymmetric detunings

In this case, we assume $\Delta _{1} = -\Delta _{2} > 0$ as an example for antisymmetric single-photon detunings of the P1 and P2 LG modes in the probe field, leading to $\Omega _{1} = \Omega _{2} = \Omega >0$. Thus, we have $\bar {\Omega }= \Omega$, $\tilde {\Omega }=0$, $\tilde {\zeta }=\varsigma =0$, and $\xi >0$. As a result, only the Russell-Saunders-type PSOC term $-\bar {\zeta } \hat {\ell } \hat {\sigma }_{3}$ with $\bar {\zeta }>0$ in the real potential is kept in Eq. (9), which can produce different phase modulations for the P1 and P2 modes. In our scheme, we can assume that the incident spinor wave function takes the form of

Accordingly, using the Dirac bra-ket notation, the incident vector beam is expressed as the nonseparable wave function $\mathcal {V}_{z=0} = (|+\ell, R \rangle + |-\ell, L \rangle )/\sqrt {2}$ where $|\pm \ell \rangle = N_{c} \mathcal {A}(r) e^{\pm i \ell \phi }$ include the radial function $\mathcal {A}(r)$ implicitly. Note that mathematically the circular polarization $|R \rangle$ and $|L \rangle$ states correspond to the pseudo-spin $|+_{z} \rangle$ and $|-_{z} \rangle$ states, respectively, in our scheme.

Moreover, when the thickness $d$ of the EIT medium is much smaller than the Rayleigh distance $z_{R}$ of the vector beam (i.e., $d \ll z_{R} = \pi w_{0}^{2}/\lambda _{p}$), the radial momentum of the beam can be dropped in Eq. (9), yielding

At the exit of the EIT medium, the transmitted spinor wave function is given by

By either changing the frequency parameter $\Omega$ (the frequency detunings $\Delta _{1} = -\Delta _{2} = + \ell \Omega$) or the control Rabi frequency $\Omega _{c}$, the PSOC coefficient $\bar {\zeta }$ can be well controlled and the nonseparable state in the transmitted vector beam can be flexibly manipulated. Without the loss of generality, we assume the OAM indices as $\pm \ell = \pm 1$, and thus the incident vector beam is given by $\mathcal {V}_{z=0} =(|+1, R \rangle + |-1, L \rangle )/\sqrt {2}$, which is also called vortex vector beam [see the inset in Fig. 1(b)].

To show the numerical results, we consider the D2 line of $^{87}$Rb atoms to construct the tripod EIT configuration. We assume the atomic number density $\mathcal {N} \approx 10^{12}$ cm$^{-3}$ and the thickness of the EIT system $d=5$ mm. The population decay rate is $\gamma _{41}=\gamma _{42}=\gamma =2 \pi \times 6$ MHz, the coherence decay rate is $\gamma _{31}=\gamma _{32}=\tilde {\gamma } = 2 \pi \times 1$ kHz, and the dipole moment is $\mu _{p} = 3.58 \times 10^{-29}$ C$\cdot$m [73]. The wavelengths of the vector probe and control fields are $\lambda _{p} \approx \lambda _{c} \approx 780$ nm.

Figure 2 indicates that the relative phase shift ($2 \bar {\zeta } \ell d$) between $| +1, R \rangle$ and $|-1, L \rangle$ is very crucial for the control over the nonseparable state, which can move the state along the equator of the HOPS. The parameter $\alpha = \mod (\bar {\zeta } \ell d / \pi )$ can be defined as the orientation angle of the linear polarization with respect to the $x$ axis at the location of $\phi =0$. And, $2 \alpha$ is the azimuthal angle in the HOPS. By changing the values of $\alpha$ from point A$_{0}$ to D$_{0}$, the polarization structure of the transmitted vector beam can be locally rotated, analogous to the spin precession of an electron. In contrast, because the $| +1, R \rangle$ and $|-1, L \rangle$ states always have the same weighting, the term $e^{-\xi \ell ^{2} d }$ has no impact on the polarization structure of the transmitted wave function $\mathcal {V}_{z=d}$.

 

Fig. 2. The HOPS with $\pm \ell = \pm 1$ for the transmitted nonseparable states is shown in the central panel. The white arrows in the four pictures around the HOPS indicate a clear variation in polarization as the spatial distribution is traversed. The relative phase shifts are $2 \bar {\zeta } \ell d = 0$ (or $2 \pi$) at point A$_{0}$, $\pi /2$ at B$_{0}$, $\pi$ at C$_{0}$, and $3 \pi /2$ at D$_{0}$ on the equator of the HOPS. Accordingly, the orientation angle is $\alpha =0$ at A$_{0}$, $\pi /4$ at B$_{0}$, $\pi /2$ at C$_{0}$, and $3 \pi /4$ at D$_{0}$. Note that the polarization structure at point A$_{0}$ also represents the incident vector beam $\mathcal {V}_{z=0}$. By changing the frequency parameter $\Omega$ or the control Rabi frequency $\Omega _{c}$, the relative phase shift $2 \bar {\zeta } \ell d$ can be modulated, leading to the motion of the transmitted nonseparable state along the equator of the HOPS. Note that, because the absorption term $e^{-\xi \ell ^{2} d }$ has nothing to do with the polarization structure and the weak absorption leads to high intensity transmission ($> 93 {\%}$) in our scheme, the intensity profiles are not crucial in the antisymmetric case and we normalize the peak intensity in the pictures.

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For numerical details, one can first fix the control Rabi frequency as, e.g., $\Omega _{c}= 5 \gamma$ and tune the frequency parameter $\Omega (=\Delta _{1} = -\Delta _{2})$ by changing a bias magnetic field along the $z$ axis to generate the Zeeman splitting. To move the nonseparable state along the equator of the HOPS, the frequency parameter can be increased from $\Omega = 0$ for point A$_{0}$ to $0.054 \gamma$ for point B$_{0}$, $0.107 \gamma$ for point C$_{0}$, and finally $0.161 \gamma$ for point D$_{0}$. Second, one can also fix the frequency parameter as, e.g., $\Omega = 0.1 \gamma$ and decrease the control Rabi frequency from $\Omega _{c} = 6.83 \gamma$ for point B$_{0}$ to $4.83 \gamma$ for point C$_{0}$, $3.95 \gamma$ for point D$_{0}$, and finally $3.42 \gamma$ for point A$_{0}$. Note that the population decay rate $\gamma$ is used as a scaling factor for the frequency parameters $\Omega$ and $\Omega _{c}$ in our numerical calculations. Besides, we also calculate the intensity transmission rates ($e^{-2 \xi \ell ^{2} d }$) using the above parameters and find quite high transmission rates ($> 93 {\%}$) in our scheme.

There is a notable difference between the above two methods. In method (i), when $\Omega =0$, the P1 and P2 modes in the probe field are exactly on EIT resonance, where no relative phase shift between the two modes can be induced (i.e., $2\alpha =0$). Therefore, the polarization of the transmitted vector beam is the same as that of the incident one. However, in method (ii), the frequency parameter (i.e., the detunings) is not zero (i.e., $2\alpha \neq 0$). To make the transmitted vector beam have the same spatial polarization distribution as the incident one, the relative phase shifts should be an integer multiple of $2 \pi$. Thus, we assume $\Omega = 0.1 \gamma$ and $\Omega _{c} = 3.42 \gamma$ to produce $2\alpha = 2 \pi$ for point A$_{0}$ in method (ii).

3.2 Asymmetric detunings

In this case, to optimize the expected results, we consider an extreme scenario and assume $\Delta _{1} = + \ell \Omega _{1} = 0$ and $\Delta _{2} = - \ell \Omega _{2} < 0$ for the extremely asymmetric single-photon detunings of the P1 and P2 LG modes in the probe field. The PSOC coefficients in the Pauli-like Eq. (9) have the relationships as $\bar {\zeta } = - \tilde {\zeta } > 0$ and $\varsigma = - \xi < 0$. Therefore, the intensity and phase of the P2 mode can be significantly modulated due to the detuning away from the EIT resonance, whereas the modulation of the P1 mode is negligible due to the on-resonance EIT response.

For $d \ll z_{R} = \pi w_{0}^{2}/\lambda _{p}$, Eq. (9) can be again simplified as

 

Fig. 3. Spatial polarization distributions of the transmitted vector beam away from the equator of the HOPS. The left panel shows points A$_{1}$–A$_{5}$ on the longitude line with $2 \alpha =0$. The corresponding relative phase shift $2 \bar {\zeta } \ell d$ and ellipticity $\varepsilon$ are given in Table 1. The right panel shows points B$_{1}$–B$_{5}$ on the longitude line with $2 \alpha = \pi /2$. The corresponding $2 \bar {\zeta } \ell d$ and $\varepsilon$ are given in Table 2.

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Fig. 4. The left panel shows points C$_{1}$–C$_{5}$ on the longitude line with $2 \alpha =\pi$. The corresponding relative phase shift $2 \bar {\zeta } \ell d$ and ellipticity $\varepsilon$ are given in Table 3. The right panel shows points D$_{1}$–D$_{5}$ on the longitude line with $2 \alpha = 3 \pi /2$. The corresponding $2 \bar {\zeta } \ell d$ and $\varepsilon$ are given in Table 4.

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Table 1. Parameters for points A$_{1}$–A$_{5}$ on the longitude line with $2\alpha =0$ in Fig. 3.^a

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Table 2. Parameters for points B$_{1}$–B$_{5}$ on the longitude line with $2\alpha = \pi /2$ in Fig. 3, where $\Omega _{2}$ or $\Omega _{c}$ can be adjusted (see the 1st or 2nd row).^a

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Table 3. Parameters for points C$_{1}$–C$_{5}$ on the longitude line with $2\alpha = \pi$ in Fig. 4, where $\Omega _{2}$ or $\Omega _{c}$ can be adjusted (see the 1st or 2nd row).^a

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Table 4. Parameters for points D$_{1}$–D$_{5}$ on the longitude line with $2\alpha = 3 \pi /2$ in Fig. 4, where $\Omega _{2}$ or $\Omega _{c}$ can be adjusted (see the 1st or 2nd row).^a

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In the left panel of Fig. 3, we show the evolution of the nonseparable states of the transmitted vector beam along the longitude line with $2\alpha =0$. To this end, compared with those in Fig. 2, stronger phase and amplitude modulations should be imparted on the transmitted probe field. We again adjust the frequency parameter $\Omega _{2}$ or the control Rabi frequency $\Omega _{c}$ to manipulate the transmitted nonseparable states (see Table 1). In details, for point A$_{1}$, it has the smallest relative phase shift ($2 \bar {\zeta } \ell d = 2 \pi$), accompanied by the largest amplitude transmission rate ($e^{2 \varsigma \ell ^{2} d} = 0.90$). Thus the ellipticity ($\varepsilon =3.1^{\circ }$) is small and point A$_{1}$ is very close to the equator ($2 \varepsilon =6.2^{\circ }$), which generates a very flat polarization ellipse. Furthermore, it can be seen that, to increase the relative phase shift ($2 \bar {\zeta } \ell d$) and ellipticity ($\varepsilon$) from points A$_{1}$–A$_{5}$, we should gradually increase the frequency parameter $\Omega _{2}$ (see the 1st row in Table 1) or decrease the control Rabi frequency $\Omega _{c}$ (see the 2nd row in Table 1), where the orientation angle $\alpha$ of the polarization ellipses remains to be zero at $\phi =0$. In the right panel of Fig. 3, we change the relative phase shift ($2 \bar {\zeta } \ell d$) to make points B$_{1}$–B$_{5}$ stay on the longitude line with $2\alpha =\pi /2$ (see Table 2), where the orientation angle $\alpha = \pi /4$ at $\phi =0$. Due to the increasing phase shift, the amplitude transmission characterized by $e^{2 \varsigma \ell ^{2} d}$ will also decrease, thus lowering the weighting of $|-\ell, L \rangle$ in the nonseparable state. As a result, the ellipticity $\varepsilon$ for points B$_{1}$–B$_{5}$ can be raised compared with the corresponding points A$_{1}$–A$_{5}$.

In Fig. 4, one can achieve similar results for the control of nonseparable states of the transmitted vector beam. The left panel of Fig. 4 shows the evolution of points C$_{1}$–C$_{5}$ along the longitude line with $2\alpha =\pi$. The right panel of Fig. 4 shows the evolution of points D$_{1}$–D$_{5}$ along the longitude line with $2\alpha =3\pi /2$. Thus, the orientation angle $\alpha$ is raised from $\pi /2$ to $3\pi /4$. All the corresponding values of the frequency parameter $\Omega _{2}$ or the control Rabi frequency $\Omega _{c}$ are given in Tables 3 and 4. Specifically, we can see that point D$_{5}$ is very close to the north pole ($2 \varepsilon = 86.8 ^{\circ }$) because the strongest phase modulation $11.5 \pi$ simultaneously accompanies the weakest amplitude transmission ($e^{2 \varsigma \ell ^{2} d}=0.03$) of the $|-\ell, L \rangle$ state, leading to the largest weighting for $|+\ell, R \rangle$ in the transmitted nonseparable state. Therefore, both the orientation angle $\alpha$ and the ellipticity $\varepsilon$ can be maximized for point D$_{5}$ amongst all the twenty points shown in Figs. 3 and 4. The polarization ellipse is very round and nearly right circularly polarized at point D$_{5}$.

Based on the above analyses, we can see that in the extremely asymmetric case, the difference between the amplitude transmission rates in the nonseparable state given by Eq. (16) (i.e., ideally the unity transmission rate for the $|+\ell, R \rangle$ state and $e^{2 \varsigma \ell ^{2} d}$ for the $|-\ell,L \rangle$ state) is inherently associated with the imaginary part $- i \varsigma \hat {\ell }^{2} \hat {\sigma }_{3}$ with $\varsigma < 0$ in the PSOC potential and plays a vital role in producing large ellipticity $\varepsilon$ to move the nonseparable state along the longitudinal lines. To make the transmitted nonseparable state reach the north pole of the HOPS, one can simply use a longer EIT medium to further enhance the amplitude absorption of the $|-\ell, L \rangle$ state beyond point D$_{5}$. For example, we can assume the medium thickness $d=10$ mm, $\Omega _{2}=0.1\gamma$, and $\Omega _{c} =\gamma$ and keep other parameters unchanged. The amplitude transmission is given by $e^{2 \varsigma \ell ^{2} d} = 0.0006$ for the $|-\ell, L \rangle$ state, whereas the medium still keeps transparent for the $|+\ell, R \rangle$ state. This means that the $|-\ell, L \rangle$ state is almost completely absorbed and thus only the $|+\ell, R \rangle$ state can pass through the EIT medium. Consequently, the EIT system demonstrates very strong circular dichroism to filter out the $|-\ell, L \rangle$ state so that the transmitted nonseparable state can reach the north pole of the HOPS (i.e., the $|+\ell, R \rangle$ state).

4. Discussions

In Section 3, we consider two typical cases to manipulate the classical nonseparability of light. By elaborately tuning the EIT parameters for the two PSOC coefficients $\bar {\zeta }$ and $\varsigma$, the nonseparable states in the vector beam can be located on the equator and the northern hemisphere of the HOPS. If we assume $\Delta _{1} = + \ell \Omega _{1} > 0$ and $\Delta _{2} = - \ell \Omega _{2} = 0$ for the P1 and P2 LG modes in the probe field, the nonseparable wave function of the transmitted beam can be given by

In general, by properly adjusting the EIT parameters, such as the detunings $\Delta _{1}$ and $\Delta _{2}$ of the probe light fields and the control Rabi frequency $\Omega _{c}$, it is reasonable to believe that the PSOC-based complex potential with the coefficients $\bar {\zeta }$ and $\varsigma$ in the EIT system can provide two degrees of freedom (i.e., the relative phase shifts and the relative weightings between $|+\ell, R \rangle$ and $|-\ell, L \rangle$) to make the nonseparable states of the transmitted beam on the entire HOPS. In particular, the coefficient $\bar {\zeta }$ associated with the phase modulation of $|+\ell, R \rangle$ and $|-\ell, L \rangle$ can lead to the local polarization rotation of the transmitted beam, analogous to the spin precession of an electron.

Based on the numerical results in Figs. 2–4, we can verify the EIT conditions used in our calculations. Let us see point D$_{5}$ in Table 4, which gives an extreme but typical numerical example. At point D$_{5}$, the EIT parameters contain the weakest control Rabi frequency $\Omega _{c} \approx \gamma$ and the largest frequency parameter $\Omega _{2}= \Delta _{2} \approx 0.1 \gamma$. Also, we assume that the population decay rate of the excited level is $\gamma =2 \pi \times 6$ MHz and the decoherence rate between the ground levels is $\tilde {\gamma } = 2 \pi \times 1$ kHz. The near-resonance EIT conditions $\Omega _{c}^{2} \gg \{ \gamma \tilde {\gamma }, \gamma \Delta _{2} \}$ can be well satisfied, where $\Omega _{c}^{2} \approx \gamma ^{2}$, $\gamma \Delta _{2} \approx 0.1 \gamma ^{2}$, and $\gamma \tilde {\gamma } \approx \gamma ^{2} /6000$. Therefore, our PSOC scheme in Section 2. is valid to describe the paraxial propagation of vector beams through the EIT system.

To construct the four-level tripod configuration, we can choose four magnetic sublevels in the D2 line of $^{87}$Rb atom, e.g., $|1 \rangle =|5S_{1/2}, F=1, m=+1 \rangle$, $|2 \rangle =|5 S_{1/2}, F=1, m=-1 \rangle$, $|3 \rangle =|5 S_{1/2}, F=2, m=-1 \rangle$ as the ground states, and $|4 \rangle =|5 P_{3/2}, F=1, m=0 \rangle$ as the excited state, respectively [73], where the control field is $\sigma ^{+}$-polarized. Because we employ the copropagation arrangement for the probe and control fields in the tripod system [see Fig. 1(b)], to remove the control field at the imaging system, we can use a $^{85}$Rb vapor cell as a hyperfine filter to absorb the spectral lines associated with $^{87}$Rb’s $|5 S_{1/2}, F=2 \rangle$ state (i.e., the control field). This technique is widely used in the rubidium atomic clock [74].

Our scheme might be extended to the EIT system with hot atoms, where the Doppler effect (i.e., the inhomogeneous broadening) induced by the atomic motion should be considered. Assuming that the cell temperature is 80 $^{\circ }$C, the most probable thermal speed of $^{87}$Rb atoms is $v_{p} = \sqrt {2 R T /M} \approx 260$ m/s, where $T$ is the Kelvin temperature, $R=8.314$ J/(mol K) is the gas constant, and $M =0.87$ kg/mol is the molar mass of the $^{87}$Rb atom. The single-photon Doppler broadening is $\Delta \omega _{D} = 2 \sqrt {\mathrm {ln} 2} k_{p} v_{p} = 3.5 \times 10^{9}$/s ($\approx 100 \gamma$). Because the hyperfine splitting between $|5 S_{1/2}, F=1 \rangle$ and $|5 S_{1/2}, F=2 \rangle$ is 6.835 GHz, the two-photon Doppler broadening can be estimated as $\delta \omega = 3.7 \times 10^{4}$/s ($\approx 0.001 \gamma$) in our scheme. The condition $\Omega _{c}^{2} \gg \Delta \omega _{D} \delta \omega \approx 0.1 \gamma ^{2}$ is consistent with the near-resonance EIT conditions mentioned above, whereby the Doppler effect can be strongly suppressed under the copropagation arrangement for the light fields [2,4]. Therefore, our theoretical derivations for the susceptibilities in Eqs. (3) and (4) should be also valid to investigate the PSOC mechanism for the nonseparable state of light in the Doppler-broadened EIT media. As a matter of fact, the thermal atomic systems provide us a low-cost and handy platform to construct EIT-based devices. Using our scheme with the hot atoms, all-optically tunable polarization shaping and filtering for the vector probe field could be realized. This scheme may also be combined with the light storage technique (see Refs. [51–53]) to perform the polarization-spatial processing for the retrieved vector beams in both the classical and quantum regimes.

In our numerical calculations, we consider the near-field propagation of the vector beam in the EIT medium and use the approximation $d \ll z_{R} = \pi w_{0}^{2}/\lambda _{p}$ to simplify Eqs. (11) and (14). In practice, such a near-field approximation of a laser beam can be conveniently accomplished. For example, we can set the waist of the vector beam as $w_{0}=250$ $\mu$m and thus the Rayleigh distance is $z_{R} \approx 254$ mm, where the medium thickness (e.g., $d =5$ mm in Figs. 2–4) is much smaller than $z_{R}$. Therefore, the radial momentum of the vector beam can be ignored, and Eqs. (11) and (14) can partially mimic the spin precession in quantum mechanics. More importantly, the presence of the imaginary PSOC term $- i \varsigma \hat {\ell }^{2} \hat {\sigma }_{3}$ in Eq. (14) controls the relative amplitude weightings between $|+\ell, R \rangle$ and $|-\ell, L \rangle$ (or, in other words, the P1 and P2 LG modes), thus moving the nonseparable states on the surface of the HOPS. Eventually, our research presents the PSOC-controlled evolution of the vector beam traveling through the EIT medium.

5. Conclusion

In conclusion, we have shown that the tripod EIT system can provide an all-optical platform to tailor the spin-orbit nonseparable state of light (i.e., a vector beam) on the surface of a HOPS. Considering both the real and imaginary parts of the susceptibilities, by quantum-optical analogy, the propagation dynamics of the probe vector beam can be described by a generalized Pauli-like equation under rotational invariance, partially mimicking the spin precession of an electron. Due to the tunable PSOC-based complex potential, the relative phase shifts and the amplitude weightings of the two LG components in the probe vector beam can be modulated by the two PSOC coefficients. In general, the change of the relative phase shifts can move the nonseparable state along the latitude line of the HOPS, whereas the change of the amplitude weightings can move the nonseparable state along the longitude line of the HOPS. Consequently, the tripod EIT system provides two degrees of freedom to completely control the nonseparable state to traverse the entire HOPS. As well as offering a comprehensive understanding of the interaction between atoms and vector beams in EIT media, our findings suggest a useful guideline for optimizing the design of EIT-based devices for space-variant polarization control of light in coherent media. Our scheme may help develop efficient and flexible strategies for all-optical manipulation of vector beams, finding potential in various EIT-based spatial-polarization applications, such as multiplexed optical communication and quantum information.