Spatial coherent manipulation of Bessel-like vector vortex beam in atomic vapor

The interaction between vector beams and atoms under a weak magnetic field could induce spatially dependent electromagnetically induced transparency (EIT). Based on such a coherence effect, we propose a method for manipulating vector beams carrying spiral varying phases in hot rubidium atoms. When a transverse magnetic field (TMF) is applied, the transparent regions of the transmitted beam are strongly depend on the beam’s polarization distribution. In addition to the intensity modulation, the alignment of the TMF is reflected in the rotation of the central symmetric transmission patterns. In theory, we discuss the physical mechanism of the spiral EIT region generated by the phase profile, and analyze the influence of system parameters on this coherent process. Our work confirms that introducing additional radial phases can also lead to spatially dependent EIT, which extends another degree of freedom to manipulate atomic polarization. This will provide potential applications in light field manipulation and multi-dimensional quantum storage.


Introduction
Following the semi-classical theory of the interaction between light and matter, quantum interference occurs between different energy levels under resonant conditions [1], such as electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA), and coherent population trapping (CPT), which are all based on quantum coherent effects [2][3][4].Because these quantum coherence effects change the polarizability of the medium and provide a transparency or absorption window for light, they have been widely applied to slow light, quantum storage [5][6][7], and ultra-narrow bandwidth perfect absorbers [8].The atomic polarizability is affected by many parameters of the system [9,10].In addition to using light, the magnetic field can also regulate the atomic polarization by changing the detuning and the relaxation rate between energy levels [11][12][13].The most well-known effect that utilizes quantum interference between transition channels under magnetic fields is the Hanle EIT, which scans the magnetic field to split the Zeeman sublevels, resulting in a Lorentz-type transmission spectrum [14].
In fact, the interaction between light and atoms under the action of a magnetic field has been studied for a long time.In 1996, Ling et al studied the CPT and EIT processes of weak probe laser fields under multiple Zeeman degenerate sublevel configurations [15], the magnetic field works by disrupting the coherent processes [16].In 2002, Budker et al discovered that applying a longitudinal magnetic field to an atomic medium causes anisotropic polarization of atoms, resulting in a magnetically induced optical rotation effect [17].Because of the vector nature of the magnetic field and the polarization of the light field, the interactions between light and atoms can be more diverse.Huss et al investigated the effect of stray magnetic fields (orthogonal directions) on the CPT and EIT processes at cross levels [18], and then Noh et al calculated the transmission spectra of arbitrarily polarized light fields in 87 Rb atoms under 3D magnetic fields [19].Adding magnetic fields offers the interaction between light and atoms an alternative quantization axis, which affects the excited transitions of different polarization states.These works inspired the measurement of longitudinal magnetic fields using atomic ensembles [20].
However, works above are based on uniformly polarized light fields, and information can only be extracted from the overall transmission spectrum of the beam.In contrast, vector beams are more widely used because of their spatial polarization distribution and high capacity to carry information [21][22][23][24][25][26].In recent years, vector beams with special structures have exhibited unique topological properties [27][28][29][30], further broadening their application in light-matter interaction [31,32].When a vector beam induces an EIT effect in an atomic medium, transparency distribution is spatially dependent, called spatial EIT (SEIT).The earliest work related to SEIT was studied by Barreiro et al in 2006 to observe the rotating Doppler effect, which inadvertently used an energy level configuration that could produce a spatially transparent distribution [33].SEIT was first studied theoretically and experimentally in cold atom ensembles [34] in detail and then used to measure the strength and 3D spatial alignment of magnetic fields [35].Subsequently, the work in hot atoms demonstrated different transmission patterns based on a similar effect [36].
In this paper, we demonstrate the spatial EIT using a vector vortex beam carrying both angular and radial phases simultaneously.Such a beam has a Bessel-type phase distribution but a Gaussian-type profile, called a Bessel-like vector vortex beam (BlVVB).Applying a weak transverse magnetic field (TMF) restructures the spatial distribution of the transparent window in the transverse plane, and the strength of the TMF could improve the contrast of interference patterns.In addition, the transmission pattern rotates with the azimuth angle of TMF.Unlike the previous work, this result reveals that the occurrence of quantum interference effects is not affected by the complex phase forms carried by the components of the coupled transition channels.Our results could extend to the spatial manipulation of atomic spiral dark polaritons [37,38] and provide more potential applications such as quantum storage, quantum communication, and the preparation of hybrid entanglement sources [39][40][41][42] based on the vectorial light-matter interaction.

Theoretical analysis
Vector beams could be represented as a coherent superposition of two orthogonal modes [43].Set the beam's propagation direction as z axis, then the BlVVB in our work takes the form where we have set the initial phase Φ R = Φ L = 1 √ 2 , R is the beam size at the initial plane, and n, l represent the radial and angular quantum numbers, respectively.Figure 1(a) shows the simulated polarization distribution of BlVVB with n = 3 and l = 2 according to this expression.
The magnetic field with three dimensions is where θ, α are the spatial and azimuth angles of the magnetic field, respectively.In order to analyze how light interacts with atoms under a magnetic field, we usually choose the light's propagation direction as the quantization axis [34].Then, the configuration of atomic energy levels which interact with the BlVVB is now shown in figure 2. The magnetic field components aligned with the quantization axis cause Zeeman splitting, while the perpendicular components couple these sublevels.Considering such a quantization axis, the linear polarization of light can be considered as two orthogonal circularly polarized components, with each component coupling a transition between the ground state and excited state and following the transition selection rule of ∆m F = ±1.Thus, using the effective Hamiltonian, we may get the full Bloch equations through the Liouville equation and solve such complex equations [44].We may also try to simplify the form of the Hamiltonian to use Fermi's golden rule [35], but this system cannot be simplified into a ladder scheme due to the cross terms in the Hamiltonian.
So, let us take such a system account in a different view by choosing another quantization axis, i.e. the direction of the magnetic field.Now, the polarization in the transverse plane can be separated into parallel  The interactions between the beam and the atomic energy level under a weak magnetic field when selecting different quantization axes.Upside row, the beam's propagation direction as quantization axis, βL and β T represent for the magnitude of Zeeman shift and the coupling strength between the ground levels respectively; downside row, the magnetic field's direction as quantization axis.We have neglected the Zeeman splitting of the upper energy level in experiment due to the short lifetime.and perpendicular components compared to the magnetic field.The linear polarization perpendicular to the quantization axis drives both σ + and σ − transitions, while the parallel component drives the π transition.The intensity of the two components can now be written as By substituting equations ( 1), ( 2) into (3), we can calculate the Rabi frequency as where is the saturation light intensity, Ω ≡ |E • d|/h is the Rabi frequency, Γ is the spontaneous emission rate.In hot atomic systems, the dephasing rate is very rapidly [45], which means the existence of the non-diagonal elements in the Hamiltonian matrix are temporary.By setting all the non-diagonal terms in the full Bloch equations equal to zero and substituting them into the rest of the diagonal terms, we get the so called population rate equations (see appendix A.1), an approximate and simplified version of the Bloch equations [46,47].In this system, we can derive: where γ is the transverse relaxation rate between the ground states [48,49], and represents the detuning due to Zeeman shift for |g −1 ⟩.The diagonal elements of the density matrix ρ e,g ±ii ≡ |e, g ±i ⟩⟨e, g ±i |(i = 0, 1, 2) represent for populations on each state and always satisfy ∑ i ρ e,g ±ii = 1 .By solving the steady-state solution of the population rate equations (5), we finally get the transmission function of the BlVVB passing through atoms as: In the following, a comparison between the simulated and experimental results is given.

Experimental setup
The experimental setup is shown in figure 3. The output of a 795 nm external cavity diode laser is divided into two parts: one part is applied for frequency locking using the saturation absorption spectrum, and the other one is sent through a single-mode fiber (SMF) to improve the spatial mode.After that, the beam goes through a half-wave plate and polarization beam splitter to control its intensity and purify its polarization.To generate the BlVVB following equation ( 1), a q-plate [50] with l = 2 and a customized liquid-crystal-based retardation wave plate [51] with n = 3 are utilized, leading to both angular and radial phases.Since the wavefront of this beam is not steady during propagation, two 4f imaging systems are employed to ensure we get the correct beam profile at the center of the 87 Rb cell and the charge-coupled device camera(CCD).In this experiment, the laser frequency is locked to the 5 2 S 1/2 , F = 2 → 5 2 P 1/2 , F = 1 transition of the 87 Rb D 1 line [52], the power of the input beam is 0.940 mW and beam waist is 3 mm.The pure 87 Rb cell is set at 67 • C with a temperature controller during the experiment which means the atomic density could achieve about 6 × 10 10 cm −3 [53].Covering around the Rb cell is a three-layer µ-metal magnetic shield to isolate the cell from ambient magnetic fields.A customized plastic frame wrapped with copper coils is installed between the shield and the Rb cell, which contains two pairs of orthogonal Helmholtz coils to generate arbitrary transverse magnetic fields precisely.The dimensions of the plastic frame are 15 cm in length, 5 cm in width, and 5 cm in height, while the cell length is only 5 cm with a cross-section radii of 1 cm, thus the TMF can be seen as uniform.After interacting with the atomic ensemble, the beam is recorded by the CCD using appropriate parameters.Moreover, we also utilize a projection measurement system consisting of an HWP, a QWP and a PBS to measure the four Stokes parameters, which represent the full polarization information of the light [54,55].
All of the results in this paper are under the magnetic field's inclination θ = π/2, i.e. a TMF is applied, but the theoretical analysis is also applicable to the magnetic field with a longitudinal component.

Experimental results and discussion
The generated BlVVB contains right-and left-handed circularly polarized components with angular and radial phase change, resulting in a spiral-variant linear polarization distribution, as shown in figure 1, which agrees with the simulated one.According to equation ( 1), the horizontal (x) and vertical (y) linear polarization components are the function of the radial and angular coordinate (r, ϕ).Due to the periodicity of polarization in space and its ability to be decomposed into two orthogonal linear polarization components, the beam can be perceived as 2 × 2l lobes (the two '2's' are arisen from orthogonality and central symmetry, respectively) in the azimuth direction.
As only linear polarization is considered, these 4l lobes couple the atomic states in two different ways, as discussed above, a π (∆m F = 0) and a σ transition (∆m F = ±1).In the absence of an external magnetic field, the hyperfine Zeeman sublevels are degenerate, and only σ transition exists, thereby inducing a transparency akin to the standard EIT effect [18].When a weak TMF is applied, the polarization perpendicular to the quantization axis leads to a coherent dark state formed by σ transitions between the ground states.However, the frequency detuning caused by Zeeman shift can disrupt this dark state, causing the absorption.Conversely, the polarization parallel to the quantization axis couples the π transitions and always tends to transmit, as the Zeeman shift can barely disrupt the optically pumped bare dark state.This implies that most atoms are pumped to the side states, where no transition occurs.[56].Thus it is conceivable that the beam profile shall be segmented by 2l lobs.
Figure 4 shows the experimental results of the spatially structured transparency by applying a weak TMF in the x(y) direction.When a very weak (18.24 mG) TMF B x is applied in the x direction, the expected 2l-lobed pattern emerges, as illustrated in figure 4(a).After interacting with the atoms, the position with horizontal polarization is essentially transmitted, whereas the vertical polarization is partially absorbed, particularly the whole pattern taking on a spiral profile.As the magnetic field strength increases uniformly, both experimental and simulation results indicate that the contrast can be further improved.The situation remains unchanged for the applying of a TMF B y in the y direction, but the transmitted or absorbed positions are opposite to the previous one.Compared to the profile of the initial input beam in figure 1, those narrow rings generated by diffraction at the periphery have vanished.This is due to the thermal motion of atoms influencing the ideal EIT process, leading to the beam not being completely transmitted in the absence of a magnetic field.It should be noted that the recorded patterns are somewhat overexposed due to specular reflection in the experiment, but this does not affect the transmission profile.
In figure 5 we show the rotation of transmission pattern with the applied TMF.The azimuth angle α of the TMF has rotated from 0 to π in our experiment by adjusting the current ratio of two orthogonal Helmholtz coils.To show the rotation of the pattern more clearly, we plotted the intensity curve of the transmission from ϕ = 0 → 2π along the radius with the largest contrast (i.e. the red dotted ring in figure 5).Although the coils were not placed strictly horizontally, resulting in a slight angle error between the theoretical and experimental results, both results showed that the transmission pattern overlapped with itself after rotating for π.Generally, the center symmetrical pattern will rotate by α/l following the TMF and reappear after a rotation of 2π/l.This characteristic might be used to calibrate the alignment of the magnetic  field [36].It should be noted that the reason for the significant discrepancy between theoretical and experimental results in the range from 3π/2 to 2π is primarily due to the non-uniformity of the input beam in the experiment (see figure 4).Additionally, the beam passing through the LCP is not strictly centered, leading to a polarization of the input beam is not strictly rotationally symmetric about the center.

Zeng et al
Regarding this process of inducing spatially structured transparency, some parameters are alternative theoretically, such as the spontaneous emission rate Γ, transverse relaxation rate γ and Rabi frequency Ω.These parameters have effect on the contour and the absorption degree of the transmission patterns.As different atomic ensembles are different with certain parameters, the same vector beam can yield distinct outcomes of interaction.This is exactly the case.In [35], the final transmission patterns exhibit 4l lobs in cold atoms, whereas our findings demonstrate 2l lobs in hot atoms.
The most significant reason for the different results of the two atomic ensembles should be the different energy level schemes.Although the tripodal energy level is simpler than what we use here, it is not feasible to directly select such scheme experimentally in hot atoms due to the Doppler broadening [57].But theoretically, changing γ in the rate equations through the method of controlling variables can help us investigate the underlying causes for the differences.Set γ in the tripodal energy level very fast, one can get a transmission pattern of 2l lobs, which align remarkably well with those prediction by the Bloch equations.Hence, it can be inferred that the slower transverse relaxation rate is accountable for yielding 4l lobs.That's because the faster transverse relaxation rate disrupts the coherent dark states, resulting in a change from transmission to absorption for the polarization components perpendicular to the quantization axis.While the utilization of our energy level scheme reveals that setting γ at a slow pace merely yields narrower 2l lobs.Thus, both the energy level scheme and the transverse relaxation rate exert an influence on the number of lobes in the transmission patterns; however, it is evident that the former holds greater significance (see appendix A.2). Other parameters, such as Γ and Ω, have less effect on the shape of the transmission lines, and they are generally unchanging for both cold and hot atomic systems experimentally.

Conclusion
In summary, we demonstrated that a beam carrying both angular and radial phase profiles can also induce a spatially dependent EIT effect.Experimentally, a q-plate and a customized LCP generate the Bessel-like vector vortex beam exhibiting a spatially structured transparency after passing through the pure 87 Rb hot atoms under a weak TMF.The contrast and rotation of the transmitted intensity patterns are closely related to the strength and alignment of the TMF.Theoretically, we propose a more straightforward method to analyze this effect relative to the full Bloch equations and Fermi's Golden Rule, namely the rate equations.The analysis shows that different energy level scheme and transverse relaxation rate are the two main factors, leading to different experimental results in hot and cold atomic systems.
The spiral-variant phase we introduce in vector beam not only brings a unique spatial absorption mapping to the atomic medium, but also provides a new variable for the programmable dispersion relations.In addition, since a single photon in a spin eigenstate can also encode different phase profiles and couple atomic coherent transition channels [58,59], our work exhibits the possibility of applications for multi-dimensional quantum storage and quantum information.Using the dipole approximation and the rotating wave approximation, we can derive the Hamiltonian for the aforementioned system within the interaction picture: then, according to the master equation of the density matrix ρ that incorporates dissipative terms: where a † i,j ≡ |i ⟩⟨j |, i ̸ = j is the spin raising operator.Assuming that the spontaneous relaxation rates from the excited state to the ground state and the metastable state are both Γ/2, we shall derive the following 6 independent Bloch equations: Seeking the steady-state solution when t → ∞, the variation of Im(ρ ba ) or ρ aa with detuning ∆ reflects the absorption of the probe light.
If the off-diagonal elements of the density matrix in the aforementioned equation dissipate more rapidly (i.e.dρ i,j /dt = 0, i ̸ = j), then we can replace the off-diagonal elements with the diagonal elements in the last three equations of equation ( 9) and substitute them into the first three equations.This yields the rate equations containing only diagonal elements, which should take the following form: (10) where W i,j , i ̸ = j represents the transition rate between |j ⟩ and |i ⟩, which can be accurately determined by equation (9).However, taking into account its intimate relationship with the Rabi frequency and detuning, we can intuitively deduce the following formulation: By doing so, we can present a contrast between the Bloch equations (BE) and the rate equations (RE) regarding the population distribution of the excited state: Figures 6(b)-(d) illustrates the absorption spectra of the two outcomes under a set of specific parameters, revealing that there are no disastrous differences between them overall.Due to the fact that a higher rate of spontaneous emission implies a faster decoherence, it can be observed from figure 6 that the agreement is better in graph (d), which also demonstrates that despite not including the explicit form of the off-diagonal elements of the density matrix, the rate equations still serve as a very good approximation to the Bloch equations.

A.2. The differences between hot and cold atoms for SEIT effect
For simplicity, let us only consider the typical vector beams carrying azimuthal vortex phase.Then, equation (1) now becomes: we can derive the Rabi frequency according to equation (2) to equation ( 4): where all symbols representing physical quantities are consistent with those in the main text.For the tripod-type energy level scheme used in [35], the following rate equations can be formulated: Here we assume that the magnetic field's spatial angle θ and azimuthal angle are both π/2, the orbital angular momentum quantum number l = 1, the spontaneous emission rate Γ = 1, the Rabi frequency for the components of the beam in all directions is 1/10, and based on the fact that there's a significant difference in the transverse relaxation rates between cold and hot atoms, figures 7(a) and (b) presents the angular transmission curves of vector beams after the SEIT process in cold and hot atoms, where γ = 10 −4 and γ = 10 −1 respectively.For the tripod-type energy level scheme, it can be clearly observed from figures 7(a) and (b) that hot atoms correspond to a number of lobes equal to 2l = 2 × 1 = 2, whereas cold atoms exhibit 4l = 4 × 1 = 4 lobes (the number of lobes can be deduced from the number of peaks in the transmission curve), which are in good agreement with experimental results.Figures 7(c) and (d) illustrate that for the eight-level scheme we employed, although the transmission curve of hot atoms also fits the experimental results well, the simulated structure corresponding to cold atoms does not exhibit the expected 4l lobes; instead, the transmission peaks become more pronounced.As discussed in the main text, the simulation results, derived from controlling variables, suggest that the energy level scheme and the transverse relaxation rate γ are the two main factors affecting the SEIT transmission patterns.It is clear that several orders of magnitude change in γ cannot fundamentally break through the limitations imposed by the energy level scheme.All structures indicate that an increase in TMF's intensity can alter the contrast of the transmission patterns, which is reassuring considering that it is the only physical quantity we can freely and finely adjust in the experiment.

Figure 2 .
Figure 2. The interactions between the beam and the atomic energy level under a weak magnetic field when selecting different quantization axes.Upside row, the beam's propagation direction as quantization axis, βL and β T represent for the magnitude of Zeeman shift and the coupling strength between the ground levels respectively; downside row, the magnetic field's direction as quantization axis.We have neglected the Zeeman splitting of the upper energy level in experiment due to the short lifetime.

Figure 5 .
Figure 5. Transmission patterns as the transverse magnetic field rotates from α = 0 to α = π per π/4.Inner arc, simulation results; intermediate arc, experimental results; outer arc, normalized transmission strength lines derived along the red dashed circle indicated in the patterns when α = 0, the red curves are simulation results, the gray square dots are experimental results.