Abstract
Based on electromagnetically induced transparency (EIT), we investigate an alloptical grating structure to realize polarizationdependent multiple beam splitting in the RamanNath limit. To optimize the grating performance, higher excited state [e.g., nS_{1/2} (n ≥ 6)] of ultracold ^{87}Rb atoms is employed to construct a fivelevel ΞΛ system sharing one common populated ground state. A principal advantage of our proposed scheme is that the σ^{±} components of a linearly polarized weak probe field can be decoupled and thus be independently diffracted with high efficiency in both one and two dimensions by exploiting different quasistanding waves as the two strong coupling fields in the Ξ and Λ configurations. Such an alloptical polarizationsensitive operation could greatly enhance the tunability and capacity of alloptical multiplexing, interconnecting, and networking in free space for both classical and quantum applications.
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Introduction
The effect of electromagnetically induced transparency (EIT) offers opportunities to control light with light, whereby alloptical devices can be implemented to significantly improve the performance of optical information processing systems in both classical and quantum regimes^{1,2,3}. As an important application of EIT effect, electromagnetically induced grating (EIG) has been actively studied for alloptical multichannel devices^{4,5,6,7,8,9,10,11}. Due to the diffraction effect, a weak probe laser can be transversely split into a series of highorder components by strong standingwave (SW) coupling fields in both one and two dimensions. To further increase the diffraction efficiency in the far field, various periodic structures, such as imageinduced blazed gratings^{12,13,14}, sinusoidal phase gratings^{15}, and volume holographic gratings^{16,17}, have been developed in atomic ensembles, which can extend the functionality of EIG for realistic applications. Also, the diffraction gives rise to other interesting optical phenomena, such as nearfield Talbot selfimaging for ultracold atoms^{18,19}.
As a fundamental property, the polarization degree of freedom can strongly influence the propagation dynamics of light in EIT media, which plays a vital role in quantum information processing. For example, alloptical polarization phase gates could be created in an atomic system to realize a conditional phase shift of the order of π for quantum logic devices^{20,21,22}. Moreover, controllable storage and retrieval of polarization qubit in EITrelated atomic systems could be a promising candidate to implement optical quantum memory, which proves to be a crucial operation for a global quantum network^{23,24,25,26,27,28,29,30}. Therefore, a further study of polarizationdependent EIG structures for multiple beam diffraction may help advance the development of alloptical multichannel processing for photon polarization information in free space.
In this article, we present an electromagnetically induced polarization grating (EIPG) scheme to realize polarizationresolved multiple beam diffraction for a weak probe laser in the RamanNath limit. By involving multiple Zeeman sublevels in 5S_{1/2}, 5P_{1/2}, and nS_{1/2} (for example n = 10) states of an ultracold ^{87}Rb atomic ensemble, a fivelevel Ξ(cascade)Λ system is examined under the condition of EIT. Two sets of thin grating structures can be simultaneously but independently induced by using different quasiSWs as the coupling fields for the Ξ and Λ configurations, respectively. By dynamically tuning the optical parameters of the EIT system, the two circularly polarized σ^{±} components of the probe field can be decoupled and flexibly controlled by diffraction, thus generating a plurality of purely polarized beams at different angles in one and two dimensions in the far field.
Methods
Lightatom interactions in EIPG
To realize independent control over the σ^{±} components of a weak probe field, we here adopt ultracold ^{87}Rb atoms as a sample medium and construct a fivelevel ΞΛ system using realistic energy levels of Rb atoms as shown in Fig. 1(a). Initially, the atomic population is optically pumped to 1〉 which serves as the common ground state of the Ξ and Λ configurations. The σ^{+} component interacts with the 1〉 ↔ 2〉 transition with a singlephoton detuning \({{\rm{\Delta }}}_{{\sigma }^{+}}\), while the σ^{−} component interacts with the 1〉 ↔ 3〉 transition with a singlephoton detuning \({{\rm{\Delta }}}_{{\sigma }^{}}\). Two coupling fields (C1 and C2) drive the 2〉 ↔ 4〉 and 3〉 ↔ 5〉 transitions with singlephoton detunings Δ_{C1} and Δ_{C2}, respectively. Thus, the 1〉 ↔ 2〉 ↔ 4〉 transitions constitute the Λ subsystem and the 1〉 ↔ 3〉 ↔ 5〉 transitions constitute the Ξ subsystem. Note that, due to the dipole selection rule, the atomic transition between 5P_{1/2}, F = 2, m_{ F } = 0〉 and 5S_{1/2}, F = 2, m_{ F } = 0〉 is forbidden and thus cannot interact with the πpolarized C1 field^{31}.
In principle, we can use standard semiclassical theory to investigate the lightatom interactions in our scheme^{32,33}. The effective interaction Hamiltonian of the system within the dipole approximation and the rotatingwave approximation can be written as
where the frequency detunings are \({{\rm{\Delta }}}_{{\sigma }^{+}}={\omega }_{{\sigma }^{+}}{\omega }_{21}={\omega }_{{\rm{p}}}{\omega }_{21}\), \({{\rm{\Delta }}}_{{\sigma }^{}}={\omega }_{{\sigma }^{}}{\omega }_{31}={\omega }_{{\rm{p}}}{\omega }_{31}\), Δ_{C1} = ω_{C1} − ω_{24}, and Δ_{C2} = ω_{C2} − ω_{53}, the Rabi frequencies are \({{\rm{\Omega }}}_{{\sigma }^{+}}={\mu }_{12}{E}_{{\sigma }^{+}}/\hslash \), \({{\rm{\Omega }}}_{{\sigma }^{}}={\mu }_{13}{E}_{{\sigma }^{}}/\hslash \), Ω_{C1} = μ_{42}E_{C1}/ħ, and Ω_{C2} = μ_{35}E_{C2}/ħ. Note that ω_{p}, ω_{C1}, and ω_{C2} are the frequencies of the light fields, ω_{21}, ω_{31}, ω_{24}, and ω_{53} are the corresponding atomic transition frequencies, and μ_{12}, μ_{13}, μ_{42}, and μ_{35} are the transition dipole moments. To describe the spontaneous emission of the excited states (2〉, 3〉, and 5〉) and the dephasing process of the ground states (1〉 and 4〉), we introduce the phenomenological decay rate Γ_{ i } for each atomic level i〉. Therefore, the equations of motion for the probability amplitudes of the atomic wave function \(\psi (t)\rangle ={\sum }_{i=1}^{5}\,{a}_{i}(t)i\rangle \) can be given by
where, for simplicity, all the Rabi frequencies are taken to be real. Because the probe light field is very weak, it is reasonable to believe that the atomic population remains in the initial state 1〉, which means a_{1} ≈ 1. Based on the steadystate solutions (\({\dot{a}}_{i}=0\), i = 1, …, 5) of the above Eqs (2)–(6), we can achieve the linear optical response of the EIT medium under the condition \({{\rm{\Omega }}}_{{\sigma }^{\pm }}\) \(\ll \) Ω_{C1}, Ω_{C2}. The linear susceptibilities read
where n_{a} is the atom number density and the nonlinear susceptibilities are ignored. It is clearly seen in Eqs (7) and (8) that the σ^{±} components of the probe field can be decoupled and thus be independently manipulated by dynamically adjusting the two coupling (C1 and C2) fields, which could greatly improve the flexibility for coherent polarization control of photons. This fact also represents a major advantage of our EIPG scheme.
Moreover, because, unlike a perfect SW, the intensity of quasiSWs in the C1 and C2 coupling fields does not vanish at the quasinodal positions [see Fig. 1(b,c)]^{34}, the weakprobefield approximation (i.e., \({{\rm{\Omega }}}_{{\sigma }^{\pm }}\) \(\ll \) Ω_{C1}, Ω_{C2}) can be maintained in the entire interacting region, which also ensures the validity of Eqs (7) and (8) for the linear susceptibilities in the atomic medium. In details, the expressions of the Rabi frequencies of quasiSW C1 and C2 fields in Eqs (7) and (8) take the forms of
for the onedimensional (1D) EIPG shown in Fig. 1(b), and
for the twodimensional (2D) case shown in Fig. 1(c), respectively. The spatial periods of the quasiSW C1 and C2 fields are G1 and G2. The terms Ω_{C1′}, Ω_{C1″}, Ω_{C2′}, and Ω_{C2″} represent the Rabi frequencies of the four laser beams that generate the quasiSW C1 and C2 fields, where we have Ω_{C1′} ≠ Ω_{C1″} and Ω_{C2′} ≠ Ω_{C2″}.
Fraunhofer diffraction of 1D EIPG
For the probe field travelling through a thin (RamanNath) 1D EIPG in our scheme, its propagation is governed by Maxwell’s equation which can be simplified as \(\partial {E}_{{\sigma }^{\pm }}/\partial z=i{k}_{{\rm{p}}}{\chi }_{{\sigma }^{\pm }}{E}_{{\sigma }^{\pm }}/2\) under the paraxial approximation, where the transverse terms \({\partial }^{2}{E}_{{\sigma }^{\pm }}/\partial {x}^{2}\) and \({\partial }^{2}{E}_{{\sigma }^{\pm }}/\partial {y}^{2}\) are ignored^{4,12}. Assuming that the probe field is a plane wave incident in the z direction (i.e., \({E}_{{\sigma }^{\pm }}^{{\rm{in}}}=1\)), for the 1D case shown in Fig. 1(b), the farfield (Fraunhofer) diffraction amplitudes can be given by 1D Fourier transform^{35}, i.e.,
where \({E^{\prime} }_{{\sigma }^{\pm }}(x)={E}_{{\sigma }^{\pm }}^{{\rm{in}}}{t}_{{\sigma }^{\pm }}(x)\) are the transmitted σ^{±} components immediately behind the grating, \({t}_{{\sigma }^{\pm }}(x)=\)\(\exp (i{k}_{{\rm{p}}}{\chi }_{{\sigma }^{\pm }}d/2)\) are the amplitude transmission functions, the susceptibilities \({\chi }_{{\sigma }^{\pm }}\) are periodic functions of x, and \({\theta }_{{\sigma }^{\pm }}\) are the diffraction angles of σ^{±} components with respect to the z direction. Note that we here ignore the grating height (h = 1 mm) in the y direction because there is no grating period in such direction in the 1D case, which has very little influence on the diffraction in the x direction.
If there are \({M}_{{\sigma }^{+}}\) (an integer) spatial periods in the grating for the σ^{+} component (i.e., the grating width \(w={M}_{{\sigma }^{+}}{G}_{1}\)), the diffraction intensity based on Eq. (13) becomes
The quantity \({E}_{{\sigma }^{+}}^{{\rm{s}}}({\theta }_{{\sigma }^{+}})\) stands for the farfield diffraction amplitude stemming from one single grating period, which reads
where \({\tilde{c}}_{{\sigma }^{+}}\) is the normalization coefficient in the absence of modulation.
To characterize the diffraction ability of EIPG for the σ^{+} component, we calculate the diffraction intensity along the 1storder diffraction angle \({\theta }_{{\sigma }^{+}}^{{\rm{1st}}}=\arcsin ({\lambda }_{{\rm{p}}}/G1)\), which is
More importantly, to thoroughly evaluate the EIPG performance for polarization manipulation, we should also derive a series of related expressions for the σ^{−} component by replacing the subscript σ^{+} with σ^{−} and the grating period G1 with G2 in Eqs (14–16). In this way, one can directly compare the diffraction properties of EIPG for different polarization components.
Results
Parameter settings
Before embarking on detailed diffraction calculations, we should prepare some necessary parameters in our scheme. In experiment, cold atomic cloud with high optical depth can be prepared in a magnetooptical trap (MOT)^{36}. Thus, we assume the atomic number density n_{a} ≈ 10^{12}/cm^{3}. Considering the finite sizes of laser beams and atomic cloud, the spatial region for effective lightatom interactions is assumed to be roughly within a cubic range of 1 × 1 × 1 mm^{3} in the atomic cloud, which also characterizes the width w (=1 mm), height h (=1 mm), and thickness d (=1 mm) of the induced grating.
Because a “thin” grating instead of a “thick” (Bragg) grating is considered in our scheme, operating in the RamanNath limit, we adopt a criteria in optical holography to determine whether the grating is thin or thick^{16,17}. For the criteria, a dimensionless factor Q = 2πλd/(n_{ i }G) < 1 is required for a thin RamanNath grating, where λ is the wavelength of the incident light field, d is the thickness of the grating, n_{ i } is the refractive index of the medium, and G is the grating period. In our EIT system, we have λ = λ_{p} ≈ 795 nm, d = 1 mm, and n_{ i } ≈ 1, which lead to a restriction for the grating periods (i.e., {G_{1}, G_{2}} > 70.7 μm). Therefore, to satisfy this restriction in the following numerical calculations, we assume that the grating periods induced by the C1 and C2 quasiSW fields are G1 = 125 μm and G2 = 100 μm, respectively.
Moreover, as an example, we here choose the Zeeman sublevel 10S_{1/2}, F = 2, m_{ F } = +1〉 as the highest excited state 5〉. For the 10s level of ^{87}Rb, we can have the decay rate Γ_{5} = 2π × 0.37 MHz corresponding to the lifetime of about 430 ns, the hyperfine splitting \({{\rm{\Delta }}^{\prime\prime} }_{{\rm{HFS}}}=112.5\) MHz, and the wavelength of C2 field (the 3〉 ↔ 5〉 transition) λ_{C2} = 532.24 nm^{37,38}. For the parameters related to other energy levels, we have the wavelength of C1 field λ_{C1} = 795 nm (the 2〉 ↔ 4〉 transition), the decay rates Γ_{1} = Γ_{4} = 2π × 3 kHz, and Γ_{2} = Γ_{3} = Γ = 2π × 6 MHz^{31}. Also, the singlephoton detunings of the σ^{±} components satisfy the relationship \({{\rm{\Delta }}}_{{\sigma }^{}}={{\rm{\Delta }}}_{{\sigma }^{+}}+2{\mu }_{{\rm{B}}}{g}_{F}B\), where μ_{B} is the Bohr magneton, g_{ F } is the hyperfine Landé gfactor, μ_{B}g_{ F } = 0.23 MHz/G for the 5P_{1/2} state of ^{87}Rb, and B is the weak magnetic field in gauss^{31}.
Additionally, for simplicity, we here assume that the coupling fields are always resonant with the corresponding atomic transitions, i.e., Δ_{C1} = Δ_{C2} = 0. If we use a weak magnetic field (e.g., B = 10 mG) to set the quantization axis of atoms, the two states 2〉 and 3〉 can be regarded to be nearly degenerate when the energy splitting (2 μ_{B}g_{ F }B = 4.6 KHz) is much smaller than the EIT linewidth estimated by \({{\rm{\Omega }}}_{{\rm{C1}}}^{2}/{{\rm{\Gamma }}}_{2}\) and \({{\rm{\Omega }}}_{{\rm{C2}}}^{2}/{{\rm{\Gamma }}}_{3}\). Under such condition, the frequency detunings of the σ^{±} components are set to be approximately equal (i.e., \({{\rm{\Delta }}}_{{\sigma }^{+}}\) ≈ \({{\rm{\Delta }}}_{{\sigma }^{}}\)) in the numerical calculations. To characterize the periodic phase and amplitude modulations, we also define the phase modulation as \({\rm{\Phi }}={k}_{{\rm{p}}}{\rm{Re}}({\chi }_{{\sigma }^{\pm }})d/2\) and the intensity transmission rate as \(T=\exp [{k}_{{\rm{p}}}\,{\rm{Im}}({\chi }_{{\sigma }^{\pm }})d]\).
Numerical results for 1D EIPG
By varying the frequency detunings of the probe field or the intensities of the quasiSW coupling (C1 and C2) fields and setting different grating periods in the C1 and C2 fields, the susceptibilities of the EIT system can be strongly modified. Therefore, it is possible to independently control and spatially separate the σ^{±} components of the probe field. In Figs 2 and 3, it is clearly seen that the highorder diffraction modes of the σ^{±} components can be separated in space because they experience different grating periods [i.e., G1 = 125 μm for σ^{+} and G2 = 100 μm for σ^{−} in Eqs (9) and (10)]. To be specific, in Fig. 2, we fix the intensities of the coupling fields as Ω_{C1′} = Ω_{C2′} = 7.5Γ and Ω_{C1″} = Ω_{C2″} = 2.5Γ in Eqs (9) and (10). When the probe frequency detunings are small (i.e., \({{\rm{\Delta }}}_{{\sigma }^{+}}\approx {{\rm{\Delta }}}_{{\sigma }^{}}=0.4{\rm{\Gamma }}\)), the phase modulation strength ΔΦ = (Φ_{max} − Φ_{min}) is slightly greater than π (1.13π for σ^{−} and 1.14π for σ^{+}) and the intensity transmission is higher than 34% for σ^{−} and 72% for σ^{+} [Fig. 2(a,b)]. Therefore, the EIT system mainly works as a phase grating, which means that more incident energy is scattered into the higherorder diffraction modes [Fig. 2(c)]. When the probe frequency detunings are increased to \({{\rm{\Delta }}}_{{\sigma }^{+}}\approx {{\rm{\Delta }}}_{{\sigma }^{}}=0.8{\rm{\Gamma }}\), the phase modulation strength ΔΦ is increased to ~2.5π (2.49π for σ^{−} and 2.51π for σ^{+}) but the intensity transmission rate is greatly lowered where the minimum of T is 9.7% for σ^{−} and 23% for σ^{+} [Fig. 2(d,e)]. Therefore, the total diffraction efficiency is decreased due to high absorption. But, the intensity distributions of the higherorder diffraction modes are more uniform due to the strong phase modulation [Fig. 2(f)].
Figure 3 illustrates that the intensities of the quasiSW coupling fields can significantly modify the diffraction patterns when we fix the probe detunings as \({{\rm{\Delta }}}_{{\sigma }^{+}}\approx {{\rm{\Delta }}}_{{\sigma }^{}}=0.5{\rm{\Gamma }}\). To change the intensities of the quasiSW coupling fields, we fix the intensities of the C1″ and C2″ laser beams in the C1 and C2 fields as Ω_{C1″} = Ω_{C2″} = 2.5Γ, but tune the intensities of the C1′ and C2′ laser beams. When Ω_{C1′} = Ω_{C2′} = 3.5Γ, the transparency window of the EIT system is relatively narrow, which leads to strong phase modulation (ΔΦ > 20π for σ^{±}) as well as severe absorption [Fig. 3(a,b)]. Therefore, the total diffraction (transmission) efficiency is low (5.7% for the σ^{+} central principal maximum and 3.4% for the σ^{−} central principal maximum), where more intensity is diffracted into the higherorder modes [Fig. 3(c)]. As a comparison, when Ω_{C1′} = Ω_{C2′} = 9.5Γ, the transparency window of the EIT system is much wider for high quasiSW intensity. Accordingly, the phase modulation becomes weak (ΔΦ ≈ 0.64π for σ^{±}) but the transmission is high [Fig. 3(d,e)]. The central principal maximum is greatly enhanced to 58.9% for σ^{+} and 48.5% for σ^{−}, and the ±1storder maximum is 16% for σ^{+} and 13% for σ^{−} [Fig. 3(f)].
Because, in general, the intensity of the 1storder diffraction mode is important to evaluate the optical performance of a grating, we also demonstrate the intensity evolution with respect to the optical parameters of EIPG using Eqs (15) and (16). In Fig. 4, we investigate the 1storder diffraction intensity as a function of Ω_{C1′} (Ω_{C2′}) with increasing probe detunings, where we fix Ω_{C1″} = Ω_{C2″} = 2.5Γ. It is seen that the 1storder diffraction intensity can be enhanced to 24.8% for σ^{+} and 19.7% for σ^{−} with higher coupling intensity and larger probe detuning [see Fig. 4(c)]. Moreover, the 1storder diffraction intensity of the σ^{+} component is always higher than that of the σ^{−} component for the same EIT parameters. Such a feature originates from the fact that, in our scheme, the σ^{+} component is manipulated in a Λtype EIT subsystem, whereas the σ^{−} component is in a Ξtype EIT subsystem. Due to the higher decoherence rate in the Ξtype subsystem, its EIT efficiency is usually lower than that of the Λtype subsystem, thereby leading to lower diffraction efficiency. A similar tendency is also presented in Fig. 5. Higher coupling intensity and larger probe detuning can raise the 1storder diffraction efficiency to 24.6% for σ^{+} and 21% for σ^{−} [see Fig. 5(a–c)]. Because under such conditions, the EIPGs act more like a phase grating than an amplitude grating, where the absorption is suppressed.
Additionally, there is one point that should be emphasized concerning our numerical results in Figs 2 and 4, where we use the weakmagneticfield approximation and nearly degenerate Zeeman sublevels (i.e., \({{\rm{\Delta }}}_{{\sigma }^{+}}\) ≈ \({{\rm{\Delta }}}_{{\sigma }^{}}\)) in the calculations. However, the data in Figs 2 and 4 can also give us a hint to study the σ^{±} diffraction intensities with moderate magnetic field by comparing the corresponding curves between different subfigures. For example, we can make a comparison between the diffraction intensity of σ^{+} with \({{\rm{\Delta }}}_{{\sigma }^{+}}=0.4{\rm{\Gamma }}\) [red solid curve in Fig. 2(c)] and that of σ^{−} with \({{\rm{\Delta }}}_{{\sigma }^{}}=0.8{\rm{\Gamma }}\) [blue dotted curve in Fig. 2(f)], where the corresponding magnetic field is give by \(B=({{\rm{\Delta }}}_{{\sigma }^{}}{{\rm{\Delta }}}_{{\sigma }^{+}})/2{\mu }_{{\rm{B}}}{g}_{F}=10.4\) G. In Fig. 4, we can also compare the 1storder σ^{±} diffraction intensities for different frequency detunings, for example, the red solid curve in Fig. 4(a) for σ^{+} with \({{\rm{\Delta }}}_{{\sigma }^{+}}=0.2{\rm{\Gamma }}\) and the blue dotted curve in Fig. 4(c) for σ^{−} with \({{\rm{\Delta }}}_{{\sigma }^{}}=0.8{\rm{\Gamma }}\). In this case, the corresponding magnetic field is B = 15.6 G. Therefore, our results not only directly compare the σ^{±} diffraction under weak magnetic field, but also offer the opportunity to assess the influence of moderate magnetic field on the diffraction of the σ^{±} components. This fact also means that magnetic field can greatly increase the tunability of polarizationselective diffraction in our EIPG scheme.
Numerical results for 2D EIPG
Because the Ξtype and Λtype subsystems are totally independent of each other, we can change the grating orientation in one subsystem to diffract different polarization components in 2D. For example, we can generate two perpendicular grating structures in Fig. 1(c), where the quasiSW in the C1 field is along the y direction described by Eq. (11). Thus, the σ^{+} component is diffracted in the y direction, whereas the σ^{−} component is still diffracted in the x direction. Consequently, the σ^{±} components can be separated far apart in space, which may show more practicability and flexibility than the 1D case. To do the numerical calculations, we employ the EIT parameters in Figs 2(a–c) and 3(a–c) and Eqs (11) and (12) to perform 2D Fourier transform^{35}, where the grating width is w = 1 mm in the x direction and the grating height is h = 1 mm in the y direction [see Fig. 1(c)]. The farfield diffraction patterns are indicated in Fig. 6. Here, for brevity, the tedious analytical expressions for the 2D Fourier integral are omitted.
It is wellknown that 2D optical diffraction is a fundamental operation for multimode spatial information processing^{35,39}. Our 2D EIPG scheme could be combined with other models in atomic EIT systems to find potential applications in polarizationdependent beam splitting and fanning^{12,13,14}, image processing^{40,41,42}, and vortex manipulation^{43,44,45}, which may further improve the capacity and speed for parallel signal processing by alloptical means.
Discussion
Although a tripod system is usually employed for polarizationdependent photon manipulation^{20,21,22}, the polarizationdependent multibeam diffraction with our EIPG scheme in the RamanNath limit cannot be simply realized using a tripod EIT system. The basic challenge is that, in a tripod EIT system, one SW coupling field can only generate one single grating period for both circular polarization σ^{±} components of a linearly polarized probe field although the refractive index modulation strengths may be different. According to the scalar diffraction theory^{35,39}, the farfield diffraction angle of a thin (RamanNath) grating (regardless of an amplitude or phase grating) only depends on the ratio between incident wavelength and grating period. Therefore, for the same diffraction order, the σ^{±} components will have the same diffraction angle for the EIG generated in a tripod system. As a result, the transmitted σ^{±} components cannot be transversely separated in free space. Namely, for each high diffraction order, one can hardly achieve a pure σ^{+} or σ^{−} component. However, by constructing a fivelevel ΞΛ system in this work, the σ^{±} components can be fully separated in high diffraction orders and the separation distance can be flexibly adjusted by independently changing the grating periods G_{1} and G_{2} of the two quasiSW coupling fields.
Recently, 2D periodic structures have been investigated to broaden the applications of EIG systems^{11,19}. The function of our 2D EIPG is totally different from these already existing proposals. In ordinary 2D EIGs, the incident probe field directly experiences a 2D grating structure without polarization sensitivity and thus the farfield diffraction pattern is not polarizationresolved. As a comparison, in our 2D EIPG, the σ^{±} components of a linearly polarized probe field actually experience different 1D grating structures having perpendicular orientations, respectively. The farfield diffraction pattern is a spatial combination of two independently tunable patterns with different polarizations, which further enriches the diffraction phenomena of EIG systems. More importantly, the 2D EIPG case also clearly shows the advantage of our fivelevel ΞΛ system over the conventional tripod system because it is hard for the tripod system with one single coupling field to independently fan out different polarization components in 2D in the RamanNath limit.
To produce the quasiSWs in the C1 and C2 coupling fields, we propose the possible experimental setups in Fig. 1(b,c). We first assume that the probe field propagates in the z direction, and a weak magnetic field is also applied in the same direction to set the quantization axis of atoms. Two πpolarized laser beams (C1′ and C1″) with unequal intensity in the xy plane can interfere to form the quasiSW C1 field whose polarization direction is parallel to the z direction. By adjusting the misalignment angle ϕ_{1} between the two lasers, we can change the grating period G_{1} = λ_{C1}/[2 sin (ϕ_{1}/2)] in the C1 field where λ_{C1} is the wavelength of C1 field. Similarly, two σ^{+} polarized laser beams (C2′ and C2″) with unequal intensity in the xz plane can form the quasiSW C2 field. Thus, the grating period of C2 field is given by G_{2} = λ_{C2}/[2 sin (ϕ_{2}/2)], where ϕ_{2} is the misalignment angle and λ_{C2} is the wavelength of C2 field. Without loss of generality, we assume that the coordinate axes y and z in Fig. 1(b) [or, x and z in Fig. 1(c)] are the bisectors of the misalignment angles ϕ_{1} and ϕ_{2}, respectively. Thus, the orientation of the quasiSW C1 field is along the x direction in Fig. 1(b) and along the y direction in Fig. 1(c), while that of the C2 field is always along the x direction. Consequently, two sets of polarizationresolved grating structures could be independently created in 1D and 2D.
In Fig. 1(b,c), the misalignment angle ϕ_{1} (ϕ_{2}) between the C1′ and C1″ (C2′ and C2″) beams is actually very small to construct the EIPG in the RamanNath regime. For the grating period G1 = 125 μm in the C1 field (λ_{C1} = 795 nm), we have ϕ_{1} = 6.36 mrad. However, the situation for the C2 field is slightly complicated. For G2 = 100 μm in the C2 field (λ_{C2} = 532.24 nm), we have ϕ_{2} = 5.32 mrad. This also means that the σ^{+}polarized C2′ and C2″ beams are not strictly parallel to the magnetic field, which can give rise to other polarization components (e.g., σ^{−} and π) in the resulting quasiSW C2 field. One can analytically derive the expression of the quasiSW C2 field and find that the peak intensity ratio between the σ^{+}, σ^{−}, and πcomponents is given by {[1 + cos (ϕ_{2}/2)]/2}^{2} : {[1 – cos (ϕ_{2}/2)]/2}^{2} : [sin (ϕ_{2}/2)]^{2}/2 = 0.9999964622 : 3.129 × 10^{−12} : 3.538 × 10^{−6}, where ϕ_{2}/2 = 2.66 mrad is the angle between each C2 beam (i.e., C2′ or C2″) and the quantization (z) axis (the bisector of ϕ_{2}). Therefore, in our scheme, the σ^{+}polarized component plays a highly predominant role in the quasiSW C2 field and other polarization components can be ignored.
In practice, to suppress the influence of MOT on the EIT system in a cold atomic ensemble, the probe light field should be turned on after the trap is switched off. Therefore, the EIT measurement can be performed using a time sequence. Such a technique has been widely exploited in the EIG experiments with cold atoms^{5,7}, where the 1/e lifetime of the MOT after switching off the trapping beams is of the order of 2 ms^{7}. Thus, the duration of the probe light field could be a few hundreds of μs, which leads to the linewidth of the probe field less than 10 kHz. Such a linewidth is much smaller than the EIT linewidth in our work. For example, the narrowest EIT linewidth is given by the data in Fig. 4, where we have Ω_{C1′} = Ω_{C2′} = 3Γ and Ω_{C1″} = Ω_{C2″} = 2.5Γ at the leftmost side of each subfigure. Based on Eqs (9) and (10), we have Ω_{C1} = Ω_{C2} = 0.5Γ at the quasinodal position and the EIT linewidth is thus estimated by \({{\rm{\Omega }}}_{{\rm{C1}}}^{2}/{\rm{\Gamma }}={{\rm{\Omega }}}_{{\rm{C2}}}^{2}/{\rm{\Gamma }}=0.25{\rm{\Gamma }}=1.5\) MHz. Therefore, the probe field can be treated as nearly monochromatic and our theoretical model is still valid in the cold atoms.
To obtain the linear susceptibilities [i.e., Eqs (7) and (8)] and thus enable the decoupling between the σ^{±} components of the probe field, the total photon number in the probe field should be much smaller than the atom number in the interacting region to maintain the atomic population in the initial state 1〉. The total photon number can be given by \({{\mathscr{N}}}_{{\rm{ph}}}={I}_{{\rm{p}}}A\tau /(\hslash {\omega }_{{\rm{p}}})\), where \({I}_{{\rm{p}}}={\varepsilon }_{0}c{E}_{{\rm{p}}}^{2}/2\) is the intensity of the probe field, ε_{0} is the vacuum permittivity, c is the vacuum light speed, A is the area of the cross section of the interacting region, and τ is the duration time of the probe field. Using the parameters in the “Parameter settings” subsection, we have A = 1 × 1 mm^{2} and the atom number \({{\mathscr{N}}}_{{\rm{a}}}={n}_{{\rm{a}}}\times V={10}^{9}\) in the interacting region where n_{a} = 10^{12}/cm^{3} and V = 1 mm^{3}. Also, we assume τ ~ 100 μs based on the linewidth analysis in the above paragraph. Due to the limit of \({{\mathscr{N}}}_{{\rm{ph}}}\ll {{\mathscr{N}}}_{{\rm{a}}}\), the range of the total Rabi frequency of the probe field can be given by \({{\rm{\Omega }}}_{{\rm{p}}}={\mu }_{{\rm{D1}}}{E}_{{\rm{p}}}/\hslash \ll 0.277{\rm{\Gamma }}\), where μ_{D1}(=μ_{12} = μ_{13}) = 2.54 × 10^{−29} C · m is the dipole moment of the ^{87}Rb D1 transition (5S_{1/2} ↔ 5P_{1/2})^{31}. Such a result is also consistent with the condition \({{\rm{\Omega }}}_{{\sigma }^{\pm }}\) \(\ll \) Ω_{C1}, Ω_{C2} which is another important prerequisite to derive Eqs (7) and (8). Hence, in our ΞΛ EIT system, the probe σ^{±} components can be decoupled under appropriate conditions.
The low absorption and high diffraction efficiency of our EIPG scheme may also find potentials in quantum information processing. For example, when a single probe photon with linear polarization is incident, it is possible to obtain the polarization and space entangled multiple Fock states in the far field, such as \({{\rm{\Psi }}}_{{\rm{far}}}\rangle ={\sum }_{i=n}^{+n}\,{b}_{i}{0}_{{\sigma }^{+},{\theta }_{n}}\rangle \cdots {1}_{{\sigma }^{+},{\theta }_{i}}\rangle \cdots {0}_{{\sigma }^{+},{\theta }_{+n}}\rangle {0}_{{\sigma }^{},{\theta }_{m}}\rangle \cdots {0}_{{\sigma }^{},{\theta }_{+m}}\rangle +{\sum }_{j=m}^{m}\,{b}_{j}{0}_{{\sigma }^{+},{\theta }_{n}}\rangle \cdots {0}_{{\sigma }^{+},{\theta }_{+n}}\rangle {0}_{{\sigma }^{},{\theta }_{m}}\rangle \cdots \) \({1}_{{\sigma }^{},{\theta }_{j}}\rangle \cdots {0}_{{\sigma }^{},{\theta }_{+m}}\rangle \) where the subscripts (σ^{+}, θ_{ i }) and (σ^{−}, θ_{ j }) denote the ithorder diffraction angle of the σ^{+} component and the jthorder diffraction angle of the σ^{−} component, respectively. Therefore, our EIPG scheme using the polarization degree of freedom can increase the capacity of multichannel optical devices for quantum information processing.
Note that we choose the 10s level only as an example to design the EIPG. Actually, other lowly excited s levels (e.g., 6S_{1/2}) can also be used as the state 5〉 in our scheme. For highly excited s levels in Rydberg atoms, the dipoledipole interactions can generate strong optical nonlinearity^{46,47}, which is beyond our theoretical model [see Eq. (1)] and will be considered in future work.
In summary, we have studied the possibility of generating a polarizationresolved RamanNath grating in an EIT medium with ultracold ^{87}Rb atoms. Considering the multiple Zeeman sublevels in ^{87}Rb, we design a fivelevel ΞΛ system and the polarizationdependent optical susceptibilities of the system are theoretically derived. By adjusting the EIT parameters, including the probe frequency detunings and the quasiSW coupling light fields, we numerically calculate the farfield (Fraunhofer) diffraction distributions of a probe light field normally incident on the EIT medium. Our results show that the probe σ^{±} components can be decoupled and independently diffracted in 1D and 2D. Such fact means that the two polarization components can be flexibly and efficiently controlled based on EIPG systems, which may increase the channels and enhance the performance of alloptical devices and networks. This idea could also be extended to complicated optical structures, such as images and vortices, offering a versatile platform for polarizationselective spatial multimode information processing in EIT media.
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Acknowledgements
L.Z. thanks F. Peng and L.J. Wang for helpful discussions. This work was supported by National Natural Science Foundation of China (NSFC) (11204154 and 11574016).
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Zhao, L. Electromagnetically induced polarization grating. Sci Rep 8, 3073 (2018). https://doi.org/10.1038/s41598018214948
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DOI: https://doi.org/10.1038/s41598018214948
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