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Abstract
We numerically investigate the transfer of optical information from a vector-vortex control beam to an unstructured probe beam, as mediated by an atomic vapour. The right and left circular components of these beams drive the atomic transitions of a double-V system, with the atoms acting as a spatially varying circular birefringent medium. Modeling the propagation of the light fields, we find that, for short distances, the vectorial light structure is transferred from the control field to the probe. However, for larger propagation lengths, diffraction causes the circular components of the probe field to spatially separate. We model this system for the D1 line of cold rubidium atoms and demonstrate that four wave mixing can lead to correlations between the optical polarization structure and the diffraction of light, generating coupled dynamics of the internal and external degrees of freedom.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past few decades, the interaction of coherent light with multilevel atomic systems has revealed many unexpected phenomena in nonlinear and quantum optics [1–4]. Of particular importance, was the discovery of electromagnetically induced transparency (EIT) [1], where the optical response of an atomic transition is dramatically altered by additional coupling to a third state. EIT is essentially the result of destructive quantum interference between these transitions, such that absorption is severely reduced for a given range of probe frequencies and eliminated entirely on resonance. Such effects are also accompanied by a sharp change in refractive index, yielding applications in all-optical communication [5], superluminal and ultraslow light [6,7], light storage [8], giant optical nonlinearities [9], optical bistability [10] and multi-wave mixing [11].
Following soon after EIT, was the discovery that light can carry orbital angular momentum (OAM). By imparting an azimuthal phase dependence around a Gaussian beam’s axis of propagation, we can ensure that the beam’s energy winds around the axis in a helical fashion. The number of windings per wavelength then provides an additional degree of freedom in manipulating optical information, but comes at the cost of dislocations in the phase profile. With strong meteorological analogy, this helical field propagation, and the resulting annular intensity profile, is known as a vortex beam [12].
Since they were first conceived and demonstrated [13,14], OAM-laden beams have found many applications: in communication [15], field manipulation [16], quantum information [17], optical tweezers [18] and, lately, a new type of compass [19]. This latter work was based on the marriage of EIT and OAM effects demonstrated by Radwell et al. [20]: where coherent interaction of optical OAM and atoms allowed for spatially dependent EIT. Central to this system, was the generation of a vector probe beam with azimuthally varying phase and polarization. Compared to scalar vortex light, a vector vortex beam has a singularity in its polarization structure, with an azimuthal gradient in the polarization direction, which has potential applications in the transmission, storage and manipulation of high-dimensional quantum information [21,22]. Further theoretical and experimental work has also demonstrated the transfer of OAM via atomic and molecular systems in the context of quantum technology [23–27].
In this paper, we analyze the co-propagation of a vectorial control and a homogeneously polarized probe beam through an extended interaction region. We model the atomic dynamics in terms of optical Bloch equations and the dynamics of the light fields via split-step propagation methods, and analyze the resulting optical fields in terms of their evolving intensity and polarization profiles on propagation.
In our simulations we find a markedly different behavior depending on the length of the interaction region: For short interaction lengths we show theoretically that vectorial light structure can be transferred from the control beam to the probe beam. For longer interaction regions, however, the probe beam with its imprinted vector structure separates into its left and right circular polarization components, which experience different diffraction, leading to a splitting of the probe beam intensity. We explain this by a spatially varying circuclar birefringence of the atomic medium, induced by the vectorial control beam.
2. Theoretical model and formulation
We consider classical control and probe beams co-propagating through a dilute atomic gas along the positive z-direction. The orthogona,l circularly polarized, components of each laser beam couple $V$-type transitions, forming a four-level double-$V$ system (Fig. 1), which has already proved to be useful in controlling the rotation of polarization [28].
Fig. 1. A schematic diagram of the closed, four-level double-$V$ system. Here, $|1\rangle$ and $|2\rangle$ are two ground states with magnetic quantum number $m_{\textrm{F}}=0$. The upper levels, $|3\rangle$ and $|4\rangle$, are magnetic sublevels of the same excited hyperfine state, with $m_{\textrm{F}}=-1$ and $+1$ respectively. The right (left) polarization component of the probe couples $|1\rangle$ $\to$ $|3\rangle$ ($|4\rangle$) with a Rabi frequency of $\Omega _{\textrm{P13}}$ ($\Omega _{\textrm{P14}}$). Similarly, the right (left) polarization component of the control couples $|2\rangle$ $\to$ $|3\rangle$ ($|4\rangle$) with a Rabi frequency of $\Omega _{\textrm{C23}}$ ($\Omega _{\textrm{C24}}$). We denote with $\delta$ and $\Delta$ the one-photon detuning and two-photon detuning, respectively, and assuming a magnetic field, the upper states are further shifted by the Zeeman effect $\pm \Delta _{\textrm{B}}$. The insets show examples of polarization profiles of the control and probe.
Such a system can be realized experimentally with $^{87}$Rb: where the atoms are approximated as a non-interacting gas and where for the two ground states, $|1\rangle =|5^{2}S_{1/2},F=1,m_{\textrm{F}}=0\rangle$ and $|2\rangle =|5^{2}S_{1/2},F=2,m_{\textrm{F}}=0\rangle$, and for upper states, $|3\rangle =|5^{2}P_{1/2},F=1,m_{\textrm{F}}=-1\rangle$ and $|4\rangle =|5^{2}P_{1/2},F=1,m_{\textrm{F}}=1\rangle$. A static magnetic field, $\vec {B}=B\hat {z}$, then changes the energy levels by 2$\hbar \Delta _{\textrm{B}}$, where $\Delta _{\textrm{B}}=g_{\textrm{F}}\mu _{\textrm{B}} B/ \hbar$ and $g_{\textrm{F}}$ and $\mu _{\textrm{B}}$ denote the Landé factor and Bohr magneton, respectively. The atoms interact with a strong vectorial control field, $\vec {E}_{\textrm{C}}$, that has a spatially varying polarization profile, and a weak, homogeneously polarized, probe field, $\vec {E}_{\textrm{P}}$. We express both laser fields in terms of their circular polarization components, with the right and left handed polarization components coupling to $|3\rangle$ and $|4\rangle$.
The electric fields of the control and probe beams are given by
The light-matter interaction is then characterized by the Rabi frequencies
Note that the transition paths $|3\rangle \to |2\rangle \to |4\rangle \to |1\rangle \to |3\rangle$ constitute a closed-loop double-$V$ coherent coupling system.
Under the rotating-wave approximation, the Hamiltonian of the atoms interacting with the control and probe fields can be expressed as:
The atomic dynamics are described by the Liouville equation:
The atomic susceptibilities, $\chi _{1j}$, denote the response of the atomic medium to the electric pump field, with the (slowly varying) atomic polarization given by $\vec {P}_{1j}=\epsilon _0 \chi _{1j} \vec {E}+{\textrm{P}}.$ This macroscopic polarization is related to the induced dipole moment on the atomic level, as $\vec {P}_{1j}=\mathcal {N} \langle \vec {d}_{1j} \rho _{1j}\rangle,$ where $\mathcal {N}$ and $\epsilon _{0}$ are the atomic density and vacuum permittivity, respectively.
In order to derive the atomic susceptibilities, we therefore have to indentify the steady state solution of the corresponding optical Bloch equations Eq. (8). We assume that our probe field (but not the control (field) is weak enough to be treated as a perturbation, with $\{ \Omega _{\textrm{P13}} / \gamma,\, \Omega _{\textrm{P14}}/\gamma \} \ll 1$. This allows us to expand the density matrix elements to first order in the probe field Rabi frequencies and to neglect all higher order terms of $\Omega _{\textrm{P}}$:
where $\rho ^{(0)}_{ij}$ is the zeroth-order and $\rho ^{(1)}_{ij}$ the first-order solution in the probe field $\Omega _{P}$. Substituting Eq. (9) into Eq. (8), the equations of motion for the first-order density-matrix elements, $\rho _{1j}$, can be expressed asWe can assume that the atomic population is initially distributed between the ground states $|1\rangle$ and $|2\rangle$, so that to zeroth-order, $\rho _{11}^{(0)}+\rho _{22}^{(0)}=1$, with all other elements $\rho _{ij}^{(0)}=0$. Ignoring higher-order components, the steady state solutions are then
In the following section, we illustrate this for the example of a fundamental Gaussian probe beam and a control beam that is composed of different Laguerre-Gaussian modes in its circular polarization components. According to Eq. (2), the Rabi frequencies are proportional to the electric field envelopes, which now take the form
The OAM carried in the control beam then plays an important role in subsequent formation of structured light. Following the simplification above, the final term in the numerator of the susceptibility is $\Omega _{\textrm{C23}}\Omega _{\textrm{C24}}^{\ast }\propto \exp [i (l_{\textrm{C}23}-l_{\textrm{C}24})\phi ]$ for $\chi _{13}$ and its complex conjugate for $\chi _{14}$, which can be interpreted as quantum interference. The difference in OAM between the two orthognal polarization components of the control beam (manifest as a spatially varying polarization) therefore leads to a spatially varying atomic susceptibility which in turn modifies the orthogonal polarization components of the probe beam upon propagation.
In order to study this effect of the control field on the probe beam, we evaluate Maxwell’s wave equations to describe the dynamic behavior of the probe field. We assume slowly varying envelopes and work within the paraxial wave approximation. The dynamics along the z-direction can then be expressed by
3. Results and discussions
We have seen that the induced susceptibility, Eq. (12), plays a vital role in describing the properties of the atomic medium. In this section we model the susceptibilities and resulting intensity and polarization profiles of the probe beam for various examples of control beam structures.
To be specific, we consider a propagation distance of up to $L=7$mm (given by the length of the atomic sample), with a beam waist of $w_{\textrm{P}}=30\mu$m, and probe wavelength of $\lambda _{\textrm{P}}=795$nm, such that $z_{\textrm{R}}=\pi w^{2}_{\textrm{P}}/\lambda _{\textrm{P}}=3.56$ mm, i.e. in a regime where diffraction effects become important. We assume a $45\times 45\mu$m span of the atomic cloud.
Figure 2 shows the imaginary and real parts of the susceptibilities. Here, each polarization component of the control beam is equally intense, $\Omega _{\textrm{C}230}=\Omega _{\textrm{C}240}$, but carries opposite topological charges: $\pm 1, \pm 2$ and $\pm 3$ in rows (a) to (c). The first and second columns are the imaginary parts of $\chi _{13}$ and $\chi _{14}$, while the third and forth columns are the real parts, corresponding to the absorption and dispersion of the two circular polarization components of the probe field, respectively. The spatially dependent absorption and dispersion show petal-like patterns with petal number $2|l_{\textrm{C23}}-l_{\textrm{C24}}|$. This is a direct consequence of the factor $\exp [\pm i(l_{\textrm{C23}}-l_{\textrm{C24}})\phi ]$ in Eqs. (11–13), which at specific azimuthal positions renders an opaque medium transparent, so that the orthogonal components of the probe field can pass through the medium, almost without loss. Therefore, the periodic oscillations in the susceptibility are responsible for the creation of structured light. We notice that the probe components’ absorption profiles are identical, whereas their dispersion profiles are mirror opposites.
Fig. 2. Linear susceptibility for control beams with different polarization structures. The first and second columns are the imaginary parts of $\chi _{13}$ and $\chi _{14}$, and the third and forth columns are the corresponding real parts. The different rows show the effect of control beams with different OAM: (a) $l_{\textrm{C23}}=-l_{\textrm{C24}}=1$, (b) $l_{\textrm{C23}}=-l_{\textrm{C24}}=2$, (c) $l_{\textrm{C23}}=-l_{\textrm{C24}}=3$. In order to satisfy the perturbation condition, we choose $\mathcal {N}=2\times 10^{10}$ atoms/cm$^{3}$, $L=7$mm, $w_{\textrm{P}}=30\mu$m, $l_{\textrm{P13}}=l_{\textrm{P14}}=0$, and $p=0$ for all control and probe beams, $\Omega _{\textrm{P130}}=\Omega _{\textrm{P140}}=0.01\gamma$, $\Omega _{\textrm{C230}}=\Omega _{\textrm{C240}}=\gamma$, $\delta =\Delta =0$, $\Delta _{\textrm{B}}=0.01\gamma$, $\gamma _{\textrm{C}}=10^{-7}\gamma$, where $\gamma$=36.1285MHz.
We then used the symmetric split-step Fourier method to calculate the intensity of the two orthogonal probe components in different transverse planes. Figure 3 shows the resulting superimposed intensities, $I_{\textrm{out}}\propto |\Omega _{\textrm{P13}}|^{2} + |\Omega _{\textrm{P14}}|^{2}$, for a control beam with topological charges $l_{\textrm{C23}}=-2$ and $l_{\textrm{C24}}=2$, as discussed in row (b) of Fig. 2. At short propagation distances we observe a four-fold petal structure, however, as the transmission distance increases each petal splits into two parts.
Fig. 3. Normalized transmission intensity distribution of the probe beam on different transverse planes with $l_{\textrm{C23}}=-2$, $l_{\textrm{C24}}=2$, corresponding to the linear susceptibility in Fig. 2(b). Other parameters are the same as those in Fig. 2.
The four-fold pattern can be explained by considering the absorption profile, which according to Eq. (12) and Fig. 2(b1) and (b2) has the same effect on the two circular polarization components of the probe field. To explain the splitting phenomenon however, we must consider dispersion, which differs for the two circular polarizations as shown in Fig. 2(b3) and (b4). This becomes apparent when we depict the individual transmitted intensity distributions of the circular components of the probe field at $z=7$mm in Fig. 4. The different dispersion profiles result in differing refractive indices, $n=\sqrt {1+{\textrm{Re}}(\chi )}\approx 1+{\textrm{Re}}(\chi )/2$, which must lead to differing diffraction. The medium can be regarded as an azimuthally varying diffractive element such that the components split and interfere spatially. With increasing propagation distance, especially under the condition of $z>z_{\textrm{R}}$, the effect is more apparent, as shown in Fig. 3. The resulting profiles are thus clearly affected by both the absorption and dispersion.
Fig. 4. Normalized transmission intensity distribution of the probe beam at $z=7$mm. (a) right-handed circular polarized component; (b) left-handed circular polarized component. Parameters are the same as those in Fig. 3
In order to give a more comprehensive description of the transmitted probe field, here we investigate the dynamic polarization. Suppose the probe is initially incident at $z=0$ and linearly polarized in the x-direction. In Fig. 5, we then see that as the propagation distance increases, the polarization of the probe field experiences spatially dependent rotation, which dynamically varies along both angular and radial coordinates. The probe field is then vectorized.
Fig. 5. Analysis of the dynamic polarization of the probe field, with the polarization of control field shown in bottom-right corner. Other parameters are the same with Fig. 3.
We have seen from Fig. 2 that the different topological charges of the control field components, $\Omega _{\textrm{C}23}$ and $\Omega _{\textrm{C}24}$, induce different refractive indices for each polarization component of the probe light: suggesting light-induced, spatially dependent, circular birefringence. The atomic medium thus acts as an optical rotation channel that structures the probe light on interaction. One may expect a Faraday rotation of $\theta \propto z[Re(\chi _{14})-Re(\chi _{13})]$. Figure 5 shows that for our atomic medium, this linear increase of rotation angle with distance holds only for very short propagation distances, and moreover is not uniform across the beam diameter. For larger distances, the change of the rotation angle, $\theta$, is decreasing, and the linear input polarization of the probe beam gradually separates into areas of right and left circular polarisation. One may be tempted to interpret the emerging increase in ellipticity as an effect of circular dichroism, i.e. a differential absorption of left- and right-handed light, but a more accurate interpretation is a differential diffraction of left- and right-handed light. At larger propagation distances, this diffraction becomes too large to accurately ‘track’ the polarization rotation, an effect that becomes more pronounced for higher-order structured light [31].
To get a better understanding of how optical rotation behaves with propagation, we plot the rotation angle and ellipticity of the probe beam as a function of z (Fig. 6.). Taking advantage of the symmetry, we consider a region just above the horizontal, for $y=0.1w_{\textrm{P}}$, $0 \leq x \leq 1.5 w_{\textrm{P}}$, and $0 \leq z \leq 2 z_{\textrm{R}}$, as indicated by the red line in the first panel of Fig. 5. According to Fig. 6(a), our simulation shows regions of clockwise (CW) and counter-clockwise (CCW) rotation, indicated in blue and red, respectively. The darker the color, the greater the rotation angle, $\theta$. However, the CCW rotation occurs only close to the optical axis for $|x|<0.1 w_{\textrm{P}}$. In this region, the intensity of the probe is extremely low as it is close to the singularity of the optical vortex, and it decreases even further due to the divergence, so that the CCW rotation becomes meaningless. In the other areas of the probe beam, optical rotation, $\theta$, varies with the transverse position, $x$, as well as propagation distance, $z$. The overall trend shows a rotation angle that decreases with distance from the optical axis and increases with propagation distance but saturates for distances larger than the Rayleigh length: $z=4$mm and above in Fig. 5.
Fig. 6. (a) The rotation angle $\theta$ and (b) ellipticity $P$, as a function of transverse $x/w_{\textrm{P}}$ and longitudinal $z/z_{\textrm{R}}$ position, respectively. The darker blue (red) correspond to a larger the rotation angle $\theta$ in Fig. 6(a), and ellipticity $P$ from 0 (linear polarization) to 1 (circular polarization), other parameters are the same as those in Fig. 3.
In addition to a rotation of linear polarization, or more generally the major axis of the polarization ellipse, also the degree of ellipticity of the probe itself can change, as evident in Fig. 5. The homogeneous, linear, polarization of the probe beam spatially separates into areas of predominantly right or left polarization for propagation distances beyond the Rayleigh length. As discussed before, the effect is reminiscent of circular dichroism, however its origin lies in circuar ‘bi-diffraction’.
The ellipticity ($P$) of the polarization of the probe light can be quantified as $P=(|\Omega _{\textrm{P14}}|-|\Omega _{\textrm{P13}}|)/(|\Omega _{\textrm{P14}}|+|\Omega _{\textrm{P13}}|)$, and its evolution is shown in Fig. 6(b). We see a similar spatial behavior of the ellipticity to that of the rotation angle in Fig. 6(a). This provides further evidence that diffraction plays an important role in the evolution of the ellipticity. At short propagation distances, $z<\frac {z_{\textrm{R}}}{4}<1$mm, the two circular components of the probe have not separated yet, so that ellipticity is negligible, while at larger distances, the separation of the two circular components, and hence the ellipticity, increases.
Here we have reported the simplest cases of control beams with identical intensity profiles, but opposite topological charges. More generally, e.g. for control beams composed of LG beams with different $|l|$ values, or different waists, the linear susceptibilities can take more complicated shapes, and in general the absorption of right and left circular light will differ. In this case, the medium will induce spatially varying circular dichroism as well as birefringence and diffraction.
4. Conclusion
In summary, we have shown, by numerical solution of propagation and optical Bloch equations, how polarization information can be transferred from a strong control to a weak probe beam in a double-$V$ atomic level structure. This initially manifests as the generation of a polarization pattern in the probe light. Although not immediately apparent in the intensity profile, the spatial optical coherence is revealed on propagation, where novel patterns expose the internal interaction. We interpret this in the context of differential diffraction or right and left polarized light, where each optical component evolves independently. We may also consider the effect from the perspective of the optical fields: the right and left circular polarization components of the control light do not interfere, but instead add to a spatially varying polarization structure across the beam profile. The atomic medium however enables quantum interference due to the nonlinear processes, generating a spatially varying susceptibility. A probe beam interacting with the atomic medium will be modified due to this light-induced susceptibility, affecting the differential phase, amplitude and propagation direction of the two circular polarization components. Throughout, we show that the susceptibility induced by a polarization structured control field is the source of these subsequent phenomena. As such, we have demonstrated the generation of controllable spatially dependent birefringence and diffraction.
Funding
National Natural Science Foundation of China (62075130); HORIZON EUROPE Marie Sklodowska-Curie Actions (721465); Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (2017-1.2.1-NKP-2017-00001); China Scholarship Council (201906280228).
Acknowledgments
SF-A acknowledge financial support from the European Training Network ColOpt, funded by the European Union Horizon 2020 program under the Marie Sklodowska-Curie Action. TWC acknowledges support by the National Research, Development and Innovation Office of Hungary (NKFIH) within the Quantum Technology National Excellence Program. JW acknowledges support by the China Scholarship Council (CSC). CF acknowledges support for the visiting reseach in Optics Group, Glasgow University, and is grateful to Sonja Franke-Arnold, Thomas W. Clark, and Jinwen Wang for the useful discussions.
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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