Muhammad Saeed and Muqaddar Abbas

1Center for Integrated Quantum Information Technologies(IOIT),School of Physics and Astronomyand State Key Laboratory of Advanced Optical Communication Systems and Networks,Shanghai Jiao Tong University,Shanghai 200240,China

2Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter(Ministry of Education),Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,School of Physics,Xi’an Jiaotong University,Xi’an 710049,China

Keywords: structure light,vortex beam

1.Introduction

Interference and atomic coherence are two basic phenomena that can influence spectrum features of a multilayer atomic system.In a broader sense,quantum interference occurs when there are two or more indistinguishable transition routes.[1-3]Normally opaque materials may become completely transparent to a probe beam via the quantum interference phenomena of electromagnetically induced transparency (EIT).The destructive interference across the probe absorption amplitudes causes such transparency.Bolleret al.initially demonstrated it experimentally in 1991,[4]EIT has resulted in increasing interests for scientific research.One of the notable consequences of EIT is the ability to slow down or completely halt light propagation many times.[5-9]

In recent years, researchers have examined that the average velocity of the group associated with a pulse of light can be reduced to 17 m/s in the Bose-Einstein condensate of sodium atom gas.[6]In literature, there has been a lot of research on EIT inV,Λ,along with cascade(ladder)atomic systems with three levels.[10-12]Comparatively minimal research in both theory and experiment concerning the Y-type atomic structure[10,13-15]has ever been carried out.Houet al.[10,13]inquired into how vacuum-induced coherence (VIC) affects the absorbance properties of Y-type atomic structures.The effect of VIC on absorption features of incoherent pumping field for Y-type four-level systems was studied.[14]Moreover, two photons absorbed in sodium vapor were suppressed by electromagnetic interference according to Gaoet al.[15]The effect for a single strong coupling field generates an EIT window inside the probe field’s absorption profile, allowing subluminal light passage, as shown by using three-level atomic systems.However, because of the extra (second) pumping field coupling with four-level atomic arrangement, an additional EIT window may exist as well as be altered as compared to a threelevel atomic structure.Several practical and theoretical studies have been conducted on such a double EIT system since it may improve non-linear optical effects when compared to single EIT structures.[16,17]

Light may have both spin angular momentum(SAM)and orbital angular momentum (OAM), with SAM related with light polarization and OAM associated with helical phase vortices eilΦ.In this case,lrepresents the topological charge,also known to be the winding number, along with the azimuthal quantum number.Allen and colleagues discovered the Laguerre-Gaussian (LG) modes, which are characteristic manifestations of light with OAM.[18]The LG beam’s pattern of intensity distribution is doughnut-shaped, exhibiting zero intensity at the center.Light having OAM (structured light)provides potential use in quantum information,[19]as well as optical communications,[20,21]optical trapping,[22,23]and optical tweezers[24]because of the intensity as well as phase structure’s specific features.Structured light interactions with atomic ensembles have been studied for a variety of applications by researchers.[25-30]Light beams with spatial structure may be employed in applications for optically sensing.In particular, the rotational Doppler effect may be used to measure the rotation of objects along with particles.[31,32]Additionally,structured light pulses having discrete phase patterns as well as intensity distributions are employed in imaging applications,such as fiber endoscopes[33]along with super-resolution imaging.[34]

EIT-based structured generation of light has been considered by employing closed three-level[35]as well as fourlevel atomic setups.[36,37]For the experimental demonstration of the vortex beam, researchers manipulated a structured probe beam with a transverse magnetic field in rubidium atoms.[36]A tripod along with Λ-type atomic setup has also been used to examine a structured light in a five-level atomic system.[38]Also, researchers considered the probing field including non-vortex, while the control field including OAM.[39-43]The phenomenon is most expected because an auxiliary field without a vortex is required for maintaining EIT when the driving fields include OAM since the optical vortex’s center has zero intensity.[44]Additionally,using twocomponent slow light[45]as well as four-wave mixing using a double-Λ configuration,[46]it has been suggested that optical vortices in a double tripod atomic structure might be transported from the control to the probe beam.Due to EIT’s success and importance in resonant atomic systems,practical and theoretical research into its use in semiconductor devices has been sparked.

In this work, we analyze a weak probing field, and two coherent coupling fields,alongside a four-level Y-type atomic system.We look at how the coupling field detuning as well as Rabi frequencies influences the probe field’s dispersion as well as absorption in such a system.We show spatially structure arising to absorption profile and how their presence can be explained by applying LG vortex to upper levels dressed state representations as well as their transitions to the lowest level.

2.Theory formulation

2.1.Y-type atomic system driven by a microwave field

Let us start by presenting the Y-type model system examined in the present study and shown in Fig.1(a).Phase profile of the LG beam withl=4 is shown in Fig.1(b).The structure of this model includes the ground level|0〉, a level that is intermediate|1〉, and two higher levels|2〉 and|3〉, which may be energy close or even(almost)degenerate.A weak and adjustable probe laser field associated with Rabi frequency?pcouples to the level|0〉 along with|1〉, whereas the level in middle|1〉connects to levels|2〉and|3〉via two distinct tunable coupling fields along with the Rabi frequencies?1and?2,respectively.

With the exception of the bottom level, each level will deteriorate by spontaneous photon emission at the showed decay rates, which areγ1,γ2, andγ3.As usual, we assume that because of their parities as well as total angular momenta, the higher levels|2〉 and|3〉 only decay to the intermediate level|1〉.This kind of Y-type model system can be approximated,for instance,in rubidium vapor through recognizing the (4p65s)2S1/2neutral rubidium at ground level using|0〉, the (4p65p)2P3/2with level|1〉 as well as the two(4p65d)2D3/2,5/2levels correspond to excited levels|2〉 and|3〉.In an interaction diagram, the total Hamiltonian for a Ytype system influenced by a microwave field reads

Fig.1.(a) Diagrammatic illustration of a four-level closed Y-type atomic system. ?p, ?1, ?2 and ?B represent the Rabi frequencies associated with the probe,LG,control and microwave fields.(b)Phase profile of the LG beam with l=4.

The detunings are defined as?p=ω10-ωp,?1=ω31-ω1, and?2=ω21-ω2.Employing the Liouville equation,we can derive the subsequent rate equations over the density matrix constituents as follows:

whereΓ10=i?p-γ1,Γ20=i(?p+?1)-γ2,andΓ30=i(?p+?2)-γ3.Also,γi(i=1,2,3)are actually the rates of spontaneous decay from an excited state|1〉pertaining to the ground state|0〉, also|2〉 to|1〉, along with|3〉 to|1〉 as shown in Fig.1(a).To resolve Eq.(2), we assume perturbation theory under a steady-state condition.In the subsequent calculations,we maintain the terms of the density-matrix equations up to the first order.When the field is inadequate enough, only the first-order term matters.We solve Eq.(2)considering ˙ρij=0 and obtainas follows:

2.2.Control vortex fields

The light beams are LG doughnut type beams.In terms of Rabi-frequencies,these LG beams are defined as

2.3.Azimuthal modulation for absorption of the probe with one control beam as a structure beam

which denotes standard EIT.Thus, to keep EIT even at the center of a vortex beam, we think of one control beam as not being a vortex.

Next,we look into the probe field’s absorption pattern for different winding numbersl1.In Figs.3(a)-3(d), we can see how the absorption profile changes in various locations when one of the control fields is used as a structure beam for (a)l1= 1, (b)l1= 2, (c)l1= 3, and (d)l1= 4.We can see the spectrum of absorption established against the standardized valuesx/wandy/w.We see shapes that look like petals,and the number of those petal shapes increases as the winding number enhances.The bright spots show where light is absorbed, and the dark show where light can transmit through.Based on the above results, it seems that the OAMl1affects how the probe field’s absorption profile changes.

Fig.2.(a) Pattern of absorption versus azimuthal angle Φ, (b) intensity pattern versus azimuthal angle Φ.Here l1 =4, ?p =0.1γ, and?B=0.1γ.

Fig.3.Pattern of absorption versus x/w and y/w for (a) l1 =1, (b)l1=2,(c)l1=3,and(d)l1=4.The rest of parameters are γ=1 MHz,γ1=0.5γ,γ2=0.01γ,γ3=0.01γ,?m=0.1γ,?p=0.1γ,?2=0.5γ,and ?p=?1=?2=0.

We did not take into account the Doppler effect when we used density matrix theory to figure out the physical response of the medium.Measurements were carried out by considering87Rb atoms in the magneto-optical trap (MOT) that is used to trap and cool atoms in this kind of setting.To decrease the Doppler broadening effect,the counter-propagating beam method can be used to analyze hot atomic media.[47]On the other hand, using cold atoms is the best way to avoid the Doppler effect.[48]

2.4.Transfer of optical vortices by considering both control fields as vortex beam

In this subsection,we show how optical vortices may be transferred through the control field to that of the probe field.We now assume that both control fields contain vortex beams as given in Eq.(4).The probing field’s time-independent transmission equation under slowly changing envelop assumptions may be given as

asαpsignifies the optical depth for the probe field,andLrepresents the atomic vapor cell length.A transmitted probing field may be expressed using Eq.(4)as

We examine control fieldsE1andE2, which both carry the OAM and have corresponding Rabi frequencies?1and?2with vorticesl1andl2.According to Eq.(7) OAMl1(l2) of pump field?1(?2) has been transferred via the transmitting probing field.At the atomic medium’s entrance,the incoming probe field does not impart any vortex.The vortex of the control beams transfers to the probe beam after some distance due to the atomic system’s closed-loop construction.We depict the intensity pattern for the transmitting probe field atz=Lin Figs.4(a)-4(d),forl1=l2=1,2,3,4 as well asαp=100.The intensity patterns depict a doughnut shape with a black zone in the middle displaying zero magnitudes.The central dark zone broadens as the winding number grows, while the associated phases change from 0 to 2πl(wèi)within the singularity point.

Fig.4.Intensity pattern versus x/w and y/w for (a) l1 =l2 =1, (b)l1 =l2 =2, (c) l1 =l2 =3, and (d) l1 =l2 =4.The rest of parameters are γ =1 MHz, γ1 =0.5γ, γ2 =0.01γ, γ3 =0.01γ, ?m =0.1γ,?p=0.1γ,?2=0.5γ,and ?p=?1=?2=0.

In Fig.5,we plot intensity versus radial distancer/wfor winding numbers(a)l1=l2=1,(b)l1=l2=2,(c)l1=l2=3,and (d)l1=l2= 4.The degree of intensity is highest atr=0.8wand zero atr>1.2wandr<0.4wforl1=l2=1 as well asl1=l2=2, as shown in Figs.5(a)-5(d).In order to acquire a physical understanding for this phenomenon, we first investigate the case in which?2is not present.

Our original Y-type system is reduced to a ladder-type three-level configuration design in this scenario.In the presence of?1,the transmission of?prises owing to EIT.Whenever?2is present,however,EIT becomes of interference and probe field absorption rises.As a result,?1pumps the population to the excited state|2〉.The magnetization field?mgenerates nonlinearity within the population, causing the intensity of its absorption profile to become spatially dependent at a particular radial distance.Assuming that?1and?2are vortex fields with decreasing intensities when approached byr>1.2wandr<0.4w, respectively, see Figs.5(a) and 5(b),the dark state formed in the presence of?1is disrupted and the conversion intensity drops.The same is true forl1=l2=3 andl1=l2=4,as illustrated in Figs.5(c)and 5(d).

Fig.5.The intensity of the spectrum of absorption against the radial distance r/w for(a)l1=l2=1,(b)l1=l2=2,(c)l1=l2=3,and(d)l1=l2=4.The remaining parameters remain same as in Fig.4.

3.Experimental realization of our proposed system

Our proposed method may be examined experimentally for the generation of structure light employing cold87Rb atoms through probe absorption[49,50]using the transitions 5S1/2-5P3/2-5D5/2,3/2.Following states may be designated as:|0〉=|5S1/2,F=2〉,|1〉=|5P3/2,F=3〉,|2〉=|5D5/2,F=4〉, and|3〉=|5D3/2,F=3〉.In the present situation, the decay rates of the excited along with intermediate states areγ1=0.5 MHz as well asγ2=γ3=0.01 MHz.In realistic investigations,?1couples with atomic transitions in an approximate wavelength of 776.2 nm,whereas?2couples in a wavelength of 775.8 nm.All of them can be generated using diode lasers.Here,the control field?1interacts via transition|5P3/2,F=3〉←→|5D5/2,F=4〉and?2interacts alongside the transition|5P3/2,F=3〉←→|5D3/2,F=3〉.Next we consider one of the control field as vortex beam that is generated by spatial light modulator,[51]and examine the generation of structured light for probe absorption.Further,we consider both control vortex beams and see the intensity profile of the generated structure beam.In each scenario, the weak probe field interacts via transition|5S1/2,F=2〉←→|5P3/2,F=3〉and hyperfine transition|5D5/2,F= 4〉←→|5D3/2,F= 3〉associated through a microwave-driven field with frequency that is around 100 GHz.We expect that our proposed system can be realized experimentally by considering the abovementioned parameters and approach.

4.Conclusion

In summary, we have explored spatial transparency in a Y-type atomic setup with two pump fields, a probe, as well as a microwave field.A vortex-based beam is utilized to periodically modulate the absorbing profile within its azimuthal region.This periodic oscillation is the main reason for structured light generation.The absorption profile exhibits 2πl(wèi)-fold symmetry, along with the OAM coupled with a vortex-type beam in a petal-like shape that varies with increasing winding number identified.It is also feasible to transfer the OAM from controlling the vortices beam to the transmitted probing beam.By considering both controlling fields as vortex beams,we see the intensity of probe field that has a doughnut shape.The size of the dark region increases with the increase of winding number.Lastly, we see the radial distance effect on the intensity of probe field.The intensity increases for a certain radial distance and then decreases as the dark state is created.Our suggested approach may pave the way for further research of applications in telecommunications and optical information processing.