Interplay of polarizations in a cascade EIT system in the presence of vortex coupling light in 87Rb atomic vapor medium

We investigate both experimentally and theoretically the cascade electromagnetically induced transparency (EIT) in 5S1/2−5P3/2−5D5/2 configuration in 87Rb atomic vapor medium in the presence of vortex coupling Laguerre Gaussian (LG) light. We demonstrated Doppler free double resonance EIT structure in cascade configuration and observed transmission spectra for |H⟩−|H⟩ and σ+−σ+ polarized probe and coupling lights. We demonstrate that the double resonance structure can be identified by two photon transition probabilities and it is found to be F′′=2 and F′′=3 . By considering coupling LG light, several polarization combinations are taken into account, and their effects on amplifying and diminishing the EIT resonances are demonstrated experimentally. In order to understand the polarization effects of vortex coupling light on EIT spectrum in a degenerate multilevel atomic system, a theoretical model is developed by considering a simplified double hut level structure. Semi-classical density matrix analysis is used to understand the dynamics and also to establish the enhancement and reduction of EIT peak height with coupling light polarizations. The impact of two photon transition probabilities, polarization combinations and relative orientations of probe and coupling lights in degenerate multilevel atomic systems leads to modification of EIT resonances significantly. We establish quantitative agreement between our theory and experimental results.


Introduction
In the quest of coherent phenomena, manipulation of atoms using laser light has led to the exciting unusual phenomena such as coherent population trapping (CPT) [1], electromagnetically induced transparency (EIT) [2] and electromagnetically induced absorption [3]. Growing interest in the field of quantum memory [4], quantum information processing [5] and quantum metrology [6] has paved the way to explore the phenomenon of EIT to deeper extent. A destruction in the interference of transition pathways leading to impetuous change in the optical properties of a medium due to a weak probe light and strong coupling light, accompanying a transparency at resonance is said to be the phenomenon of EIT [2,7]. Lasing without inversion [8], four wave mixing [9], slow light [10] are the applications emerging out of EIT. Harris et al, pioneered EIT in the Λ configuration [2] and Banacloche and co-authors made a seminal contribution in cascade type configuration [11]. Considerable interest has grown for exploring cascade EIT configuration for its potential application in Rydberg states [12], atomic filters [13] and metrology [6].
Cascade EIT configuration has been explored thoroughly in the literature. Two photon effects on cascade EIT and its application in the high resolution spectroscopy is established [14]. Effect of strong probe in cascade system give rise to enhanced probe absorption [15]. The effect of probe and coupling polarization on cascade EIT spectra has been demonstrated by McGloin et al, and also showed that change in EIT spectra is due to degenerate Zeeman levels and their relative strengths [16]. Moon and co-authors put forth the idea that two photon transition leads to DROP effect in the cascade configuration and its utility for frequency stabilization [17]. In relation to above, authors also establish the effects of DROP on EIT spectra and emphasizes the effect caused by combination of probe and coupling polarizations on DROP and EIT resonances [17][18][19][20][21]. EIT is usually modeled in three level system results in ignoring the effects of DROP arising due to optical pumping and coherence. Simplistic three level model cannot accommodate the effects arising due to the involvement of degenerate Zeeman levels [22]. Realistic atomic systems are more complicated and merely three level treatment does not suffice and detailed calculation is required. Dong et al, reported the suppression of DROP signal when probe is weak and mixture of DROP and EIT arises when probe power is moderate [23]. Light field can be structured based on phase, amplitude and polarization leading to wide class of beams with various properties. One such class of beam is the Laguerre Gaussian (LG) beam which carries orbital angular momentum (OAM) and a distinct phase singularity(optical vortices) at center with a helical wavefront. When radial index(p) is zero in LG modes, leads to doughnut structure intensity tranverse profile with inhomogeneous intensity distribution [24,25]. Due to helical phase dependence and also phase singularity, they are also called as vortex beams. EIT in the presence of vortex coupling light is established experimentally and theoretically [26][27][28][29][30].
LG beam carrying OAM as a coupling light introduces narrowing of EIT resonance at higher l values [26,28,31] due to the reduction in effective Rabi frequency of the coupling light. Narrow EIT spectrum has a direct consequence in the group velocity of probe light enabling quantum memory [5,29] and also has applications in the field of precision measurements [6,32]. Polarization effects on the probe light in the presence of vortex coupling light has remained unexplored in the context of EIT.
In this paper, we study experimentally and theoretically cascade EIT configuration for 87 Rb in following two aspects : (i) two photon transition probability is considered to identify the experimentally obtained EIT peaks and (ii) to study the effects of vortex coupling light polarization on the linearly polarized probe beam and its contribution in the enhancement of EIT resonances. Vortex beam with the value of l = 10 is generated for coupling light, probe and coupling lights are interacting in counter propagation direction giving rise to Doppler free EIT signal. Semi-classical density matrix approach is applied to explore the dynamics of double hut system. Enhancement and diminishing of EIT resonances is established quantitatively between experiment and theoretical results.
The manuscript begins with introduction in section 1, followed by experimental technique in section 2. Section 3 comprises of detailed theoretical analysis by considering simplification of degenerate multilevel atomic system to double hut model. Results and Discussion are reported in section 4. Finally, concluding remarks are included at the end.

Experimental technique
The experimental atomic system with relevant energy levels for 87 Rb atoms is depicted in figure 1. In our three level cascade system, probe laser light is addressing the D2 transition of Rb from |5S 1/2 , F = 2⟩ to |5P 3/2 , F ′ = 1, 2, 3⟩ and it is scanned over the entire excited state. Vortex coupling laser light is resonant to |5P 3/2 ⟩ to |5D 5/2 ⟩ transition forming a cascade EIT configuration. Corresponding hyperfine and degenerate magnetic levels are also shown in the figure 1. The spontaneous relaxation rates from 5P 3/2 to 5S 1/2 state is 6 MHz, and that from the 5D 5/2 state is 0.97 MHz [18].
The experimental apparatus developed for our study is shown in the figure 2. Apparatus comprises mainly diode lasers DL1 and DL2, set of accompanying optical elements such as half wave plates, quarter wave plates, Rubidium atomic vapor cell, spatial light modulator and the detector. Laser light emerging out of Sacher Lasertechnik Micron VBG laser DL1 with a wavelength of 780 nm passes through HWP1 and PBS1 combined configuration to control the power. The reflected s-polarized (as per the plane of incidence it is |V⟩ polarized in our case) component is used for wavelength monitoring using Highfinesse WS6 wavelength meter. Whereas, the transmitted p-polarized (as per the plane of incidence it is |H⟩ polarized in our case) part is used as a probe light for the experiment, which is directed to an atomic vapor cell containing Rubidium species via a lens L1 for expansion of the probe beam. Likewise, Coupling light is derived out of Sacher Lasertechnik external cavity diode laser DL2 with a wavelength of 775 nm and it passes through the HWP3 and PBS4 combination. The s-polarized component is used for wavelength monitoring and the p-polarized component is further directed to lens pinhole combination for expansion of the coupling beam. An expanded Gaussian beam of the order of 3 mm hits the Holoeye reflective SLM and it is characterized by the wavelength of 775 nm as discussed in the [31]. Computer generated hologram is displayed on the surface of SLM which introduces the phase shift to the input Gaussian light and transforms it into vortex LG light of defined l and p values. We chose p = 0 and l = 10 mode and vortex beam with a wide singularity at the center gets generated which is captured in a CCD camera by splitting one of the components in PBS3. Further, the transmitted light is taken to the experiment as a coupling light after passing through the mirror M1 and PBS2. The transmitted component of PBS2 is horizontally polarized. QWP1 is placed in the path of the coupling LG 10 0 Figure 1. Energy level scheme of 87 Rb for cascade EIT configuration under consideration. Hyperfine and degenerate Zeeman levels are also shown. Probe light is addressing |5S 1/2 , F = 2⟩ to |5P 3/2 , F ′ = 1, 2, 3⟩ with a wavelength of 780 nm. Vortex coupling light with l = 10 is resonant to |5P 3/2 ⟩ to |5D 5/2 ⟩ with a wavelength of 775 nm. vortex beam such that its fast axis is perpendicular to the incident horizontally polarized light at first. The effect of vortex polarized coupling light on probe light spectrum is observed using photodiode PD.
To investigate the effects of coupling light polarization on probe spectrum, QWP1 angles are changed for introducing various polarizations which is placed in the path of vortex coupling LG light. Throughout the experiment, it is ensured that probe light has horizontal linear polarization and it is in counter-propagating configuration for the coupling LG light to achieve the Doppler free EIT spectra. The power of the probe laser is 2.1 mW and the coupling laser is 3.6 mW and both the powers are maintained constant throughout the experiment with careful attention.

Theoretical analysis
To establish the effects of vortex coupling light on the probe spectrum, we developed a relevant theoretical model for cascade system in 87 Rb atomic vapor medium. Energy levels involved in the experiment is shown are the figure 1. Our focus is to develop the effect of various polarizations of coupling LG light on the EIT spectra. To capture the essence of polarization, merely considering the fine structure will not justify the experimental results. We specifically chose hyperfine transitions in the aforementioned levels. We further go beyond hyperfine transitions and consider all the associated magnetic level transitions to establish the QWP angle variation with respect to coupling LG light polarization as shown in the figure 3(i). In our experimental setup, probe is maintained as horizontally polarized(|H⟩), whereas the polarization state of coupling light controlled by changing the QWP1 angle. Initial polarization before the QWP1, the polarization of coupling light is |H⟩. After passing through the QWP1, the general state of polarization |P⟩ is elliptical in nature, for convenience it can be written as the linear superposition of |σ + ⟩ and |σ − ⟩ polarization and it is given by, where, θ is the angle between the QWP1 fast axis and coupling polarization state before passing through QWP1 (which is |H⟩ polarized throughout the experiment).
We chose simplified double hut model as shown in figure 3(ii) to understand the dynamics of atoms. We adopted semi-classical density matrix approach in order to obtain the atomic dynamics. As shown in the where, H 0 is the bare state Hamiltonian and it is given by, H I is the interaction Hamiltonian and it can be written as, where, ∆ p and ∆ c are the probe and coupling detuning respectively. Ω p is the probe Rabi frequency and Ω c1,c2 are the coupling Rabi frequencies for σ + and σ − light respectively, H.C is the hermitian conjugate. The total Hamiltonian in equation (2) can be written more transparently in the following matrix form, Here, ∆ p = ω p − ω 12 and ∆ c = ω c − ω 23 are the probe and coupling detunings. ω p,c are angular frequency corresponding to probe and coupling laser while ω 12,23 are angular frequencies corresponding to energy gap between state |1⟩ − |2⟩ and state |2⟩ − |3⟩ respectively. Rabi frequency for probe and coupling transition which characterizes the strength of the transition is given by, E p and E c1,c2 are the electric field corresponding to probe and coupling transition. d p and d c are the dipole moments of probe and coupling transitions respectively. Although, E p is fixed for the experiment, E c1 and E c2 are changing its value based on the θ according to equation (1). E c1,c2 is the electric field amplitude of the coupling LG light for l = 10 and p = 0 and it is given in the reference [24]. Time evolution of coherence that exist between driven atomic states and population can be obtained by solving a set of partial differential equations obtained by Liouville equation, where, ρ is the density operator, L ρ is the Lindblad superoperator to account for the decay mechanism present in the system. We numerically solved equation (8) in steady state situation by considering the inhomogeneity arising out of vortex coupling light. Since, we are concerned in the coherence established between the states addressed by probe transition, we calculated ρ 12 , ρ 54 and ρ 87 . We can write the linear susceptibility of the probe transition as, where, N is the number of atoms per unit volume, 5, 9 and 5 are multiplied to account for transition strengths of the corresponding transitions. Transparency is observed in the absorption profile which is given by imaginary part of susceptibility and real part gives the index of refraction. Imaginary part of χ is plotted at various angles of coupling light and their heights are calculated. Variation of calculated peak height with the QWP angle is discussed in section 4.2.

Results and discussion
Probe light in the experiment is addressing D2 transition and is scanned over entire range of excited state. Coupling vortex light is kept stable at a wavelength of 775 nm and is monitored through out the experiment. In this section, experimental observations and their theoretical validation is established in two aspects : (i)  Initially, EIT signal is obtained using |H⟩ − |H⟩ polarized light for both probe and coupling beams(in the absence of QWP1 and QWP2 in figure 4) and later, σ + − σ + polarized light for both probe and coupling beams (in the presence of QWP1 and QWP2 at 45 • to the horizontal as shown in the figure 4) and spectrum of EIT is obtained. Also, spectrum is identified using the theoretical consideration of two photon transition probability. (ii) Experimental observation on the effect of vortex coupling light polarizations on EIT spectra is discussed and using the theoretical model developed in section 3 enhancement and reduction in peak height at various QWP angles are established.

Observation of EIT spectra for |H⟩ − |H⟩ and σ + − σ + polarized light
Firstly, in the experiment both probe and coupling lights are maintained as horizontally polarized. We obtained EIT spectrum by resolving two peaks in the hyperfine manifold as shown in the figure 5. It is evident from the obtained spectrum that both the peaks are comparably of same height and very narrow due to the presence of vortex LG beam l = 10 and p = 0 as established in the [31]. Further, to understand the effect of polarization on the EIT spectrum, we modified the polarization by placing QWP1 in the path of coupling light and QWP2 in between L1 and VC for the probe light at an angle of 45 • from the horizontal as shown in the figure 4. This arrangement of QWPs at a particular angle modifies the incoming |H⟩-polarized light to σ + polarized light for both the beams. The transmission spectrum obtained for σ + − σ + polarized light combination is shown in the figure 6. The spectrum demonstrates that the left peak is substantially lower in height than the right peak. We further conducted a theoretical study employing two photon transition probabilities to explain and identify the spectra obtained at different polarizations, as discussed in the following sub section.

Peak identification for |H⟩ − |H⟩ and σ + − σ + polarization of probe and coupling lights
Two photon transition probability study is carried out over magnetic levels in order to identify the peaks corresponding to different polarization combination of probe and coupling light. To calculate the In the figure 7, arrow indicates hyperfine transition, of course there are allowed magnetic transitions between these hyperfine levels according to the selection rule q = m F − m F ′ = 0, ±1 respectively for π, σ + , σ − transition. Transition strengths between level m F (initial) and m F ′ (final) which belongs to hyperfine manifold F and F ′ respectively are given by Wigner-Eckart theorem as [12,16], where, dq is the dipole moment operator between initial and final state which can be characterized by quantum number J angular momentum, total angular momentum F and nuclear spin I. (J||d||J ′ ) is the reduced dipole moment which is independent of magnetic quantum number m F and m F ′ . Terms inside the parenthesis on the right hand side of equation (10) is Wigner 3j symbol while the terms in curly bracket is Wigner 6j symbol. With the help of equation (10) relative transition strength for different combination of m F state can be evaluated with following expression, Relative transition strengths for each F state and polarization is calculated using the equation (11), and it is normalized for initial values of F for both probe and coupling lights. Later, individual transition probabilities of probe and coupling lights are summed over all the transitions and multiplied it with each other. In the table 1, transition 2,1,1 refers to the path chosen by atoms to satisfy the two photon resonance condition and path being the ground state F = 2, intermediate state F ′ = 1 and the excited state F ′ ′ = 1 and the corresponding transition probability is 12 600. Similarly, two photon transition probabilities are calculated for all the transition pathways and it is listed in tables 1 and 2 for both linear and σ + polarized combination of probe and coupling lights respectively. From the two photon transition probabilities calculated as shown in table 1, it is clearly evident that for |H⟩ − |H⟩ combination, the F ′ ′ = 2 and F ′ ′ = 3 has comparably similar values. From this analysis, we identified that the peaks obtained are F ′ ′ = 2 and F ′ ′ = 3, yet it is unclear which peak corresponds to F ′ ′ = 2 and F ′ ′ = 3. Further, table 2 shows that for the above mentioned peaks, the values of two photon probability for F ′ ′ = 2 is almost three times smaller compared to F ′ ′ = 3 in case of σ + − σ + lights. From the experimentally obtained spectra and by analysing the two photon transition probability tables, we identified that the peak on the left is F ′ ′ = 2 and the right peak corresponds to F ′ ′ = 3.

Effect of coupling light polarization on the EIT spectrum
In this section, effect of coupling light polarization on the probe spectrum is studied. During this experiment, probe light is maintained as |H⟩-polarized throughout. Vortex coupling LG light is initially at |H⟩-polarization, QWP1 placed in the path of coupling light as shown in figure 2 and the angle in QWP1 is varied at 10 • interval. From |H⟩-polarized light, the light beam transforms to elliptical nature as established in the equation (1). QWP1 angle is further increased and EIT spectra is obtained at 10 • , 20 • . . . , 90 • . From the obtained EIT transmission spectra at each angle, peak heights for both F ′ ′ = 2 and F ′ ′ = 3 are measured. The plot of peak height vs QWP1 angle is displayed in the figure 8. Both the peaks shows the similar trend, as Table 2. Two photon transition probability for probe σ + and coupling σ + polarized light.  QWP1 angle is increased, peak height increases and reaches maximum at an angle of 40 • . Further, increase in QWP1 angle decreases the peak height.
To establish the experimentally obtained enhancement and reduction in the peak heights of transmission spectra (figure 8) using the theory, we considered the double hut model as discussed in the section 3. Equation (1) accounts for the various combinations of polarization for vortex coupling light and at θ = 0 the state of polarization represents the |H⟩-state as considered in the experiment. At different θ's, weightage for σ + and σ − is changed as per the equation (1), giving rise to elliptically polarized light similar to the polarization state chosen in the experiment. At each angle, Rabi frequency of the vortex coupling light is calculated and linear susceptibility is obtained by using equation (9).
Imaginary part of the susceptibility is plotted and further peak height is measured for all the angles. The variation of theoretically calculated peak height versus QWP angle is shown in the figure 9. From the plot, it is noticeable that the peak height increases until 40 • and further increase in the QWP angle, decreases the peak height. The above theoretically obtained result quantitatively agrees with the experimentally obtained results. After careful observation of the model developed, the enhancement and reduction in the EIT peaks is due to the combined effect of polarization states of the coupling light considered and the relative transition strengths between the states. It is also evident that the degenerate multilevel atomic structure present in the system involved in the formation of EIT subsystems leading to change in the amplitude of absorption peaks because of variation in the polarization of probe and coupling lights [22,33]. The relative orientation of both probe and coupling lights are also the major contributors for polarization dependent enhancement and reduction in the amplitudes of EIT peaks. By selecting the right polarization for the probe and coupling fields, EIT can be improved or diminished. It has been demonstrated that changes in the EIT spectrum is a direct consequence from the relative two-photon transition from the ground state to the upper state connected by the coupling field [16]. Improving EIT contrast using polarization has significant application in the field of magneto optic rotation. These effects are more prevalent when magnetic field is employed in the system and has potential application in precision measurements.

Conclusion
In conclusion, three level cascade EIT configuration in 87 Rb atomic vapor medium is investigated in the presence of vortex coupling LG light. Doppler free narrow EIT signal is observed due to the presence of vortex LG coupling light with l = 10 and p = 0. Transmission spectra are obtained for |H⟩ − |H⟩-polarized and σ + − σ + probe and coupling light configuration with a double resonance structure. Obtained peaks are identified as F ′ ′ = 2 and F ′ ′ = 3 by theoretically considering the two photon transition probabilities. Various combinations of polarizations are considered for vortex coupling LG light and its effect on enhancing and diminishing the EIT resonances are experimentally achieved. To capture the essence of polarization dependent EIT in a degenerate multilevel atomic system, a simplified double hut model is considered and dynamics are captured using the semi-classical density matrix approach. Enhancement and diminishing effects on EIT resonances are a direct consequence of polarization combination of probe and coupling lights and relative two photon transition probabilities. Additionally, the degenerate Zeeman levels play a significant role in modifying the EIT contrast. This study has prime significance in the field of magneto optic rotation and precision measurements, further gives scope for exploring polarization dependent spectroscopy in degenerate atomic systems.