## Abstract

The dependencies of the polarization rotation on the probe ellipticity and the longitudinal magnetic field have been studied both experimentally and theoretically in a V-type electromagnetically induced transparency for ^{87}Rb vapour. The angle of rotation varies periodically with the change in the probe ellipticity and non-linearly with the variation of the magnetic field. We have observed that the plane of polarization is rotated maximum for linearly polarized light and have obtained angle of rotation 0.32^{°} ± 0.01^{°} for *B* = 0.036 ± 0.001 mT while it was 0.0267^{°} ± 0.0002^{°} without magnetic field. Thus our measurement becomes sensitive to the low magnetic field. A four-level system is considered and the corresponding density matrix equations have been solved analytically to explain these observations theoretically with the help of degenerate and non-degenerate magnetic sub-levels.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The rotation of the plane of polarization of an optical field can be induced by the intrinsic helicity of the atoms or molecules in the medium or can be generated by applying external electric, magnetic and optical fields. When the original direction of linear polarization of an optical field is rotated by introducing another circularly polarized optical field then this phenomenon is called polarization rotation (PR) [1] or optical rotation (OR) [2] or optical Faraday rotation (OFR) [3]. Again, if the external magnetic field is applied to rotate the polarization of the optical field then the phenomenon is known as the magneto-optical rotation (MOR) [4]. Now, if the magnetic field is applied in the direction of light propagation, the phenomenon is called the Faraday effect [5,6], while for a transverse magnetic field, the phenomenon is known as the Voigt effect [7,8]. On the other hand, when the pump beam produces coherence among the relevant atomic states in a three level system, the medium can manifest the phenomenon of electromagnetically induced transparency (EIT) [9–11] for the probe beam. This quantum interference phenomenon raises an intriguing possibility that a single coupling beam can execute the medium both anisotropic and transparent to the probe beam. Under this EIT condition, the medium becomes highly dispersive to the probe field. As a result, we get a normal dispersion in the EIT region. Moreover, the quantum coherence or the quantum interference plays a major role in the modification of the optical properties of the system and has been extensively studied both theoretically and experimentally because of its numerous applications in optical physics [12]. Bridging these phenomena gives us the opportunity to investigate how the PR and MOR can be controlled coherently in an EIT medium.

Polarization spectroscopy with the probe and the pump beams had been studied using the ladder type configuration of Cs atom at room temperature [13], where the power dependency of the slope of the observed signal was investigated. The rotational signal in cascade configuration of Rb atom was used as an error signal to lock the probe beam [14]. Effects of polarization in the interaction between multi-level atoms and two optical fields has also been studied in a $\Lambda$ - type and a ladder type configuration of $^{85}$Rb [15]. Recently Moon et al. [16] has presented an experimental and theoretical study of polarization rotation or the optical rotation in $^{87}$Rb with D$_2$ transition. Veisi et al. [3] investigated the nonlinear polarization rotation of a probe field passing through a double V-type closed-loop atomic system in the absence of an external static magnetic field. The effect of quantum interference due to the spontaneous emission on the linear MOR has been studied [17]. Coherent control of MOR has also been investigated in a cascade system theoretically [4,18] and experimentally [19]. Besides, there are several studies on the Faraday rotation in cold atoms [20], hot atomic vapour [2], in a tripod system [21] and in multi-Zeeman levels [22] of room temperature atomic vapour.

This polarization rotation found several applications in optical locking [23], linear and non-linear magneto-optical rotation [24], squeezing of light [25–27], magnetometry [24,28,29], detection of slow light [30], birefringence lens [31] etc. Hence, to improve the efficiency of all of the above applications we need to enhance the birefringence effect in the medium. In our earlier studies [32,33], we have investigated how the EIT phenomenon influences the polarization rotation of the probing field by forming a V-type system for both $^{87}$Rb and $^{85}$Rb with $D_1$ and $D_2$ transitions. The non-linear dependency of polarization rotation with electromagnetically induced transparency (PREIT) condition on the pump beam intensity has been studied [32]. We have also systematically investigated, both experimentally and theoretically, how the change in the angular mismatch between the pump and the probe beams, the change in the optical depth (OD) of the medium and the change in the spot size or the radius of the pump beam can modify the angle of PREIT of the probe beam in $^{87}$Rb atomic medium for a V-type system having $D_1$ and $D_2$ transitions [33]. Apart from these, the ellipticity of the beam and the external magnetic field are the two important parameters in the case of study of the coherent phenomena.

The effect of incident field ellipticity on the coherent phenomena in Hanle configuration have been well studied both experimentally and theoretically [34,35]. In our present work, we represent how the angle of PREIT can be controlled by changing the ellipticity of the linearly polarized probe beam in absence of the external magnetic field. This work also involves an observation of the modification of polarization rotation due to the application of static longitudinal or axial magnetic field in the EIT medium. To observe these phenomena we have considered a V-type system of $^{87}$Rb with $D_1$ and $D_2$ transitions in room temperature. To explain the experimentally observed phenomena theoretically, we have considered a four-level system combing the magnetic sub-levels of the corresponding hyperfine transition. We have also considered a semi-classical approach to get the probe response of the medium.

## 2. Experiment

Figure 1(a) describes our experimental setup to study how the polarization rotation of the probe beam in an EIT medium was affected by changing the ellipticity of the probe beam and by also applying an external magnetic field in the medium. As shown in Fig. 1(a), the probe and the pump beams were taken from an external cavity diode laser (ECDL) and a distributed feedback diode laser (DFBL) respectively. To avoid back reflections to both the lasers we used optical isolators (O.I.). The probe beam was made linearly polarized by a glan-laser polarizer (GL) and a neutral density filter (ND) was used to reduce the intensity of the beam. The reflected part of the pump beam from a 10:90 (R:T) beam splitter (BS) was used to lock the pump beam to a particular hyperfine level using the saturation absorption spectroscopy setup (SAS setup) and the transmitted part was taken for the polarization spectroscopy experiment. The pump beam was made circularly ($\sigma ^{+}$) polarized using the combination of a polarizing beam splitting cube (PBS1), a half-wave plate (HWP1) and a quarter wave plate (QWP).A variable neutral density filter (VND) was used in the path of the pump beam to choose appropriate intensity of this beam. The probe and the pump beams were mixed in a 50:50 non-polarizing beam splitting cube (CBS). Then both of the beams were co-linearly co-propagated through a cylindrical Rubidium vapour cell (Rb cell) of length $0.05$ m with $0.025$ m diameter. Our Rb cell contained both $^{85}$Rb and $^{87}$Rb in natural abundance without any buffer gas under $\sim 10^{-7}$ Torr pressure at room temperature. To study the effect of the external magnetic field, the Rb cell was placed inside a solenoid. To avoid interference due to the stray magnetic field of the earth, the entire arrangement of the Rb cell along with the solenoid was covered by three layers of $\mu -$metal shields. A band-pass optical filter (O.F., Thorlabs FBH780-10) placed after the Rb cell was used to block the pump beam ($795$ nm) so that only the probe beam ($780$ nm) can be detected. Then the probe beam was detected with the help of balanced polarimetric technique [1,36] to get the polarization rotational signal. To study the phenomena in an EIT or coherently prepared medium, we have chosen a V-type level scheme with D$_1$ and D$_2$ hyperfine transitions of $^{87}$Rb as shown in Fig. 1(b). The probe beam was scanned from $F=2 \longrightarrow F^{\prime }=3$ of D$_2$ transition and the pump beam was locked at $F=2 \longrightarrow F^{\prime }=2$ of D$_1$ line. Throughout our experiment the intensities of the probe and the pump beams before the Rb cell were kept constant at $0.47$ mW/cm$^{2}$ and at $5.09$ mW/cm$^{2}$ respectively.

## 3. Results and Discussions

#### 3.1 Dependency on the Ellipticity

To study the behaviour of the angle of polarization rotation in a coherently prepared atomic medium due to the ellipticity of the probe beam, we have varied the ellipticity of the probe beam by rotating QWP2 placed in the path of the linearly polarized probe beam as shown in the Fig. 1(a)).The polarization rotation signals with EIT (PREIT signals) for different ellipticities with normal dispersion is shown in the Fig. 2(a). From this figure we have seen that the intensities, as well as the slopes of the signals, are changing periodically as the ellipticity of the probe beam ($\beta$) was changed. This indicates that the birefringence created in the medium due to difference in the contributions of the $\sigma _+$ and $\sigma _-$ components of the probe beam is changing periodically as we changed the probe ellipticity. This is because the intensities of both the components were changing as the ellipticity was varied. The output intensity difference ($\Delta I (\omega _p$)), detected by the balanced photo-detector (B.Det in Fig. 1(a)), depends on the phase shift due to the anisotropy in the refractive indices ($\Delta n (\omega _p) = n_{+}(\omega _p)-n_{-}(\omega _p$)) created in the medium corresponding to the two circular polarization components $(\sigma _+$ and $\sigma _-)$ of the probe beam. Following the derivation in Appendix A, it can be written as [32],

The detected angle of polarization rotation is defined [30,33] as,

So, the angle of rotation in the vicinity of the EIT resonance is calculated by Taylor series expansion around $\omega _p\approx \omega _0$ of Eq. (1) as shown in Appendix A,

From the slope of the rotation signal we have calculated the angle of polarization rotation at the vicinity of EIT i.e. the angle of PREIT using Eq. (3). In the Fig. 2(b), we have plotted angle of PREIT ($\theta (\beta$)) as a function of the probe beam ellipticity ($\beta$). From this plot we have observed that $\theta (\beta )$ shows a periodical behaviour with the variation of $\beta$. It shows maxima at $\beta = m\frac {\pi }{4}$ while the minima occur at $\beta =(m+1)\frac {\pi }{4}$, where $m = 0,2,4,\ldots$ . To explain the physics behind this phenomenon let us express the electric field of the probe beam with the ellipticity as,

where $E_p$ is the complex amplitude and $\hat {e}$ is the unit polarization vector. The polarization vector can be written as,Now from Eq. (4) and Eq. (5) we can say that when $\beta = m\frac {\pi }{4}$, our probe beam became linearly polarized and both $\sigma _+$ and $\sigma _-$ components of the beam were equal in intensities. So the birefringence created due to the difference between the contributions of the $\sigma _{+}$ and the $\sigma _{-}$ components of the probe beam in the medium was maximum. As a result, we got maximum rotation in the medium. Again for $\beta = (m+1)\frac {\pi }{4}$ the probe beam became circularly polarized. Depending on the value of $\beta$, it had either $\sigma _+$ or $\sigma _-$ component in the medium and we got minimum rotation as the birefringence was minimum in this condition. In the intermediate value of $\beta$, the intensities of $\sigma _+$ and $\sigma _-$ components were not equal and we got an intermediate value of rotation. Therefore, we got a periodical variation of the angle of PREIT with the probe beam ellipticity with a $\pi /2$ period. In another way, depending on the value of rotation we can find the corresponding ellipticity from this variation. Getting the value of $\beta$ from Eq. (5) we can say whether the beam is linearly or circularly or elliptically polarized. So, our study becomes important to find the unknown state of polarization of the probe beam. In our previous studies [32,33] we showed that from the angle of PREIT, one can calculate the refractive index of the medium. So our study become useful when one uses this experimental technique to find the group velocity of light as the refractive index is directly proportional to the angle of rotation. Our study also confirms that to get the most reduced group velocity of the probe beam we need to use linearly polarized probe with circularly polarized pump in the polarization spectroscopy technique as the angle of rotation for linearly polarized beam is maximum.

#### 3.2 Dependency on the longitudinal magnetic field

To venture how the longitudinal magnetic field affects the angle of PREIT, we kept the probe beam as linearly polarized and the pump beam as circularly polarized. We have recorded the polarization rotation signal first without an external magnetic field under EIT condition. Then we varied the magnetic field inside the solenoid. For each magnetic field we took the data corresponding to the PREIT signal of the probe beam.In this case the anisotropy in the refractive index was also created due to the applied magnetic field ($\vec {B}$). Therefore, the output intensity difference is written as (see Appendix A),

This gives the angle of rotation at the EIT resonance condition as,

In the Fig. 3(a) the PREIT signals have been shown for different longitudinal magnetic fields. We have calculated the angles of PREIT $(\theta (B))$ using Eq. (7) from these signals as a function of the magnetic field.

The variation of the angle of PREIT with the applied magnetic field is shown in the Fig. 3(b). We have observed that at the beginning, $\theta (B)$ increased with the increment of $B$ field till a certain value. After that $\theta (B)$ decreased for further increased $B$ field. A qualitative explanation behind this phenomenon is that when we apply the longitudinal magnetic field, the degeneracy among the magnetic sub levels is removed (see Fig. 6(b)). Since our medium is interacting with both the circularly polarized pump beam and the external magnetic field, both of these created a circular birefringence in the medium due to the non-uniform distribution of the population among the magnetic sub-levels of the atom [1,13,19,32]. Again, due to the fixed intensity of the pump beam, in this case the applied longitudinal magnetic field is the only reason for the enhancement of the circular birefringence for more non-uniform population redistribution. This increased the anisotropy of the refractive indices for the $\sigma _+$ and $\sigma _-$ components of the probe beam which in turn augmented the angle of PREIT as we increased the magnetic field. For this reason, we got an increasing behaviour in $\theta (B)$ with increasing $B$ field till a certain value where the population redistribution saturated. After this, as we have seen from the Fig. 3(b), further increment in the $B$ field worsen the birefringence effect in the medium due to the decreasing coherence as the population redistribution got saturated. So, $\theta (B)$ shows a decreasing trend with increasing $B$ in this region. Thus we observed a non-linear variations of $\theta (B)$ with the $B$ field.

Further, we can also comment that the angle of polarization rotation in the EIT medium is sensitive towards the magnetic field. From our experiment, we got the angle of rotation as $0.0267^{\circ }\pm 0.0002^{\circ }$ without any external magnetic field, while the angle of rotation was $0.32^{\circ }\pm 0.01^{\circ }$ for $B = 0.036\pm 0.001$ mT. This got maximized as $8.79^{\circ }\pm 0.05^{\circ }$ at $0.900\pm 0.001$ mT. So, the angle of rotation was increased by one order of magnitude on introducing a magnetic field comparable to the earth’s magnetic field from the value of rotation without any magnetic field and it get maximized with two order of magnitude using our atomic system. Thus our study becomes important when one uses the polarization spectroscopy technique to measure an unknown magnetic field.

## 4. Theoretical analysis

To describe the experimentally observed phenomena for the dependency of the angle of PREIT on the probe beam ellipticity and the magnetic field, we have considered a four-levels system as shown in the figures Fig. 4(a) and Fig. 6(a) with degenerate and non-degenerate excited states respectively. As shown in these figures, the $\sigma _+$ and $\sigma _-$ components of the probe beam interact between $| {1}\rangle \longrightarrow | {3}\rangle$ and $| {1}\rangle \longrightarrow | {4}\rangle$ respectively, while the $\sigma _+$ pump beam couples between $| {1}\rangle \longrightarrow | {2}\rangle$ states. To calculate the angle of PREIT theoretically, we need the susceptibilities ($\chi _+$ and $\chi _-$) corresponding to the $\sigma _+$ and $\sigma _-$ components of the probe beam since the refractive index of an atomic medium is related to the real part of the susceptibility of the medium. Therefore, we have to calculate the probe coherence terms $\rho _{31}$ and $\rho _{41}$ of the density matrix component, where these two terms carry information about the contribution of the $\sigma _+$ and $\sigma _-$ components of the probe field to the medium respectively. To get these terms we need to solve the master equation [37,38],

where $H = H_0 + H_p$ is the total Hamiltonian and $\Gamma _{relax}$ contains phenomenologically included decay terms. $H_0 = \sum _{n=1}^{n=4}\hbar \omega _{nn}| {n}\rangle\langle {n}|$ is the unperturbed Hamiltonian for both the cases of study. The perturbed Hamiltonian ($H_p$) will be different for the two cases. In Eq. (8), $\rho$ is a $4\times 4$ density matrix, whose diagonal terms contain the information of the population for the individual energy levels and the off-diagonal terms give the information about the coherence contributions corresponding to the respective electric fields.#### 4.1 Dependency on the ellipticity

To explain how the rotation phenomenon occurred in the study of dependency of the angle of PREIT on the ellipticity of the probe beam, we considered that both the $\sigma _+$ and $\sigma _-$ components of the probe beam contributes two photon coherence contribution in the medium along with the $\sigma _+$ pump beam. But there was a mismatch between the contributions of the $\sigma _+$ and the $\sigma _-$ probe beams depending on their amplitudes of the electric fields as the ellipticity of the probe beam was varied. Thus anisotropy was created between the refractive indices of the two circular polarization components of the probe beam leading to the rotation of its plane of polarization. Fig. 4(b) represents the energy level diagram with $\sigma _+$ and $\sigma _-$ polarization components of the probe beam and $\sigma _+$ component of the pump beam, where all the possible degenerate sub-level couplings for both the components of the probe beam and that for the $\sigma _+$ pump beam have been shown. From this figure it is also clear how the contributions can differ due to different intensities of the components. For simplicity, we have taken only four sub-levels (see Fig. 4(a)) to explain the phenomenon theoretically.

In this case, the perturbation in the system was only due to the electric fields of the applied beams so the perturbed Hamiltonian can be written as, $H_p =H_E = -\hbar \lbrace \Omega _{21} cos(\omega _c t-\vec {k}_{c}.\vec {r})| {2}\rangle\langle {1}|+\Omega _{31}(\beta ) cos(\omega _p t-\vec {k}_{p}.\vec {r})| {3}\rangle\langle {1}|+\Omega _{41}(\beta ) cos(\omega _p t-\vec {k}_{p}.\vec {r})| {4}\rangle\langle {1}|+ c.c\rbrace$. We have considered the probe field as in Eq. (4) and the pump field to be $\vec {\varepsilon _c}(\vec {r},t)= E_c \hat {e}_{z} cos(\omega _c t - \vec {k}_{c}.\vec {r})$ where, $E_{c}$ being the amplitude of the electric field of frequency $\omega _{c}$, $\vec {k}_{c} = 2\pi /\lambda$ is the wave vector and $\vec {r}$ is the propagation direction of the fields. Here, $|\Omega _{31}(\beta )| = \frac {\mu _{31}E_{p} cos(\beta -\frac {\pi }{4})}{\hbar }=\Omega _p cos(\beta -\frac {\pi }{4})$, $|\Omega _{41}(\beta )| =\frac {\mu _{41}E_{p} cos(\beta +\frac {\pi }{4})}{\hbar }=\Omega _p cos(\beta +\frac {\pi }{4})$ are the Rabi frequencies for the $\sigma _+$ and $\sigma _-$ components of the probe field respectively and $|\Omega _{21}|= \frac {\mu _{21}E_{c}}{\hbar }=\Omega _c$ is the pump Rabi frequency.

Using the rotating wave approximation (RWA) and the first order perturbation for the probe field, solving Eq. (8) under steady state condition, we obtained the two photon contribution terms $(\rho _{31})$ and $(\rho _{41})$ as,

The parameters used in Eq. (9) , Eq. (10), Eq. (11) and Eq. (12) are described as follows,

The probe resonance frequency is $\omega _{21} = \omega _0$ and the probe detuning is $\delta _{p}(\omega _p) = (\omega _0 - \omega _p)$. As we have taken the propagation direction of the probe beam along the $z-$axis, the Doppler detuning of the probe is $\Delta _{p}(\omega _p,v) = (\delta _p(\omega _p) + \vec {k_p}\cdot \vec {v})= (\delta _p(\omega _p) + k_p v)$ (as, $\vec {v}=v \hat {z}$). Similarly, we have defined the detuning of the pump beam as $\delta _{c}(\omega _c) = (\omega _{31} - \omega _c)$ with the Doppler detuning of the pump beam $\Delta _{c}(\omega _c,v) = (\delta _{c}(\omega _c) + k_c v)$. Since in our experiment pump beam was locked so in our calculation we take $\omega _{31} = \omega _c$. Therefore, in our calculation, $\delta _{c} = 0$ and $\Delta _{c}(v) = k_c v$.Convoluting the velocity distribution for the atoms, the susceptibilities can be written as,

In the above equation, $N(v) = \frac {N_0}{\sqrt {\pi u^{2}}}\exp [-\frac {v^{2}}{u^{2}}]$ is Maxwell-Boltzmann (MB) velocity distribution, where $u$ is the most probable velocity of the atoms and $N_0$ is the number density of the atoms at temperature ‘T’K. The dipole moment, $\mu _{31(41)} = 1.73\times 10^{-29}$ C.m [39]. The simulated polarization rotation spectra with positive slope in the vicinity of the EIT resonance are shown in the Fig. 5(a) for different values of $\beta$. Here also we have observed the same periodical variations in the intensity of the rotational spectra with changing $\beta$ as we had observed in our experiment.

With help of Eq. (16) we have calculated the angle of rotation according to (see Appendix A),

Therefore, $\theta (\beta )$ at the EIT resonance position can be calculated by expanding the Eq. (17) in Taylor-series upto the first order term as,

In the above equations $\chi ^{\prime }$ is the real part of the susceptibility. In Eq. (18), $\chi _{31}^{\prime }(\omega _{0})=\chi _{41}^{\prime }(\omega _{0})=0$. Therefore, $\theta (\beta )$ entirely depends on the derivatives of $\chi _{31}^{\prime }(\omega _p,\beta )$ and $\chi _{41}^{\prime }(\omega _p,\beta )$ at the resonance position. Since the susceptibilities depend on $\beta$ as shown in the Eq. (16), the angle of PREIT is also a dependent of $\beta$. Using Eq. (18) we have calculated $\theta (\beta )$ for different values of $\beta$. In the Fig. 5(b) we have shown the theoretical variation of the angle of PREIT with the probe beam ellipticity $\beta$. We got similar periodical variation of $\theta$ as a function of $\beta$ with a period of $\frac {\pi }{2}$.

#### 4.2 Dependency on the longitudinal magnetic field

In the Fig. 6(a), a simplistic level scheme has been shown where the $\sigma _+$ and $\sigma _-$ components of the probe beam and the $\sigma _+$ component of the pump beam coupled the non-degenerate magnetic sub-levels. In this case, we have considered that the $\sigma _+$ component of the probe beam along with the $\sigma _+$ pump beam created the EIT condition which contributed two photon coherence effect in the medium. While, the $\sigma _-$ component of the probe beam gave only one photon contribution to the medium. Thus, there was a mismatch between the contributions of the $\sigma _+$ and the $\sigma _-$ probe beams due to the non-uniform population distribution. Further, as we applied the magnetic field along the direction of propagation of the pump beam, here both the pump beam and the longitudinal magnetic field created circular birefringence in the medium. Thus an anisotropy was generated between the refractive indices of the two polarization components of the probe beam leading to the rotation of the plane of polarization of the probe beam. In the Fig. 6(b) we have shown the effect of the magnetic field ($B$) , which is responsible for shifting the magnetic sub-levels by an amount of $g_F \mu _{B} m_{F} B$. For simplicity, we have taken only four sub-levels (Fig. 6(a)) to describe the phenomenon theoretically.

To get the probe response and the angle of PREIT ($\theta (B$)) under effect of the magnetic field we have solved the master equation (Eq. (8)) analytically similar to the previous case. In this case, both the external magnetic field and the electric fields of the applied beams caused the system to be perturbed. Therefore, $H_p$ becomes $H_p = H_B + H_E = \sum _{n=1}^{n=4}\hbar g_{F_n}\mu _B m_{F_n} B| {n}\rangle\langle {n}| -\hbar \lbrace \Omega _{21} cos(\omega _c t-\vec {k}_{c}.\vec {r})| {2}\rangle\langle {1}|+\Omega _{31} cos(\omega _p t-\vec {k}_{p}.\vec {r})| {3}\rangle\langle {1}|+\Omega _{41} cos(\omega _p t-\vec {k}_{p}.\vec {r})| {4}\rangle\langle {1}|+ c.c\rbrace$, where $g_{F_n}$ is the Lande g-factor, $\mu _{B}$ is the Bohr magneton, $m_{F_n}$ is the magnetic quantum number corresponding to the hyperfine state $F$. For $n = 3,4$, we write $g_{F_n}$ as $g_{p}$ and for $n=2$ we write $g_{F_n}$ as $g_{c}$. According to the Fig. 6(a), $m_{F_1}=0$, $m_{F_2}=+1$, $m_{F_3}=+1$, and $m_{F_4}=-1$. Since, in this case our probe beam is linearly polarized ($\beta = 0$), so the probe Rabi frequencies are no longer a function of ellipticity. Therefore, the probe Rabi frequencies can be written as, $|\Omega _{31}|= \frac {\mu _{31} E_p}{\hbar }=\Omega _p$ and $|\Omega _{41}|= \frac {\mu _{41} E_p}{\hbar }=\Omega _p$. The probe coherence terms are given by,

The zeroth order populations are given by,

The parameters in Eq. (19) , Eq. (20), Eq. (21) and Eq. (22) are described as follows,

The Doppler detunings for both the probe and the pump beams have been changed to $\Delta _{p+}(\omega _p,B,v) =\{ \Delta _{p}(\omega _p,v) + g_{p}\mu _{B} B\}$, $\Delta _{p-} = \{\Delta _{p}(\omega _p,v) - g_{p}\mu _{B} B\}$ and $\Delta _{cB} = \{\Delta _{c}(\omega _c,v) + g_{c}\mu _{B} B\}$ respectively. Since our real atomic system is a V-type system of $^{87}$Rb with the combination of D$_1$ and D$_2$ transitions, we have taken $g_{p} =\frac {2}{3}$, $g_{c} = \frac {1}{6}$ and $\mu _{B} = 1.4h$ MHz/G [39]. In this case, $\Delta _{cB} = \{\Delta _{c}(v) + g_{c}\mu _{B} B\}$ since $\delta _{c}=0$. All the values of decay parameters are the same as in the earlier case. Therefore, integrating over all the velocity range, the susceptibilities are given by,

After calculating $\theta (B)$ we have numerically plotted it as a function of the magnetic field($B$). The theoretical characteristic variation of $\theta (B)$ with respect to $B$ is shown in the Fig. 7(b), where it can be seen that $\theta (B)$ increases with $B$ field upto a certain value. Then further increase in B decreases the value of $\theta (B)$ similar to what we observed in the experiment.

## 5. Conclusion

In this work we have investigated how the angle of polarization rotation of the probe beam at the EIT region can be modified by changing the ellipticity of the probe beam without the external magnetic field in $^{87}$Rb atomic vapour at room temperature. We have also studied how the polarization rotation in the EIT region can be affected by applying longitudinal magnetic field when the probe beam was linearly polarized and the pump beam was circularly polarized. We have observed that the angle of PREIT shows a periodical behaviour of periodicity $\frac {\pi }{2}$ when the ellipticity of the probe beam was varied. The angle of PREIT became maximum when the probe beam was linearly polarized and became minimum when the probe beam was circularly polarized. It has been noticed that the angle of PREIT depends non-linearly on the applied longitudinal magnetic field. The circular birefringence was increased due to the increment in the applied magnetic field upto a certain value; after which, the circular birefringence decreases due to decreasing coherent effect. Therefore, we can say that one can control the enhancement of polarization rotation occurring in the atomic medium at the EIT condition by controlling the ellipticity of the probe beam as well as by controlling the longitudinal magnetic field. Thus we can also enhance the anisotropy, as well as the chiral behaviour, of the atomic medium.

Figure 4(b) and Fig. 6(b) represent all the degenerate and non-degenerate Zeeman sub-levels (seventeen in total), which are present in our experimental system. For the shake of simplicity we reduced the seventeen-level system into a four-level system. To understand the experimental observation we have analytically solved the density matrix equation for this four-level system. The analytical expression of the probe coherence term helps us to understand the phenomenon in detail, i.e., which term is responsible for each observation (dependency on ellipticity, magnetic field, etc.). We have considered the excited states of the system to be degenerate for the case of ellipticity variation as there was no external magnetic field. While the excited states were considered to be non-degenerate for the purpose of magnetic field variation. From our theoretical calculation, we got a good agreement in the characteristic variation of the angle of rotation with the experimental observations. If we consider the seventeen-level system, a full solution can be obtained numerically but, in that case, the analytical expression of the probe coherence term will be missed. However, the numerical simulation of the seventeen-level system may help us to remove the mismatch between the numerical values from the experiments and theoretical results.

As discussed in the section 3, we can find the ellipticity of an unknown polarized probe beam from our study of dependency of the angle of PREIT on the ellipticity of the probe beam. Therefore, we can use the plot of $\theta (\beta )$ vs $\beta$ as a calibration curve for this purpose. From our observation we can also say that the polarization rotation is highly dependent of the magnetic field as the angle of polarization rotation increased by one order on the magnitude when $0.036\pm 0.001$ mT magnetic field was applied compared to the angle of rotation without the magnetic field. Therefore, this experimental technique of detecting rotation spectrum can be useful to detect unknown magnetic field. From this study it is clear that if we want to use the polarization spectroscopy technique to reduce the group velocity of the probe beam, we need to take the probe beam as linearly polarized because for this polarization of the probe beam the angle of rotation, as well as the refractive index, became maximum as we observed from the Fig. 2(b). Moreover, by applying an appropriate magnetic field for which the birefringence effect is dominating, the group velocity can also be reduced. Besides this, we can also say that as the angle of polarization rotation can be enhanced by controlling the polarization of the probe field and by applying the longitudinal magnetic field, our study plays a significant role in the applications of the polarization rotation.

## A. Appendix

For a linearly polarized light ($\beta =0$) after passing through the atomic medium of length L, the electric field amplitude vector can be written as [32]

Here, $n_+$ and $n_-$ are the refractive indices, $\alpha _+$ and $\alpha _-$ are the absorption coefficients of the atomic medium corresponding to the $\alpha _+$ and $\alpha _-$ components of the beam.

When the light has an ellipticity $\beta$, then the refractive indices, absorption coefficients of the medium due to $\alpha _+$ and $\alpha _-$ components and the amplitude of the beam become a function of $\beta$. Therefore, the above Eq. (30) changes accordingly,

Now, the intensity difference detected by the BDet can be written as [32],

Here, $\Delta n=n_+ - n_-$ measures the anisotropy or the relative refractive index of the medium and $\Delta \alpha =(\alpha _+ + \alpha _-)/2$ is average absorption coefficients of the medium.

The angle of polarization rotation is defined as [30,33],

Therefore, from Eq. (33) and Eq. (34) we can write it as,

The angle of rotation at the EIT resonance can be derived from the Taylor series expansion of Eq. (35). Considering terms up to 1st order derivative we obtained,

Since at $\omega _p=\omega _0$, $\Delta I(\omega _0,\beta )=0$, therefore the above equation becomes,

Further, if there exist an external magnetic field of strength $B$ and $\beta =0$, then we can write Eq. (30) as,

Here, the refractive indices, absorption coefficients of the medium due to $\sigma _+$ and $\sigma _-$ components become a function of $B$. Therefore, following the same calculation like the previous one we obtained the intensity difference detected by the BDet as,

From Eq. (39) we can write the angle of polarization rotation at the EIT condition following the same definition as in Eq. (34),

Now, for calculation of the polarization rotation angle theoretically, we need to calculate the susceptibilities of the medium corresponding to $\sigma _+$ and $\sigma _-$ components of the beam. It should be done because the refractive index of the medium depends on the susceptibility of the medium as,

Here, $\chi ^{\prime }$ is the real part of the complex susceptibility of the medium. Using Eq. (41), the relative refractive index becomes,

Therefore, following Eq. (34), Eq. (37), and Eq. (43) for an elliptically polarized light, the angle of polarization rotation at the vicinity of the EIT resonance can be written as,

Here also, $\chi ^{\prime }_{+}(\omega _0,\beta )=\chi ^{\prime }_{-}(\omega _0,\beta )$.

Further, due to presence of an external magnetic field, the angle of rotation at the EIT resonance can be calculated theoretically by following Eq. (34), Eq. (40) and Eq. (43) as,

Therefore, we have used Eq. (37) and Eq. (40) to calculate the angle of PREIT experimentally corresponding to the two separate cases. The calculation of the same has been done using Eq. (44) and Eq. (45) in our theoretical analysis. In our theoretical model, $\chi _{+(-)}^{\prime }$ corresponds to $\chi _{31(41)}^{\prime }$.

## Funding

Department of Atomic Energy, Government of India (1503/4/2019/R&D-II/DAE/13368); Science and Engineering Research Board (TAR/2018/000710).

## Disclosures

The authors declare no conflicts of interest.

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