1 Introduction

When a strong coupling laser field is used to resonantly drive an atomic transition in a three-level \(\varLambda\)-type atomic system, the absorption of a weak-probe laser field can be significantly reduced or even eliminated. This effect is well known as electromagnetically induced transparency (EIT) [1]. It allows controllable coherent manipulation on the interacting light fields and the optical properties of the atomic media, which serves as a key step toward advanced ultra-fast and quantum information technology [1–5]. Numerous EIT-based applications have been proposed and demonstrated, including precision measurement [6, 7] and optical buffers [8–10], and they offer novel ways for optical information storage and quantum information processing [5, 10–17]. Optical information can be encoded in different freedoms of photons, yet polarization is a robust and most commonly used one [18], so manipulation of light polarization in quantum coherence processes is of great importance and has been widely studied [19–23]. Wielandy et al. [21] employed the quantum coherence to provide an efficient control over the polarization state of a probe field, making an initially isotropic atomic vapor behave as a linearly or circularly birefringent material. Large coherently controlled optical polarization rotation was also achieved via asymmetry in EIT based on a multi-level system in rubidium atoms [22], and coherent control on the light polarization based on nonlinear magneto-optical rotation was demonstrated under EIT condition in rubidium vapor [23]. Most of these polarization control experiments studied in coherent process so far were based on atomic vapors, taking advantage of their particular level selection rules.

However, rare earth ion-doped crystals (REIC), which have practical advantages of robustness, compactness, no atomic diffusion and good spectral selectivity, have been largely ignored in the studies of polarization effect in the quantum coherence process. The main reasons for that are the crystalline birefringence and the polarization-dependent absorption of the doped ions, making the polarization control difficult and complicated. Recently, several groups reported efficient polarization-based quantum qubit memory by means of atomic frequency comb with high fidelity in various REIC by using different compensation techniques [24–26]. Clausen et al. [24] used two identical quantum memories \(M_A\) and \(M_B\) of the same length, and the principal axis of \(M_B\) was rotated by \(90^{\circ }\) with respect to that of \(M_A\). In this configuration, both polarization components will experience the same total absorption and global phase shift after passing through \(M_A\) and \(M_B\), and therefore the system is polarization preserving. Zhou et al. [25] designed a sandwiched structure consisting of two pieces of \(\hbox {Nd}^{3+}:\hbox {YVO}_4\) crystals with a \(45^{\circ }\)-oriented half-wave plate (HWP) between them. The sample shows nearly equal absorption depth for two orthogonal polarization components of light and therefore can be used for arbitrarily polarized photons. Gündoĝan et al. [26] employed a beam displacer in front of a \(\hbox {Pr}^{3+}:\hbox {Y}_2 \hbox {SiO}_5\,(\hbox {Pr}^{3+}:\hbox {YSO})\) crystal to split the two orthogonal polarization components of the incident light into two parallel spatial modes, and a HWP was inserted to rotate the horizontally polarized spatial mode by \(90^{\circ }\) such that both spatial modes launched into the crystal were of the same polarization.

In this article, we investigated the EIT-induced birefringence effect in \(\hbox {Pr}^{3+}\):YSO crystal. We showed that two orthogonal components of the probe beam experience different refractive index changes in general when the crystal is coherently co-driven by a strong coupling beam. Particularly, a polarization rotator was demonstrated at the two-photon resonant condition based on the EIT-induced birefringence in \(\hbox {Pr}^{3+}\):YSO crystal. Our results suggest that the EIT-induced birefringence in REIC can be used to coherently manipulate the polarization of light, and it has potential applications in polarization-based classical and quantum information processing such as all-optical switching, polarization preserving light pulse memory and polarization qubits.

2 Methods of experiment and characterization

Fig. 1
figure 1

(Color online) Schematic diagram of experimental setup. \(\varOmega _\mathrm{C}\) and \(\varOmega _\mathrm{P}\): the Rabi frequencies of the coupling and the probe fields, respectively; \(\frac{\lambda }{2}\): HWP; \(\frac{\lambda }{4}\): QWP; L1 and L2: lens; P: polarizer; PC: polarization compensator; D1 and D2: photodetectors. The inset in dotted box is the schematic definition of the orientation angle \(\varphi\) and the ellipticity \(\varepsilon\) of an elliptically polarized light in the Oxy coordinate system

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In our experiments, we used the orientation angle \(\varphi\) and the ellipticity \(\varepsilon\) (see the dotted box in Fig. 1) to characterize the polarization state of the probe beam [27], in which \(\varphi\) is defined as the crossing angle between the major principle axis of an elliptically polarized light and the crystalline x-axis (here we set an Oxy coordinate system with its two orthogonal axes being the crystalline x- and y-axes of \(\hbox {Pr}^{3+}\):YSO crystal, respectively), and the ellipticity \(\varepsilon\) satisfies

$$\begin{aligned} \tan \varepsilon =\pm \frac{b}{a}, \end{aligned}$$
(1)

where a and b are half of the major- and minor-axis length of the polarization ellipse, respectively, and the handedness of the ellipse is determined by the sign of \(\varepsilon\), in which the signs \(+\) and \(-\) represent the right-handed rotation and left-handed rotation, respectively.

A balanced polarimeter [28, 29], consisting of a quarter-wave plate (QWP), a Wollaston prism and a balanced detecting system (including two detectors) (see Fig. 1), was employed to measure the polarization state of the output probe beam. In general, an elliptically polarized light with orientation angle \(\varphi\) and ellipticity \(\varepsilon\) of unit light field amplitude [denoted for simplicity as (\(\varphi\), \(\varepsilon\)) in the following] can be expressed in the form of Jones matrix as

$$\begin{aligned} \hat{J}(\varphi ,\varepsilon )=\left( \begin{array}{c} \cos \varphi \cos \varepsilon -i\sin \varphi \sin \varepsilon \\ \sin \varphi \cos \varepsilon +i\cos \varphi \sin \varepsilon \\ \end{array}\right) . \end{aligned}$$
(2)

After the light transmits through a QWP with its principle optical axes oriented at \(45^{\circ }\) with respect to the x-axis, one can write the two light field components as

$$\begin{aligned} \left( \begin{array}{c} E_{x}\\ E_{y} \\ \end{array} \right) =\frac{\sqrt{2}}{2}\left( \begin{array}{cc} 1 &{} i \\ i &{} 1 \\ \end{array} \right) \left( \begin{array}{c} \cos \varphi \cos \varepsilon -i\sin \varphi \sin \varepsilon \\ \sin \varphi \cos \varepsilon +i\cos \varphi \sin \varepsilon \\ \end{array} \right) . \end{aligned}$$
(3)

Note that the two principle optical axes of the Wollaston prism are initially set to be parallel to the x- and y-axis of the Oxy coordinate system. Then, the normalized intensity difference between the two components is simply related to the ellipticity \(\varepsilon\) of the polarization state:

$$\begin{aligned} I_{y}-I_{x}=\sin (2\varepsilon ), \end{aligned}$$
(4)

where \(I_x\) and \(I_y\) are the normalized intensity of the light field component along the x-axis and y-axis, respectively. So the ellipticity \(\varepsilon\) can be determined according to

$$\begin{aligned} \varepsilon =\frac{1}{2}\arcsin (I_{y}-I_{x}). \end{aligned}$$
(5)

By removing the QWP and then rotating the Wollaston prism around the light propagation direction by \(45^{\circ }\), i.e., one of the principle optical axis of the Wollaston prism is tilted at \(45^{\circ }\) with respect to the x-axis, one can measure the orientation angle \(\varphi\) of the output probe beam. In this configuration, the two output light fields linearly polarized along the two principle optical axes of the Wollaston prism are given by

$$\begin{aligned} \left( \begin{array}{c} E_{x'}\\ E_{y'} \\ \end{array} \right) =\frac{\sqrt{2}}{2}\left( \begin{array}{cc} 1 &{} 1 \\ -1 &{} 1 \\ \end{array} \right) \left( \begin{array}{c} \cos \varphi \cos \varepsilon -i\sin \varphi \sin \varepsilon \\ \sin \varphi \cos \varepsilon +i\cos \varphi \sin \varepsilon \\ \end{array} \right) , \end{aligned}$$
(6)

and the normalized intensity difference between the two components is

$$\begin{aligned} I_{y'}-I_{x'}=-\cos (2\varepsilon )\sin (2\varphi ). \end{aligned}$$
(7)

Hence, the orientation angle \(\varphi\) is given by

$$\begin{aligned} \varphi =\frac{1}{2}\arcsin \left( \frac{I_{x'}-I_{y'}}{\cos (2\varepsilon )}\right) . \end{aligned}$$
(8)

Thus the two parameters \(\varphi\) and \(\varepsilon\) we use to characterize the polarization state can be determined by using Eqs. (5) and (8).

Our experiments were performed in a 3-mm-thick \(\hbox {Pr}^{3+}\)-doped YSO crystal (0.05 %), which is an excellent candidate for quantum optical experiments due to its small inhomogeneous broadening, long coherence time of ground states and large oscillator strength of hyperfine transitions at cryogenic temperature [30, 31]. The experimental setup is illustrated in Fig. 1. Two lower levels \((|b\rangle\) and \(|c\rangle )\) and one upper level \((|a\rangle )\) of the transition \(^3H_4\leftrightarrow {}^1D_2\) (\(\sim\)605.78 nm) of \(\hbox {Pr}^{3+}\) ions in YSO crystal were chosen to form a typical three-level \(\varLambda\)-type configuration [10, 16], as shown in Fig. 2. Two continuous-wave beams from a Coherent 899 dye ring laser operating at \({\sim }605.78\) nm with a spectral bandwidth of \(\sim\)500 kHz were served as the probe beam \(\varOmega _\mathrm{P}\) and the coupling beam \(\varOmega _\mathrm{C}\), respectively, where each beam could be temporally modulated in intensity and shifted in frequency by the corresponding acousto-optic modulators (not shown here). The intensity and polarization of the input pulses were controlled by a combination of high-quality HWP and polarizer. The \(\hbox {Pr}^{3+}\):YSO crystal was placed in the confocal point of a confocal system and was kept at  3.4 K in a cryostat. The crystal was oriented in such a way that the two principal axes of the index of refraction were in the input facet of the crystal and parallel to the horizontal x-axis and vertical y-axis of the Oxy coordinate system, respectively. The probe and coupling beams propagated along the crystalline z-axis, and the crossing angle between the coupling and the probe was 0.025 rad in air. The size of the probe beam inside the crystal was 200 \(\upmu\)m. The polarization state of the output probe beam after transmitting through the crystal was measured by using the balanced polarimeter described above.

Fig. 2
figure 2

(Color online) The hyperfine energy-level structure of the transition \(^3H_4\leftrightarrow {}^1D_2\) of Pr\(^{3+}\) ion in YSO crystal, where \(\varOmega _\mathrm{C}\) and \(\varOmega _\mathrm{P}\) are the Rabi frequencies of the coupling and the probe fields, respectively. One upper level \(|a\rangle\) and two lower levels \(|b\rangle\) and \(|c\rangle\) form the typical \(\varLambda\)-type three-level configuration employed in the experiment

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3 Theoretical model, results and discussions

Fig. 3
figure 3

(Color online) The measured absorption coefficient \(\alpha\) of \(\hbox {Pr}^{3+}\):YSO crystal as a function of orientation angle \(\varphi\) at 3.4 K. The angle coordinate origin corresponds to the horizontally polarized light

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In general, the light propagating in a \(\hbox {Pr}^{3+}\):YSO crystal will be decomposed into two orthogonal components. These two components will be of different EIT effect due to the polarization-dependent absorption effect (see Fig. 3) and therefore will experience different refractive index change. So when the probe beam travels inside the crystal, besides a constant crystalline birefringence, it will also experience an extra birefringence induced by the EIT effect, which can be tuned by adjusting experimental parameters such as frequency detuning and intensity of the probe and coupling beams. By combining the absorption anisotropy and the EIT-induced birefringence, it is possible for one to control the polarization state of the output probe beam, which is very important for practical polarization-based applications such as all-optical switching, polarization preserving light pulse memory and polarization qubits.

The hyperfine energy-level structure scheme of \(\hbox {Pr}^{3+}\) ion in YSO crystal is depicted in Fig. 2. Two hyperfine energy levels \(|b\rangle\) and \(|c\rangle\) of \(^{3}H_{4}\) and one hyperfine energy level \(|a\rangle\) of \(^{1}D_{2}\) form the typical \(\varLambda\)-type three-level configuration used in the experiments. The coupling field with a Rabi frequency \(\varOmega _\mathrm{C}=\mu _{ac}E_\mathrm{C}/2\hbar\) couples the transition \(|a\rangle \leftrightarrow |c\rangle\), while the transition \(|a\rangle \leftrightarrow |b\rangle\) is probed by a weak coherent field with a Rabi frequency \(\varOmega _\mathrm{P}=\mu _{ab}E_\mathrm{P}/2\hbar\), where \(\mu _{ac}\) and \(\mu _{ab}\) are the corresponding matrix elements of the dipole moment, and \(E_\mathrm{C}\) and \(E_\mathrm{P}\) are the amplitude of the coupling and probe fields, respectively. The motion for the density matrix element \(\rho _{ij}\) (\(i,j=a,\, b,\, c\)) can be described as [32]

$$\begin{aligned} \dot{\rho }_{ab}= & {} -\varGamma _{ab}\rho _{ab}-i\varOmega _\mathrm{P}(\rho _{aa}-\rho _{bb})+i\varOmega _\mathrm{C} \rho _{cb}, \end{aligned}$$
(9)
$$\begin{aligned} \dot{\rho }_{cb}= & {} -\varGamma _{cb}\rho _{cb}-i\varOmega _\mathrm{P}\rho _{ca}+i\varOmega _\mathrm{C}\rho _{ab}, \end{aligned}$$
(10)
$$\begin{aligned} \dot{\rho }_{ac}= & {} -\varGamma _{ac}\rho _{ac}-i\varOmega _\mathrm{P}\rho _{bc}-i\varOmega _\mathrm{C}(\rho _{aa}-\rho _{cc}), \end{aligned}$$
(11)
$$\begin{aligned} \dot{\rho }_{cc}= & {} -w_{cb}\rho _{cc}+w_{ac}\rho _{aa}+w_{cb}\rho _{bb}-i\varOmega _\mathrm{C}(\rho _{ca}-\rho _{ac}), \end{aligned}$$
(12)
$$\begin{aligned} \dot{\rho }_{aa}= & {} -(w_{ab}+w_{ac})\rho _{aa}-i\varOmega _P(\rho _{ab}-\rho _{ba})-i\varOmega _\mathrm{C}(\rho _{ac}-\rho _{ca}), \end{aligned}$$
(13)
$$\begin{aligned}&\rho _{aa}+\rho _{bb}+\rho _{cc}=1. \end{aligned}$$
(14)

where the complex decay rate \(\varGamma _{ij}\) is defined as \(\gamma _{ij}+i\Delta _{ij}\), \(\gamma _{ij}\) is the decay rate of the transition \(|i\rangle \rightarrow |j\rangle\) and one sets \(\gamma _{ab}=\gamma _{ac}=\gamma\). \(w_{ij}\) is the population relaxation rate, \(\Delta _{ab}\) and \(\Delta _{ac}\) are the frequency detuning between the atomic transition (\(\omega _{ab}\) and \(\omega _{ac}\)) and the corresponding driving field (\(\omega _\mathrm{P}\) and \(\omega _\mathrm{C}\)), respectively, while \(\Delta _{bc}\) is the frequency detuning between the transition \(|c\rangle \rightarrow |b\rangle\) (\(\omega _{cb}\)) and the frequency difference between the coupling and probe fields.

The above density matrix equations can be solved analytically in the steady-state and under the weak-probe approximation, and the density matrix element \(\rho _{ab}\) can be expressed as

$$\begin{aligned} \rho _{ab}= & {} \frac{-i\varOmega _\mathrm{P}}{B}\left\{ -(\gamma _{cb}+i\Delta _{cb})\left( (\gamma ^2+\Delta _{ac}^2)w_{cb}+\varOmega _\mathrm{C}^2\gamma \right) \right. \nonumber \\&\left. +\frac{w_{cb}\varOmega _\mathrm{C}^2(\gamma ^2+\Delta _{ac}^2)}{\gamma -i\delta \omega _{ac}}\right\} \end{aligned}$$
(15)

with

$$\begin{aligned} B= & {} \left[ (\gamma +i\Delta +i\delta \omega _{ab})(\gamma _{cb}+i\Delta _{cb})+\varOmega _\mathrm{C}^2\right] \nonumber \\&\times \left[ 2w_{cb}(\gamma ^2+\Delta _{ac}^2)+\gamma \varOmega _\mathrm{C}^2\left( 1+\frac{3w_{cb}}{w_{ac}}\right) \right] , \nonumber \end{aligned}$$

where \(\delta \omega _{ac}=\omega _{ac}-\omega _{ac0}\) and \(\delta \omega _{cb}=\omega _{cb}-\omega _{cb0}\) are the frequency deviations of the atomic transitions \(|a\rangle \rightarrow |c\rangle\) and \(|c\rangle \rightarrow |b\rangle\) from the corresponding inhomogeneous line centers \(\omega _{ac0}\) and \(\omega _{cb0}\), respectively, and \(\Delta\) is the frequency detuning of the probe beam with respect to the inhomogeneous line center \(\omega _{ab0}\) corresponding to the atomic transition \(|a\rangle \rightarrow |b\rangle\). The susceptibility \(\chi\) should be averaged over the entire range of the inhomogeneous spectral profiles [32]

$$\begin{aligned} \chi =\int \mathrm{d}\omega _{ab}f(\omega _{ab})\int \mathrm{d}\omega _{cb}f(\omega _{cb})\frac{N_\mathrm{eff}\mu _{ab}^2}{2\hbar }\frac{\rho _{ab}}{\varOmega _\mathrm{P}}, \end{aligned}$$
(16)

where \(N_\mathrm{eff}\) is the effective Pr\(^{3+}\) ion density, \(f(\omega _{ij})\) (\(ij=ab\) or cb) is the inhomogeneous frequency distribution function in the Lorentzian line shape with a full width at half maximum \(2W_{ij}\), and it is expressed as

$$\begin{aligned} f(\omega _{ij})=\frac{1}{\pi }\frac{W_{ij}}{\delta \omega _{ij}^2+W_{ij}^2} \, . \end{aligned}$$
(17)
Fig. 4
figure 4

(Color online) The frequency-detuning spectra of the absorption coefficient \(\alpha\) (a) and the EIT-induced refractive index change \(\Delta n\) (b) experienced by the probe beam at two orthogonally polarized eigencomponents of \((\varphi _{0}, \varepsilon _{0})=(0,0)\) (black solid curves) and \((\varphi _{0}, \varepsilon _{0})=(90^{\circ },0)\) (red dashed curves), respectively. The simulation parameters are \(W_{ab}=4.2\times 10^{5}\) Hz, \(W_{cb}=3.0\times 10^{4}\) Hz, \(N_\mathrm{eff}=2.4\times 10^{14} \hbox {cm}^{-3}\), \(\varOmega _\mathrm{C}=200\) kHz, and other parameters are the same as those in Ref. [32]

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The refractive index and the absorption spectra experienced by the probe light can be obtained through \(n=\sqrt{1+{\mathrm{Re}}(\chi )}\) and \(\alpha =\frac{4\pi {\mathrm{Im}}(n)}{\lambda }\), respectively. Figure 4 shows the typical anisotropic spectra of the refractive index change \(\Delta n\) and the absorption coefficient \(\alpha\) for two orthogonally polarized eigencomponents of light fields with \((\varphi _{0}, \varepsilon _{0})=(0,0)\) (black solid curves) and \((\varphi _{0}, \varepsilon _{0})=(90^{\circ},0)\) (red dashed curves), respectively. The typical simulation parameters are the half width of the effective inhomogeneously broadened optical transition \(W_{ab}=4.2\times 10^{5}\) Hz and \(W_{cb}=3.0\times 10^4\) Hz, the effective \(\hbox {Pr}^{3+}\) ion density \(N_\mathrm{eff}=2.4\times 10^{14} \hbox {cm}^{-3}\) and the Rabi frequency of the coupling beam \(\varOmega _\mathrm{C}=200\) kHz. Here, we set the ratio of the absorption coefficient between the two orthogonal components to be 5 according to the measured data in Fig. 3. Other parameters are the same as those in Ref. [32]. One sees that the EIT effect is anisotropic and the EIT-induced birefringence is observed, which will result in a polarization state change in the transmitting probe beam. Interestingly, one notes that there are three positions marked with A, B and C, especially at the two-photon resonant condition corresponding to position B in Fig. 4b where the EIT-induced birefringence disappears. This means it is possible to keep the output probe beam to be linearly polarized with a linearly polarized input probe beam.

The orientation angle of the output probe beam can be calculated through

$$\begin{aligned} \varphi =\frac{1}{2}\arctan \left( \frac{2E_x E_y}{E_x^2-E_y^2}\cos \Delta \phi \right) , \end{aligned}$$
(18)

where \(E_x=E_{x0}e^{-\alpha _x d/2}\) and \(E_y=E_{y0}e^{-\alpha _y d/2}\) are the amplitudes of two orthogonal eigencomponents of the output probe field with \(E_{x0}\) and \(E_{y0}\) being the amplitudes of the corresponding eigencomponents of the input probe field, respectively, \(\Delta \phi =2\pi (\Delta n_x - \Delta n_y) d/\lambda\) is the phase difference between the two orthogonal components of the probe field due to the EIT-induced birefringence, d is the thickness of the crystal and \(\lambda\) is the probe wavelength. Here, the quantities with subscript x (or y) are the ones for the field component along x (or y) axis of the Oxy coordinate system. Once \(\varphi\) is determined, the ellipticity \(\varepsilon\) can then be obtained through Eq. (1) with

$$\begin{aligned} a=\sqrt{E_{x}^{2}\cos ^{2}\varphi +E_{y}^2\sin ^{2}\varphi +E_{x}E_{y}\sin 2\varphi \cos \Delta \phi } \end{aligned}$$

and

$$\begin{aligned} b=\sqrt{E_{x}^{2}\sin ^{2}\varphi +E_{y}^2\cos ^{2}\varphi -E_{x}E_{y}\sin 2\varphi \cos \Delta \phi }, \end{aligned}$$

respectively.

Figure 5 shows a typical example for the polarization state of the output probe beam with an initial input polarization state \((\varphi _{0},\varepsilon _{0})=(20^{\circ },0)\), where the polarization state change induced by the crystalline birefringence has been omitted. The simulation parameters are the same as those in Fig. 4. As expected, when the frequency of the probe beam is scanned from red-detuned to blue-detuned positions, there are three states (A, B and C in Fig. 5a), at which the probe beam remains to be linearly polarized but with different orientation angle as compared to that of the input probe beam. Particularly, one notes that, at the two-photon resonant condition (i.e., under the EIT condition), the coherently prepared \(\hbox {Pr}^{3+}\):YSO crystal serves as a polarization rotator for a linearly polarized probe beam. This is because the EIT-induced birefringence, therefore the phase difference \(\Delta \phi\), disappears at these three positions (see Fig. 4b), while the residual absorption coefficient is still anisotropic as shown in Fig. 4a, which leads to a change in the orientational angle of the probe beam, as predicted by Eq. (18). Nevertheless, large optical depth or high effective ion density may be required to achieve large rotation angle, which could be the major obstacle in doped solids, as we will discuss lately in this paper.

Fig. 5
figure 5

The frequency-detuning spectra of the EIT-induced polarization changes in the ellipticity (a) and the orientation angle (b) of the output probe beam with an initial input polarization state \((\varphi _{0},\varepsilon _{0})=(20^{\circ },0)\) of the probe beam. The simulation parameters are the same as those in Fig. 4

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In the experiments, we first measured the EIT spectrum of the probe beam. A spectrally isolated \(\varLambda\)-type three-level ensemble in an initial state with all population in level \(|b\rangle\) was prepared following the procedures in Refs. [10, 31, 33] by combining the optical pumping and spectral hole-burning techniques. This technique permits selective addressing of well-defined transition in a single ensemble of ions with an effective density \(N_\mathrm{eff}\). For convenience, the frequency offsets in the following were given with respect to the resonant frequency of the atomic transition \(|a\rangle \rightarrow |b\rangle\). The Pr\(^{3+}\):YSO crystal was pumped optically by a 90-ms, 6.4-mW preparation pulse beam, sweeping over a frequency range of \(-12.7\)–5.3 MHz for ten times to create a broad spectral hole. This was followed by a 50-\(\upmu s\), 6.4-mW repumping pulse with a relative frequency shift of 17.3 MHz to create spectral antiholes within the just burned spectral hole. After a 100-\(\upmu s\) delay, a series of 200-\(\upmu s\), 1.2-mW cleaning pulses with a periodical time of 400-\(\upmu s\) and a relative frequency offset of \(-10.2\) MHz were applied for 8 ms to prepare the initial state with all population in level \(|b\rangle\). The EIT spectrum of the probe beam can be obtained by scanning the frequency of the probe beam from \(-1.5\) to 1.5 MHz in 30 \(\upmu s\) while keeping the coupling beam at resonant with the transition \(|a\rangle \rightarrow |c\rangle\). A typical example with a coupling beam of 3.7 mW and a probe beam of 150 \(\upmu\)W is shown in Fig. 6, where the initial input polarization state for both the probe and the coupling beams is set to be (\(\varphi _0\), \(\varepsilon _{0}\)) = (20\(^{\circ }\), 0).

Fig. 6
figure 6

The frequency-detuning spectrum of the probe beam in a Pr\(^{3+}\):YSO crystal with a coupling beam of 3.7 mW and a probe beam of 150 \(\upmu\)W, respectively. Both the probe and the coupling beams are at the same input polarization state \((\varphi _0\), \(\varepsilon _{0})=(20^{\circ }, 0)\)

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Fig. 7
figure 7

The frequency-detuning spectra of the EIT-induced polarization changes in the ellipticity \(\varepsilon\) (a) and the orientation angle \(\varphi\) (b) of the output probe beam. The initial input polarization state of both the probe and the coupling beams is set to be \((\varphi _{0},\varepsilon _{0})=(20^{\circ },0)\), and the coupling beam is of 3.7 mW and the probe beam is of 150 \(\upmu\)W, respectively

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To observe the purely EIT-induced birefringence effect, we compensated for the crystalline birefringence firstly. In this case, the frequency of the probe beam was first tuned to a nearby non-absorptive position, and a linearly polarized light would change into an elliptically polarized state in general after transmitting through the crystal due to the crystalline birefringence. We used a polarization compensator (PC: New Focus, Model 5540) to compensate for the crystalline birefringence so that the probe beam transmitting through the crystal was of the same polarization state as that of the input probe beam. Next, while keeping the frequency of the coupling beam at the resonant position, the frequency of the probe beam was scanned from \(-1.5\) to 1.5 MHz with respect to the two-photon resonant frequency. By using the methods described in Sect. 2, a QWP was inserted after the polarization compensator but in front of the Wollaston prism to measure the ellipticity \(\varepsilon\) of the output probe beam. The orientation angle \(\varphi\) of the polarization state of the output probe beam was later determined by removing the QWP and then rotating the Wollaston prism around the light propagation direction by \(45^{\circ }\) with respect to the Ox axis. The experimental results are shown in Fig. 7, where 7a, b are the spectra of the EIT-induced change in ellipticity \(\varepsilon\) and orientation angle \(\varphi\), respectively, of the polarization state of the probe beam. As expected, the polarization state of the probe beam changes as its frequency-detuning scans, and the probe beam remains to be linearly polarized at the two-photon resonant condition. The experimental results are in qualitative agreement with the theoretical simulation shown in Fig. 5. The deviation is mainly from the refractive index dispersion, the laser jitter and the vibration of the cryostat.

Fig. 8
figure 8

The rotation angle \(\Delta \varphi\) (a), the change in ellipticity \(\Delta \varepsilon\) (b) and the residual absorption coefficient \(\alpha\) (c) experienced by the probe beam, respectively, at different power of the coupling beam under the two-photon resonant condition. The blue empty circles are the experimentally measured data, and the solid curves are the theoretical fits with the same parameters as those in Fig. 4

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Fig. 9
figure 9

The calculated rotation angle \(\Delta \varphi\) (a) and the change in ellipticity \(\Delta \varepsilon\) (b) of the output probe beam at different power of the coupling beam under the two-photon resonant condition with an effective \(\hbox {Pr}^{3+}\) ion density \(N_\mathrm{eff}=2.4\times 10^{15}\,\hbox {cm}^{-3}\). Other parameters are the same as those in Fig. 8

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From Figs. 5 and 7, one sees that the output probe beam remains to be linearly polarized because of the disappearance of the EIT-induced birefringence at the two-photon resonant condition, and therefore the EIT media serves as an effective polarization rotator. The rotation angle \(\Delta \varphi =\varphi -\varphi _{0}\) of the probe beam, where \(\varphi\) and \(\varphi _0\) are the orientation angles of the output and input probe beams, respectively, can be manipulated by adjusting the experimental conditions such as the power of the coupling beam while keeping the coupling and the probe beams at the two-photon resonant condition. Figure 8a shows the measured rotation angle \(\Delta \varphi\) when one changes the power of the coupling beam while keeping the probe beam at 150 \(\upmu\)W. Both the probe and the coupling beams were initially linearly polarized at (\(\varphi _0\), \(\varepsilon _{0}\))=(20\(^{\circ }\), 0). One sees that the rotation angle \(\Delta \varphi\) decreases with the increase in the coupling beam power. As predicted by Eq. (18), this is because the EIT effect is imperfect especially at low coupling beam power as shown in Figs. 4a and 8c, and the residual absorption anisotropy experienced by the probe beam is larger at lower coupling beam power. One may also note that the measured rotation angle \(\Delta \varphi\) and residual absorption of probe beam \(\alpha\) are deviated obviously from the theoretically calculated curves when the coupling beam power is lower than \({\sim }1\) mW. This is because the weak-probe approximation in the theoretical calculation is no longer satisfied with a 150-\(\upmu\)W probe beam when the coupling beam power is lower than \({\sim }1\) mW. Although the demonstrated tuning range of \(\Delta \varphi\) is only around \(4^{\circ }\) in the current conditions, it can be significantly improved simply by increasing the interacting length and the \(\hbox {Pr}^{3+}\)-doping concentration. As an example, Fig. 9a shows the theoretically calculated rotation angle \(\Delta \varphi\) as a function of the coupling beam power when the effective \(\hbox {Pr}^{3+}\) ion density is set to be \(N_\mathrm{eff}=2.4\times 10^{15} \hbox {cm}^{-3}\) at the two-photon resonant condition. One sees that a tuning range of the rotation angle \(\Delta \varphi\) as large as \(40^{\circ }\) is possible. Nevertheless, high doping concentration of \(\hbox {Pr}^{3+}\) ions may result in a degradation of crystal quality, a possible increase in the inhomogeneous spectral linewidth and a decrease in the coherence time of the atomic ensemble. Therefore, it would be a good choice to improve the tuning range of \(\Delta \varphi\) through a combination of a longer crystal length and an appropriate higher \(\hbox {Pr}^{3+}\)-doping concentration. Note that no change in ellipticity was observed both experimentally and theoretically, as shown in Figs. 8b and 9b, where \(\Delta \varepsilon =\varepsilon -\varepsilon _0\) is the change in polarization ellipticity of the probe beam after transmitting through the crystal.

4 Summary

In summary, we characterized both theoretically and experimentally the EIT-induced birefringence in a \(\hbox {Pr}^{3+}\):YSO crystal by using the balanced polarimeter technique. Such an EIT-induced birefringence in a rare earth ion-doped solid was found to be controllable and therefore can be used to control the polarization state of the interacting light fields. We demonstrated that, at the two-photon resonant condition, the ellipticity will be kept unchanged whereas the orientation angle will rotate for a linearly polarized input probe beam, which means that the rare earth ion-doped solids can serve as a controllable polarization rotator. Our results show that the EIT-induced birefringence can provide a novel way to coherently manipulate the polarization state of an interacting light field and therefore will be useful for polarization-based applications such as all-optical switching, polarization preserving light pulse memory and polarization qubits based on rare earth ion-doped solids.