1. Introduction

Phenomena associated with atomic coherence effects are counter-intuitive and interesting. Electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) are representative phenomena due to atomic coherence effect [1,2]. The common characteristic of EIT and EIA is a narrow sub-natural spectral width of spectrum, which finds various applications in, for example, atomic clocks, atomic magnetometers, the manipulation of light pulse, light storage, and quantum teleportation [3–7].

The EIA, which is the opposite phenomenon of EIT, is a consequence of the transfer of coherence (TOC) due to the atomic coherence between degenerate excited levels transferring spontaneously to degenerate ground levels [8,9]. The spectral width of the EIA spectrum is related to atomic coherences of excited states and ground states, and interaction time of atoms with the laser [8–15]. In a pure atomic vapor cell, atomic coherence between ground states is limited by wall-collisions and interaction time is limited by laser beam size. However, in an atomic vapor cell with buffer gas, the buffer gas may increase atomic coherence of the ground state and interaction time due to diffusive atomic motion by the collision with the buffer gas without decoherence of the ground states. However, observation of EIA is difficult in an atomic vapor cell with a buffer gas, as EIA resonances are suppressed when the coherence of the excited state is destroyed by collisions [16,17]. To date, almost all demonstrations of EIA have been investigated in pure atomic vapor cells, and the spectral width of EIA has been limited by wall-collisions and transit-time-broadening [8–15]. EIA spectra with sub-kHz spectral width have not been observed.

An anti-relaxation coated atomic vapor cell without buffer gas can maintain atomic coherence between ground states in spite of wall collisions [17–23]. For EIT, anti-relaxation coated cells, such as a paraffin-coated alkali vapor cell, have been successfully used to increase the decay time of atomic coherence [21–23]. Spectral width narrowing in anti-relaxation coated cells can be explained by the wall-induced Ramsey effect [22]. However, EIA has not been observed in anti-relaxation coated cells due to spontaneous atomic coherence transfer and the limitation of spectral width of EIA in pure vapor cells has not been overcome.

This work reports EIA with narrow sub-kHz spectral width in Hanle configuration of the ⁸⁷Rb D₂-line using a paraffin-coated Rb vapor cell. To the best of our knowledge, this is the first reported observation of a double-structure spectrum composed of a very narrow and a broad EIA spectrum in a paraffin-coated Rb vapor cell. The feature of the EIA spectrum in a paraffin-coated cell was compared with that in a pure cell. Also, the narrow EIA spectra were examined as a function of the angle between the transverse magnetic field and the direction of the linear polarization of the laser.

2. Experimental setup

Figure 1(a) shows an energy level diagram of the 5S_1/2-5P_3/2 transition of ⁸⁷Rb atoms. The natural linewidth of the 5P_3/2 state is approximately 6.07 MHz. Many studies related to EIA have been performed in the closed atomic system of the F_g→ F_e= F_g+1 transition [9–15]. Our experiment for Hanle EIA was also achieved in the F_g=2 → F_e= 3 transition of the D₂ line of ⁸⁷Rb. When the magnetic field used as the quantization axis is parallel to the propagation of the laser in an Rb atomic sample, the σ⁺-transition (Δm=+1) and the σ^--transition (Δm=−1) between the Zeeman sublevels are possible by the linearly polarized laser decomposing into left-circularly polarized laser (σ^-) and the right-circularly polarized laser (σ⁺). As shown in Fig. 1(a), the W-type configuration is generated between the Zeeman sublevels of the 5S_1/2(F=2)-5P_3/2(F’=3) transition [24].

 

Fig. 1 (a) Level diagram for Zeeman sublevels of the 5S_1/2(F=2)-5P_3/2(F’=3) transition of the ⁸⁷Rb atom. (b) Experimental setup for Hanle EIA in a paraffin-coated Rb vapor cell; longitudinal magnetic field (${\overrightarrow{B}}_{\text{s}}$), transverse magnetic field (${\overrightarrow{B}}_{\text{t}}$), laser polarization vector ($\overrightarrow{E}$), and angle (θ _H) between ${\overrightarrow{B}}_{\text{t}}$ and $\overrightarrow{E}$.

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Figure 1(b) shows the experimental setup for EIA in Hanle configuration of the ⁸⁷Rb D₂-line. A 5-cm-long paraffin-coated Rb vapor cell and a 5-cm-long pure Rb vapor cell at room temperature were used to examine EIA in the Hanle configuration. An external cavity laser diode used for the experiment was monitored using the conventional technique for saturated absorption spectroscopy (SAS) in the pure Rb vapor cell. The effect of the Earth’s magnetic field was minimized by a μ-metal chamber. The atomic vapor cell was surrounded by a solenoid coil in the μ-metal chamber. The longitudinal magnetic field (${\overrightarrow{B}}_{\text{s}}$) parallel to the laser’s propagation (z-axis) was scanned around the zero value to obtain the spectrum in Hanle configuration. The residual magnetic field in the μ-metal chamber was measured to be sub-mG. There was a residual transverse magnetic field (${\overrightarrow{B}}_{\text{t}}$) perpendicular to the laser’s propagation in the μ-metal chamber. In our experiment, the dependence of the angle (θ _H) between ${\overrightarrow{B}}_{\text{t}}$ and the linear polarization ($\overrightarrow{E}$) of the incident laser was investigated by varying the direction of the linear polarization of the laser with a half-wave plate. The laser intensity was also adjusted by neutral density (ND) filters.

3. Experimental results & discussion

Figure 2 shows the double structure EIA spectrum of the 5S_1/2(F=2)-5P_3/2(F’=3) transition of ⁸⁷Rb in the paraffin-coated Rb cell. The horizontal axis of the figure is ${\overrightarrow{B}}_{\text{s}}$, which was scanned in the region near the zero value. The laser power was 9.5 μW and the laser polarization was linear. The laser beam diameter was approximately 2 mm and θ _H was zero.

 

Fig. 2 Hanle EIA spectrum of the 5S_1/2(F=2)-5P_3/2(F’=3) transition of ⁸⁷Rb in the paraffin-coated Rb vapor cell, where the horizontal axis of the figure is ${\overrightarrow{B}}_{\text{t}}$ generated by the solenoid coil.

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The double structure EIA spectrum shown in Fig. 2 is due to the spontaneous TOC among the degenerate states from the excited levels to the ground levels. This is first observation of the EIA spectrum with sub-kHz spectral width in an anti-relaxation coated. The spectral width of the narrow EIA spectrum was measured to be 0.55 mG (390 Hz), as shown in Fig. 3(a) . The double structure EIA spectrum is composed of narrow part and broad part of EIA spectrum. The paraffin-coated cell allowed maintenance of not only atomic coherences between the ground states in spite of wall collisions, but also the decoherence of the excited states was free from buffer gas collision [17–22].

 

Fig. 3 Hanle EIA spectra of the 5S_1/2(F=2)-5P_3/2(F’=3) transition of ⁸⁷Rb in (a) the paraffin-coated cell and (b) the pure Rb vapor cell.

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In particular, the narrow EIA resonance was due to the preserved atomic coherence with atoms returning to the laser beam after many collisions with the paraffin coated wall. Atomic coherence due to TOC can be maintained after many collisions with the paraffin-coated wall. The spectral width of the narrow EIA resonance is limited by the dephasing time of atomic coherence among the Zeeman sublevels of the 5S_1/2(F=2) state after many wall collisions.

The broad background EIA resonance in Fig. 2 is due to TOC for the single interaction of atoms with the laser. This is the same as observed for EIA in a pure atomic vapor cell. This EIA spectrum is limited by the interaction time (τ _p) of atoms with the laser. However, we can see the transparency at near the narrow EIA resonance. This transparency is associated with changing the atomic magnetic momentum around ${\overrightarrow{B}}_{\text{s}}$ =0. This is due to the residual ${\overrightarrow{B}}_{\text{t}}$ and the incompletely linearly polarized laser. The residual ${\overrightarrow{B}}_{\text{t}}$ effect according to θ _H is discussed in detail below.

The Hanle EIA spectrum in the paraffin-coated vapor cell was compared with one obtained in a pure atomic vapor cell under similar conditions. Figures 3(a) and (b) show the narrow EIA spectrum in the paraffin-coated vapor cell and the typical EIA spectrum in the pure Rb vapor cell, respectively. The spectral widths of the two spectra were measured to be 0.55 mG (390 Hz) and 147 mG (104 kHz), respectively. Also, the contrasts of the two spectra of Figs. 3(a) and (b) were estimated approximately 0.5% and 3%, respectively. The number of collisions (n _c) is related to the dephasing time (τ _d), and the spectral width of the narrow EIA spectrum decreased proportionally with 1/τ _d. The spectral width in the paraffin-coated vapor cell was measured to be 0.55 mG (390 Hz) and τ _d was estimated to be approximately 2.6 ms. When the diameter of the vapor cell was 25 mm and the mean velocity of the Rb atoms was 270 m/s, the minimum number of collisions with the paraffin-coated vapor wall to retain the atomic coherence was estimated to be approximately 30.

The spectral width of EIA resonance observed in the pure Rb vapor cell was affected the laser power and the interaction time (τ _p) of atoms with the laser. Weakening laser power decreased the spectral width of the EIA resonance proportional to 1/ τ _p. With a 2 mm diameter laser beam and the 270 m/s mean velocity of Rb atoms, τ _p was estimated to be approximately 7.4 μs.

We investigated the spectral width of the narrow EIA spectra in the paraffin-coated vapor cell under the same conditions of Fig. 2. Figure 4 shows the spectral width of the narrow EIA spectra as function of the laser power between 2.1 μW and 14 μW. The power broadening of the spectral width was not observed before the laser power reached approximately 10 μW, as shown in Fig. 4. This means that the spectral width of the narrow EIA spectrum is limited to the dephasing time (τ _d) of the atomic coherence at less than laser power 10 μW. We accomplished our experiment under the condition without broadening spectral width due to laser power in order to obtain the narrow EIA spectrum. Also, as the laser power was increased, the magnitude of the narrow EIA spectra was increased, but the interesting change of the spectral feature was not observed.

 

Fig. 4 Spectral width of the narrow EIA spectra in the paraffin-coated cell as a function of the laser power.

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Recently, the effects of the transverse magnetic field and the polarization of incident laser on EIT and EIA resonances have been reported in pure vapor cells and buffer-gas cells, respectively [25–28]. This paper reports a dramatic change of the narrow EIA peak when the linear polarization of the laser changed from perpendicular to horizontal by a half-wave plate (shown in Fig. 1). Figure 5 shows the narrow EIA spectra according to the angle (θ _H) between the direction of linear polarization ($\overrightarrow{E}$) and ${\overrightarrow{B}}_{\text{t}}$. Intuitively, EIA spectra were not expected to vary according to the angle (θ _H) because used as the quantization axis was parallel to the propagation direction of the laser. However, in the presence of a weak transverse residual magnetic field, the atomic magnetic momentum did vary around ${\overrightarrow{B}}_{\text{s}}$ =0. In our experiment, the residual ${\overrightarrow{B}}_{\text{t}}$ in the μ-metal chamber was estimated to be less than 0.6 mG.

 

Fig. 5 Hanle EIA spectra of the 5S_1/2(F=2)-5P_3/2(F’=3) transition of ⁸⁷Rb in the paraffin-coated Rb vapor cell according to the angle of linear polarization. θ _H is angle between${\overrightarrow{B}}_{\text{t}}$ and $\overrightarrow{E}$.

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When the angle (θ _H) is changed by rotating the direction of linear polarization ($\overrightarrow{E}$) with the HWP, the magnitude of the narrow EIA peak at θ _H = 0 was maximum, as shown in Fig. 5. When the direction of linear polarization ($\overrightarrow{E}$) is parallel to ${\overrightarrow{B}}_{\text{t}}$, quantization axis at ${\overrightarrow{B}}_{\text{s}}$ =0 is taken along ${\overrightarrow{B}}_{\text{t}}$. In this configuration, there is the only π-transition (Δm =0) between the Zeeman sublevels of the 5S_1/2(F=2)-5P_3/2(F’=3) transition. J. Dimitrijević, et. al. reported that the magnitude of EIA in a pure Rb vapor cell is increased, when the transverse magnetic field is parallel to the laser polarization [25]. This is why the atoms of the 5S_1/2(F=2) state are optically pumped into the m_g=0 sublevel of the 5S_1/2(F=2) state. Although there is ${\overrightarrow{B}}_{\text{t}}$ , the m_g=0 sublevel is not shifted by Zeeman effect and the absorption in the m_g=0 sublevel is increased at ${\overrightarrow{B}}_{\text{s}}$=0.

On the other hand, for θ _H = 90°, the the direction of linear polarization ($\overrightarrow{E}$) is perpendicular to ${\overrightarrow{B}}_{\text{t}}$, there is the σ ^± -transition (Δm=±1) between the Zeeman sublevels of the 5S_1/2(F=2)-5P_3/2(F’=3) transition. The W-type configuration with the σ ^± -transitions occurred between the Zeeman sublevels of the ground level (F=2) and the excited level (F’=3). The atoms of the 5S_1/2(F=2) state are optically pumped into the m_g=±2 sublevels of the 5S_1/2(F=2) state. As ${\overrightarrow{B}}_{\text{t}}$ is increased, the m_g=±2 sublevels are shifted by Zeeman effect and the absorption in the m_g=±2 sublevels are decreased. Near θ _H =45°, the N-type EIA configuration can be generated by the superposition of the π-transition and the σ ^± -transition. Therefore, the transverse magnetic field’s direction and magnitude should be considered when analyzing EIA in the paraffin-coated cell.

4. Conclusion

We experimentally demonstrated the EIA with sub-kHz spectral width in Hanle configuration of the ⁸⁷Rb D₂-line in a paraffin-coated Rb vapor cell. The EIA spectrum in a paraffin coated Rb vapor cell had a double structure composed of a very narrow and a broad EIA. The spectral width of the narrow EIA resonance was measured to be 390 Hz. The narrow EIA resonance arose through the maintenance of atomic coherence after many collisions with the paraffin coated walls. Also, the narrow Hanle EIA spectra near ${\overrightarrow{B}}_{\text{s}}$=0 were significantly affected by the angle between the weak transverse residual magnetic field and the direction of linear polarization. The narrow EIA resonance was related not only to the maintenance of atomic coherence but also to the change of the atomic magnetic momentum due to the weak residual transverse magnetic field (${\overrightarrow{B}}_{\text{t}}$) around the zero value of the longitudinal magnetic field (${\overrightarrow{B}}_{\text{s}}$). Narrow EIA resonance is expected to be applied to superluminal pulse propagation with higher speed.