1. Introduction

The representative phenomena of the atomic coherence effect owing to the interaction of a multi-level atom with two lasers are electromagnetically induced transparency (EIT) [1] and electromagnetically induced absorption (EIA) [2–4]. The optical properties of an atomic medium due to atomic coherence are dramatically changed, for example, to transmittance or absorption with a narrow sub-natural spectral width. The narrow spectral characteristic of atomic coherence phenomena can be applied to atomic clocks, atomic magnetometers, frequency stabilization of lasers, and the manipulation of light pulse speed [5–9]. These applications have been used in the two-photon resonance condition as a classical view.

However, coherently laser-driven atomic systems exhibit large nonlinearity and rich quantum interference effects [10]. The atomic coherence is interpreted by quantum interference among alternate pathways for the transition. The quantum effects of EIT have been theoretically and experimentally studied in the field of quantum optics [11–13]. The correlation between Stokes and anti-Stokes photons due to four-wave mixing under EIT has been demonstrated [14–21]. Alzar et al. have demonstrated the intensity fluctuation between the coupling and probe lasers after the interaction of the lasers with an EIT medium [22]. Scully and associates have experimentally and theoretically studied the intensity fluctuation in coherently prepared Rb vapor [23–25]. The quantum coherence effects have been applied to quantum memory, atomic quantum repeaters, squeezed photons, and paired-photon generation [11–21].

EIA, which is a phenomenon opposite to EIT, is also an atomic coherence effect that leads to the substantial absorption enhancement of a probe field. The fundamental mechanism responsible for EIA is the transfer of coherence (TOC), which is the atomic coherence between degenerate excited levels transferring spontaneously to degenerate ground levels [2–4,26–31]. As is noted above, the TOC of EIA should be also interpreted as the quantum interferences among competing multi-photon pathways owing to the interaction of atoms with photons [32]. Although many studies about have investigated the intensity fluctuations in EIT have been performed, the quantum effects of EIA have not been experimentally studied.

In the present work, we focused on the intensity fluctuation between two components of coupling and probe fields in the EIA Hanle configuration of the ⁸⁷Rb D₂-line using a Rb vapor cell. To the best of our knowledge, this is the first reported observation of the intensity fluctuation between coupling and probe fields in an EIA medium. The second-order correlation between the two intensities of coupling and probe fields was investigated based on the longitudinal magnetic field.

2. Experimental setup

Figure 1(a) shows an energy-level diagram of the transitions among the magnetic sublevels of the 5S_1/2 (F_g = 2)-5P_3/2 (F_e = 3) transition of ⁸⁷Rb atoms. The cycling transition for EIA is the closed atomic system of the F_g → F_e = F_g + 1 transition [2–4]. The longitudinal magnetic field parallel to the propagation of the laser in a Rb atomic vapor cell has been used for the Hanle configuration. When the longitudinal magnetic field is used as the quantization axis, the σ⁺ transition (Δm = + 1) and the σ^– transition (Δm = −1) occur between the magnetic sublevels. After interaction with the EIA medium, the left circularly polarized laser (σ^–) and right circularly polarized laser (σ⁺) are expected to correlate strongly due to the spontaneous coherence transfer and quantum interference effects.

 

Fig. 1 (a) Level diagram for Zeeman sublevels and (b) the typical Hanle EIA spectrum of the 5S_1/2(F_g = 2)-5P_3/2(F_e = 3) transition of ⁸⁷Rb atom.

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When the longitudinal magnetic field was scanned in the region near the zero value, the typical Hanle EIA spectrum of the 5S_1/2(F_g = 2)-5P_3/2(F_e = 3) transition of the ⁸⁷Rb atom was as shown in Fig. 1(b). The horizontal axis in Fig. 1(b) represents the longitudinal magnetic field, where the linearly polarized laser power was 9.5 μW. The spectral width and contrast of the EIA spectrum were measured to be 147 mG and 3%, respectively. Compared to the natural linewidth 6.07 MHz of the 5P_3/2 state, the spectral width of the EIA spectrum of Fig. 1(b) was very narrow and estimated to be 104 kHz.

Figure 2 shows the schematic of our experimental setup for the intensity fluctuation in EIA. An external cavity diode laser (ECDL) is tuned to the D₂ line (λ = 780nm) of ⁸⁷Rb. The frequency of the laser was fixed on the 5S_1/2(F_g = 2)-5P_3/2(F_e = 3) transition in the ⁸⁷Rb D₂ line and monitored using a conventional technique for a saturated absorption spectroscopy (SAS). Before incidence onto the Rb atomic vapor cell, the laser beam was linearly polarized at 45° to the normal axis using a polarizing beam-splitter (PBS) and a half-wave plate (HWP) in order to adjust the balance between avalanche photodiodes (APDs) 1 and 2. Comparing two experimental schemes, in the case of the previous study [23], the phase fluctuation between two orthogonally polarized laser beams can be generated due to mechanical vibration and air flow. Therefore, the phase fluctuations of incident laser include the phase fluctuations in the combined process and the intrinsic phase fluctuation of laser. In our experiment, the intensity fluctuation was considered the effect due to the intrinsic phase fluctuation of a incident laser.

 

Fig. 2 A schematic of the experimental setup for intensity fluctuation in an EIA medium; ECLD: external cavity laser diode, IS: isolator, PBS: polarizer beam splitter, HWP: half wave plate, APD: avalanche photodiode (frequency bandwidth of 30kHz to 1.2 GHz), QWP: quarter wave plate.

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The atomic vapor cell was surrounded by a solenoid coil in a μ-metal chamber. The longitudinal magnetic field parallel to the laser beam was generated by a solenoid coil and scanned around the zero value to obtain the spectrum in the Hanle configuration. The effect of the Earth’s magnetic field was minimized by a μ-metal chamber. A heating system was also installed to control the atomic density of the atomic vapor cell. The laser was propagated through a 2-mm-diameter aperture and the Rb vapor cell. After it passed though the Rb vapor cell and interacted with the atoms, the laser was passed through a quarter-wave plate (QWP) and a second PBS to discriminate the intensity fluctuations between the counter-rotated circular polarization components. The two separated beams were detected by two avalanche photodiodes (APDs) (ET2030A; frequency bandwidth of 30 kHz to 1.2 GHz). We used the RF-amplifiers (ZFBT-4R2G-FT + ) to amplify the small signals obtained by two APDs. The amplified signals are enough larger than the noise of the two APDs. To collect all the light passing through the medium, we put two lenses (f = 50mm) in front of the two APD, individually. The optical path lengths for both the beams were kept approximately the same to avoid a time delay between the signals in the two channels. Signals from the amplifiers were recorded by a NI-5114 digitizer with a bandwidth of 250 MHz. The intensity fluctuations between the two beams could be calculated from the recorded data.

3. Experimental results and discussions

Figures 3(a) and 3(b) show the measured intensity fluctuations of the two opposite circular-polarization components of EIA under the conditions of on-resonance and off-resonance. The laser power was 3 mW, the laser beam diameter was approximately 2 mm, and the temperature of the Rb vapor cell was 50 °C. In the on-resonance case for the graph shown in Fig. 3(a), the intensity fluctuations of both $\delta I_{\text{C}}$ and $\delta I_{\text{p}}$were estimated to be ± 10 mV, where $\delta I_{\text{C}}$ and $\delta I_{\text{p}}$ represent the intensity fluctuations measured by APDs 1 and 2, respectively. However, when the longitudinal magnetic field was 200 mG, the intensity fluctuations for off-resonance were estimated to be approximately ± 50 mV, which is almost five times higher than those for on-resonance. Interestingly, the signal behaviors of the intensity fluctuations on- and off-resonance were opposite to each other.

 

Fig. 3 Measured intensity fluctuations of two opposite circular-polarization components of EIA under (a) on-resonance and (b) off-resonance conditions. Second-order correlation function g⁽²⁾(τ) for intensity fluctuations of two components of EIA as a function of time delay τ under (c) on-resonance and (d) off-resonance conditions.

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The intensities of the two circular-polarization components (I_C, I_p) after they have passed through the EIA medium can be expressed as

Here, $〈I_{\text{C},\text{p}}〉$ and $\delta I_{\text{C},\text{p}}$ represent the average value of intensity during the measurement time and the magnitude of fluctuation from the average value, respectively. In our experiment, we used APDs and measured only the second term in Eq. (1). To confirm the intensity-intensity fluctuation between the two circular-polarization components in the EIA medium, we calculated the second-order correlation function g⁽²⁾(τ) for the intensity fluctuations of the two components as a function of the time delay τ. The second-order correlation function g⁽²⁾(τ) is described as follows:

To calculate g⁽²⁾(τ) in our experiment, the data acquisition time of intensity fluctuations was 4 μs, and the average number of calculations of g⁽²⁾(τ) was 200. In the case of on-resonance of EIA, Fig. 3(c) shows the calculated g⁽²⁾(τ) from the results of Fig. 3(a). The value of g⁽²⁾(0) was calculated to be 0.35. Interestingly, the oscillation of the g⁽²⁾(τ) background was observed near 0.23. The amplitude of the oscillating background in on-resonance of EIA is larger than that in off-resonance of EIA. In the case of on-resonance of EIA, as the temperature of the vapor cell and the laser intensity are increased, the amplitude of the oscillating background is increased, but the period is not significantly changed.

The width of the correlation peak was estimated to be approximately 17.5 ns. When it was detuned from EIA resonance, the value of g⁽²⁾(0) was calculated to be −0.93, as shown in Fig. 3(d). A negative value of g⁽²⁾(0) implies anti-correlation between $\delta I_{\text{C}}$ and $\delta I_{\text{p}}$. The width of the anti-correlation peak was estimated to be approximately 15 ns. The width of the correlation peak is inversely proportional to the population decay rate of the excited state and the Rabi frequency, which is related to the spectral width (Δν) of the excited state. The width of the correlation peak is 1/Δν. The width of the correlation peak was narrower than the lifetime of the 5P_3/2 state, which was approximately 26 ns (natural linewidth of 2π × 6.07 MHz). In general, as the laser intensity and the atomic density are increased, the spectral width (Δν) of the excited state in broaden. Therefore, the causes of correlation peak narrowing are the collision broadening due to vapor cell heating and the power broadening due to high laser intensity.

Under the same experimental conditions as those used to obtain the data in Fig. 3, g⁽²⁾(τ = 0) was measured as a function of the applied longitudinal magnetic field, as shown in Fig. 4 . The value of g⁽²⁾(τ = 0) was continuously changed from 0.35 (correlation) to −0.93 (anti-correlation) according to the applied magnetic field. We can explain the mechanism for the correlation and the anti-correlation of the intensities of two lasers in EIA medium as the similar with that of EIT [23]. The intensities of two lasers are correlated in on-resonance of EIA is because the phase fluctuations of two lasers lead to the same detuning. So, the direction of the two intensity fluctuations is the same. On the other hand, the anti-correlation of the intensities of two lasers in off-resonance of EIA is because of the different detuning of the two lasers, that is, the one laser frequency is tuned closer to the resonance but another is tuned out of resonance. The distribution of g⁽²⁾(τ = 0) according to the applied longitudinal magnetic field showed a resonance-like Lorentzian curve. The width of g⁽²⁾(τ = 0) was measured to be 22 mG, which was narrower than the EIA spectral width. As is shown in Fig. 4, the anti-correlation of EIA off resonance was stronger than the correlation of EIA on resonance. This result can be explained with the help of the EIA process, in which the absorption of laser energy is maximal in EIA resonance. Despite the attenuation of the incident laser interacting with EIA medium, we could measure the correlation and anti-correlation of EIA due to the quantum interference effect.

 

Fig. 4 Measured g⁽²⁾(τ = 0) values as a function of the applied longitudinal magnetic field.

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Comparing the intensity fluctuation for EIA with EIT, the maximum value of g⁽²⁾(0) calculated in on-resonance of EIA is smaller than that of the previously reported EIT [23]. As reported in fluctuation of EIT, two mechanisms for generation of intensity fluctuation in an atomic medium are the phase fluctuations of laser and the four-wave mixing of vacuum modes [23]. However, the four-wave mixing effect in EIA medium is relatively weaker than that in EIT, because of the enhanced absorption of EIA. In contrast, the nonlinear process in EIT medium is stronger at higher intensity. Therefore, the intensity fluctuation in on-resonance of EIA is smaller at the maximum of the absorption. The difference between the maximum values of g⁽²⁾(0) in EIT and EIA is because the effect of phase noise conversion to intensity noise is dominant in EIA medium. On the other hand, as shown in Fig. 3(a) and 3(b), the intensity fluctuation in off-resonance of EIA is larger than that in on-resonance of EIA. This is the reason why the phase fluctuations of laser induce a large intensity fluctuation at the maximum slope of the absorption curve.

We investigated the dependence of g⁽²⁾(0) on the temperature of the vapor cell. The atomic density of a vapor cell depends on its temperature. As the atomic density is increased, the interaction with atoms is increased. The temperature of the vapor cell affects the correlation or anti-correlation, because of the EIA medium. Figure 5(a) shows g⁽²⁾(0) as a function of the applied magnetic field according to the temperature of the vapor cell, for which the power of the incident laser is 2.0 mW. Generally, when the temperature of the vapor cell is low, the decay rate between atomic states is decreased because of the low collision rate of atoms. However, the width of g⁽²⁾(0) increased with decreasing temperature of the vapor cell, as shown in Fig. 5(a). Upon comparing g⁽²⁾(0) values at 20 °C with those at 50 °C for the same applied magnetic field of 30 mG, the anti-correlation g⁽²⁾(0) at 50 °C was found to be stronger than the anti-correlation g⁽²⁾(0) at 20 °C, because the interaction with atoms was increased.

 

Fig. 5 (a) g⁽²⁾(0) values as a function of the applied magnetic field; (b) spectral width (solid circles) and contrast (open circles) of EIA spectra, according to the temperature of the vapor cell.

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The g⁽²⁾(0) values of the EIA medium (B = 0) on resonance, according to the temperature of the vapor cell, are not significantly different, as shown in Fig. 5(a). As the atomic density is increased, the transmitted laser power is decreased because of the enhanced absorption of EIA resonance, but the interaction with atoms is increased. The reason is a compensation between the two effects of the attenuation of the incident laser and the increase in atomic density. In the case of off-resonance, for EIA (B = ± 200 mG), as the temperature of the vapor cell is increased from 20 °C to 50 °C, the g⁽²⁾(0) magnitude of anti-correlation increases from −0.4 to −0.9. The transmitted laser power of EIA for off resonance is higher than that for on resonance. Therefore, the increase in the g⁽²⁾(0) value of anti-correlation according to atomic density is due to the increase of the atoms interacted with lasers.

The properties of EIA spectrum may be related to the values of second-order correlation function g⁽²⁾(0) as a function of the applied longitudinal magnetic field. Figure 5(b) represents the spectral width (solid circles) and the contrast (open circles) of EIA spectra according to the temperature of the vapor cell for which the power of the incident laser is 2.0 mW. As the temperature of the vapor cell was increased, the spectral width of the EIA spectra slightly increased because of the high collision rate of atoms and its contrast increased because of the high atomic density. The high contrast of EIA can strongly induce the intensity fluctuations interacted with atoms. Therefore, the width of the g⁽²⁾(0) value is closely related to the interaction degree with atoms and the contrast of EIA spectra according to the temperature of a vapor cell.

We also measured the dependence of g⁽²⁾(0) on the incident laser power. Figure 6 shows g⁽²⁾(0) as a function of the applied magnetic field according to the incident laser power, and in this case, the temperature of the vapor cell was fixed at 50 °C. The g⁽²⁾(0) values of EIA medium on resonance (B = 0), according to the laser power, changed from zero to 0.32, as shown in Fig. 6(a). As the incident laser power was decreased, the transmitted laser power was dramatically decreased because of the enhanced absorption of EIA resonance. When the incident laser power was less than 0.8 mW, the g⁽²⁾(0) values were estimated to be zero because the incident laser was nearly attenuated. In the case of off-resonance, for EIA (B = ± 200 mG), as incident laser power was increased from 0.3 mW to 3.0 mW, the g⁽²⁾(0) value of anti-correlation changed from −0.2 to −0.93, as shown in Fig. 6(a).

 

Fig. 6 (a) g⁽²⁾(0) values as a function of the applied magnetic field; (b) spectral width (solid circles) and contrast (open circles) of EIA spectra, according to the incident laser power.

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Figure 6(b) shows the spectral width (solid circles) and the contrast (open circles) of EIA spectra according to the power of the incident laser for which the temperature of the vapor cell is 50 °C. As the power of the incident laser was increased, the spectral width of the EIA spectra increased because of spectral power broadening and its contrast increased nonlinearly. As the incident laser power is increased, the intensity fluctuation increases because of the contrast of EIA.

Figure 7(a) shows g⁽²⁾(0) values at B = + 200 mG as a function of the incident laser power. The experimental data was fitted with a first-order exponential decay as shown by the gray curve in Fig. 7(a). From the analyzed result of Fig. 7(a) under the condition of off-resonance, for EIA, the g⁽²⁾(0) value of anti-correlation increased exponentially according to the incident laser power. As the incident laser power is increased, the intensity fluctuation increases because of the contrast of EIA, as shown in Fig. 6(b). Also, as the incident laser power was decreased, the transmitted laser power was dramatically decreased and the anti-correlation g⁽²⁾(0) values was also decreased. Comparing Fig. 6(a) and Fig. 7(a), we can see that the contrast of EIA spectra is significantly related to the anti-correlation g⁽²⁾(0) values in off-resonance of EIA. In addition, we analyzed the increase in the width of g⁽²⁾(0) with the incident laser power, as shown in Fig. 7(b). As was already known from Fig. 6, the width of g⁽²⁾(0) values did not change according to the incident laser power. The average width of g⁽²⁾(0) values was estimated to be 24.7 mG. Therefore, the width of g⁽²⁾(0) did not affect the incident laser power, but rather the temperature of the vapor cell.

 

Fig. 7 Under the condition of off-resonance, for EIA, (a) g⁽²⁾(0) values at B = + 200 mG as a function of the incident laser power and (b) the width of g⁽²⁾(0) distribution as a function of the incident laser power.

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4. Conclusion

We experimentally demonstrated the intensity-intensity fluctuation of EIA in a Rb vapor cell in the Hanle configuration of the 5S_1/2(F = 2)-5P_3/2(F' = 3) transition of ⁸⁷Rb atoms. The correlation and anti-correlation between the intensity fluctuations of the two components of circular polarization for EIA were first observed according to the conditions of on-resonance and off-resonance. In particular, we showed that the atomic coherence between degenerate excited levels transferring spontaneously to degenerate ground levels can generate the intensity fluctuations between the coupling and probe fields. These fluctuations of EIA could be explained as the conversion of phase noise to amplitude noise in the atomic medium, similar to the fluctuation mechanism of EIT. The distribution of g⁽²⁾(τ = 0) values as a function of the applied longitudinal magnetic field has a resonance-like Lorentzian curve. The value of g⁽²⁾(τ = 0) changed from 0.35 (correlation) to −0.93 (anti-correlation), and the width of g⁽²⁾(τ = 0) was measured to be 22 mG. In the EIA medium, the anti-correlation of EIA off resonance was stronger than the correlation of EIA on resonance. Further, we investigated the dependence of g⁽²⁾(0) on the temperature of the vapor cell and the incident laser power. The value and the width of g⁽²⁾(0) changed according to the atomic density and the incident laser power. Recently, the atomic coherence has been studied for application to quantum optics and nonlinear optical processes, for which a strong correlation between coupling and the probe field is important. We believe that our results will help to understand the quantum properties of the atomic coherence.