1. Introduction

Electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) are representative phenomena of two-photon quantum coherence [1,2]. The important characteristics of EIT and EIA are a narrow spectral width and a steep dispersion. The steep dispersion property of the coherent atomic medium has been useful for controlling the group velocity of the light pulse, and by tuning the normal dispersion of EIT and the anomalous dispersion of EIA, it is possible to slow down or speed up, respectively, the group velocity of the light pulse. Slow light has been intensively studied for optical buffer and optical quantum memory applications [3–11], but fast light is attractive because of its physical mechanism and the information velocity of fast light, even though the fast light pulse does not violate causality [12–20].

The change of the group velocity due to dispersion in the coherent atomic medium can be easily understood from a classical view of light waves. Because the phase of the light wave is a fundamental wave property, an investigation of how the phase changes by passing through the dispersive medium is important. In the case of EIT, Yu and associates have measured the phases of the storage and retrieved light pulses in an EIT medium and reported preservation of the phases of the slow and retrieved pulses; no phase jump was detected [11, 21]. Furthermore, there have been many studies about the optical properties of slow and retrieved light in the EIT medium, such as the phase, polarization, photon number statistics, and quantum state [21–25].

Fast light has been studied using various proposals to demonstrate fast light and the fundamental issues surrounding the information velocity [12–20]. Fast light propagation with a laser pulse has been demonstrated using anomalous dispersion in various systems [26–29], and fast light in an EIA medium has the advantage that the group velocity advancement can be easily controlled by an adjustment of the coupling laser intensity or atomic density [30]. Although the optical properties of the slow and retrieved light in the EIT medium have been intensively studied, the phase information of a fast light pulse before and after the pulse passes through an EIA medium has not yet been investigated. The group velocity of the fast light pulse is understood as a consequence of superposition of many different frequency components of the light pulse, which are differently phase-shifted due to the anomalous dispersive of EIA. In principle, each frequency component of the light pulse is spread in the infinite time range from negative to positive time. In this paper, we report the direct measurement of phase evolution of the fast light pulse in EIA medium.

This work investigates the phase of the fast light pulse in an EIA medium by using a beat-note interferometer method. We measure the phase of the probe pulse before and after passing through an EIA medium in the 5S_1/2 (F = 2)–5P_3/2 (F′ = 3) transition of ⁸⁷Rb atoms. To remove the relative phase difference of the coupling and probe lasers, the probe pulse, coupling laser, and assist laser for the beating-note interferometer are generated from a single-laser system. Furthermore, by changing the atomic density, the group velocity of the pulse changes, and we can estimate the classical fidelity of the fast light pulse at EIA resonance.

2. Experimental setup

In general, EIA experiments have been achieved in closed atomic systems, which the total angular momentum of the ground state is lower than that of the excited state [29]. We consider the closed atomic system of the 5S_1/2 (F_g = 2)–5P_3/2 (F_e = 3) transition of ⁸⁷Rb, as shown in Fig. 1(a). The Rabi frequencies of the coupling and probe light are denoted by Ω_C and Ω_p, respectively, and Δ_p is the detuning frequency of the probe laser from the resonance of the 5S_1/2 (F_g = 2)–5P_3/2 (F_e = 3) transition. The coupling and probe lasers were orthogonal linearly polarized. Taking the quantization axis to be parallel to the linear polarization of the probe laser, the probe laser can induce the π-transition (Δm = 0) between the Zeeman sublevels, while the coupling laser induces σ⁺ (Δm = + 1) and σ⁻ (Δm = −1) transitions between the Zeeman sublevels. The cause of the EIA is well known as the spontaneous transfer of atomic coherence from the excited states to the ground state.

 

Fig. 1 (a) Energy level diagram of the Zeeman sublevels of the 5S_1/2 (F_g = 2)–5P_3/2 (F_e = 3) transition of the ⁸⁷Rb atom. (b) Experimental setup for the phase measurement of the fast light probe pulse in the EIA medium.

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Figure 1(b) shows the experimental setup for the phase measurement of the fast light probe pulse in the ⁸⁷Rb D₂-line EIA medium. To stably measure the phase difference of the probe laser pulse in the EIA medium, the probe pulse, coupling laser, and assist laser were generated from a single-laser system. The frequency of the external cavity diode laser (ECDL) used for the experiment was locked to the 5S_1/2 (F_g = 2)–5P_3/2 (F_e = 3) transition by using the conventional technique for saturated absorption spectroscopy (SAS). A 5-cm-long pure Rb vapor cell was used to obtain EIA spectra by scanning the probe laser frequency. The Rb cell was installed inside a heating oven for temperature control purposes. The output of the ECLD is split into two parts, which are the coupling and other (probe and assist) lasers. One of the ECLD outputs was used as the coupling laser. After another output of the ECLD is frequency-shifted −80 MHz by AOM1, the frequency-shifted light was separated into the probe and assist lights, again. The probe laser was designed to be a Gaussian pulse by passing the beam through acousto-optic modulator 2 (AOM2), which was driven by Gaussian RF pulse of 80 MHz. To measure the phase of the probe laser pulse using the beat-note interferometer method, we added an assist beam that was generated by AOM3; the driving RF-frequency of AOM3 was adjusted to 82.8 MHz. The probe beam by AOM2 and the assist beam by AOM 3 were combined on the beam splitter (BS). When the electric field amplitudes of the probe and assist laser beams were E_p and E_a, respectively, the beat-note signal was represented as E_p² + E_a² + 2E_pE_a cos(ω_bt + Δϕ), where ω_b and Δϕ are beating frequency and the phase difference between the probe and assist laser beams, respectively. Δϕ is changed by the only phase shift due to the atomic medium, because of the fixed optical paths of the probe and assist laser beams. The probe light pulse (one of the BS outputs) passed through the EIA medium and was detected by avalanche photodiode 1 (APD1), while the reference light (another of the BS outputs) passed through air and was detected by APD2. In this way, we were able to compare the phases of the fast light pulse and the free-space light pulse.

3. Experimental results and discussion

When the RF-frequency of AOM2 was scanned over the range 80 ± 2 MHz, we observed the single-laser system EIA spectrum shown in Fig. 2(a). The frequency is detuned from the resonance frequency of the 5S_1/2 (F_g = 2)–5P_3/2 (F_e = 3) transition. The powers of the probe and coupling lasers were 31 μW and 0.3 mW, respectively, and the laser beam diameter was approximately 2 mm. The temperature of the Rb vapor cell was maintained at 60 °C. The EIA spectrum shown in Fig. 2(a) is due to the spontaneous transfer of coherence among the degenerate states from the excited levels to the ground levels. The spectral width of the EIA spectrum was measured to be 0.7 MHz, which includes the power broadening of the coupling laser. The contrast of EIA spectrum, defined as the ratio of the EIA peak to the magnitude of EIA off-resonance, was estimated to be approximately 80%. The narrow absorption characteristic of the EIA is related to the steep anomalous dispersion property of the coherent atomic medium; the group velocity of the pulse is increased due to steep anomalous dispersion.

 

Fig. 2 (a) EIA spectrum of the 5S_1/2 (F = 2)–5P_3/2 (F′ = 3) transition of ⁸⁷Rb in the pure Rb vapor cell. (b) The reference pulse (black solid curve) and the fast light pulse (red solid curve) under the conditions in (a).

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We could observe the profile of the fast light due to the anomalous dispersion of the EIA, as shown in Fig. 2(b). Figure 2(b) shows normalized pulses of the reference pulse (black solid curve) and fast light pulse (red solid curve). Here, the measured light pulses were averaged 126 times. The Gaussian-shaped probe pulse had a full width at half maximum (FWHM) of 5.4 μs, and the advancement of the fast light pulse was measured to be 0.45 μs. As shown in Fig. 2(b), the signal-to noise ratio (SNR) of the fast light pulse was sufficiently large and the error of advancement time was estimated to be 0.01 μs.

To investigate the phase variation of the continuous-wave (CW) laser probe light induced by the EIA medium, we used the beat-note interferometer method [21]. Figure 3 shows the beat-note interferometer signals of the CW-reference light (black solid curve) in free-space and the CW-probe light (red solid curve) that passes through the on-resonance EIA medium. The beating signals in Fig. 3 measured the interference between the probe and reference light. The frequency of the reference light was detuned by 2.8 MHz from the EIA resonance, giving a beating signal with a frequency difference of 2.8 MHz. Although the reference laser interacted with the Rb atoms, the phase of the reference light was not shifted by the EIA effect because the spectral width of the EIA is 0.7 MHz, as shown in Fig. 2(a). To compare the phases of the reference light and the probe light that passes through the on-resonance EIA medium, we estimated the classical fidelity as follows [11];

 

Fig. 3 Signals of the beat-note interferometer of the CW-reference light (black solid curve) in free-space and the CW-probe light (red solid curve) passing through the on-resonance EIA medium.

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We then measured the phases of the reference and fast light pulses using the beat-note interferometer method with AOM3 turned on and observed the reference light pulse (gray solid curve) and fast light pulse (red solid curve) signals shown in Fig. 4(a). In particular, we used the low beating frequency of 2.8 MHz for the phase measurement of the fast light pulse to observe not only the phase maintenance but also the phase-shift of the fast light pulse. The beating signal of the probe light pulse is generated by the interference between the probe pulse light and the reference light in free-space, and the envelope of the beating signal is consistent with the pulse shape of the probe light. The temperature of the Rb vapor cell was maintained at 50 °C. The beating signal of the fast light pulse reveals that the envelope of the beating signal shifted to a negative time, and the advancement of the fast light pulse was measured to be 0.201 μs, which corresponds to approximately one half period of the beat signal in Fig. 3.

 

Fig. 4 (a) Measured phases of the reference (gray solid curve) and fast light (red solid curve) pulses using beat-note interference. (b) The magnified beating signals at the front (upper) and posterior (lower) of both pulses.

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By comparing the phases of the reference and fast light pulses, we find that the phase of the fast light pulse is the same as that of the reference light pulse. Although the group velocity of the fast light pulse is faster than that of the reference light pulse, there is no phase difference relative to the reference light pulse. To compare both phases in detail, Fig. 4(b) shows magnified views of the front and posterior of both pulses in Fig. 4(a); no phase difference can be seen. The classical fidelity was estimated to be approximately 99.5%. However, if the phase of the fast light pulse is shifted with the envelope of the beating signal, the classical fidelity would be approximately less than 20%, because the phase difference of the beat signal between the reference and fast light pulses is π, which corresponds to approximately one half period of the beat signal. The group velocity of the fast light pulse is understood as being a consequence of the superposition of many different frequency components in the light pulse, which are phase-shifted by a different amount due to the anomalous dispersion of EIA. In principle, each frequency component of the light pulse is dispersed in an infinite time range, from negative to positive time, and therefore, the phase information of the fast light pulse does not differ from that of the probe pulse light in the negative time region.

The group velocity of the fast pulse is affected by the atomic density, which is related to the temperature of the atomic vapor cell. When the atomic density is increased, the contrast in the EIA spectrum mainly increases before the absorption effect saturates. To control the advancement of the fast light pulse, we investigated the effects of varying the atomic density. Figure 5(a) shows the advancement of the fast light pulse as function of the temperature of the Rb vapor cell. The measured light pulse signals were averaged 126 times and each measurement was repeated more than five times. The powers of the probe and coupling lasers were 0.70 μW and 0.11 mW, respectively. As the cell temperature increased to up to 65 °C, the advancement of the fast light pulse also increased. The errors of the data points of Fig. 5(a) were estimated to be from 0.01 μs to 0.02 μs. The reason of the error increase is the attenuation effect of the pulse amplitude due to the high atomic density. When the classical fidelity of the fast light pulses was estimated, as shown in Fig. 5(b), the classical fidelity was estimated to be more than 98%. However, when the cell temperature is more than 70 °C, the magnitude of the EIA peak decreases and the advancement of the fast light pulse decreased because of the high atomic density. The main cause of the classical fidelity decrease is the attenuation effect of the pulse amplitude due to the high atomic density. Therefore, if the advancement of the fast light pulse is sufficiently large, it is possible to retrieve the phase of the light pulse in the negative time region. From these results, we were able to confirm that the phase of the original probe light was maintained in the negative time region.

 

Fig. 5 (a) Advancement of the fast light pulse and (b) estimation of the classical fidelity as function of the temperature of the Rb vapor cell.

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4. Conclusion

We have experimentally demonstrated the phase measurement of a fast light pulse in an EIA medium consisting of the 5S_1/2 (F = 2)–5P_3/2 (F′ = 3) transition of ⁸⁷Rb atoms. When the phase of the fast light pulse was investigated using the beat-note interferometer method with the single-laser system, although the envelope of the beating signal in the EIA medium was advanced to one half period of the beat signal, no phase difference between the reference and fast light pulses was observed. The classical fidelity was estimated to be approximately 99.5%. This result indicates that the phase of the fast light pulse at EIA resonance was maintained to be the same as that of the free-space light pulse; there was no phase shift from the interaction with the EIA medium. We also found that the classical fidelity was relatively constant and nearly independent of the advancement of the fast light pulses. From our results, we were able to verify that the phase of the fast light pulse was maintained in the negative time region.