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We report the essential condition for three-photon electromagnetically induced absorption (TPEIA) in a Doppler-broadened ladder-type atomic system. When the two coupling lasers operate at different frequencies, we observed Doppler-free TPEIA resonance at a counterintuitive frequency, which is the almost half-frequency detuning of the frequency difference between the two coupling fields. Considering three-photon coherence in a Doppler-broadened ladder-type three-level atomic system, the TPEIA due to the atomic group of non-zero velocity was in good agreement with the calculated TPEIA spectrum under the one-photon resonance condition of all three optical fields. From the results, we found that an atomic group with a proper velocity for ladder-type TPEIA should satisfy the one-photon resonances of all three optical fields.
© 2015 Optical Society of America
1. Introduction
Multi-photon atomic coherence generated by the interaction of an atom with coherent electromagnetic fields, such as electromagnetically induced transparency (EIT) due to two-photon coherence and three-photon electromagnetically induced absorption (TPEIA) due to three-photon coherence, is a counter-intuitive process. Such multi-photon coherence phenomena can be understood as the consequence of quantum interference effects between quantum transitions [1–4]. Various applications based on multi-photon coherence have been studied, and these include nonlinearity in atomic medium, all-optical switch, and photon-pair generation [5–10]. Many studies on multi-photon coherence have been performed in a Doppler-broadened three-level Λ-type atomic system [11–13]. Recently, three-photon atomic coherence in ladder-type atomic systems has been experimentally and theoretically investigated in detail [14–18].
In a given system, to understand the conditions required for multiphoton atomic coherence occurring is important, regarding the application as well as the fundamental physics. Thus, the Doppler-free configuration for the observation of TPEIA in a Doppler-broadened atomic system is of significance because it is more complex than the one for EIT due to two-photon coherence. To observe the three-photon atomic coherence effect due to quantum interference, not only the condition for three-photon resonance with three coherent fields but also that for two-photon resonance with the probe and coupling fields should be satisfied to maintain the atomic coherences [16]. Under both these conditions, TPEIA has been clearly observed when the two coupling fields have the same frequency.
As shown in Fig. 1(a), TPEIA resonance has been observed for the →→→ transition, because the conditions for both two- and three-photon resonances are satisfied. In the case of two coupling fields with the same frequencies (Fig. 1(a)), when both these conditions are satisfied, the three optical fields automatically satisfy the one-photon resonance condition. However, when the frequencies of the two coupling fields are different (ωC1≠ωC2), we consider the transition configuration for TPEIA without one-photon resonance condition in the presence of two-and three-photon resonance, as shown in Fig. 1(b). As mentioned previously, in the case of Fig. 1(b), we can expect a TPEIA signal because both the two-photon and three-photon resonance conditions are satisfied, even though the optical fields do not satisfy the one-photon resonance condition.
Fig. 1 TPEIA configurations of a three-level ladder-type system for the two cases of (a) identical and (b) nonmatching frequencies (frequency difference = δ) of the two coupling fields.
However, when we experimentally demonstrated TPEIA under the condition of Fig. 1(b) in the 5S1/2–5P3/2–5D5/2 transitions of 87Rb, the TPEIA peak frequency was different from the expected resonance frequency. In this backdrop, in this study, we investigate the spectral features of TPEIA according to the detuning frequency of the two coupling lasers in the case of an 80 MHz frequency difference (δ) between them. To understand the difference between the expected and the measured resonant conditions for TPEIA, the essential condition for TPEIA for the three-photon atomic coherence is investigated via the numerically calculated result by considering three-photon coherence in a Doppler-broadened ladder-type three-level atomic system.
2. Experimental setup
Figure 2 shows the energy-level diagram of the ladder-type atomic system of the 5S1/2–5P3/2–5D5/2 transition of 87Rb and the experimental schematic for TPEIA in Doppler-broadened ladder-type atomic systems interacting with a probe field (Ωp) and two coupling fields (ΩC1 and ΩC2). The frequency of Ωp is scanned around the 5S1/2(F = 2)–5P3/2 transition and those of both ΩC1 and ΩC2 are fixed around the 5P3/2 –5D5/2 transition. In Fig. 2(a), parameters δp, δ C1, and δC2 represent the detuning frequencies of the probe and the two coupling fields, respectively.
Fig. 2 (a) Energy-level diagram of the ladder-type atomic system of the 5S1/2–5P3/2–5D5/2 transition of 87Rb. (b) Experimental schematic for TPEIA. The probe field (Ωp) and two counter-propagating coupling fields (ΩC1 and ΩC2) traverse in a Rb vapor cell (AOM: acoustic optic modulator, PBS: polarization beam splitter, HWP: half-wave plate, IF: interference filter, M: Mirror, and APD: avalanche photodiode).
The fields ΩC1 and ΩC2 counter-propagate through a 5-cm-long natural Rb atomic vapor cell, as shown in Fig. 2(b). The polarizations of the probe and coupling lasers are linear and orthogonal, respectively. The Ωp, ΩC1, and ΩC2 fields well overlap in the Rb atomic vapor cell with the use of two polarization beam splitters (PBSs). The temperature of the vapor cell is maintained at room temperature. The residual magnetic field effect including the earth’s magnetic field is shielded by a three-layer μ-metal chamber. The beam diameters of the probe and coupling fields are about 2 mm. The intensities of the probe and coupling fields are 5.1 μW/mm2 and 9.9 mW/mm2, respectively.
To observe a high-quality TPEIA signal, we ensured complete overlapping of the three laser fields and removed the coupling field (ΩC2) using an interference filter (IF) with a transmittance bandwidth of 1.5 nm and center wavelength of 780 nm. Further, the weak probe field was utilized for the observation of the narrow TPEIA signal with the use of an avalanche photodiode (APD). To measure the TPEIA signal in the case of nonmatching frequencies of ΩC1 and ΩC2, the frequency of ΩC2 could be shifted from that of ΩC1 using an AOM (acoustic optic modulator).
3. Experimental results and discussion
When the frequencies and intensities of ΩC1 and ΩC2 are equal, we observe three TPEIA spectra (Fig. 3, black, red, and blue curves) corresponding to the frequency detunings of the coupling lasers for the 5S1/2(F = 2)–5P3/2–5D5/2 transition of 87Rb. The blue and black curves in Fig. 3(a) represent the ladder-type TPEIA signals of the 5S1/2(F = 2)–5P3/2(F′ = 2)–5D5/2(F″ = 3) and 5S1/2(F = 2)–5P3/2(F′ = 3)–5D5/2(F″ = 4) transitions, respectively. The Doppler-free ladder-type TPEIA peaks are observed at the hyperfine resonances of the 5S1/2–5P3/2 transition [15,16]. The slight asymmetry of absorption profile on the TPEIA spectra originates from the multilevel structure in 5P3/2 state [19]. As shown in Fig. 1(a), this ladder-type TPEIA condition is satisfied only at the resonance of each transition in the case of the two coupling fields having the same frequency. The TPEIA peaks appear only at the resonance frequency for the 5S1/2–5P3/2 transition because of the velocity-selective effect for the three-photon resonance conditions.
Fig. 3 (a) TPEIA spectra as a function of the detuning frequency of the probe laser over each section for three cases (black, red, and blue curves) of the different detuning frequencies of the coupling laser of the 5S1/2(F = 2)–5P3/2–5D5/2 transition. (b) Crossover transition configurations for counter-propagating probe (Ωp) and coupling (ΩC1 and ΩC2) lasers interacting with two counter-propagating velocity groups of atoms presented using simple energy diagrams of the 5S1/2(F = 2)–5P3/2(F′ = 2 and 3)–5D5/2(F″ = 3) transition.
However, we clearly observed a small TPEIA peak at the crossover transition between the 5S1/2(F = 2)–5P3/2(F′ = 2 and 3) transitions, as indicated by the red curve in Fig. 3(a). This TPEIA peak appears to correspond to the peak for the non-resonant case. To explain how the TPEIA peak appears at a non-resonant frequency, we examine the Doppler-shifts of the three lasers considering the direction of the laser propagation and their corresponding velocity groups, as shown in Fig. 3(b). If an atom co-propagates with ΩC1 and counter-propagates with respect to Ωp and ΩC2, the atomic group with a velocity of 103.9 m∕s exhibits Doppler shifts of + 133.2 MHz and + 133.9 MHz with respect to the counter-propagating probe (Ωp) and coupling (ΩC2) lasers, and of −133.9 MHz with respect to the co-propagating coupling (ΩC1) laser. When δp and δC1 are detuned redward to −133.2 MHz for the 5S1/2(F = 2)–5P3/2(F′ = 3) transition and to + 133.9 MHz for the 5P3/2(F′ = 3)–5D5/2(F″ = 3) transition, atoms with a velocity of 103.9 m∕s satisfy the three-photon resonance condition for the four-level atomic system of the 5S1/2(F = 2)–5P3/2(F′ = 2)–5D5/2(F″ = 3)–5P3/2(F′ = 3) transition. In the case of atoms moving along the opposite direction, under the same condition, the configuration for the three-photon resonance condition of atoms with a velocity of –103.9 m∕s corresponds to the 5S1/2(F = 2)–5P3/2(F′ = 3)–5D5/2(F″ = 3)–5P3/2(F′ = 2) transition. The TPEIA signal at the crossover transition can be understood as three-photon resonance due to the unique atomic group in the four-level atomic system with different intermediate states. Therefore, although the velocity of the atomic group for the TPEIA at the crossover transition is different from those corresponding to the black and blue curves in Fig. 3(a), all the TPEIA peaks in Fig. 3 satisfy the three-photon resonance condition, and the three optical fields are resonant at each transition of the one-photon resonance.
So far, TPEIA signals have been observed at the resonance of the three optical fields, as shown in Fig. 1(a), because this configuration satisfies the one-, two-, and three-photon resonance conditions. It is well known that the narrow EIT transmittance due to two-photon coherence has no bearing on one-photon resonance. However, let us consider the case of nonmatching frequencies (ωC1≠ωC2) of the two coupling fields, as shown in Fig. 1(b). In this case, we may assume that TPEIA may be observed under the condition of two- and three-photon resonances in the absence of the one-photon resonance condition, because TPEIA originates due to quantum interference. The black curve in Fig. 4(a) indicates the narrow EIT signal in the absence of ΩC2 for the two-photon resonance of the 5S1/2(F = 2)–5P3/2(F′ = 3)–5D5/2(F″ = 4) transition.
Fig. 4 Absorption spectra for a frequency difference of 80 MHz between δC1 and δC2, EIT spectrum (black curve) of the 5S1/2(F = 2)–5P3/2–5D5/2 transition and the absorption spectra of the probe laser according to several detunings of δC1 and δC2.
To investigate TPEIA for nonmatching frequencies of the two coupling fields, we measured the absorption spectra when the frequency of ΩC2 was shifted by −80 MHz with respect to that of ΩC1 by using an AOM. First, when another coupling laser (ΩC2) shifted by −80 MHz is added under the two-photon resonance condition between the Ωp and ΩC1 fields for the 5S1/2(F = 2)–5P3/2(F′ = 3)–5D5/2(F″ = 4) transition, the blue curve in Fig. 4 does not indicate TPEIA but a distorted EIT spectrum. In this case, the absence of TPEIA is natural, because the shift in ΩC2 by −80 MHz does not satisfy the three-photon resonance condition. Next, to observe the TPEIA signal, we set the detuning frequencies of the three lasers to −80 MHz (δp), 80 MHz (δC1), and 0 MHz (δC2) for Ωp, ΩC1, and ΩC2, respectively. This means that the conditions for two-photon resonance (δp + δC1 = 0) and three-photon resonance (δp + δC1 − δC2 = 0) are satisfied. However, we observed not a TPEIA but an EIT signal under this condition, as indicated by the green curve in Fig. 4. It is not easy to explain the absence of the TPEIA signal in Fig. 4, because the atoms of zero velocity satisfy the conditions for both two-photon resonance (δp + δC1 = 0) and three-photon resonance (δp + δC1 − δC2 = 0).
Interestingly, when δp, δC1, and δC2 are set to −40 MHz, 40 MHz, and −40 MHz, respectively, we observe the TPEIA peak, which is indicated by the red curve in Fig. 4. The observation of the TPEIA signal under this condition is counter-intuitive because there is no three-photon resonance (δp + δC1 − δC2 = −40 MHz).
To understand the Doppler-free TPEIA conditions for nonmatching frequencies of the two coupling fields in a Doppler-broadened ladder-type three-level atomic system, we illuminated the TPEIA signal (red curve in Fig. 4) using a three-level atomic system with a frequency difference of 80 MHz between the two coupling fields in the Doppler- broadened ladder-type system, which is shown in Fig. 5. Figure 5(a) shows the configuration corresponding to the TPEIA signal (red curve in Fig. 4) in the zero-velocity atomic group. In this configuration, although Ωp and ΩC1 satisfy two-photon resonance (δp + δC1 = 0), Ωp, ΩC1 and ΩC2 do not satisfy the three-photon resonance condition.
Fig. 5 Ladder-type three-level atomic system with model considering a frequency difference of 80 MHz between ΩC1 and ΩC2 in (a) zero-velocity atomic group and (b) atomic group with a special velocity of 31.1 m∕s.
However, when this configuration is reconsidered for a moving atomic group, as shown in Fig. 5(b), we observe that this configuration satisfies both the two- and three-photon resonance conditions upon taking into account the Doppler shifts due to the atomic group with a velocity of 31.1 m∕s. This atomic group has a Doppler shift of + 39.9 MHz with respect to the counter-propagating Ωp laser and of −40.1 MHz with respect to the co-propagating ΩC1 laser. When δp is detuned to −39.9 MHz and δC1 is detuned blueward to + 40.1 MHz with respect to the − transition, Ωp and ΩC1 satisfy the two-photon resonance condition. In addition, the Doppler shift of the counter-propagating ΩC1 laser is + 40.1 MHz, and Ωp, ΩC1, and ΩC2 satisfy the three-photon resonance condition when δC2 is detuned redward to −40.1 MHz with respect to the − transition. This ladder-type TPEIA in the case of nonmatching frequencies of the two coupling fields satisfies the one-photon resonance condition due to the presence of the atomic group with the special velocity.
On the other hand, a TPEIA signal is not observed in Fig. 4 (green curve) even though the two- and three-photon resonance conditions are satisfied by the zero-velocity atomic group. Technically, the TPEIA resonance should be observed on the −39.9MHz detuned frequency of the probe field, because the wavelength of probe field (780nm) is not same as that of coupling field (776nm) and that point makes the slightly different Doppler shift by probe and coupling fields for moving atoms. Although we couldn’t rigorously claim that the TPEIA resonance is presented when the frequency of probe field is exactly −39.9MHz detuned, because our typical external cavity diode laser does not have linewidth with the precision of less than 0.1MHz, it is obvious that we only find the TPEIA resonance at the almost half-frequency detuning of the frequency difference between the two coupling fields. This means that an additional condition is necessary for TPEIA to be observed. From the experimental results depicted in Fig. 4 and the energy diagram in Fig. 5, we confirmed that the ladder-type TPEIA conditions should be satisfied for one-photon resonance as well as two- and three-photon resonances.
To theoretically investigate the Doppler-free TPEIA for nonmatching frequencies of the two coupling fields in ladder-type atomic systems, we employed a modified three-level atomic system considering three-photon coherence, as shown in Fig. 5. For the three-level atomic system with a frequency difference of 80 MHz between the two coupling fields, we calculated the density matrix considering the Maxwell–Boltzmann velocity distribution in Doppler-broadened ladder-type atomic systems. Figure 6(a) shows the resulting calculated absorption spectra according to the Doppler shift of the atomic group with different velocities, where the calculation parameters are Ωp = 0.2 MHz, ΩC1 = 10 MHz, and ΩC2 = 8 MHz with a frequency difference of 80 MHz between the detuning frequencies (δC1 = + 40 MHz and δC2 = −40 MHz) of the two coupling fields and relaxation rates of the intermediate and excited states are γ1 = 6 MHz and γ2 = 0.97 MHz, respectively. Considering the decay channel of the 6P3/2 state, for the 5S1/2(F = 2)–5P3/2(F′ = 3)–5D5/2(F″ = 4) transition, the branching ratios of the intermediate and excited states are set to b1 = 1 and b2 = 0.75, respectively. The calculated absorption spectra are symmetric at the center of the TPEIA spectrum, corresponding to the Doppler shift of 40 MHz (velocity of 31.1 m/s). We observe absorption peaks at a probe detuning of nearly −40 MHz, as indicated by the gray dashed line.
Fig. 6 (a) Numerically calculated absorption spectra according to the Doppler shift of the atomic group with velocities varying from 7.8 m/s (10 MHz) to 54.6 m/s (70 MHz). (b) Numerically calculated TPEIA spectrum for a frequency difference of 80 MHz between the ΩC1 and ΩC2 fields.
When the absorption peaks were integrated over the Maxwell–Boltzmann velocity distribution, we calculated the TPEIA with a frequency difference of 80 MHz between ΩC1 and ΩC2, which result is shown in Fig. 6(b). The TPEIA spectrum corresponding to the red curve in Fig. 4 shows good agreement with the numerical calculation. However, upon calculating the spectrum under the condition (δp = −80 MHz, δC1 = + 80 MHz, and δC2 = 0 MHz) corresponding to the green curve in Fig. 4, we obtained the calculated result of the distorted EIT spectrum corresponding to the experimental result. Although our three-level atomic model differs from the real atomic system with hyperfine structures, the experimental TPEIA spectrum for nonmatching frequencies of the two coupling fields appears to be in good agreement with the calculated spectrum. Therefore, we confirmed that the Doppler-free TPEIA in the ladder-type atomic system can be observed for one-photon resonance of the three optical fields in a unique atomic velocity group.
4. Conclusion
We investigated the conditions for ladder-type TPEIA due to presence of the atomic group with non-zero velocity in the cases of the crossover transition of the 5P3/2(F′ = 2 and 3) intermediate state and nonmatching frequencies of the two coupling lasers. The TPEIA peak was clearly observed in the non-resonant case of the crossover transition for the 5S1/2(F = 2)–5P3/2(F′ = 2 and 3) transition. The TPEIA signal at the crossover transition could be attributed to the three-photon resonance due to the presence of atomic groups with a velocity of ± 103.9 m∕s in a four-level atomic system with different intermediate states. It is necessary for observation of the TPEIA of the crossover transition that the three optical fields are one-photon resonant at each transition for the special non-zero velocity atomic groups. When we investigated TPEIA for nonmatching frequencies of the two coupling lasers, Doppler-free TPEIA resonance was interestingly observed at the half-frequency detuning of the frequency difference between the two coupling fields. To illuminate the TPEIA with a frequency difference of 80 MHz between the two coupling fields, we theoretically investigated this case using a Doppler-broadened ladder-type three-level atomic system. Considering the Doppler shifts, the atomic group with a special velocity of 31.1 m∕s satisfied the one-photon resonance condition of the three optical fields. From the experimental and theoretical results, we verified that the TPEIA condition in the ladder-type atomic system should be satisfied with the one-photon resonance of all three optical fields. We believe that our results can contribute to a better understanding of the conditions required for multiphoton atomic coherence occurring due to quantum interference and photon-pair generation from a ladder-type atomic system.
Acknowledgment
This work was supported by the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant No. 2015R1A2A1A05001819).
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