1. Introduction

Ladder-type atomic systems have come to prominence in various fields such as frequency up-conversion [1,2], multi-wave mixing [3], atomic filters [4], and Rydberg gases [5–7]. Spectroscopy of ladder-type atomic systems has been conducted by a variety of methods including optical–optical double-resonance (OODR) [8], double-resonance optical pumping (DROP) [9], two-photon spectroscopy (TPS) [10], and electromagnetically induced transparency (EIT) [11]. Recently, ladder-type EIT is of a growing interest for a variety of applications including the search for Rydberg states [12], slow light [13], and photon-pair generation [14].

EIT is the well-known transparency of the probe laser at two-photon resonance due to two-photon coherence based on three-level atomic systems, while two-photon spectroscopy (TPS) is a representative high-resolution spectroscopy due to two-photon absorption (TPA) in a dipole-forbidden transition between a ground state and an excited state of the three-level atomic system. The non-resonant TPA based on three-level atomic systems is an absorption phenomenon caused by the two-photon coherence between the ground and excited states, where one-photon detuning is far from the transition between the ground and intermediate states [15,16]. Ladder-type TPA due to two-photon coherence is different from two-step excitation due to the one-photon coherence. In the case of the two-step excitation, the population of atoms is excited step-by-step from a ground state into an excited state via an intermediate state. Although both EIT and non-resonant TPA are due to two-photon coherence, they are apparently opposite phenomena (absorption and transmittance) in three-level ladder-type atomic systems.

Under the condition of one-photon resonance, resonant TPA can be observed in open atomic systems with population leakage [17,18]. Recently, Hayashi et al. [19] reported experimental observation of the EIT and two-step excitation in ladder-type three-level systems of a hot sodium atom. As mentioned above, two-step excitation is due to one-photon coherence but TPA is due to two-photon coherence. The two-photon coherence in a three-level atomic system simultaneously contributes to both EIT and TPA. Thus, our question is what determines the occurrence of EIT or TPA. In this paper, we focused on the transformation between absorption and transmittance due to the resonant TPA and EIT in one hyperfine structure to understand the dynamics of the two-photon coherence in TPA and EIT. Although the cause of both resonant TPA and EIT was two-photon coherence effect, the role of two-photon coherence between TPA and EIT has not been clearly explained.

In this paper, we investigate the crossover between the resonant TPA and EIT in the three-level ladder-type atomic system. We experimentally demonstrate that the narrow TPA resolves the hyperfine states (F″ = 1, 2, 3) of the 5D_5/2 state and investigate the transformation of TPA into EIT as the coupling laser intensity increases. The crossover between TPA and EIT is numerically calculated for various coupling laser intensities in the open ladder-type atomic system without any cycling transition. To clarify the relationship between the TPA and EIT, the two-photon coherence is decomposed into a pure two-photon coherence term contributing to the EIT and a mixed term contributing to the TPA. In addition, the effects of the different intermediate state contributions and the relative phase noise between the two lasers are numerically investigated.

2. Experimental results

The TPA spectrum was investigated for the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0, 1, 2)-5D_5/2 (F″ = 1, 2, 3) transitions in ⁸⁷Rb; an energy level diagram is shown in Fig. 1 [16,20]. Although ladder-type TPA has been previously studied in a simple three-level configuration, the energy diagram of a real ladder-type atom is complicated because of the large number of hyperfine states and Zeeman sublevels. To analyze the openness of the real atomic system, optical pumping effects due to spontaneous decays of various paths should be considered [21]. When the weak probe laser is scanned over the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0, 1, 2) transition, atoms in the 5S_1/2 (F = 1) ground state can be optically pumped into the other 5S_1/2 (F = 2) ground state. As shown in Fig. 1, the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition is an open ladder-type atomic system that includes the decays into the other ground state 5S_1/2 (F = 2). The natural linewidths of the 5P_3/2 and 5D_5/2 states are approximately 6.0 MHz and 0.67 MHz, respectively. Our experimental setup is the same as that for typical ladder-type EIT in an atomic vapor cell, i.e., the probe and coupling lasers overlap and counter-propagate through an Rb atom vapor cell [21].

 

Fig. 1 Energy level diagram for the 5S_1/2-5P_3/2-5D_5/2 transitions of ⁸⁷Rb (I = 3/2). The total angular momentum values are given on the left.

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Figure 2(a) shows the resonant TPA peaks of the hyperfine states in the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transitions when the frequency of the probe laser was scanned over the 5S_1/2 (F = 1)-5P_3/2 transition and that of the coupling laser was free-run near the resonance of the 5P_3/2-5D_5/2 transition. The detuning frequency of the probe laser from resonance between 5S_1/2 (F = 1) and 5P_3/2 (F′ = 2) is shown on the horizontal axis. The two lasers were linearly polarized in the perpendicular direction, and the intensities of the probe and coupling lasers were 6.4 μW/mm² and 1.0 mW/mm², respectively. In Fig. 2(a), the black curve is the saturated absorption spectrum of the probe laser of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0, 1, 2) transition, and we can observe clearly the 5D_5/2 hyperfine states (F″ = 1, 2, 3) in the resonant TPA spectrum. The spectral width of the observed TPA spectrum was extremely narrow because of the two-photon coherence contribution.

 

Fig. 2 The spectra in the open and closed ladder-type atomic systems: (a) TPA spectrum in the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition of ⁸⁷Rb (open) and (b) EIT spectrum in the 5S_1/2 (F = 2)-5P_3/2-5D_5/2 transition of ⁸⁷Rb (closed).

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The EIT spectra of the 5S_1/2 (F = 2)-5P_3/2-5D_5/2 transitions in the case of the ladder-type atomic system of the 5S_1/2 (F = 2)-5P_3/2 cycling transition, under the same conditions as those in Fig. 2(a), are shown in Fig. 2(b). The main difference in the experimental conditions for the two cases is the resonant transition of the probe laser. In the case of the TPA, the frequency of the probe laser is resonant to an open atomic system with population leakage, such as the 5S_1/2 (F = 1)-5P_3/2(F′ = 0, 1, 2) transition. However, that of the probe laser for the EIT is resonant to the closed atomic system of the 5S_1/2 (F = 2)-5P_3/2 (F′ = 3) cycling transition. From the results in Fig. 2, we can confirm that the decay rate of the population leakage between the ground and intermediate states significantly affect whether TPA or EIT occurs in the ladder-type atomic system.

To understand the resonant TPA and EIT in detail, we investigated the dependence of the TPA spectra on the coupling laser intensity, as shown in Fig. 3 . Here, the probe laser intensity was fixed at 2.7 μW/mm². In the case of a weak coupling intensity of 0.32 mW/mm², we obtained a narrow TPA spectrum, and the spectral width of the TPA was measured to be 3.9 MHz. In the case of the ladder-type EIT, the spectral width is limited to the Γ_e natural linewidths of the 5D_5/2 excited state [22]. This EIT spectral width is estimated to be 0.67 MHz. The spectral width of the time-averaged beating signal at 1 s was measured to be approximately 1.8 MHz. If the relative phase noise of the two independent lasers used in our experiment is considered, the measured TPA spectral width is close to this limit.

 

Fig. 3 Transformation of TPA into EIT in the 5D_5/2 (F″ = 3) state as a function of coupling laser intensities (with a probe laser intensity of 2.7 μW/mm²).

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As shown in Fig. 3, the shape of the TPA spectra changed significantly as the coupling laser intensity changed. In particular, the absorption of the 5D_5/2 (F″ = 3) state dramatically transformed into transparency when the coupling laser intensity was increased. This means that there is crossover between the TPA and EIT according to the coupling laser intensity. We hypothesize that the two-photon coherence contribution to the EIT is stronger than that to the TPA under conditions of a weak probe and strong coupling. The cause for the transformation from TPA to EIT is discussed in detail in section 3.

In the case of the 5D_5/2 (F″ = 1, 2) hyperfine states, we can see that the TPA spectral shape is distorted and broadened as the coupling laser intensity changes, as shown in Fig. 3. In the distorted TPA signal at a coupling laser intensity of 3.7 mW/mm² there appear to be TPA multipeaks but no EIT is observed. The distortion is due to the atomic coherence effects that depend on the coupling laser intensity. The different behavior of the three hyperfine signals for different coupling laser intensities comes from the difference in the branching ratios of each transition between the hyperfine states. In section 3, the main cause of the change in the TPA spectrum is analyzed using the density matrix equations considering all the degenerate magnetic sublevels under the same conditions as those in our experiment.

3. Theoretical results

To elucidate the coupling-laser-intensity-dependent crossover between the TPA and EIT in the open ladder-type atomic system, we numerically calculated the density matrix equations considering all the degenerate magnetic sublevels of the 5S_1/2-5P_3/2-5D_5/2 transition of ⁸⁷Rb [23]. After calculating the density-matrix elements at given a time and an atomic velocity, the absorption coefficient was averaged over the velocity distribution and transit time. The calculated spectra were averaged over the Maxwell-Boltzmann velocity distribution in order to consider the Doppler-broadened atomic medium. Figure 4 shows the calculated spectra of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0, 1, 2)-5D_5/2 (F″ = 1, 2, 3) transitions for various coupling laser intensities, where the parameters for the numerical calculations are the same as the experimental conditions.

 

Fig. 4 Numerically calculated spectra of the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition for various coupling laser intensities (same conditions as those in Fig. 3).

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A comparison of the calculated and experimental spectra revealed that the calculated spectra were complicated but distinct because the relative phase noise between the two independent lasers was not considered. The spectral profiles of the three hyperfine states in Fig. 3 were seen to be different; in particular, the signal of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 2)-5D_5/2 (F″ = 3) transition is considered as the simple three-level ladder-type atomic system. The TPA and EIT spectra of the 5D_5/2 (F″ = 3) state is simpler than those of the 5D_5/2 (F″ = 1, 2) states, as shown in Fig. 4. In the case of the 5D_5/2 (F″ = 1) state, it is possible to transit from the 5S_1/2 (F = 1) ground state to the 5D_5/2 (F″ = 3) excited state by passing through three intermediate states (F′ = 0, 1, 2). The signals of the 5D_5/2 (F″ = 2) state, however, were two component peaks because two transitions passed through two intermediate states (F′ = 1, 2).

To understand the crossover between the EIT and TPA in detail, we decomposed the calculated spectra into a TPA component and an EIT component as described in previous three-level atomic system research [24]. Figures 5(a) and 5(b) show the EIT component due to the pure-two-photon coherence term and the TPA component due to the mixed term, respectively. In the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition, the EIT component can be calculated by ignoring all the populations of the intermediate and excited states except for those of the ground state, which are set by the thermal equilibrium value.

 

Fig. 5 Decomposition of the spectra in Fig. 4 into the TPA and EIT terms: (a) the calculated pure-two-photon coherence term and (b) the calculated mixed term for various coupling laser.

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As the coupling laser intensity increased, we see increased EIT transmittance signals, as shown in Fig. 5(a). The calculated TPA component (Fig. 5(b)) using the same calculation parameters as those in Fig. 4, is due to the mixed term, which is given by subtracting the one-photon and pure-two-photon coherence terms from the total absorption coefficient. The mixed term gives the two-photon coherence with population change between the ground and excited states [24]. The TPA is different from the two-step excitation which does not require two-photon coherence. The influence of the optical pumping due to one-photon coherence can be negligible, because the probe laser intensity is weak. We assert that the main causes of the absorption and transparency signals in the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition are TPA and EIT due to two-photon coherence.

Comparing the magnitudes of the EIT and TPA components, for a coupling laser intensity of more than 2.4mW/mm², the magnitude of the TPA is nearly constant but that of the EIT increased. Therefore, we concluded that the TPA is dominant for low coupling intensities and the EIT is dominant for high coupling intensities, as shown in Figs. 4 and 5. However, in the case of the closed three-level ladder-type atomic system, the crossover between TPA and EIT also exists but is not observed when the probe laser intensity is weak, because EIT is more dominant than TPA.

To investigate the origin of the multipeaks in the calculated spectra in Fig. 4, we separate the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 spectrum into three spectra with different transition routes according to the 5P_3/2 (F′ = 0, 1, 2) intermediate hyperfine states, as shown in Fig. 6(a) . However, the frequencies of the TPA and EIT in each transition with different intermediate states were not consistent, as shown in Fig. 6(b). This frequency difference is due to the different Doppler shifts of the probe and coupling lasers and the different atomic velocity groups. In the case of the route via the 5P_3/2 (F′ = 0) state, the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0)-5D_5/2 (F″ = 1) transition is allowed because of the selection rule between hyperfine states ($\Delta \text{F}=0,\pm 1$). We can see in Fig. 6(a) the asymmetric spectrum of the F′ = 0 transition, which is composed of an absorption peak and transmittance dip with a different central frequency. The two transitions, 5D_5/2 (F″ = 1 and 2), in the route via the 5P_3/2 (F′ = 1) state and the three transitions, 5D_5/2 (F″ = 0, 1 and 2), in the route via the 5P_3/2 (F′ = 2) state are also allowed, as shown in Fig. 6(b). Comparing the three spectra via each 5P_3/2 (F′ = 0, 1, 2) intermediate hyperfine state, the magnitudes and asymmetry of the calculated signals are different because of the different transition probabilities. When the three spectra via different routes are added, the total spectrum has the complicated structure shown in Fig. 6(a). The multipeaks of the 5D_5/2 (F″ = 1) state are composed of three signals due to three ladder-type transitions via different intermediate states. In the cases of the 5D_5/2 (F″ = 2 and 3) states, two and one transition contributed to the total spectrum, respectively.

 

Fig. 6 (a) Decomposition of the calculated spectrum of the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transitions according to the F′ = 0, 1, 2 intermediate hyperfine states of the 5P_3/2 state. The calculated spectra of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 2)-5D_5/2(F″ = 1, 2, 3) transition (i, ii, and iii), the 5S_1/2 (F = 1)-5P_3/2 (F′ = 1)-5D_5/2 (F″ = 1, 2) transition (iv and v), and the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0)-5D_5/2 (F″ = 1) transition (vi). (b) Transition configurations for counter-propagating probe and coupling lasers interacting with different velocity groups of atoms in the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition.

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However, we can see in Fig. 6 that the central frequencies of the three ladder-type transition signals are slightly different because of the two-photon Doppler shifts of the probe and coupling lasers due to the difference in the respective wavelengths. When the wavelengths of the probe (780 nm) and coupling (776 nm) lasers are different, the Doppler shift frequencies of the probe and coupling lasers in a moving atom are also different. The two-photon resonance condition changes according to the different atomic velocity groups. In our case, the difference in the wavelengths is 4 nm and thus two-photon Doppler shift is present.

The two-photon Doppler shift and corresponding velocity groups were determined by the resonance conditions for the probe and coupling laser frequencies. The two-photon Doppler shift (${\delta }_D$) is given by ${\delta }_D=\frac{k_C-k_p}{k_C}{\Delta }_{2F\text{'}},$where $k_p$ and $k_C$ are the wave number of the probe and coupling lasers, respectively, and ${\Delta }_{2F\text{'}}$ represents the frequency spacing of other hyperfine state from the 5P_3/2 (F′ = 2) state. In the case of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 1)-5D_5/2 (F″ = 1) transition, when the frequency of the coupling laser was fixed at the 5P_3/2 (F′ = 2)-5D_5/2 (F″ = 1) transition and the probe laser was scanned around the 5S_1/2 (F = 1)-5P_3/2 transition, ${\Delta }_{2F\text{'}}$was 156.9 MHz and ${\delta }_D$ was estimated to be 0.89 MHz. In addition, ${\delta }_D$ of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0)-5D_5/2 (F″ = 1) transition was estimated to be 1.3 MHz, with ${\Delta }_{2F\text{'}}$ equal to 229.1 MHz. Therefore, when the three spectra of the three routes via the three intermediate hyperfine states are added, the spectrum of the 5D_5/2 (F″ = 1) state has the complicated multipeak structure shown in the total spectrum in Fig. 6(a). However, because the spectrum of the 5D_5/2 (F″ = 3) state was contributed by the only 5S_1/2 (F = 1)-5P_3/2(F′ = 2)-5D_5/2(″ = 3) transition, the TPA and EIT spectra of the 5D_5/2(F″ = 3) state are useful for understanding the atomic coherence in the open ladder-type atomic system.

However, when the lasers interacted with real atoms, the laser phase noise prevented the observation of the narrow and complicated multipeaks. The relative phase noise between the two independent lasers affects the spectral width and shape, but it is difficult to phase-lock two lasers with a wavelength difference of 4 nm for our experiment. To determine the spectral broadening effect in the experimental results in Fig. 3, we numerically calculated the density matrix equations considering a relative phase noise of 0.8 MHz between coupling and probe lasers. Figure 7 shows the calculated spectrum and the experimental spectrum with the laser spectral broadening at a coupling laser intensity of 3.7 mW/mm². Comparing the two spectra, we see that the calculated spectrum (theory1 in Fig. 7) are in quite good agreement with the experimental result (experiment in Fig. 7). However, we did not consider the laser spectral broadening effect in the calculated results of Figs. 4, 5, and 6 to correctly interpret the origin of the observed TPA spectra, as shown in the green curve (theory2) of Fig. 7.

 

Fig. 7 Comparison the experimental spectrum (red curve) with the calculated spectrum (blue curve; theory1) considering the relative phase noise between the two lasers and the calculated spectrum (green curve; theory2) without the laser spectral broadening effect at a coupling laser intensity of 3.7 mW/mm².

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4. Conclusion

We have investigated the highly resolved TPA and EIT in the open ladder-type atomic system of the 5S_1/2 (F = 1)-5P_3/2 (F′ = 0, 1, 2)-5D_5/2 (F″ = 1, 2, 3) transitions in ⁸⁷Rb. The characteristic of the open ladder-type atomic system is not a cycling transition, and the system is easily optically pumped into the other ground state. The resonant TPA spectrum due to the two-photon coherence was observed in the open ladder-type atomic system, and the TPA spectral width was measured to be 3.9 MHz. The openness of the ladder-type atomic system is an important factor for determining whether TPA or EIT occurs. When the coupling laser intensity was increased, the absorption of the 5D_5/2 (F″ = 3) state dramatically transformed into transparency because of the crossover between the TPA and EIT. Using the experimental parameters and the density matrix equations considering all the degenerate magnetic sublevels of the 5S_1/2-5P_3/2-5D_5/2 transition, we could numerically calculate the TPA and EIT spectra for different coupling laser intensities, which were then decomposed into a TPA term due to the mixed term and an EIT term due to the pure-two-photon coherence term. The observed spectra were shown to be in good agreement with the numerical results that considered the relative phase noise between the two lasers. When the calculated spectrum of the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 transition was separated into three spectra via different transition routes according to the F′ = 0, 1, 2 intermediate hyperfine states of the 5P_3/2 state, the multipeaks of the 5S_1/2 (F = 1)-5P_3/2-5D_5/2 (F″ = 1) transition could be clarified as a result of the two-photon Doppler shifts with three different atomic velocity groups. Our experimental and calculated results are expected to assist in the understanding of the two competing phenomena (EIT and TPA) in three-level ladder-type atomic systems.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant#2012R1A2A1A01006579) and (2011-0009886).