A two-level system driven by a strong bichromatic laser field would lead to novel features which are not present in the monochromatic case. Complicated dynamics in new regimes of light–matter interactions, such as interference-induced spectral line elimination and the fluorescence or probe absorption spectra, have also been studied extensively both theoretically and experimentally. Previous works in these regions have been reported by many groups.^[1–6] There are also some meaningful researches for resonant bichromatic laser fields.^[7,8] The very recent work by He et al.^[9] is the first demonstration of spontaneously induced spectral cancelation using the transitions of a solid-state quantum emitter. In 2014, Ruan et al.^[10] measured the spectroscopic properties of singly dressed states in a strong monochromatic field, but no quantitative theoretical calculation was given. This paper is the further work of Ruan et al.^[10] We concentrate on the cases where analytical solutions for a doubly driven two-level atom can be found and give a quantitative comparison of theoretical and experimental data.

It is well known that the analysis formula for the energy level diagram of a singly driven two-level system can be given by the two-level dressed-atom approach.^[11,12] Therefore, the analytical solutions for a system of a two-level atom driven by a strong bichromatic laser field would be expected if the light–matter interactions in such system can be simplified as a two-step process: the first step is that the first coupling field acts on the bare atomic levels, and then leads to the singly dressed states. The second step is that the second coupling field acts on the singly dressed states and results in doubly dressed states. The quantum system of the singly dressed states in a quasi-resonant coupling field is similar to a quantum system of atoms in a coupling field, i.e., a natural generalization of the two-level dressed-atom approach. Therefore, the analysis formula of an energy level diagram for a two-level system exposed to a strong near-resonant bichromatic laser field can be obtained. More specifically, the Hamiltonian of a two-level atomic system interacting with a strong near-resonant bichromatic laser field and decomposed as a two-step process is as follows:

Figure 1 gives a simple and intuitive picture of the energy exchanges in the coupled atom within a drive/coupling bichromatic laser field system for the two-step process. In the drive/coupling bichromatic laser field, the first coupling field frequency ω₁ is quasi-resonant with the transition between the two bare atomic levels, and the second coupling field frequency ω₂ is quasi-resonant with the transition connecting two singly dressed levels. A natural generalization of the two-level dressed-atom approach for the singly dressed states in a quasi-resonant coupling field is shown in Figs. 2–4. The energy formulas for the doubly dressed levels are listed in Table 1.

Fig. 1.

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	Fig. 1. A quantum system of a two-level atom driven by a strong bichromatic laser field, which can be simplified as a two-step process.

Fig. 2.

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Fig. 2. Manifolds , and of uncoupled states of the singly dressed states plus laser photons system. The energy difference between the two levels of a given manifold is very small compared with the gap between two adjacent manifolds. The manifold formed by these two singly dressed states and (N is the number of laser photons of the first coupling field and M is the number of laser photons of the second coupling field) is written as .

Fig. 3.

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	Fig. 3. Uncoupled levels and perturbed levels of the manifolds . , where is the Rabi frequency of the coupling field for the corresponding singly dressed levels and represents the detuning of the second coupling field frequency from the transition connecting two singly dressed levels.

Fig. 4.

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	Fig. 4. Doubly dressed levels for a two-level system exposed to a strong near-resonant bichromatic laser field. Here , where ω1, , and represent the frequency, detuning, and Rabi frequency for the first coupling field acting on the bare atomic levels, respectively. N and M are the numbers of laser photons of the first and the second coupling fields, respectively.

Table 1.

Table 1.

Table 1. The theoretical energy formulas for the doubly dressed levels. . Level Energy formula 1 2 3 4 5 6 7 8

Table 1. The theoretical energy formulas for the doubly dressed levels. .

These energy level diagram properties can be probed by a third laser field, i.e., record transmitted intensity as a function of frequency for a low-intensity probe beam focused through the atom sample. As shown in Fig. 4, our theoretical analysis predicts that the absorption spectrum of the probe beam for such a system is composed of three characteristic triplet structures, formed by central lines at ω₁, ω₂, and , respectively, and two sidebands symmetrically distributed at . Here , where ω₂, , and represent the frequency, detuning, and Rabi frequency for the second coupling field acting on the singly dressed levels, respectively. While the probe absorption spectrum of a singly driven two-level atom is composed of a dispersion-like structure located at the coupling field frequency and two symmetrically distributed side-band spectral features termed the absorption peak and the gain peak.

As for the experimental verification, cold atoms produced by a magneto-optical trap (MOT)^[13] offer great opportunities for the studies of precise spectroscopic measurements due to significantly reduced Doppler broadening and low collision rates. In this paper and the work of Ruan et al.,^[10] we take the and states of ⁸⁷Rb atoms as a two-level atom system. Cold ⁸⁷Rb atoms are produced in a standard MOT. Details of the apparatus were described in our previous work.^[14–16] Thus, the and states of ⁸⁷Rb atoms as well as the cooling/trapping beams and an additional coupling field constitute the degenerate two-level atomic system in two strong monochromatic coupling fields. The cooling/trapping beams play the role of the first coupling field which acts on the bare atomic levels, and the second coupling field acting on the dressed levels is provided by an independent frequency-stabilized diode laser of Toptica DL100. The probe beam is derived from the Toptica DL100 laser and is independently tuned over a limited range in frequency by acousto-optic modulators (AOM). Following the method adopted by Ruan et al.,^[10] we have measured the probe spectrum in the presence of two coupling fields and the probe spectrum with only one coupling field, aiming to compare and identify these spectral features resulting from the transition connecting one pair of doubly dressed levels. In this scheme, the detuning of the first coupling field frequency from the transition is fixed at −12 MHz. The frequency of the second coupling field is quasi-resonant with the gain peak formed only by the first coupling field and the frequency of the second coupling field varies by an AOM. Note that the second coupling field propagates through the cold ⁸⁷Rb atoms in two cases. (i) Similar to the work of Ruan et al.,^[10] the second coupling field passes the atom sample once. A typical probe absorption spectrum is shown in Fig. 5. (ii) The second coupling field is a counterpropagating beam acting on the atom sample by adding a totally reflecting mirror. A typical probe absorption spectrum is shown in Fig. 6. The present uncertainties in frequency calibration mainly result from the frequency jitter MHz of lasers in this work, and the errors (less than 1%) from the frequency calibration procedure.

Fig. 5.

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Fig. 5. The probe absorption spectrum measured in a two-level atomic system with two coupling fields. The frequency of the second coupling field is red-shifted with respect to the energy position of the gain peak formed only by the first coupling field. The peaks labeled as , , and represent the spectroscopic properties resulting from the transition connecting one pair of doubly dressed levels. The thick curve is the probe absorption spectrum in the presence of two coupling fields. The detunings of the two coupling fields are −12 MHz and −63 MHz with respect to the transition , respectively. The second coupling field passes through the atom sample once. The medium thickness curve is the pump–probe spectrum measured in the presence of the first coupling field. The first coupling field is provided by the cooling/trapping beams with the detuning of −12 MHz with respect to the transition . The thin curve is the pump–probe spectrum measured in the presence of the second coupling field. In this case, the cooling/trapping beams are switched off and the second coupling field is provided by an independent frequency-stabilized diode laser. The detuning of the coupling field is MHz with respect to the transition .

Fig. 6.

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Fig. 6. The probe absorption spectrum measured in a two-level atomic system with two coupling fields. The frequency of the second coupling field is red-shifted with respect to the energy position of the gain peak formed only by the first coupling field. The peaks labeled as , , and (G, D, and A) represent the spectroscopic properties resulting from the transition connecting one pair of doubly dressed levels. The thick curve is the probe absorption spectrum in the presence of two coupling fields. The detunings of the two coupling fields are −12 MHz and −63 MHz with respect to the transition , respectively. The second coupling field is a counterpropagating beam acting on the atom sample. The medium thickness curve is the pump–probe spectrum measured in the presence of the first coupling field. The first coupling field is provided by the cooling/trapping beams with the detuning of −12 MHz with respect to the transition . The thin curve is the pump–probe spectrum measured in the presence of the second coupling field. In this case, the cooling/trapping beams are switched off and the second coupling field is provided by an independent frequency-stabilized diode laser. The detuning of the coupling field is −63 MHz with respect to the transition .

Figures 5 and 6 display that these spectral features resulting from the transition connecting one pair of doubly dressed levels agree with the present theoretical prediction. More specifically, we observe a characteristic triplet structure formed by a central line at ω₂, and two sidebands symmetrically distributed in the case that the second coupling field passes the atom sample once. While in the case that the second coupling field is a counterpropagating beam acting on the atom sample, we observe two characteristic triplet structures formed by central lines at ω₂ and ω₁, respectively, and two sidebands symmetrically distributed. Although the present theoretical analysis predicts that the absorption spectrum of the probe beam for a such system is composed of three characteristic triplet structures, the intensities of these spectral features are determined by nonzero transition matrix elements and the corresponding populations, thus it is understandable that not all of the spectral features can be observed. In the two cases, while the intensity of the second coupling field is the same, the Rabi frequency is different, thus the probe absorption spectrum is different. Furthermore, we give a comparison of quantitative theoretical and experimental data in Table 2 and Figs. 7–9. Table 2 gives experimental energy levels for these spectral features resulting from the transition connecting one pair of doubly dressed levels as a function of the second coupling field frequency and the deduced Rabi frequency of the second coupling field. Certainly, Rabi frequency is independent of if the intensity of the second coupling field is fixed. In the present work, we fix the intensity of the second coupling field at 344 mW/cm² in the two cases, while the intensity of the second coupling field is about 254.8 mW/cm² in the work of Ruan et al.^[10] As shown in Table 2 and Figs. 7–9, the deduced Rabi frequency from the experimental energy levels is independent of . It is consistent with our theoretical prediction. In addition, as shown in Table 2 and Fig. 7,

Fig. 7.

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	Fig. 7. The present Rabi frequency as a function of the detuning in the case that the second coupling field passes through the atom sample with a counterpropagating beam.

Fig. 8.

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	Fig. 8. The present Rabi frequency as a function of the detuning in the case that the second coupling field passes through the atom sample once.

Fig. 9.

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	Fig. 9. The Rabi frequency as a function of the detuning in the case that the second coupling field passes through the atom sample once for the work of Ruan et al.[10]

Table 2.

Table 2.

Table 2. Experimental energy levels (estimated errors in parentheses) for these spectral features resulting from the transition connecting one pair of doubly dressed levels. The detuning of the first coupling field (cooling/trapping beams) is −12 MHz with respect to the transition . The frequency ω2 of the second coupling field is quasiresonant with the gain peak formed only by the first coupling field. . The second coupling field passes the atom sample once a /(2π × MHz) /(2π × MHz) /(2π ×MHz) /(2π × MHz) /(2π × MHz) This workb −20(2.8) −24.1(2.5) − − 13.4(2.7) −23(2.8) −25.9(2.5) − − 11.9(2.7) −26(2.8) −28.3(2.5) − − 11.2(2.7) Ruan et al.[10]c −6(2.4) −27.4(2.1) −0.4(1.3) +30.3(1.6) 28.2(2.6) −8(2.4) −29.8(2.1) −1.6(1.3) +27.2(1.6) 27.4(2.6) −10(2.4) −30.9(2.1) −1.4(1.3) +28.1(1.6) 27.8(2.6) −21(2.4) −36.7(2.1) −2.1(1.3) +33.4(1.6) 28.1(2.6) −24(2.4) −33.8(2.1) +1.3(1.3) +37.0(1.6) 26.0(2.6) This work, the second coupling field is a counterpropagating beam c /(2π × MHz) /(2π × MHz) /(2π × MHz) /(2π × MHz) /(2π × MHz) −20(2.8) −24.0(2.5) 0.7(1.3) 33.3(1.6) 20.5(3.0) −23(2.8) −36.7(2.5) 0.9(1.3) 25.3(1.6) 20.8(3.0) −26(2.8) −28.0(2.5) 1.1(1.3) 35.6(1.6) 18.3(3.0) c /(2π × MHz) /(2π × MHz) /(2π × MHz) /(2π × MHz) −20(2.8) −29.9(2.5) 1.6(1.3) 22.0(1.6) 16.5(3.0) −23(2.8) −32.8(2.5) 0.8(1.3) 23.1(1.6) 15.9(3.0) −26(2.8) −35.9(2.5) 0.3(1.3) 23.4(1.6) 14.2(3.0) a Obtained from the present theoretical prediction that a central line is at ω2 and ω1 respectively, and the two sidebands termed the absorption peak and the gain peak are symmetrically distributed at with . b No data for the central line and absorption peak due to too larger error. is obtained from the energy position of the gain peak. c is obtained from the energy difference between the gain peak and the absorption peak.

Table 2. Experimental energy levels (estimated errors in parentheses) for these spectral features resulting from the transition connecting one pair of doubly dressed levels. The detuning of the first coupling field (cooling/trapping beams) is −12 MHz with respect to the transition . The frequency ω2 of the second coupling field is quasiresonant with the gain peak formed only by the first coupling field. .

deduced from the triplet structure with a central line at ω₂ is consistent with that deduced from the triplet structure with a central line at ω₁ within the experimental errors in the case that the second coupling field is a counterpropagating beam. The from the case that the second coupling field is a counterpropagating beam is larger than that for the other case. It is reasonably and displayed in Table 2 and Figs. 7 and 8.

In summary, as shown in Table 1 and Fig. 4, we have deduced analytical solutions of energy level diagram of the doubly driven/dressed atom for a two-level system exposed to a strong near-resonant bichromatic laser field in the special case that such a system can be simplified as a two-step process. The present theoretical analysis predicts that the absorption spectrum of the probe beam for such a system is composed of three characteristic triplet structures, formed by central lines at ω₁, ω₂, and , respectively, and two sidebands symmetrically distributed at . We take and states of ⁸⁷Rb atoms in an MOT as well as the cooling/trapping beams and an additional coupling field as a quantum system of a two-level atom driven by a strong bichromatic laser field. We have measured the probe absorption spectra of the doubly driven two-level atom. The comparison of theoretical and experimental results shows good qualitative agreement between the predications and experimental data, as shown in Table 2 and Figs. 7–9.