Cascade splitting of two atomic energy levels due to multiphoton absorption

We have theoretically and experimentally studied the spectroscopic properties of dressed levels in a strong monochromatic field, and propose a model of cascade splitting of two atomic energy levels. In this model two related dressed levels can be split into four levels, and transitions connecting four new levels will constitute spectroscopic structures. Two types of proof-in-principle experiments are performed to verify the model. One experiment measures the probe absorption spectra of a degenerate two-level atomic system with two strong monochromatic coupling fields. The system consists of $5^2{S}_{1/2},F=2$ and $5^2{P}_{3/2},F^{\prime }=3$ states of ${}^{87}\mathrm{Rb}$ atoms in a magneto-optical trap (MOT) as well as the cooling beams and an additional coupling field. New spectral features are observed and proven to be due to the transitions of new levels generated by splitting of the dressed levels. The other experiment measures the pump-probe spectra in a degenerate two-level atomic system with one strong monochromatic coupling field. The system consists of $5^2{S}_{1/2},F=2$ and $5^2{P}_{3/2},F^{\prime }=3$ states of the ${}^{87}\mathrm{Rb}$ atom in a magneto-optical trap and one coupling field. We have observed spectral features that obviously differ from the prediction that comes from the two-level dressed-atom approach. They cannot be explained by existing theories. The model of cascade splitting of two atomic energy levels is employed to explain the observations in these two types of experiments.

Figure 1

Cascade splitting of two atomic energy levels in a two-level atomic system with two strong monochromatic coupling fields. The frequency of the second coupling field is near the energy position of the gain peak. Here, ${\omega }_1,{\Delta }_1$, and ${\Omega }_1$ represent the frequency, detuning, and Rabi frequency for the first coupling field acting on the bare atomic levels. ${\omega }_2,{\Delta }_2$, and ${\Omega }_2$ represent the frequency, detuning, and Rabi frequency for the second coupling field acting on the dressed levels. ${\Omega }_1^{\prime }=\sqrt{{\Omega }_1^2+{\Delta }_1^2}$. ${\Omega }_2^{\prime }=\sqrt{{\Omega }_2^2+{\Delta }_2^2}$.

Figure 2

Cascade splitting of two atomic energy levels in a two-level atomic system with two strong monochromatic coupling fields. The frequency of the second coupling field is near the energy position of the absorption peak. The meanings of physical quantities are the same as those in the caption of Fig. 1.

Figure 3

Cascade splitting of two atomic energy levels in a two-level atomic system with one strong monochromatic coupling field. The meanings of physical quantities are the same as those in the caption of Fig. 1. In this case, ${\Delta }_2=0$ MHz. The transitions connecting the levels formed by splitting of the dressed levels correspond to our observed spectral features under the condition that the pump and probe polarizations are cc shown in Fig. 10.

Figure 4

The probe absorption spectrum measured in degenerate two-level atomic systems with two coupling fields. The frequency of the second coupling field is red-shifted with respect to the energy position of the gain peak. (a) (Thick curve) The probe absorption spectrum in the presence of two coupling fields. The detunings of the two coupling fields are $-12$ and $-63$ MHz with respect to the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=3$, respectively. (b) (Medium thickness curve) The pump-probe spectrum measured in an operating MOT. The coupling field is provided by the cooling beams with the detuning of $-12$ MHz. (c) (Thin curve) The pump-probe spectrum measured in the case that cooling beams are switched off and the coupling field is provided by an independent frequency-stabilized diode laser. The detuning of the coupling field is $-63$ MHz. (Inset) The detail near the energy position of the gain peak.

Figure 5

The probe absorption spectra measured in degenerate two-level atomic systems with two coupling fields. The frequency of the second coupling field is red-shifted with respect to the energy position of the gain peak. The detunings of the second coupling field are $-48,-52$, and $-66$ MHz with respect to the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=3$, respectively.

Figure 6

Energy positions of the new spectral features when the probe absorption spectra are measured in degenerate two-level atomic systems with two coupling fields and also obtained from the proposed model. Solid inverted triangles, circles, and squares represent the energy positions of the new absorption peak, dispersionlike structure, and gain peak, respectively. Open triangles represent the energy positions symmetrical with the new absorption peak and these energy positions should agree well with the energy positions of the new gain peak predicted by our model. Open circles represent the energy positions of the new dispersionlike structure predicted by our model.

Figure 7

The probe absorption spectrum measured in degenerate two-level atomic systems with two coupling fields. The frequency of the second coupling field is blue-shifted with respect to the energy position of the absorption peak. (Thick curve) The probe absorption spectrum in the presence of two coupling fields. The detunings of the two coupling fields are $-12$ and +50 MHz with respect to the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=3$, respectively. (Medium thickness curve) The pump-probe spectrum measured in an operating MOT. The coupling field is provided by the cooling beams with the detuning of $-12$ MHz. (Thin curve) The pump-probe spectrum measured in the case that cooling beams are switched off and the coupling field is provided by an independent frequency-stabilized diode laser. The detuning of the coupling field is +50 MHz. (a) The whole spectrum around the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=3$ when the scale of the digital oscilloscope is big. (b) Amplification of the spectrum around the dispersionlike structure to show the weak gain peak G′.

Figure 8

A typical experimental pump-probe spectrum of cold ${}^{87}\mathrm{Rb}$ measured in an operating magneto-optical trap with the detuning ${\Delta }_1/2\pi =-12$ MHz and the cooling laser intensity $I=170.9 {\mathrm{mW}/\mathrm{cm}}^2$. The saturated absorption spectrum of an ${}^{87}\mathrm{Rb}$ vapor cell is presented for frequency calibration (curve in the top). The energy positions are calculated for these spectral features in cc and coc spectra based on the theoretical model of Lipsich et al. [3], in which the Rabi frequency of the coupling field is 36.9 MHz that is obtained from the Autler-Townes doublets for the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=2$. (Inset) The detail near the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=3$.

Figure 9

Experimental pump-probe spectra of cold ${}^{87}\mathrm{Rb}$ atoms measured in an operating magneto-optical trap with various detunings ${\Delta }_1$ and the fixed cooling laser intensity $I=153.4 {\mathrm{mW}/\mathrm{cm}}^2$.

Figure 10

Experimental pump-probe spectra of cold ${}^{87}\mathrm{Rb}$ atoms measured in the well-defined polarization configurations of the coupling and probe pump fields, i.e., pump and probe polarizations are circular and equal (cc). (a) ${\Delta }_1/2\pi$ = $-12$ MHz; (b) ${\Delta }_1/2\pi$ = $-8$ MHz; (c)${\Delta }_1/2\pi$ = $-4$ MHz; (d) ${\Delta }_1/2\pi =0$ MHz. The Rabi frequency of the coupling field ${\Omega }_1\sim 19$ MHz is estimated from the Autler-Townes doublet (labeled as A) for the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=2$. (e) Energy positions of spectral features. Solid squares and circles represent the energy positions of the absorption peak and the gain peak, respectively. Open squares represent the energy positions symmetrical with the absorption peak and these energy positions should agree well with the energy positions of the gain peak predicted by our model. (f) Comparison between ${\Omega }_1$ obtained from the Autler-Townes doublet for the transition $5^2{S}_{1/2},F=2\to 5^2{P}_{3/2},F^{\prime }=2$, and ${\Omega }_{1,\mathrm{fit}}$ calculated using the energy level formulas predicted by our model shown in Fig. 3. Solid circles and open squares represent ${\Omega }_1$ and ${\Omega }_{1,\mathrm{fit}}$, respectively.