1. Introduction

Rydberg atom has attracted more and more attentions on account of its extraordinary physical properties [1], such as the long lifetime of hundred of microsecond, the energy level interval of $\thicksim$ GHz, the large transition matrix element and the big polarizability. These properties make Rydberg atom widely used in various physics domains for the dipole interaction [2–4], the quantum information [5], the precise measurement of microwave field [6–9], and preparation of Rydberg molecules [10–13] and so on.

Quantum nonlinear effect based on the strong coherent interaction between an atom and laser is a promising application in quantum physics and all-optical information detection, among of them, an electromagnetic induced transparency (EIT) [14–16] and Autler-Townes (AT) splitting [17] are the typical examples. The Rydberg cascade three-level EIT plays an important role for an optically detection of Rydberg states and precisely measuring of the microwave field based on EIT-AT splitting. Both the Rydberg EIT and AT splitting have been carried out extensively experimentally and theoretically. Teo $et~al.$ [18] studied a nonlinear optical effect in the laser excitation of Rb Rydberg states and observed high contrast AT spectra; Weatherill $et~al.$ [19] showed the EIT of a weakly interacting cold Rydberg gas; Zhang $et~al.$ [20] investigated the AT splitting of a cascade three-level system in ultracold Cs Rydberg atoms. In addition to the basic research, the Rydberg EIT and AT splittings are also employed to measure the microwave field [6–9] and the vector microwave electrometry [21,22]. Further, Rydberg EIT was used to realize the parallelized CNOT gate [23], single-photon source [24], multi-wave mixing [25,26] and so on. In the past few years, the multi-photon scheme EIT becomes an another alternative to study EIT and AT splitting. Three-photon Rydberg EIT and EIA in a Rb or Cs vapor [27–30] were studied and the three-photon EIT of cesium was employed to achieve the terahertz (THz) near-field imaging [31,32]. These works have been done in the room-temperature cell and the probe transmission of EIT signal was detected.

Different from the previous literature [28–30], in this work, we investigate the three-photon AT spectra and splitting of the cold cesium Rydberg gas by measuring the Rydberg population ionized with the field ionization technique. The field ionization detection has the advantage of the high sensitivity, specially for the cold atomic system with a large frequency scanning range of this work. The three-photon AT spectroscopy as a function of the Rabi frequency and detuning of the excitation laser are investigated. Three-peak AT spectra and related AT splittings are obtained, which are found to have a qualitative dependence on the Rabi frequency and the detuning of two coupling lasers. Experimental results are agreement with simulations of the four-level theoretical model accounting for decay rates of Rydberg system and the spectral line broadening. The three-photon spectroscopy of the cold Rydberg atom can be used to investigate the ultracold P-type molecules and the application of Rydberg-atom-based field measurement [33] and so on.

2. Experimental setup

The four-level diagram of our experiment is displayed in Fig. 1(a). The first photon of 852 nm laser drives the lower transition $|6\mathrm {S}_{1/2}, F=4\rangle \rightarrow |6\mathrm {P}_{3/2}, F'=5\rangle$ with a detuning $\delta _{852}$ and a Rabi frequency $\Omega _{852}$. The second photon of 1470 nm laser couples the middle transition $|6\mathrm {P}_{3/2}\rangle \rightarrow |7\mathrm {S}_{1/2}\rangle$ with a Rabi frequency $\Omega _{1470}$. The third photon of 780 nm laser achieves the Rydberg excitation of the up transition $|7\mathrm {S}_{1/2}\rangle \rightarrow |n\mathrm {P}_{3/2}\rangle$ with the detuning $\Delta _{780}$ and the Rabi frequency $\Omega _{780}$. In our AT atomic system, the first 852 nm and second 1470 nm lasers are two coupling lasers, whereas the third 780 nm laser is the probe laser due to that, i) $\Omega _{780} \ll \Omega _{852},\Omega _{1470}$, and ii) we detect the Rydberg populations here. The first-photon frequency is varied by changing the modulation voltage of the acoustic-optical modulator (AOM). All lasers are external cavity diode lasers from Toptica with a linewidth less than 500 kHz. $\Gamma _{i} (i=1\sim 4)$ is the spontaneous radiation decay rate of energy levels $|i\rangle$, where $\Gamma _{1} = 0$.

 

Fig. 1. (a) Energy level schematic of the three-photon excitation (left panel) and dressed state (right panel). The first-photon 852 nm laser (red) with a Rabi frequency $\Omega _{852}$ drives the lower transition of $|6\mathrm {S}_{1/2}\rangle \rightarrow |6\mathrm {P}_{3/2}\rangle$, the second-photon 1470 nm laser (purple) with a Rabi frequency $\Omega _{1470}$ couples the middle transition of $|6\mathrm {P}_{3/2}\rangle \rightarrow |7\mathrm {S}_{1/2}\rangle$, and the third-photon 780 nm laser (dark red) with a Rabi frequency $\Omega _{780}$ couples the Rydberg transition of $|7\mathrm {S}_{1/2}\rangle \rightarrow |nP_{3/2}\rangle$. $\delta _{852}$ and $\Delta _{780}$ display the detunings of the first and third photon, respectively. $\Gamma _{i} (i=2,3,4)$ is the decay rate of the related energy level. (b) Schematic diagram of an experimental setup. Three excited lasers are crossed through the MOT center, Rydberg atoms are ionized by the field ionization method and resultant Rydberg ions are detected with a MCP detector, and analysed with a boxcar and recorded with a computer. Three pairs of girds are placed on either sides of the MOT (the other two pairs do not show here) that can be used to apply a potential for ionizing Rydberg atom or compensating a stray electric field. Schematic not shown to scale.

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The three-photon AT experiment is performed in a standard magneto-optical trap (MOT) with the temperature $\thicksim 100~\mathrm {\mu }$K and a peak density $\thicksim 10^{10}~\mathrm {cm}^{-3}$ and MOT diameter $\thicksim 600 \mu$m. The experimental setup is shown in Fig. 1(b). An 852 nm laser and a 780 nm laser with the relative waist of 80 $\mu$m and 100 $\mu$m are set to co-propagate through the MOT center along the $x$-direction, whereas a 1470 nm laser with a waist of 800 $\mu$m is crossed with the 852 nm laser at the MOT center, forming a cigar-shaped Rydberg excitation region. All lasers are linearly polarized along the $z$-direction. Frequencies of the 852 nm and 1470 nm coupling lasers are locked with the technology of saturated absorption spectrum (SAS) and the optical-optical double-resonance spectrum (OODRS), respectively. The Rydberg AT spectrum is attained by scanning the 780 nm laser frequency, monitored with a calibrated wavelength meter (HighFinesse-Angstrom WS-U). A pair of girds are placed on either side of the MOT center in the $z$-direction. After switching off the 1-$\mu$s duration Rydberg excitation, a ramped ionization electric field which has a ramp time 3 $\mu$s is applied on girds to ionize Rydberg atoms. The Rydberg ions are accelerated to the microchannel plate (MCP) detector and analyzed with the boxcar integrator and recorded with a computer. In addition, another two pairs of girds are placed in $x$- and $y$-directions, not shown in Fig. 2(b), where we apply the dc field to compensate the stray electric field and responding stray field less than 30 mV/cm [34].

 

Fig. 2. Measurement (red dot) and simulation (black line) of the three-photon AT splitting spectrum of $|60\mathrm {P}_{3/2}\rangle$ Rydberg state with $\Omega _{852}=2\pi \times 110$ MHz, $\Omega _{1470}=2\pi \times 85$ MHz, $\Omega _{780}=2\pi \times 0.25$ MHz and $\delta _{852}=2\pi \times 110$ MHz. Vertical dot-dashed lines mark the center of peaks A, B and C. The $\gamma _{\mathrm {AB}}$ ($\gamma _{\mathrm {BC}}$) displays the space between peaks A and B (peaks B and C). Zero detuning of $\Delta _{780}=0$ MHz indicates the resonant position of the three-photon excitation when $\delta _{852}=\delta _{1470}=0$.

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3. Results and discussions

The three-photon AT spectrum is obtained by measuring the Rydberg population of the cascade four-level system in Fig. 1(a). In Fig. 2, we present the $|60\mathrm {P}_{3/2}\rangle$ Rydberg AT spectroscopy with the detuning $\delta _{852}$ = 2$\pi \times 110\;\textrm {MHz}$ and the Rabi frequency of three photons $\Omega _{852}=2\pi \times 110\;\textrm {MHz}$, $\Omega _{1470}=2\pi \times 85\;\textrm {MHz}$ and $\Omega _{780}=2\pi \times 0.25\;\textrm {MHz}$, respectively. Zero detuning indicates the resonant position of Rydberg three-photon excitation for $\delta _{852}=\delta _{1470}=0$. We can clearly observe the three peaks marked by letters A, B and C, which located at $x_{\mathrm {A}}=-134.4\pm 0.1 \;\textrm {MHz}$, $x_{\mathrm {B}}=-28.9\pm 0.1\;\textrm {MHz}$ and $x_{\mathrm {C}}=52.9\pm 0.1\;\textrm {MHz}$, respectively, signing with black dot-dashed lines. The three-peak spectrum is attributed to the three-photon AT splitting, explained with the dressed state, see the right panel of the Fig. 1(a). It is also found that the AT spectroscopy displays the asymmetric spectral profile due to the detuning $\delta _{852}$ = 2$\pi \times 110\;\textrm {MHz}$ of the first coupling interaction. Furthermore, the three-photon AT splitting $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$, defined by the interval of the peak to peak, are attained to be $\gamma _{\mathrm {AB}}/2\pi =105.6\pm 0.1\;\textrm {MHz}$ and $\gamma _{\mathrm {BC}}/2\pi =81.8\pm 0.2\;\textrm {MHz}$, respectively. The AT splitting $\gamma _{\mathrm {AB}}$ mainly comes from the strong coupling between the first 852 nm laser and the atom, whereas the splitting $\gamma _{\mathrm {BC}}$ mainly from the coupling between the second 1470 nm laser and the atom. The maximum Rydberg population is $N\approx 2000$ and atomic density of $\sim 1.1\times 10^{8}~\mathrm {cm}^{-3}$. The internuclear distance is $12.95~\mathrm {\mu m}$ that is much larger than the blockade radius of $6.47~\mathrm {\mu m}$ for $|60\mathrm {P}_{3/2}\rangle$ state, which means that we can ignore the interaction between Rydberg atoms.

To understand the spectral feature of Fig. 2, we consider the cesium cascade four-level system in Fig. 1(a). In the rotating-wave approximation and the field picture, the Hamiltonian of our system can be written as,

From Eqs. (1)–(3), the optical Bloch equations for elements $\rho _{mn}$ of one-atom density matrix are given by [33],

Considering $\delta _{1470}=0$ in our experimental scheme, and setting $\dot {\rho }_{mn}=0$ (the calculation of the evolution process shows that the atomic system has already reached the steady state in 1 $\mu$s of the excitation duration), we numerically solve the Eq. (4) to obtain the steady-state solutions of $\rho _{44}$, the Rydberg population, as a function of the laser detuning $\Delta _{780}$ for a range of value of $\Omega _{852}$ and $\Omega _{1470}$ and $\delta _{852}$.

In order to better simulate measured AT spectrum, the calculated $\rho _{44}$ is convoluted with a Gaussian function accounting for the spectral broadening caused by the laser linewidth, an external field induced inhomogenous broadening and so on [35,36]. The Gaussian profile function is written as,

To further investigate the dependence of the three-photon AT splitting on the detuning and Rabi frequency of the coupling lasers, we take a series of measurements of the three-photon AT spectra for the different detuning $\delta _{852}$ by changing the modulation voltage of the AOM in the light path of 852 nm laser and for the different Rabi frequency $\Omega _{852}$ and $\Omega _{1470}$ via varying the laser power. In Fig. 3(a), we present the contour plot of the measured $|60\mathrm {P}_{3/2}\rangle$ Rydberg AT spectra as a function of the $\Delta _{780}$ for the $\delta _{852}$ range from $2\pi \times 75$ MHz to $2\pi \times 148$ MHz. The measured spectra show a good agreement with the simulations of Fig. 3(b). It is seen, from the Fig. 3, that intensities of three peaks decrease and the asymmetry of the three-photon spectra enhance when the detuning $\delta _{852}$ increases, which is similar to the case of the two-photon AT spectra of three-level atom [20]. In addition, we find that AT splittings of $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$ show increasing with the detuning $\delta _{852}$.

 

Fig. 3. Contour plots of measurements (a) and theoretical simulations (b) of the three-photon Rydberg AT spectra for Cs $|60\mathrm {P}_{3/2}\rangle$ state as a function of $\Delta _{780}$ and $\delta _{852}$ for $\Omega _{852}=2\pi \times 90$ MHz, $\Omega _{1470}=2\pi \times 70$ MHz, $\Omega _{780}=2\pi \times 0.15$ MHz and $\delta _{1470}=0$.

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We fit the AT spectra of the Fig. 3 with the Gaussian function, yielding the peak center of three peaks and obtaining the interval $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$. Figure 4(a) presents the measured $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$ as a function of $\delta _{852}$. It is found that both $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$ increase with the first photon detuning $\delta _{852}$, but the increase of the $\gamma _{\mathrm {AB}}$ is faster than that of $\gamma _{\mathrm {BC}}$. Therefore, the splitting of $\gamma _{\mathrm {AB}}$ exhibits stronger dependence on the first photon coupling. According to the diagonalized Hamiltonian in Eq. (1), we can attain the splitting between two peaks, reads as,

 

Fig. 4. (a) Measured AT splitting $\gamma _{\mathrm {AB}}$ (blue squares) and $\gamma _{\mathrm {BC}}$ (red circles) (extracted from Fig. 3(a)) as a function of the detuning $\delta _{852}$. Dashed lines show the fitting by the Eq. (6). (b) Measurements of the $\gamma _{\mathrm {AB}}$ (blue squares) and $\gamma _{\mathrm {BC}}$ (red circles) as a function of the Rabi frequency $\Omega _{852}$ with $\Omega _{1470}=2\pi \times 80\;\textrm {MHz}$, $\Omega _{780}=2\pi \times 0.25\;\textrm {MHz}$ and $\delta _{852}= 2\pi \times 110\;\textrm {MHz}$. Dashed lines are the fittings by the Eq. (6).

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The AT splitting is proportional to the Rabi frequency of the coupling laser for the three-level atom [14,20]. To investigate the dependence of the three-photon AT splitting on the coupling Rabi frequency, we fix the detuning of $\delta _{852}= 2\pi \times 110$ MHz and change the Rabi frequency $\Omega _{852}$ by changing the laser power and do a series of measurements. In Fig. 4(b), we present the measured AT splitting as a function of the Rabi frequency of the first excitation photon for $\Omega _{1470}=2\pi \times 80$ MHz, and $\Omega _{780}=2\pi \times 0.25$ MHz. We can see that with the increasing of $\Omega _{852}$, $\gamma _{\mathrm {AB}}$ shows a increasing from $\sim$92 MHz to $\sim$121 MHz, on the contrary, $\gamma _{\mathrm {BC}}$ shows a decreasing from $\sim$87 MHz to $\sim$ 70 MHz. The dashed lines in Fig. 4(b) are the fittings of Eq. (6) to the measured data, corresponding coefficients are $\beta _{\mathrm {AB}}=1.46\pm 0.08$ and $\beta _{\mathrm {BC}}=-0.89\pm 0.08$. The coefficient of $\gamma _{\mathrm {AB}}$ ($\gamma _{\mathrm {BC}}$) is positively (negatively) correlated with the first photon Rabi frequency in the range of $\Omega _{852}/2\pi <$ 100 MHz. It is seen, from Fig. 4, that the splitting $\gamma _{\mathrm {AB}}$ mostly comes from the coupling of the first photon.

 

Fig. 5. (a-b) Calculated positions of three peaks (a) and the AT splitting $\gamma _{\mathrm {AB}}$, $\gamma _{\mathrm {BC}}$ and $\gamma _{\mathrm {AC}}$ (b) as a function of the Rabi frequency $\Omega _{852}$ with the fixed $\delta _{852}/2\pi$=110 MHz and $\Omega _{1470}/2\pi$ = 80 MHz. Inset is the enlargement of the shadow part in (b). (c-d) Calculated positions of three peaks (c) and the AT splitting $\gamma _{\mathrm {AB}}$, $\gamma _{\mathrm {BC}}$ and $\gamma _{\mathrm {AC}}$ (d) as a function of the Rabi frequency $\Omega _{1470}$ with the fixed $\delta _{852}/2\pi$=110 MHz and $\Omega _{852}/2\pi$ = 90 MHz.

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To understand the effect of two coupling lasers on the AT splitting, we do more calculations of the three-photon AT splitting for the large range of coupling Rabi frequency. Figure 5 presents the position of the three peaks and related AT splittings as a function of the $\Omega _{852}$ [Figs. 5(a) and (b)] and the $\Omega _{1470}$ [Figs. 5(c) and (d)]. We clearly see that AT splittings $\gamma _{\mathrm {AB}}$ and $\gamma _{\mathrm {BC}}$ are attributed to two coupling lasers, and the splitting of $\gamma _{\mathrm {AB}}$ mainly comes from the first photon that couples the $|6\mathrm {S}_{1/2}\rangle \rightarrow |6\mathrm {P}_{3/2}\rangle$ lower transition. The $\gamma _{\mathrm {AB}}$ displays the increase with $\Omega _{852}$, whereas the $\gamma _{\mathrm {BC}}$ shows the decrease for $\Omega _{852}/2\pi \lesssim 100\; \textrm {MHz}$(see the inset of Fig. 5(b), showing the agreement with the measurement of Fig. 4(b)), and the increase for $\Omega _{852}/2\pi >$ 100 MHz. However, the splitting of $\gamma _{\mathrm {BC}}$ mainly comes from the second photon that couples the $|6\mathrm {P}_{3/2}\rangle \rightarrow |7\mathrm {S}_{1/2}\rangle$ intermediate transition, the $\gamma _{\mathrm {AB}}$ ($\gamma _{\mathrm {BC}}$) displays the decrease (increase) with increasing of the $\Omega _{1470}$. The total AT splitting $\gamma _{\mathrm {AC}}$= $\gamma _{\mathrm {AB}}$ +$\gamma _{\mathrm {BC}}$ displays the increase both with the first photon and the second photon Rabi frequency.

4. Conclusion

We have investigated three-photon AT spectra and AT splittings of the cold cesium Rydberg gas by detecting the Rydberg population. In the four-level atom, two intermediate states are coupled by two coupling lasers (852 nm and 1470 nm lasers) to form dressed states, resulting in three effective transition channels and forming the three-peak AT spectroscopy. The spectral profile and corresponding AT splitting strongly depend on the detuning and Rabi frequency of the two coupling lasers. The AT splitting $\gamma _{\mathrm {AB}}$ mainly comes from the first photon coupling, whereas the $\gamma _{\mathrm {BC}}$ from the second photon coupling with atoms. The steady-state solution $\rho _{44}$ of Bloch equations of the four-level system is numerically simulated, which agree with the measured AT spectra and splitting. The coefficient $\beta$ is defined with the Eq. (6) to qualitatively characterize the dependence of the AT splitting on the two coupling lasers. The three-photon AT spectroscopy of the cold Rydberg atom is of great significance for fully understanding the interaction between multi-photon and atoms and the application of Rydberg-atom-based field measurement [28] and Terahertz imaging [32] .