1. Introduction

Investigating interaction between highly excited Rydberg atoms and radiation fields has a long history and continues to provide fertile ground for new physics. Rydberg atom, with a principal quantum number n $\gg$ 1, has a great polarizability ($\sim$ $n^{7}$) and large microwave transition dipole moment ($\sim$ $n^{2}$) [1]. The microwave coupling Rydberg neighbour states leads to the electromagnetically induced transparency and Autler-Townes (EIT-AT) spectroscopy, which is widely employed to measure the microwave field [2–26], including microwave electric field [2–5], millimeter wave detection [6] and sub-wavelength imaging [7–10] and microwave polarization measurement [11] and so on. Rydberg atoms are also used as radio frequency receivers to retrieve both amplitude modulated (AM) signals and frequency modulated (FM) signals [27–31]. In addition, the interaction of Rydberg atoms with microwave field have recently caused extensive research on quantum entanglement [32–35] and encoded Rydberg qubit [36]. Two-photon microwave transitions and strong-field effects in a room-temperature Rb gas have been investigated [37], where the microwave electric-field range is extended up to about 40 V/m.

In this work, two-photon microwave coupled Rydberg spectroscopes are investigated in room-temperature cesium atoms via Rydberg-EIT-AT. Compared with the Ref. [37], we use a similar technology, but different atoms to investigate the microwave two-photon spectroscopy. The main differences include: (i) As we know that each atom has its own characteristic levels and spectra. Comparing with a rubidium atom, a cesium atom has different quantum defects of low-l (l is angular momentum quantum number) state, and therefore the different energy level structures, which present the special spectral characteristics that a Rb atom does not have. (ii) The line crossing point appears earlier for the cesium than for the rubidium atom, which means that state mixture appears in weak field for cesium and in strong field for rubidium. Our work together with the Ref. [37] will provide the theoretical data for application of the atom based microwave field measurements. In our system, the three-level atom of $|6{S}_{1/2}\rangle {\rightarrow } |6{P}_{3/2}\rangle {\rightarrow } |68{D}_{5/2}\rangle$ is used to probe a two-photon $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition. This transition involves the near-resonant intermediate $|{P}_{J}\rangle$ and $|{F}_{J}\rangle$ states, forming the microwave EIT spectroscopy including a rich of information like AT-splitting and level shifting and so on. The Floquet theoretical simulations reproduce the experimental measurements well. The microwave two-photon spectroscopy of $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$ transition is also presented for comparing with the case of $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition.

2. Experimental setup

Two-photon microwave experimental scheme of cesium Rydberg atom is shown in Fig. 1(a). A probe laser with a wavelength of $\lambda _p$ = 852 nm (Toptica DLpro) and a coupling laser with $\lambda _c$ = 510 nm (Toptica DLpro + Precilasers YFL-SHG-509-1.5) are overlapped in a counter propagating geometry along $x$-axis through the cesium room-temperature cell. The coupling laser drives the Rydberg transition, $|6{P}_{3/2}\rangle {\rightarrow } |68{D}_{5/2}\rangle$, while the probe laser couples the lower transition of $|6{S}_{1/2}(F = 4)\rangle {\rightarrow } |6{P}_{3/2}(F'= 5)\rangle$ and probe the Rydberg level and microwave two-photon spectroscopy, see the level scheme in Figs. 1(b) and (c). The waist of the probe and coupling lasers at the cesium cell center are $\omega _{p}$ = 90 $\mu$m and $\omega _{c}$ = 135 $\mu$m, corresponding Rabi frequencies are $\Omega _{p}/2\pi$ = 17.2 MHz and $\Omega _{c}/2\pi$ = 4.4 MHz, respectively. A standard gain horn is placed on 94 cm far away from the cell center, and the polarization of an emitted microwave is set along the $z$ axis, which is perpendicular to the polarization of the probe and coupling lasers setting along the $y$ axis.

 

Fig. 1. (a) Sketch of the experimental setup. A probe laser $\lambda _p$ = 852 nm ($\Omega _{p}$) and a coupling laser $\lambda _c$ = 510 nm ($\Omega _{c}$) are counter-propagated through a cesium room-temperature cell, the polarizations of the probe and coupling lasers are set along the $y$ axis. The horn antenna is placed 94 cm far away from the cell. The microwave field couples the Rydberg transition, and polarization of microwave is set along the z axis. DM: dichroic mirror; LB: laser beam block for a green laser; PBS: polarizing beam splitter; PD: photodiode detector. (b) Energy-level diagram for the four-level configuration. The coupling laser couples the Rydberg transition of $|6{P}_{3/2}\rangle {\rightarrow } |68{D}_{5/2}\rangle$, while the probe laser, coupling the lower transition $|6{S}_{1/2}(F = 4)\rangle {\rightarrow } |6{P}_{3/2}(F'= 5)\rangle$, probes the Rydberg level and two-photon microwave spectroscopy by Rydberg EIT. The $\nu _{DD}$ = 11.42865 GHz microwave field drives the $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition forming the two-photon microwave spectroscopy. (c) Energy-level diagram for the $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$ transition with $\nu _{SS}$ = 11.73503 GHz.

Download Full Size | PDF

The ground state, $|6{S}_{1/2}\rangle$, an excited state, $|6{P}_{3/2}\rangle$ and Rydberg state form Rydberg-EIT that is used to detect the microwave two-photon spectroscopy. The microwave field is produced with an analog signal generator (Agilent N5183B) with frequency $\nu _{DD}$ = 11.42865 GHz coupling two-photon transition of $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$, [see Fig. 1(b)], and $\nu _{SS}$ = 11.73503 GHz coupling the two-photon transition of $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$, [see Fig. 1(c)].

3. Experimental results and discussions

In the experiment, the frequency of probe laser is locked at $|6{S}_{1/2}(F'= 4)\rangle {\rightarrow } |6{P}_{3/2}(F'=5)\rangle$ transition, and the frequency of coupling laser is tuned to the vicinity of the $|6{P}_{3/2}\rangle {\rightarrow } |68{D}_{J}\rangle$ (J=3/2, 5/2) or $|6{P}_{3/2}\rangle {\rightarrow } |69{S}_{1/2}\rangle$ Rydberg transition with the frequency sweep range of 1 GHz, which covers the transition from the hyperfine structure energy level of the intermediate state $|6{P}_{3/2}\rangle$ to the Rydberg state $|68{D}_{J}\rangle$ or $|69{S}\rangle$. Rydberg EIT spectrum without microwave field is shown in Fig. 2(a), the dominant peak at zero detuning corresponds to the EIT spectrum formed by $|6{S}_{1/2}(F'=4)\rangle {\rightarrow } |6{P}_{3/2}(F'= 5)\rangle {\rightarrow } |68{D}_{5/2}\rangle$, related linewidth of main EIT spectrum is 2$\pi \times$14 MHz. The small peak at red-detuned 168 MHz is the Rydberg EIT formed by the hyperfine structure 6P$_{3/2}(F'=4)$ of the intermediate state, as indicated by the red dashed line in the Fig. 2(a). The interval between two hyperfine EIT peaks, 168 MHz, is attributed to the Doppler mismatch factor, $\lambda _p/\lambda _c$-1 = 0.67, which is used to calibrate the frequency scales of all spectra in this work. The peak at about −210 MHz is the Rydberg EIT formed by the fine structure of Rydberg $|68D_{3/2}\rangle$ state, marked with a triangle.

 

Fig. 2. (a) Rydberg-EIT spectrum without microwave field. The main peak is formed by the cascade three-level atom of $|6{S}_{1/2}(F = 4)\rangle {\rightarrow } |6{P}_{3/2}(F'= 5)\rangle {\rightarrow } |68{D}_{5/2}\rangle$, the small peak at red-detuned 168 MHz is the Rydberg EIT formed by the $|6{S}_{1/2}(F = 4)\rangle {\rightarrow } |6{P}_{3/2}(F'= 4)\rangle {\rightarrow } |68{D}_{5/2}\rangle$. The interval between two hyperfine EIT peaks is used to calibrate the coupling frequency. The peak at about −210 MHz is the Rydberg EIT coming from the fine structure of Rydberg $|68{D}_{3/2}\rangle$ state, marked with a triangle. (b) and (c) Two-photon microwave spectroscopies for the frequency of $\nu _{DD}$ = 11.42865 GHz coupling $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition and the microwave electric fields are 0.09 V/cm (b) and 0.28 V/cm (c), respectively. Values of the microwave field are obtained by comparing the spectra of the Floquet model calculations, see text. The spectral curve in (b) displays the two-photon AT splitting and ac Stark shift. The spectroscopy in (c) presents the complicated spectral profile, where the level degeneracy is lifted.

Download Full Size | PDF

In Figs. 2(b) and (c), we present two-photon microwave spectra of $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition for the microwave field of 0.09 V/cm (Fig. 2(b)) and 0.28 V/cm (Fig. 2(c)). The value of the microwave field is obtained by comparing the calculated spectra of the Floquet model, described later. The two-photon microwave spectrum in Fig. 2(b) illustrates the two-photon AT splitting and microwave-induced ac Stark shifts, corresponding AT splitting is 16.2 MHz. For the case of $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$ transition, the nearest dipole-allowed intermediate states are 70$P_{3/2}$ and 66$F_J$ states from which the microwave frequency is detuned by $\Delta$, shown in Fig. 1. $\Delta$ = 9.35 GHz for 70$P_{3/2}$ state, and $\Delta$ = 1.41 GHz for 66$F_{5/2}$ and $\Delta$ = 1.40 GHz for 66$F_{7/2}$ state, respectively. Considering an intermediate state as shown in Fig. 1(b), the two-photon Rabi frequency can be expressed as $\Omega _{2MW}$ = $\Omega _1\Omega _2/(2\Delta )$, where $\Omega _1$ and $\Omega _2$ are the single photon Rabi frequencies of dipole allowed transition from intermediate state.

To investigate two-photon microwave spectrum of $|68{D}_{5/2}\rangle {\rightarrow } |69{D}_{5/2}\rangle$, we perform a series of measurements such as in Fig. 2 for different microwave source power, displayed on a linear gray scale in Fig. 3. The microwave electric field the atom experienced is calibrated through Floquet theoretical calculation, varying from 0.06 V/cm to 0.4 V/cm. The calculated two-photon spectra with Floquet model are displayed with semitransparent symbols in Fig. 3, showing agreement with measurements. It is seen that the two-photon spectral line displays the ac Stark blue shifts and AT splitting of two peaks when microwave field less than 0.15 V/cm. As the field increase further, spectra demonstrate the ac Stark splitting of $m_j$=1/2, 3/2, 5/2 AT line pairs. When the microwave field larger than 0.25 V/cm, spectral lines also present the state mixing of 68D$_{5/2}$ and 68D$_{3/2}$ Rydberg atoms. The state mixing appears at the specific field value, which is accurate and can be used as a standard to calibrate the microwave field [38]. The uncertainty comes mainly from the linewith of EIT spectrum.

 

Fig. 3. Two-photon microwave spectra of 68D$_{5/2}{\rightarrow } 69{D}_{5/2}$ transition for the microwave electric field range from 0.06 V/cm to 0.4 V/cm and microwave frequency of $\nu$$_{DD}$ = 11.42865 GHz. Experimental EIT line strength is displayed on a linear gray scale. Colored dots display the simulation of the Floquet calculation for $m_j$=1/2 (blue), 3/2 (red) and 5/2 (green), related circle areas proportional to the probability of two-photon microwave excitation. The two-photon spectra show energy-level shifts and $m_j$ splittings due to the ac Stark effect and two-photon AT splittings. The weak spectra at about −200 MHz comes from EIT-AT spectra of the 68D$_{3/2}$ state. The electric field is calibrated with the Floquet calculation. The rectangular box marks the state mixture of two-photon spectra.

Download Full Size | PDF

To model the level shifts and splitting observed in Fig. 3, we consider off-resonant transitions through intermediate $66F_{J}$ and $70P_{3/2}$ levels. For the possible intermediate levels and related dipole allowed transitions, we calculate single-photon transition matrix elements and Rabi frequencies at the microwave field of 1 V/m. For 66$F_J$ intermediate states, the calculated Rabi frequencies are $2\pi \times$12 MHz ($|66F_{5/2}\rangle {\rightarrow } |68D_{5/2}\rangle$), $2\pi \times$209 MHz $(|66F_{7/2}\rangle ){\rightarrow } |68D_{5/2}\rangle$), $2\pi \times$10 MHz ($|66F_{5/2}\rangle {\rightarrow } |69D_{5/2}\rangle$) and $2\pi \times$170 MHz ($|66F_{7/2}\rangle {\rightarrow } |69D_{5/2}\rangle$), respectively. For the 70$P_{3/2}$ intermediate state, the calculated Rabi frequencies are $2\pi \times$75 MHz ($|70P_{3/2}\rangle {\rightarrow } |68D_{5/2}\rangle$) and $2\pi \times$254 MHz ($|70P_{3/2}\rangle {\rightarrow }|69D_{5/2}\rangle$), respectively. Taking account into the detuning of microwave from the intermediate states, effective two-photon Rabi frequencies are $\Omega _{eff}$/ $2\pi$ = 0.042 MHz (66$F_{5/2}$), 12.68 MHz (66$F_{7/2}$) and 1.02 MHz (70$P_{3/2}$), respectively. The dominant coupling is the $66F_{7/2}$ level. The two-photon Rabi frequency through this intermediate level is about a factor of 100 larger than the one through the $66F_{5/2}$ and a factor of 10 larger than the one through the 70$P_{3/2}$ level. In the calculation below, we consider three possible intermediate states for accuracy. We perform a Floquet analysis, which accounts for the effect of the microwave field in an exact nonperturbative manner, details of Floquet analysis see Ref. [17,37,38]. As shown in Fig. 3, the colored circles display the simulations of the Floquet calculations for $m_j$=1/2, 3/2 and 5/2, related circle areas proportional to the probability of two-photon microwave excitation. Viewing the Fig. 3, simulations reproduce the experimental two-photon spectroscopy well. According to the spectral characteristics, the spectra of Fig. 3 are divided into three regions. The first weak microwave field region, $E\lesssim$ 0.15 V/cm, the spectra mainly present the minor energy shift and the two-photon AT splitting. The second middle microwave field region, 0.15 V/cm $\lesssim E \lesssim$ 0.25 V/cm, the spectra display both two-photon AT splitting and microwave induced Stark splitting, $m_j$ = 1/2, 3/2 and 5/2 line pairs. The third large field region, $E\gtrsim$ 0.25 V/cm, except AT splitting and Stark splitting, the two-photon spectra demonstrate the line mixture, marked with rectangular box of Fig. 3. We can see that $m_j$ = 1/2 and 3/2 Stark lines of 68$D_{3/2}$ mix with 68$D_{5/2}$ lines at field $E \backsim$ 0.27 V/cm and $\backsim$ 0.32 V/cm, respectively. The mixture of two-photon spectra provides a rich of the information, which can be used to calibrate the microwave field. It is noted that the spectra at about −200 MHz of Fig. 3 is the two-photon microwave spectra produced due to the 68$D_{3/2}$ state, also see Figs. 2(b) and (c). Due to the smaller excitation probability (about a factor of 9) of $D_{3/2}$-type than $D_{5/2}$-type Rydberg state and detuning coupling of the microwave field, the two-photon spectra of 68$D_{3/2}$ state is much weaker than one of 68$D_{5/2}$ state.

The two-photon microwave spectra in Fig. 3 demonstrate complicated spectral features, we also measure the two-photon coupled $|69S_{1/2}\rangle {\rightarrow } |70S_{1/2}\rangle$ transition with microwave frequency $\nu _{SS}$ = 11.73503 GHz for comparison, related energy diagram is shown in Fig. 1(c). Figure 4 presents the measured and simulated two-photon spectra for microwave field coupling $|69S_{1/2}\rangle {\rightarrow } |70S_{1/2}\rangle$ transition. Similar to Fig. 3, the electric field is calibrated by Floquet calculations. For the $|69S_{1/2}\rangle {\rightarrow } |70S_{1/2}\rangle$ transition, there are two intermediate states $|69{P}_{3/2}\rangle$ and $|69{P}_{1/2}\rangle$, corresponding detunings $\Delta$ = 94 MHz and 858 MHz, respectively. At the field of 1 V/m, the calculated Rabi frequencies are $2\pi \times$ 181 MHz ($|69{P}_{3/2}\rangle {\rightarrow } |69{S}_{1/2}\rangle$) and $2\pi \times$177 MHz ($|69{P}_{3/2}\rangle \rightarrow |70S_{1/2}\rangle$), respectively. While Rabi frequencies are $2\pi \times$ 133 MHz ($|69{P}_{1/2}\rangle {\rightarrow } |69{S}_{1/2}\rangle$) and $2\pi \times$119 MHz ($|69{P}_{1/2}\rangle \rightarrow |70S_{1/2}\rangle$), respectively. Considering the detuning, the effective two-photon microwave Rabi frequency is $\Omega _{eff} /2\pi$=163 MHz (9.2 MHz) for the intermediate state of 69P$_{3/2}$ (69P$_{1/2}$) level. The two-photon microwave Rabi frequency of the intermediate $|69{P}_{3/2}\rangle$ is 16 times larger than that of $|69{P}_{1/2}\rangle$ state. Therefore the dominant near-resonant intermediate state is 69P$_{3/2}$, which results in the main feature of the two-photon microwave spectroscopy.

 

Fig. 4. Measurements and simulations of two-photon microwave spectroscopy of $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$, the area of dots is proportional to the excitation probability. Two-photon microwave spectra show microwave induced ac Stark shift and the two-photon AT splittings and state mixing. The Floquet simulation reproduces the experimental measurements well.

Download Full Size | PDF

Similar to Fig. 3, two-photon spectra of Fig. 4 display ac Stark shifts and two-photon AT splittings and state mixing. The blue circles in Fig. 4 represent Floquet simulations, which agree well with measured two-photon spectroscopes. It is found that the two-photon spectroscopy of Fig. 4 presents the one more spectral line at about −100 MHz, which is attributed to the intermediate state $|69{P}_{3/2}\rangle$, also showing the frequency shift as the microwave field strength increases. The two-photon microwave spectra of $|69S_{1/2}\rangle {\rightarrow } |70S_{1/2}\rangle$ also display the state mixing at about 0.25 V/cm, this is because $n$S state of cesium atom has small fraction quantum defect leading to the mixture of $n$S state and ($n$-4) manifolds.

In order to investigate the dependence of two-photon microwave AT splitting on the microwave field, we extract two-photon AT splittings from the Floquet simulations in Figs. 3 and 4. Figure 5(a) presents the two-photon AT splitting as a function of the microwave field $E^2$ for microwave coupled transition of $|68D_{5/2}\rangle \rightarrow |69D_{5/2}\rangle$ with the electric field range E < 0.5 V/cm. We find that two-photon AT splittings for all $m_j$ pairs are proportional to the microwave fields $E^2$ at the field range of 0.15 V/cm $\lesssim E\lesssim$ 0.25 V/cm (the middle microwave field region) due to the two-photon Rabi frequency $\Omega _{2MW}$ = $\Omega _1\Omega _2/(2\Delta )$. In this region, two-photon AT splitting can be used to measure the microwave field. But in the range of $\lesssim$ 0.15 V/cm (the first weak field region), the AT splitting nonlinearly increase with $E^2$. This nonlinear dependence of $E^2$ may be attributed to multi-intermediate states that would have an effect to the two-photon microwave coupling excitation and AT splitting being the EIT regime [39]. When the field E > 0.25 V/cm, the two-photon spectra enter the strong field region where the spectra display state mixture features. In Fig. 5(b), for comparison, we present two-photon EIT-AT splittings for the $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$ transition. Similar to Fig. 5(a), AT splittings for $|69{S}_{1/2}\rangle {\rightarrow } |70{S}_{1/2}\rangle$ transition also display nonlinear and linear regions.

 

Fig. 5. Two-photon microwave AT splitting extracted from the simulation in Figs. 3 and 4 as a function of the microwave field $E^2$, (a) for $|68D_{5/2}\rangle \rightarrow |69D_{5/2}\rangle$ transition of $m_j =$ 1/2 (blue), 3/2(red), 5/2(green). (b) for $|69S_{1/2}\rangle \rightarrow |70S_{1/2}\rangle$ transition.

Download Full Size | PDF

4. Conclusion

In conclusion, we have presented the two-photon microwave spectra in a room temperature cesium atom using Rydberg-EIT. The two-photon spectra display a rich of information including microwave ac Stark shift, and two-photon microwave AT splitting, and the state mixture in the strong microwave field range. The two-photon EIT-AT spectra can be divided weak nonlinear region, middle linear region and strong mixing region according to their spectral features. In regard to the Rydberg atom-field interactions, Floquet treatment was employed to simulate the two-photon microwave spectra. Within the measurement uncertainty, Floquet theory reproduced the observed level shifts and AT splittings well. This work exhibits the utility of room-temperature Rydberg-EIT to investigate the Rydberg atom-field interactions. The rich of information provided in this work suggests the method that may be employed to an atom-based technique for precision field measurements.