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Rydberg atoms hold special attraction in electric applications due to their large transition electric dipole moments and huge polarization, which leads to a strong response of atom to electric fields. In radio-frequency (RF) fields, the Rydberg levels are AC Stark shift and splitting, which can realize the study of high-sensitivity electric field sensor of Rydberg atoms. In this work, we use the simpler Shirley's time-independent Floquet Hamiltonian model to calculate the AC Stark energy spectrum of Cs Rydberg atoms. This model can reduce the basic Hamiltonian into such a Hamiltonian that includes only those Rydberg states that have direct dipole-allowed transitions with the target state, thereby significantly improving the speed of computation. The accuracy of the calculation is proved by fitting with the calculated frequency shift of DC Stark energy levels in the weak fields, and the polarizability of 60D5/2 and 70D5/2 Rydberg atomic states are obtained by fitting with the measured ion spectra of DC Stark Cs ultra-cold Rydberg atoms in magneto-optical trap. In addition, we calculate the AC Stark shift of Cs Rydberg atom $ \left| {60{{\text{D}}_{5/2}},{m_j} = 1/2} \right\rangle $ state in electric fields with different frequencies with ε = 100 mV/m. Rydberg atoms provide a structured spectrum of sensitivity to electric fields due to strong resonant interaction and off-resonant interaction with many dipole-allowed transitions to nearby Rydberg states. This kind of the frequency response structure is of significance to a broadband sensor. And we calculate the sensitivity and the scaling of the signal-to-noise ratio (SNR), β, varying with detuning from the$ \left| {60{{\text{D}}_{5/2}}} \right\rangle \to \left| {61{{\text{P}}_{3/2}}} \right\rangle $ transition. The value of β allows one to use the result for any Rydberg state sensor to determine the SNR for any Ε in a 1 s measurement. Therefrom, Rydberg sensor can preferentially detect many RF frequencies spreading across its carrier spectral range without modification while effectively rejecting large portions where the atom response is significantly weaker, and the signal depends primarily on the detuning of the RF field to the nearest resonance which does not convey the RF frequency directly.-
Keywords:
- AC Stark /
- Rydberg sensor /
- Floquet method
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图 1 理论计算 Cs 原子 (a) $ \left| {60{{\text{D}}_{5/2}}, {m_j} = 5/2} \right\rangle $和(b) $ \left| {70{{\text{D}}_{5/2}}, {m_j} = 5/2} \right\rangle $态的 Stark 能谱, 红色虚线为校准后的 AC Stark 能谱, 黑色实线为 DC Stark 能谱
Figure 1. Calculated Stark spectra for Cs Rydberg atom of (a) $ \left| {60{{\text{D}}_{5/2}}, {m_j} = 5/2} \right\rangle $and (b) $ \left| {70{{\text{D}}_{5/2}}, {m_j} = 5/2} \right\rangle $ state. The red dotted line is the AC Stark spectrum, and the black solid line is the Stark map in a DC electric field on a field axis scaled such that the rms fields.
图 2 实验测的 x 方向 70D5/2 里德伯原子 Stark 谱, 黑、红和绿色点线分别为理论计算的 mj = 1/2, 3/2和5/2的Stark 谱
Figure 2. Measurements of the Stark spectrum for 70D5/2 Rydberg atom in the x-direction, and the black, red, and green dotted lines show the calculation Stark spectra of mj = 1/2, 3/2, and 5/2, respectively.
图 3 Cs里德堡原子60D5/2态在 ε = 100 mV/m不同频率射频场中的AC Stark频移, 其中临近的共振态用黑色虚线标定
Figure 3. AC Stark shifts of the 60D5/2 state in RF field with ε = 100 mV/m. The adjacent resonant state is labeled with a black dashed line.
图 4 在低频DC场中, 测量时间t = 1 s, 原子数N = 103 (104, 105, 106)的$ \left| {60{{\text{D}}_{5/2}}, {m_j} = 1/2} \right\rangle $Cs里德伯原子的最小可检测场与射频频率的关系
Figure 4. In a quasi-DC field, the minimum detectable field in a 1 s measurement versus RF frequency using a $ \left| {60{{\text{D}}_{5/2}}, {m_j} = 1/2} \right\rangle $ target state with N = 103 (104, 105, 106) Cs Rydberg atoms.
图 5 测量时间为1 s的最小可检测场和信噪比缩放因子β(红色实线)与$ \left| {60{{\text{D}}_{5/2}}} \right\rangle \to \left| {61{{\text{P}}_{3/2}}} \right\rangle $射频共振失谐量的关系, 其中三角形、正方形和五边形分别对应β = 1, 1—2和 2
Figure 5. Measurement time of 1 s for the minimum detectable field and the scaling of the SNR, β, (red solid) scaling versus detuning of RF transition from the $ \left| {60{{\text{D}}_{5/2}}} \right\rangle \to $$ \left| {61{{\text{P}}_{3/2}}} \right\rangle $. The triangle, square, and pentagon symbols match correspond to β = 1, 2 or somewhere in between, respectively.
表 1 实验测量极化率与计算60D5/2和70D5/2态极化率的比较
Table 1. Comparisons of the measurement and calculation of the polarizability for 60D5/2 and 70D5/2 Rydberg atom.
状态 极化率实验值/
(MHz·cm2·V–2)极化率理论值/
(MHz·cm2·V–2)$ 60{{\text{D}}_{5/2, 1/2}} $ –5007.11 –4984.80 $ 60{{\text{D}}_{5/2, 3/2}} $ –3620.61 –3633.78 $ 60{{\text{D}}_{5/2, 5/2}} $ 281.19 280.93 $ 70{{\text{D}}_{5/2, 1/2}} $ –15455.87 –15432.29 $ 70{{\text{D}}_{5/2, 3/2}} $ –11160.68 –11171.34 $ 70{{\text{D}}_{5/2, 5/2}} $ 835.18 833.77 
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