We have systematically investigated the influence of the gas temperature (T), the Rabi frequencies of the probe laser (Ωp), the coupling laser (Ωc) and the radio-frequency (ΩRF) on the Rydberg electromagnetically induced transparency (EIT) and Autler-Townes (AT) splitting (Δf) by defining a general Doppler mismatch factor Dg=ΩRF/Δf in the Rydberg atom-based microwave electrometry. The effect of T on Dg is studied in detail from 0 to 1000 K, the results show that Dg is insensitive to T when T<10 μK or T>10 K, while Dg changes significantly with 10 K>T>10 μK. Then the effects of Ωp, Ωc and ΩRF on factor Dg at T=300 K (typical room temperature) and T=10 μK (typical temperature of cold atom by laser cooling) are studied in detail, respectively. The results show that the linewidth of Rydberg EIT (ΓEIT) can be used as a key parameter to characterize the dependence of Dg on Ωp and Ωc in both cases. Dg is insensitive to T, Ωp and Ωc when ΩRF>3ΓEIT which means that ΓEIT determines the lower limit of the linear region of the radio-frequency (RF) electric field strength measured by EIT-AT splitting. More interesting, the range where Dg is insensitive to Ωp and Ωc can be greatly expanded by lowering the gas temperature to 10 μK. The ranges of parameters where Dg is insensitive to T, Ωp, Ωc and ΩRF are given, and such relationship can be easily scaled to other atomic systems. The results can help the selection of various parameters in the experiments and specific applications to ensure the accuracy of measuring the RF electric field.

In this paper, we have theoretically and experimentally studied a quantum microwave electrometry in a cold atomic system using Rydberg electromagnetic induction transparency (EIT) and Autler-Townes splitting (EIT-AT splitting). We obtained spindle-shaped cold atomic clouds in a magneto-optical trap and then pumped cold atoms to quantum state 5S_1/2，F=2，m_F=2 by using the optical-pump laser. We obtained the Rydberg EIT spectrum peak with narrow linewidth by taking the advantages of the low temperature and small residual Doppler broadening. The results show that the typical EIT linewidth with 16 μK cold atoms is about 460 kHz which is 15 times narrowed than that of 7 MHz obtained in the thermal vapor cell. The microwave electric field amplitude is measured by EIT-AT splitting in the cold atoms for frequencies of 9.2 GHz, 14.2 GHz and 22.1 GHz , receptively. The results show that there is a good linear relationship between the EIT-AT splitting interval and the microwave electric field amplitude. The lower limit of the microwave electric field amplitude that can be measured in the linear region can reach as low as 222 μV/cm, which is enhanced about 22 times than that in the traditional thermal vapor cell about of 5 mV/cm. The improvement of the lower limit by EIT-AT splitting method is roughly scaled as the narrowing EIT line width by cold atom samples. This demonstrate that, benefiting from the smaller residual Doppler effect and the narrower EIT linewidth in cold atoms, the cold atom system is more advantageous in the experiment of measuring the weak microwave electric field amplitude by using the EIT-AT splitting method. This is of great benefit to the absolute calibration of very weak microwave electric fields. Furthermore, the lower limit of the microwave electric field amplitude that can be measured smaller than 1 μV/cm by using the change of transmittance of the prober laser at the EIT resonance, and the corresponding sensitivity can reach 1 μV/cm Hz^-1/2. These results demonstrate the advantages of cold atomic sample in microwave electric field measurement and its absolute calibration.

We theoretically and experimentally studied quantum microwave electrometry in a cold atomic system using Rydberg electromagnetic induction transparency (EIT) and Autler--Townes splitting ( EIT-AT splitting). In cold atoms, a spectral linewidth of ~ 500 kHz for EIT was achieved owing to a significant reduction in residual Doppler width, i.e., by at least an order of magnitude, compared to that in vapor cells at room temperature. Therefore, the minimum microwave electric field intensity EMW that can be measured is 430 μV/cm, which is one order higher sensitivity in the EIT-AT regime than that in vapor cells at room temperature. Unlike microwave electrometry in atomic vapor cells, EIT-AT splitting cannot be observed if EMW is so large that the EIT-AT splitting interval Δ fm exceeds the absorption peak width of the cold atom while scanning the frequency of the probe laser (ωp). Moreover, EIT-AT splitting can be observed if Δ fm exceeds the natural linewidth Γeg of the intermediate states while scanning the coupling laser (ωc) and maintains a high spectral resolution with a high signal-to-noise ratio. While scanning ωc, the upper microwave electric field intensity is limited by the scanning range of our setup. Using our system, we measure the maximum field to be 21.6 mV/cm, nearly three times higher than that of 6.8 mV/cm while scanning ωp. The results indicate that the linear range of EMW measured using EIT-AT splitting considerably improves in cold Rydberg atoms.

Recently, a Rydberg atom-based mixer was developed to measure the phase of a radio frequency (RF) field. The phase of the signal RF (SIG RF) field is down-converted directly to the phase of a beat signal created by the presence of a local RF (LO RF) field. In this study, we propose that the Rydberg atom-based mixer can be converted to an all optical phase detector by amplitude modulation (AM) of the LO RF field; that is, the phase of the SIG RF field is related to both the amplitude and phase of the beat signal. When the AM frequency of the LO RF field is the same as the frequency of the beat signal, the beat signal will further interfere with the AM of the LO RF field inside the atom, and then the amplitude of the beat signal is related to the phase of the SIG RF field. The amplitude of the beat signal and the phase of the SIG RF field show a linear relationship within the range of 0 to π/2 when the phase of the AM is set with a difference π/4 from the phase of the LO RF field. The minimum phase resolution can be as small as 0.6 degree by optimizing the experimental conditions according to a simple theoretical model. This study will expand and contribute to the development of RF measurement devices based on Rydberg atoms.

A Rydberg atom-based mixer has opened up a new method to characterize microwave electric fields such as the precise measurement of their phase and strength. This study further demonstrates, theoretically and experimentally, a method to accurately measure the polarization of a microwave electric field based on a Rydberg atom-based mixer. The results show that the amplitude of the beat note changes with the polarization of the microwave electric field in a period of 180 degrees, and in the linear region a polarization resolution better than 0.5 degree can be easily obtained which reaches the best level by a Rydberg atomic sensor. More interestingly, the mixer-based measurements are immune to the polarization of the light field that forms the Rydberg EIT. This method considerably simplifies theoretical analysis and the experimental system required for measuring microwave polarization using Rydberg atoms and is of interest in microwave sensing.