An all-optical phase detector by amplitude modulation of the local field in a Rydberg atom-based mixer

Xiu-Bin Liu(刘修彬) Feng-Dong Jia(贾凤东) Huai-Yu Zhang(张怀宇) Jiong Mei(梅炅)Wei-Chen Liang(梁玮宸) Fei Zhou(周飞) Yong-Hong Yu(俞永宏) Ya Liu(刘娅)Jian Zhang(张剑) Feng Xie(谢锋) and Zhi-Ping Zhong(钟志萍)

1School of Physical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

2Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China

3National Time Service Centre,Chinese Academy of Sciences,Xi’an 710600,China,University of Chinese Academy of Sciences,Beijing 100049,China

4Institute of Nuclear and New Energy Technology,Collaborative Innovation Center of Advanced Nuclear Energy Technology,Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education,Tsinghua University,Beijing 100084,China

5CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences,Beijing 100190,China

Keywords: quantum sensor, phase detector, Rydberg atoms, micorwave, electromagnetically induced transparency,amplitude modulation

1. Introduction

The Rydberg atom-based radio-frequency (RF) sensor utilizes electromagnetically induced transparency (EIT) and Autler-Townes (AT) splitting to enable an optical readout of the RF electric field.[1-4]This approach offers a direct international system of units(SI)traceable,a broad frequency range,and a self-calibrated measurement of the RF electric field.[1-6]The Rydberg atom-based RF sensor has made rapid progress in the past decade and provided new detection capabilities beyond the traditional antenna and other RF detectors.[7-15]The Rydberg atom-based RF sensor has also been used as atomic receivers to demonstrate RF communication.[16-19]

The technique for precisely measuring the phases of RF fields is important and applied broadly in antenna metrology,radar, remote sensing, and various other fields. Simonset al.successfully converted the Rydberg atom-based sensor to a mixer and directly measured the phase of the signal RF(SIG RF)electric field.[20]The phase of the SIG RF field is downconverted directly to the phase of the intermediate frequency(IF)beat signal(on the order of kHz),which is created by the presence of a local RF(LO RF)field.[20-22]In the present Rydberg atom-based mixer schemes, the amplitude of the beat signal remains constant,and the phase of the SIG RF field can be read out by comparison with another same frequency reference waveform,[20-22]for example,combines an RF combiner or a spectrum analyzer with the mixer.

In this study, we propose the conversion of the Rydberg atom-based mixer to an all-optical Rydberg atom-based phase detector. Through the amplitude modulation(AM)of the LO RF field with the frequency of the beat signal in the Rydberg atom-based mixer, the amplitude of the beat signal was changed and related to the phase of the SIG RF field. We present a more convenient readout for RF phase measurements after studying the relationship between the output voltage of the phase detector and the phase of the SIG RF field. Furthermore, we use the Rydberg atom-based phase detector to realize a sub-degree phase resolution.

2. Theory

Figure 1(a)shows the atomic energy levels applied in this study. The energy levels are87Rb,5S1/2(F=2),5P3/2(F＇=3), 53D5/2(F=4) and 54P3/2(F=3) corresponding to|1〉,|2〉,|3〉,|4〉, respectively. Rydberg EIT occurs when the frequency of the weak probe field is scanned near the resonant transition of|1〉to|2〉in the presence of a strong coupling field coupled with|2〉to|3〉transition. The LO RF field resonates with the Rydberg transition|3〉and|4〉. The frequency difference between the LO RF field and SIG RF field is Δω.The electric field experienced by the atoms is defined asEatomand the low frequency termEmodis given by[20]

whereELOandESIGare the electric field amplitudes of the LO RF and SIG RF field,respectively. Δφ=φSIG-φLOis the difference between the SIG RF field phaseφSIGand the LO RF field phaseφLO,andφLOis set to zero for simplicity.

Fig.1. Basic principle of the Rydberg atom-based phase detector. (a)Levels scheme. (b)Block diagram of the Rydberg atom-based“phase detector”,which combines the Rydberg atom-based“mixer”with AM of the LO RF field;the phase of the SIG RF is carried in the oscillation amplitude of the transmittance of the probe laser(known as the beat signal).

The RF field causes AT splitting of the EIT signal,and the transmittance of the probe lightTpis proportional to the amplitude of the RF field within a certain range.[23]For a weak SIG RF field,whereESIG≪ELO,Eq.(1)can be written as[20]

It can be seen that we can directly extractφSIGthroughABeatin the experiment.WhenELOis modulated,only the amplitude of the beat signal is affected,and the phase information of the SIG RF field is completely inherited.It can be seen from Eq.(3)that in addition toφSIG,all ofDAM,φAM,ELOandESIGwill affect the amplitude of the beat signal. Therefore, these quantities should remain stable and this is relatively easy to realize in practical applications. For example,ESIGcan be attenuated to a fixed value and then sent to the atomic sensor.Consequently, the relationship betweenφSIGand output voltage of the phase detector can be obtained. Thus, our method is to directly compare the reference phase withφSIGinside the atomic mixer. It avoids the perturbation that may be encountered in the transmission process of the beat signal to the RF combiner in the traditional phase detector scheme.

3. Experimental setup

Figure 2 schematically shows the experimental setup,which is based on the Rydberg atomic sensor and mixer. A weak probe light withλp=780 nm was generated using an external cavity semiconductor laser(DL100,Toptica),and the frequency was locked on the resonance transition of|1〉to|2〉using the Zeeman modulation saturated absorption spectrum(SAS).[24]A relatively stronger coupling light withλc=480 nm was generated using a frequency-doubled diode laser(TA-SHG-Pro,Toptica),and the frequency was locked on the resonance transition of|2〉to|3〉using the Zeeman modulation Rydberg EIT spectrum.[25]The linewidths of all the lasers were estimated to be less than 500 kHz using the linewidth of the Rydberg EIT with 494 kHz in cold atom samples.[25]The minimum Allen varianceσ(τ)laserof lasers was approximately 7.3×10-11at 8 s for laser stabilization.[26]The probe light was focused to an 800µm 1/e2diameter using an achromatic lens with powerIp=25 µW. The coupling light with powerIc=30 mW was focused to a 900 µm 1/e2diameter.The probe light and coupling light were both linearly polarized and were counter-propagating in the Rb cell. Then, the intensity of the probe beam passing through the cell was detected using a photodiode (PD) and sent to the lock-in amplifier(LI5640, NF Corporation), and finally recorded by the oscilloscope.

Fig.2.Schematic of the experimental setup for the Rydberg atom-based phase detector. PD denotes the photodetector, purple lines denote the rubidium clock synchronization signals, orange line denotes the reference signal of the lock-in amplifier,and the red dotted line denotes the signal received by the PD and the output of the lock-in amplifier. OSC denotes the oscilloscope.

Here, we used two signal generators to produce the two microwave fields. The first signal generator (8340 B,Keysight Technologies)was used to generate the LO RF field at 14.233 GHz to drive the Rydberg transition|3〉to|4〉. The second signal generator (E8257 D, Keysight Technologies)was used to apply the SIG RF field at 14.233 GHz-Δω(where Δωcan vary from 1 kHz to 100 kHz). The outputs of the two signal generators were connected to two separate horn antennas(LB-62-10-C-SF)that radiate the microwaves toward the cylinder Rb cell(diameter 25 mm and length 75 mm),and the antennas were placed far away from the Rb cell to satisfy the far-field condition. The two signal generators were triggered by the same GPS (Global Positioning System)-tamed 10 MHz Rb clock (FS725, Stanford) to keep the frequency drift of both signal generators to a very low level. The tested minimum Allen varianceσ(τ)clockof the 10 MHz Rb clock was 1.44×10-14after being tamed for 24 h by the GPS.The frequency drifts of the two microwaves were 10-2Hz within 2 h(compared with 1 Hz frequency drifts after being triggered by the Rb clock inside the signal generator within 2 h),which was sufficiently stable for our experiments.[23]The polarizations of the two microwave electric fields were the same as those of the probe and coupling beams and propagated in a vertical direction to the two laser beams.

Next, we describe the conversion of the Rydberg atombased mixer to a phase detector. A function generator generated a sinusoidal signal with the same frequency as the beat signal to modulate the amplitude of the LO RF field. The amplitude and phase of the modulation signal can both be controlled. In addition,the output signal of the function generator was also sent to the lock-in amplifier as a reference signal to demodulate the amplitude of the beat signal and then recorded by the oscilloscope. Another function generator applied a triangular wave signal or a transistor-transistor logic(TTL)signal to trigger a phase shifter to change the phase of the SIG RF fieldφSIG. Specifically,when the voltage of the triangular wave or the TTL signal changes from-0.5 V to 0.5 V,φSIGcan change linearly within the set range.

4. Results and discussion

First,we discuss the effect ofELOandESIGonTp. Then,we describe the influence of the amplitude modulation of the LO RF field on the performance of the phase detector,including the phaseφAMand depthDAMof the amplitude modulation of the LO RF field. Furthermore, we study phase resolution capability of the phase detector.

We briefly discuss the relationship between the transmittance of the probe lightTpand the RF electric field amplitude experienced by the atomsELO(orESIG)when the probe light and the coupling light are locked at the EIT resonance frequency. The results show thatTpandELO(orESIG) satisfy a monotonic proportional relationship when 0.8 mV/cm≤ELO(orESIG)≤20 mV/cm. WhenELO(orESIG) is less than 0.8 mV/cm or greater than 20 mV/cm,Tpwill not change significantly withELO(orESIG). Therefore, in the following experiments, we will study and optimize the performance of phase detector by choosingELOandESIGin the range of 0.8-20 mV/cm.

To study the relationship betweenφAMandφSIGunder LO RF field amplitude modulation,we used a phase shifter to changeφSIGcontinuously over 4πand setφLO=0 for simplicity. The data were obtained with a beat frequency Δω=1 kHz. The black, blue, and red solid lines in Fig. 3 represent the experimental results withφAM= 0,π/4, andπ, respectively.The dotted lines represent the corresponding theoretical calculation results in Eq. (5). Therefore, for the convenience of comparison,a coefficient calibration was performed on the theoretical calculation results. As shown in Fig.3,it is found that the amplitude of the beat signalABeatandφSIGshow a monotonic relationship. Furthermore,the output amplitude of the beat signal andφSIGshow a approximate linearity from 0 toπ/2 whenφAM=π/4,while the traditional commercial phase detector can realize the linear region of 2π. Therefore,φSIGcan be directly converted into a voltage signal, and then the Rydberg atom-based mixer can act as a phase detector. From Fig.3,we can also find that if we changeφAM,the amplitude of the beat signal will be only related to the difference betweenφSIGandφAM. For example, when the amplitude of the beat signal is the positive largest, the difference betweenφSIGandφAMis alwayskπ,k=0,±2,±4,...,as shown in Fig.3. This is also consistent with Eq. (5). Therefore, from this method,we can clearly know that even ifφAMchanges,φSIGcan also be deduced from the amplitude of the beat signalABeat. The calculation result of Eq. (5) is in agreement with the experimental results. The slight difference may owe to the oversimplification of Eq.(5), for example,Tpis not strictly linear withELOorESIG. However,this does not affect the use of the phase detector because the output of the phase detector can be calibrated before the application. The results in Fig.3 can be repeated when Δωvaries from 1 kHz to 100 kHz, where the upper limit of 100 kHz is limited by the bandwidth of the lock-in amplifier. At 100 kHz, the signal is still clear, which shows that the applicable frequency range of this method is very large.

Next,we study the influence of the amplitude modulation depth of the local RF fieldDAMon the output signal of the phase detector,and the results are given in Fig.4. The results show that when ΔφSIGis fixed at 10°and Δω=100 kHz,the change in the phase detector output is proportional to the modulation depthDAMwithin 10%-90%. WhenDAMis less than 10%, the signal cannot be demodulated, and the error of the system at eachDAMis very small. The results show that the output of the phase detector is proportional toDAM,and it has a wide range ofDAMand high repeatability and stability. The red dotted line was calculated using Eq.(3)with the same parameters as in the experiment and was qualitatively consistent with the experimental results. Noted that the theoretical calculation was normalized to facilitate comparison with the experimental results.Although the phase detector can work in a wide range ofDAM, in order for both to satisfyELO≫ESIGandELO×DAM≈ESIGat the same time,we chooseDAM=20%in the following experiments.

Fig. 3. Examples of the Rydberg atom-based phase detector; experimental parameters: ELO = 10.14 mV/cm, ESIG = 2.75 mV/cm,Δω =1 kHz, and the modulation depth DAM is 20%; assuming that φLO=0. The black,blue,and red solid lines represent the experimental results with φAM=0,π/4,and π,respectively;each curve corresponds to 32 averages to improve the signal-to-noise ratio. The dotted lines represent the corresponding theoretical calculation results in Eq. (5).Theoretical calculation amplitude is normalized to facilitate the comparison with the experimental results.

Fig. 4. Relationship between the beat frequency oscillation amplitude and the modulation depth DAM,where ΔφSIG=10°. Black solid circles denote the experimental results with parameters ELO =10.14 mV/cm,ESIG =2.75 mV/cm, and Δω =100 kHz; red dotted line is calculated using Eq.(3). The theoretical calculation is normalized to facilitate the comparison with the experimental results, and the calculation parameters are the same as the experimental parameters.

Finally, we demonstrate the phase resolution capability of the Rydberg atom-based phase detector by changing the phase difference of the SIG RF field ΔφSIGand measuring the output voltage of the phase detector. Based on Eq. (3),ABeatis most sensitive toφSIGwhenDAM×ELO≈ESIG, therefore we choose the experimental parameters:ELO=10.14 mV/cm,ESIG=2.75 mV/cm, and the modulation depthDAM=20%.In the meantime,φAMwas set toπ/4 to obtain the maximum slope ofABeattoφSIGnearφSIG=0 as shown in Fig. 3. The data obtained with the beat frequency Δω=1 kHz and averaged over 128 sets of data to enhance the signal-to-noise ratio(SNR) of the readout. Figure 5(a) shows the output voltage of the corresponding phase detector when ΔφSIGis taken as 12°, 10°, 8°, 5°, 2°, and 1°, respectively. It can be seen that the phase detector still has a good readout when ΔφSIGis set to 1°. Figure 5(b)shows the corresponding analysis results in Fig. 5(a). The horizontal axis represents ΔφSIG, and the vertical axis corresponds to the amplitude of output signal of the phase detector. The black hollow circles are the experimental results corresponding to Fig. 5(a), and the red solid line is the result of a linear fitting that crosses the 0 points. After analyzing the average value and error of the corresponding phase detector output voltage at high level and low level in the experiment,the achievable phase resolution is 0.6°,while Ref.[13]obtained a resolution of 0.8°using the superheterodyne method,and a traditional commercial phase detector can achieve the phase resolution of about 1°.[27]

Fig. 5. Relationship between the beat frequency oscillation amplitude and the set phase difference. (a)Phase control and reaction of the Rydberg atombased phase detector. Black solid line is the control voltage of the phase shifter;-0.5-0.5 V corresponds to the set phase difference. Red line,green line, blue line, cyan line, magenta line and dark yellow line represent the output voltage of the corresponding phase detector when ΔφSIG is 12°, 10°,8°, 5°, 2° and 1°, respectively. Each curve corresponds to 128 averages to enhance SNR.By setting ΔφSIG,φSIG can jump back and forth with the value of ΔφSIG. Specifically, 0.5 V phase control corresponds to φSIG =ΔφSIG/2,and -0.5 V phase control corresponds to φSIG =-ΔφSIG/2; phase difference between the amplitude modulation of the local field and the phase of the local field is fixed to π/4; experimental parameters: ELO =10.14 mV/cm,ESIG =2.75 mV/cm, Δω =1 kHz, and the modulation depth DAM =20%.(b)Relationship between the beat frequency oscillation amplitude and the set phase difference. Black hollow circle denotes the experimental result corresponding to(a);red solid line denotes a linear fitting that crosses the 0 points.The experimental data are not collected in the same day,so the data of Figs.3 and 5 cannot be directly compared.

5. Conclusion

We have converted the Rydberg atom-based mixer to an all-optical Rydberg atom-based phase detector by the amplitude modulation of the LO RF field; that is, the phase of the SIG RF field can be directly converted into a voltage signal.The amplitude of the beat signalABeatand the phase of the SIG RF fieldφSIGshow a monotonic relaionship when the amplitude of the local RF field is modulated by the frequency of the beat signal. The output voltage of the phase detector andφSIGshow a linear relationship within the range of 0-π/2 of the SIG RF field phase whenφAMis set with a difference ofπ/4 from the phase of the local RF field. A sub-degree phase resolution is also achieved by optimizing the experimental conditions according to a simple theoretical model. Because the Rydberg atom-based phase detector is an optical readout,it is better than the traditional phase detector in noise suppression.This work will expand and contribute to the development of RF measurement devices based on Rydberg atoms.

Acknowledgements

Project supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0304900 and 2017YFA0402300), the Beijing Natural Science Foundation (Grant No. 1212014), the National Natural Science Foundation of China (Grant Nos. 11604334, 11604177,and U2031125), the Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB08-3), the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Grant No. KF201807), the Fundamental Research Funds for the Central Universities,and Youth Innovation Promotion Association CAS.