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Abstract
Researchers are interested in the sensor based on Rydberg atoms because of its broad testing frequency range and outstanding sensitivity. However, the discrete frequency detection limits its further employment. We expand the frequency range of microwaves using Rydberg atoms under the Zeeman effect. In such a scheme, the magnetic field is employed as a tool to split and modify adjacent Rydberg level intervals to realize tunable frequency measurement over 100 MHz under 0-31.5 Gauss magnetic field. In this frequency range, the microwave has a linear dynamic variation range of 63 dB, and has achieved a sensitivity of 11.72 µV cm-1Hz-1/2 with the minimum detectable field strength of 17.2 µV/cm.. Compared to the no magnetic field scenario, the sensitivity would not decrease. By theoretical analysis, in a strong magnetic field, the tunable frequency range can be much larger than 100 MHz. The proposed method for achieving tunable frequency measurement provides a crucial tool in radars and communication.
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1. Introduction
Researchers have given a lot of attention on the new method based on optical detection technology due to its distinct benefits over conventional microwave testing techniques [1–4]. With benefits of high sensitivity, self-calibration and intrinsic accuracy, microwave detection technology based on the Rydberg atom has achieved significant advancements in recent years [5–12]. These include the detection of weak field sensitivity of microwave [13,14], detection of strong fields [4], measuring the phase of a radio frequency wave [15], polarization measurement [16], measurement of incident angle [5], microwave imaging [17], all optical detection [8,18], multi-band microwave communication [19], etc. Up to now, the minimum detectable microwave electric field strength is 55 nV cm-1Hz-1/2[14]. Researchers have completed modulation signals testing on FM/AM [20,21] and PSK digital signals for communication [22]. Due to the abundant Rydberg energy levels, the measurement frequency of microwave can range from MHz to THz, successfully addressing the issue of narrow frequency range in conventional antenna receivers. However, this measurement of Rydberg atoms is limited to discrete frequencies within a small bandwidth less than 10 MHz, and they are related to the transitions of nearby Rydberg levels [15,23]. Recently, many researchers have utilized far off resonant AC stark effect [9], two off-resonant microwave [12], and waveguide coupling [10] to achieve tunable frequency measurement microwave. The above techniques rely on electric field to achieve tunability.
In this paper, we utilize the sensitivity of Rydberg atoms to magnetic fields and modulate the Rydberg state levels, achieving a tunable frequency measurement of 100 MHz under maximum magnetic field strength of 31.5 Gauss. The fundamental idea is that the Zeeman splitting, which lengthens the selection path for energy level transitions, allows the degenerate Rydberg level to be split into symmetrically dispersed energy levels. The magnetic field can adjust the energy intervals of adjacent Rydberg energy levels, therefore the measurement frequency of microwave is tunable rather than a fixed frequency. This technique simply requires the Helmholtz coil and does not require complicated waveguides or expensive power sources. We choose Rydberg state mj = 5/2 or mj =-5/2 to detect microwave, because they have more obvious EIT-AT effect than other hyperfine structure levels. The incident strong signal field is read out by Rydberg-EIT-AT and the weak signal can be read by superheterodyne [14]. The microwave signal has a linear dynamic range of 63 dB under 31.5 Gauss magnetic field and the amplitude of beat frequency signal is as a function of the square root of microwave source power P, which is comparable with the measurement of 60 dB without magnetic field. The laser linewidth and laser intensity fluctuations and electronic circuit constraint results in a minimum observable field strength of 17.2 µV/cm and sensitivity of 1.72 µV cm-1Hz-1/2 under magnetic field. The minimum detectable field strength is also comparable with the resonant frequency sensor without magnetic field which is 18.4 µV/cm and sensitivity is 0.823 µV cm-1Hz-1/2. The frequency range can be further broadened with high sensitivity when the magnetic field strength is strong enough.
2. Experimental principles
The energy levels will be modulated by the magnetic field [24,25]. This necessitates a thorough analysis of the Zeeman splitting and the hyperfine energy levels. According to perturbation theory, the system's total Hamiltonian can be expressed as follows:
where Hhfs represents the Hyperfine Hamiltonian [26–29], HBis the Hamiltonian of the interaction between atom and magnetic field. The magnetic interaction is much stronger than the hyperfine interaction (HB≫Hhfs) for the Rydberg state. Meanwhile, Rydberg atoms are in the strong field, which has the Paschen-Back effect. As a result, the strong field eigenstates |JmJ, ImI› are undisturbed state, and the Hyperfine Hamiltonian Hhfs is a perturbation term. The following equation [30] can be used to determine the Zeeman splitting of hyperfine levels:Because of the linear relationship as shown below, the Hyperfine structure parameters ${{A}_{{hfs}}}$ and ${{B}_{{hfs}}}$ are produced for theoretical analysis of the Zeeman splitting by fitting the known values of various states [31] according to equations (3):
3. Experimental setup
A typical atomic superheterodyne receiver is set up for microwave detection, as seen in Fig. 1. The Fig. 1 (a) shows an electric field dressed four-level EIT system with the Rydberg atom transition and causes the AT splitting effect in the Rydberg atom to detect an electric field of interest. The levels used are 6S1/2, 6P3/2, 52D5/2, and 53P3/2, which are labeled as |1>, |2>, |3 > and |4 > respectively. The energy level interval is also shown under Zeeman splitting effect. There are many possible transitions between nondegenerate Rydberg Zeeman sublevels. With the strength of magnetic field increase, the differences of transitions paths can be reflected in EIT and EIT-AT spectra under microwave. The probe laser at 852 nm is generated by an external cavity semiconductor laser, and the frequency of probe laser is locked by polarization spectroscopies [8], which is resonant with the levels |1 > and |2 > . In order to control the strength of the probe laser and keep it stable during the experiment, we utilize an attenuator (made by ThorLabs) and PID controller. The output of 509 nm coupling laser is through the Topic's frequency doubling laser and the coupling laser frequency is locked using a EIT spectrum [18], which corresponds to the transition between the levels | 2 > and | 3 > . Collimating lens at the fiber ports of the coupling laser and probe laser are connected to adjust the spot sizes of the two beams of lasers, and the waist radius of coupling laser and probe laser are 0.563 mm and 1 mm, respectively. The power of coupling laser and probe laser are 5.89 mW and 72.31 µW. A coupling laser and a probe laser counter propagate through a cylindrical room-temperature cesium (Cs) cell with a length of 30 mm and a diameter of 10 mm. In experiment, the polarization of the coupling laser and the probe laser are adjusted to be a linear polarization through polarized beam splitter (PBS), perpendicular to the ground. The EIT spectrum is obtained by scanning coupling laser to eliminate Doppler background.
Fig. 1. Experimental setup diagram for typical atom superheterodyne receiver. (a) The four levels diagram of the Rydberg atom (left) shows the distribution of the Zeeman energy levels under a magnetic field (right). The interval of frequency between the Zeeman energy levels has been given. For simplicity, only the splitting level has been drawn for the Rydberg energy levels. (b) The experimental schematic diagram shows that the coupling laser (green line) and the probe laser (red line) counter propagate through the cesium (Cs) atom vapor cell in a line. The transmission of the probe laser is detected using a balanced photodiode (the reference laser is not drawn) to offset the drift of energy. The signal field and LO field are generated by two horn antennas (DH8912) and the Helmholtz coils generates a uniform magnetic field to work on the cesium atom vapor cell.
The signal field and local field (LO field) are generated by two antennas connected to two signal generators (E8257D, Keysight Technologies) as shown in Fig. 1 (b). Two signal generators are also connected through coaxial lines to ensure the stability and synchronization of the two microwave signals. The microwave frequency is resonant the transition between the |3 > and |4 > levels, corresponding to a resonant frequency of 5.045 GHz without magnetic field. The antennas are placed in the far-field area of the antenna, and the main lobe width of the antenna is far larger than the length of the vapor cell to ensure the uniformity of electric field in the vapor cell. The polarization directions of the microwave are both perpendicular to the ground direction and the propagation directions are perpendicular to the propagation direction of the two laser beams.
In experiment, a homogenous magnetic field is produced using Helmholtz coils, which are two circular copper coils spaced apart. The cesium atom vapor cell is placed at the central axis of the two coils. The distance between the two coils is L = 15 cm and the diameter of coils is d = 25 cm. The size of coils is far larger than the volume of the cesium atom vapor cell, so that the reflection of the coil to microwave can be minimized. By measuring the magnetic field around cesium atom vapor cell using magnetic Hall elements, it is found that the magnitude of the magnetic field is uniform and has a variation of less than 1%. The direction of the magnetic field is consistent with the propagation direction of the probe laser. To guarantee the magnetic field's stability, a constant current stabilized power supply feeds the coil. Due to the limitation of coil current, the maximum magnetic field in the experiment is 35 Gauss.
4. Results and discussion
The transmission spectrum of a probe laser can be modulated by magnetic fields [18]. The EIT spectra under different magnetic fields are produced to understand the precise impact of magnetic fields on the Rydberg level. The Fig. 2 (a) and (b) show the EIT spectra under different magnetic fields. To facilitate the explanation of peak splitting, the Fig. 2. (c) shows the naming of the split peaks. The modulation of the magnetic field manifests mainly on the degeneracy and shift on Rydberg levels. The main peak is a relatively sharp single peak corresponding to the transition of 6P3/2 to 52D5/2 in the EIT without magnetic field. With the introduction of the magnetic field, the transmission peak gradually widens and decreases, causing the degenerate energy level at the 52D5/2 position to degrade, split and shift. For example, when the magnetic field strength is up to 3 Gauss, the splitting phenomenon of the main peak in the EIT manifests clearly, indicating that the energy level | 3 > has split obviously in Fig. 1(a). When the magnetic field strength is up to around 10 Gauss, the small splitting peaks appear obviously. The frequency difference between the peaks of the EIT splitting spectrum as a function of magnetic field strength in experiment and in theory is shown in in Fig. 2. (d). As is shown in in Fig. 2. (d), the findings of the experiment are displayed as solid spheres, and the theoretical calculations results are shown as the black line. In the linear zone of the Zeeman effect, the peaks divide symmetrically and linearly alter as the magnetic field increases. It can be seen in Fig. 2 (d), the results of experiment and calculations are close to each other. Due to different population distribution and transition energy levels of each peak, the sensitivity and polarizability to microwave of each peak are different. In experiment, we selected the 3-left peak or 3-right peak (mj = -5/2 or mj = 5/2) with the obvious EIT-AT effect to obtain the sensitivity of this system for microwave.
Fig. 2. Illustration of the EIT transmission spectra and energy levels under different magnetic fields. (a) The EIT transmission spectra under different magnetic field strength. There is a total of six splitting peaks as the magnetic field increases. (b) The EIT splitting frequencies under the magnetic. (c) The naming of each peak in EIT under magnetic field. (d) The frequency difference of the six peaks in the EIT spectrum under the magnetic field. The dots sphere is data for the experiment and black lines are data for the theoretical calculation results. The peaks shift and split symmetrically are relative to resonant level without magnetic field.
The Fig. 3 displays the EIT-AT spectra curves under different magnetic fields corresponding to different resonant frequencies. The EIT-AT effect is occurred at either the 3-left peak or 3-right peak. Under the same strength of magnetic field, they have nearly same shifting of frequency under different magnetic field. Since the resonant frequency is 5.045 GHz, when there is no magnetic field. Therefore, the tunable frequencies of microwave are symmetric to 5.045 GHz. For example, EIT-AT spectra curves at 5.098 GHz and 4.998 GHz under 31.5 Gauss in Fig. 3(b) and (c) have been demonstrated, and their EIT-AT effect occurred on two side splitting peaks respectively relative to the resonant EIT peak without magnetic field.
Fig. 3. EIT-AT splitting curves under different magnetic fields. (a) EIT-AT without magnetic field at resonant frequency of 5.045 GHz. (b) EIT-AT spectra at a resonant frequency of 4.998 GHz under magnetic field strength of 31.5 Gauss (c) EIT-AT spectra at resonant frequency of 5.098 GHz under magnetic field strength of 31.5 Gauss (d) EIT-AT spectra at resonant frequency of 5.076 GHz under magnetic field strength of 21.5 Gauss.
In the superheterodyne test, the LO (local) field is fixed at an optimal operating point [9] and the beat frequency δ is set to 5 kHz, which reduces the harmonic frequency interference from the noise signal in the detector and oscilloscope. The relationship curves between the amplitude of the signal and the square root of microwave source power P are presented in Fig. 4. The amplitudes of the beat frequency signal are obtained by oscilloscope and analyzed through the Fast Fourier Transform (FFT). The amplitude value of the FFT as a function of the square root of microwave source power P is represented in the figure by the black square dots. The red circle dots are the measurement electric field result by EIT-AT spectrum, and the calculated results are plotted with a red solid line as the reference line to calibrate the measurement results. The following equation, which is based on the modified radar equation at the vapor cell, can be used to determine the calibration value for the electric field:
where E is the electric field, G is the gain of the antenna, d is the distance between the antenna port and the center of the cesium cell and a is calibration factor of the vapor cell. The Fig. 4(a) shows the microwave measurement at f = 5.045 GHz without magnetic field. And the minimum field strength is 18.4 µV/cm and the sensitivity is 0.823 µV cm-1Hz-1/2. As is shown in the Fig. 4(a), the amplitude of signal has a good linear variation of 60 dB with the square root of microwave source P, which provides a good solution for microwave amplitude modulation FM communication and electromagnetic metrology. The linearity breaks down at high signal field because the condition Ωsig ≪ΩLO is no longer suitable, and it also breaks down at low signal field since the noise is compatible to the signal field.Fig. 4. The measurement results of the microwave at different frequencies under different magnetic field strength. The data are taken with the resonant superheterodyne (black squares) and EIT-AT in strong field regime (red circles dots). The red solid line displays the electric field strength calculated by the equation of E = a·(30·P·g)1/2/d. The dash line shows the minimum measured electric field. (a) The measurement results of the microwave at 5.045 GHz without magnetic field. (b) The measurement results of the microwave at 5.098 GHz under magnetic field strength of B = 31.5 Gauss (c) The measurement results of the microwave at 4.998 GHz under magnetic field strength of B = 31.5 Gauss. (d) The measurement results of the microwave at 5.018 GHz under magnetic field strength of B = 21.5 Gauss.
To illustrate our ability to extend the range of measurement frequency, we adjust the magnetic field strength to realize tunable frequency of microwave by superheterodyne detection. The amplitude of signal is the function of square root of microwave power. The measurement curves corresponding to 5.098 GHz, 4.998 GHz, and 5.018 GHz under different magnetic fields are shown in the Fig. 4. (b), (c), (d), respectively. And their sensitivities are 1.72 µV cm-1Hz-1/2, 2.04 µV cm-1Hz-1/2 and 1.92 µV cm-1Hz-1/2, respectively. Through the test, tunable frequency of 4.997-5.098 GHz with a range of 100 MHz under 0-31.5 Gauss magnetic field and a minimum detectable field strength of 17.2 µV/cm at a magnetic field strength of 31.5 Gauss are achieved. Considering the influence of experimental errors and uncertainties in experiment, the sensitivities at different detection frequency of microwave under different magnetic field are comparable with the one at resonant frequency without magnetic field. According to the experimental principle in Fig. 2, the tunable frequency range can be further extended when the strength of the magnetic field is large enough. It should be emphasized that magnetic field effects are consistent regardless of magnetic field direction.
5. Conclusion
In this paper, we have demonstrated the tunable frequency microwave measurement of 4.998-5.098 GHz using Zeeman splitting under magnetic field in hot cesium atomic vapor cell. The nD Rydberg EIT spectra exhibit magnetic field-dependent Zeeman splitting and shifting. We have achieved the minimum electric field 17.2 µV/cm, the sensitivity of 1.72 µV cm-1Hz-1/2 and a linear dynamic range of 63 dB microwave signal, which are comparable to the resonant microwave frequency at 5.045 GHz without magnetic field. We also investigate the influence of magnetic field on EIT/EIT-AT spectra and analyzed them theoretically. Under the same strength of magnetic field, their EIT-AT effect occurred on two side splitting positions respectively relative to the resonant EIT peak without magnetic field. They have nearly same shifting of frequency under different magnetic field. Since the resonant frequency is 5.045 GHz, when there is no magnetic field. Therefore, the tunable frequencies of microwave are symmetric to 5.045 GHz. It should be mentioned that this work provides a novel method to detect the microwave. This could contribute to the further development of quantum microwave sensing technology based on Rydberg atom for better implementation.
Funding
National Natural Science Foundation of China (62071040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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