1 Introduction

With special properties not available in classical radio frequency (RF) microwave (MW) technology such as self-calibration [1, 2], high sensitivity [3, 4], and a wide range of operating frequencies [5–10], Rydberg atomic quantum metrology has been extensively pursued over the past decade, e.g., receivers for communication signals (amplitude modulated [8, 11–13], frequency modulated [14], phase modulated signals [15–17]), stereo players [18], spectrum analyzers [19, 20], Rydberg microwave-frequency-comb spectrometer [21], imaging [22, 23]. To further improve the measurement sensitivity, it has been proposed solutions including atomic superheterodyne receiver [24], enhanced metrology at the critical point of a many-body Rydberg atomic system [25], and resonator [26–31]. In the design of microwave aperture, some control is needed, such as metasurface[32, 33] control of electromagnetic wave, resonant cavity. These methods require local oscillator (LO) fields and mostly use free-space illumination. However, the mobility and integration in practical applications are constrained owing to the striking attenuation of the LO field during long-distance propagation, as well as multipath interference caused by scattering in real environments.

To effectively tackle these concerns and enhance the mobility, previous research [30] used the Rydberg superheterodyne system inside a parallel-plate waveguide(PPWG) configuration, utilizing the PPWG as a means to transmit the LO signal. However, it is noted that the electric field measurement improvement is merely 7.9 dB, suggesting a limited capability for enhancement. Paper [34] proposed an LO embedded field measurement enhancement resonator employing the PPWG antenna radiation and utilized cesium 60D state electromagnetically induced transparency(EIT) signal to detect 638 MHz MW field, where different \(m_{j} = 1/2\), 3/2, 5/2 have different polarizability [35, 36]. If no polarizability value and analysis are used, this method may run the risk of generating mistakes in the computation of the E-field. The experimental setup enabled a significant E-field measurement enhancement of 34 dB and an electrical size of 0.342λ. However, it should also be mentioned that the S11 could only reach −0.11 dB, showing poor impedance matching ability. As a result, efficiently directed to the resonator configurations that can overcome attenuation with no multipath and improve S11 remain elusive.

In this paper, we propose an innovative design that integrates a PPWG with a split-ring resonator (SRR) resonant cavity. The proposed design comprises a resonator that has 32 dB electric(E) field improvement capacity and can effectively guides the LO signal to the resonator by PPWG. Here, we comprehensively analyze the PPWG transmission line LO port integrated methods of split-ring resonator. E field enhancement ability, S11-parameters and E field homogeneity of integrated resonator are investigated, and it is shown that S11-parameters results from numerous complicated factors, including the impedance matching [37] between the coax adapters and the PPWG, and feeding method for two ports. To overcome the gap in S11-parameters, we optimize the structure impedance matching and utilize a common mode signal to supply the LO port as the common mode impedance provides a better match for the port. Experimental measurements show that, the value of S11 is −38.2 dB, which is significantly lower than the previously reported S11 value of −0.11 dB in the literature [34]. Furthermore, employing PPWG transmission line integration without PPWG antenna radiation, the electrical size of the resonator is determined to be smaller 0.088λ. The findings may have implications for the mobility in practical scenarios.

2 Experimental setup

Figure 1 displays the main experimental setup. The energy level diagram in Fig. 1(a) illustrates the utilization of the EIT ladder approach [3, 38–40] to produce the Rydberg states of cesium (Cs) atoms within a vapor cell through two-photon transitions. This paper uses off-resonant Rydberg detection. Two laser sources are used: 852 nm has \(1/e^{2}\) beam diameter of 800 μm with Rabi frequencies of about 2π × 2.56 MHz and approximately 509 nm has \(1/e^{2}\) beam diameter of 1.4 mm with Rabi frequencies of about 2π × 63.39 MHz. The 852 nm laser frequency is locked to the \(6S_{1/2}\rightarrow 6P_{3/2}\) transitions with Rabi frequency \(\Omega _{p}\). The 509 nm laser frequency is locked to the \(6P_{3/2}\rightarrow 50S_{1/2}\) transitions with Rabi frequency \(\Omega _{c}\).

Figure 1
figure 1

(a) Energy level diagram. δ represents the energy level shift caused by the external MW field according to the AC Stark effect. (b) Geometrical details of the injection LO ports integrated resonator are presented. The material is copper with silver plating on the outer surface to increase conductivity. (c) Overview of the experimental setup

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Figure 1(b) illustrates the configuration of the integrated resonator. The LO signal is effectively guided to the resonator via a planar parallel waveguide (PPWG). The integrated resonator is made of copper with silver plating on the outer surface to increase conductivity, and the loop has side lengths of \(28\times 10 \text{ mm}^{2} \) and thickness of 2 mm. The PPGW is \(47.6\times 10 \text{ mm}^{2} \). The vapor cell has an outer dimension of \(10\times 10\times 10 \text{ mm}^{3}\) and 1 mm thick.

Figure 1(c) presents that a vapor cell is inserted into the LO ports integrated SRR. The probe laser, coupling laser are counter-propagating and coincide in the cesium vapor cell, and probe laser transmission is to be monitored on a photodetector at room temperature. The MW field is produced by using a weak signal generator (SIG) with a maximum output power of 25 dBm and is emitted via a horn antenna operating in the frequency range of 0.3-3 GHz. The LO signal is fed by two coax adapters using a common mode signal. The antenna is placed 20 cm in front of the vapor cell. SIG and LO MW fields irradiate into the atomic vapor cell with vertical polarization and propagate perpendicular to the propagation direction of the probe and coupling lasers. To stabilize the laser frequency, two Cs vapor cells are applied in the experiment. The probe laser is stabilized on the saturated absorption spectral signal of the Cs D2-line [41, 42]. Similarly, the coupling laser can be stabilized by the EIT signal generated by another vapor cell. The spectrum analyzer measures photodetector output voltage with a 1 Hz resolution bandwidth. Superheterodyne sensitivity is theoretically defined as the minimum detectable power when the Signal Noise Ratio(SNR) decreases to 1. The LO MW signal frequency is f, while the frequency of the weak SIG signal is detuned to be f+100 kHz. The 100 kHz detuned frequency means that once the coupling laser is set, the probe laser intensity will oscillate at a beat-note frequency of 100 kHz. The results indicate that the Rydberg atom can achieve a function as a mixer [43], accurately down-convert the MW signal according to the frequency of the LO signal. The overall experimental scheme is illustrated in Fig. 2.

Figure 2
figure 2

Overview of the experimental setup. We have used the following notations: DL PRO: external-cavity diode lasers, HWP: half-wave plate, PBS: polarizing beam splitter, QWP: quarter-wave plate, BS: beam splitter, HR: dielectric mirror, DM: dichroic mirror, SA: spectrum analyzer, RPD: 2-way microwave resistive power divider, SIG: a weak signal generator, LO: local oscillator, BD: balanced detector, PD: photodetector

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By comparison with and without cell scenario in Fig. 3(a), the resonant frequency experiences a downward trend from without cell to with cell regardless of considering ambient factors. In general, the relative dielectric constant \(\varepsilon _{r}\) of cell wall is not equal to 1, which renders much larger electrical size. Changes, with and without cell in FDTD (Finite Difference Time-Domain Methods) simulations, the cell walls refractive index n (\(n=\sqrt{\varepsilon _{r}}\)) parameter can greatly impact the resonant frequency, where \(\varepsilon _{r}\) is relative permittivity. Due to the small size of vapor-cell compared with the wavelength of the E field, the effects of the vapor cell on the measured MW E field are not significant [44].

Figure 3
figure 3

(a) The comparison of E field versus frequency curves with the notched. The red solid line represents the presence of a cell, while the black dotted line represents the absence of a cell, both within a structure of identical dimensions. (b) E-field strength distribution with cell and notch with a 1 V/m single-frequency continuous wave excitation source. (c) Cumulative histogram of E field within the beam. The green solid line denotes the resonator without notches, whereas the brown dotted line represents the resonator with notches, both having the same structural dimensions and cell

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The fundamental purpose of the LO ports integrated resonator is to ensure a significant rise in the E field at the selected cell position, minimize reflection, and achieve a uniform distribution of E-field. To acquire a major enhancement in the E field, we choose to integrate the PPWG at the open-ended of the SRR, which minimises the impact on the resonant cavity and transmission line. Additionally, to improve S11-parameters, it is also vital to optimise the impedance matching [37] between the coax adapters and the PPWG. Because the common mode impedance matches the port better, two coax adapters are fed the LO signal by utilizing a common mode signal with identical amplitude and phase to both feed ends simultaneously. Furthermore, achieving a uniform distribution of E field intensity for the LO signal throughout the PPWG is necessary. Homogeneous E field distribution represents a transmission line with Transverse Electric and Magnetic Field(TEM) wave uniform propagation for specified frequency range. To get homogeneous E field distribution, special structural design or a vapor cell perforated walls [45] can be used. Here we consider to add bilateral symmetrical notches on the top and bottom of the structure. Figure 3(b) illustrates the E field distribution with cell and notches in EastWave FDTD [46] simulation with a 1 V/m single-frequency continuous wave excitation source and linear polarization along the vertical direction. In the bottom panels, we present the cumulative histogram (Fig. 3(c)) generated by sampling a cylinder region of ten million samples in cell. The cylinder region with a length of 8 mm and a diameter of 0.5 mm, which is a reasonable size for the probe laser. Figure 3(c) shows the slopes(K) of with and without notches is 0.2 and 0.13, respectively. The FDTD simulation results clearly demonstrate the differences between the two scenarios, with notches distinctly improving the homogeneity of E-field.

3 Results and discussion

Off-resonant Rydberg atomic heterodyne approach [34–36, 47–50] has been employed. The presence of an applied MW E field causes an AC Stark shift in the Rydberg atomic energy level within the off-resonant region. The electric field strength E and the spectral shift δ follow quadratic correspondence [51]:

$$ \delta =-\frac{E^{2}}{4}\alpha _{0}(\omega _{RF}) $$
(1)

where \(\alpha _{0}(\omega _{RF})\) is called as the AC dynamic scalar polarizability in the SI units of MHz/(V/m)2. In this study, we employ the cesium \(50S_{1/2}\) \(m_{j} = 1/2\) state Electromagnetically Induced Transparency (EIT) signal to detect a MW field. The E field in the far off-resonant regime is primarily influenced by the polarizability \(\alpha _{0}(\omega _{RF})\). Consequently, the usage of the 50D state EIT [34] has been precluded due to the significant variations in polarizability among different \(m_{j}\) values, namely 1/2, 3/2, and 5/2 [35, 36, 52]. The antenna radiation formula can be expressed as \(|E|=\sqrt{\eta P \alpha G/4\pi d^{2}}\) to calculate experimental E-fields, where \(\eta =377 \Omega \) is the free space wave impedance, P is the output power of the signal generator, \(\alpha =-0.8\) dB is the insertion loss between the analog signal generator and the antenna, \(G=7\) dBi is the gain of the antenna, and \(d=0.3 m\) is the distance from the antenna to the atomic-vapor cell. In the experiment, all the above parameters are constant, except for P. Consequently, the power P can be directly converted to the E-fields amplitudes. We calculate the \(|50S_{1/2}>\) Rydberg \(\alpha _{0} = 57 \text{MHz}\cdot \text{cm}^{2}/\text{V}^{2}\) by fitting a quadratic relationship of the AC shift and the given RF E-field amplitudes as shown in Fig. 4(a). It is worth noting that the enhancement factor is calculated by comparing the SIG output power under the identical energy level shift with and without the resonator, with polarizability removed. As a result, the magnetic substates has virtually no effect on the enhancement factor.

Figure 4
figure 4

(a) Rydberg state \(|50S_{1/2}>\) AC Stark shift δ fitted to E field. (b) Measured EIT signals acquired without MW field (black dotted curve), with an applied 946 MHz −7 dBm MW field and resonator (green solid curve) and with an applied 946 MHz 25 dBm MW field (yellow dotted curve). The figure shows green and yellow curve have same AC shift \(\delta _{l}\) of about 19.9 MHz. (c) Measured enhancement factor versus MW E-field frequency. (d) Measured S11 versus MW E-field frequency. Red solid curve is without cell and green dotted curve is with cell

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Figure 4(b) shows Rydberg state EIT spectra(black dotted curve) in the state \(|50S_{1/2}>\) incident MW fields at 946 MHz, 25 dBm (yellow dotted curve) without resonator and −7 dBm (green solid curve) with resonator. Note that applied microwave field at 946 MHz induces a red-shift. By attenuating to 32 dBm after loading the resonator, the green and yellow curves showing the same AC shift \(\Delta _{l}\) of 19.9 MHz, indicating the same observable E-field intensity. Thus, the resonator provides electric-field enhancement factor of 39.8(\(10^{(32/20)}=39.8\)).

Figure 4(c) shows the evolutions of enhancement factor with the increment of microwave frequency where dots correspond to experimental results. We can clearly find that the resonator has a maximum measured E-field enhancement factor at 946 MHz and exhibits a fairly narrow frequency response. The resonant frequency shift from 962 MHz(FDTD simulation) to 946 MHz(experiment) can be attributed to the following: first, the cell inner surface has nonzero conductivity due to the adsorption of alkali-metal atoms [53] electric-field-screening effect. Second, the resonator metallic losses and minor differences in manufacturing dimensions may also lead to a reduction in the resonant frequency. Furthermore, scattering from the surroundings can alter the resonant frequency and E-field distribution.

Figure 4(d) shows S11 of the LO coax adapters versus MW frequency. By comparing S11 curves with and without cell, the resonant frequency falls in the presence of the cell. This transformation trend exhibits similarity to the one depicted in Fig. 3(a). By optimizing the impedance matching [37] between the coax adapters and the PPWG, S11 is improved to −38.2 dB. Thus, the optimized integrated resonator needs considerably less input power for the same field intensity within the specified frequency spectrum.

In order to improve the sensitivity of the E-field measurement when the SIG signal amplitude is fixed, we optimise various parameters [54, 55]. These include the LO signal amplitude (which is related to the EIT spectrum shifts shown in Fig. 5(a)) and the LO signal detuned frequency (Fig. 5(b)), as well as the power of the probe and coupling laser. As shown in Fig. 5(a), the SNR of beat-note signal curve has optimal points when EIT spectrum shift equals to 2.6 MHz. Figure 5(b) shows that SNR first rises and then plateaus as the detuned frequency increasing. We choose the 100 KHz detuned frequency.

Figure 5
figure 5

(a) Measured SNR versus EIT specturm shift under 100 KHz detuned frequency. (b) Measured SNR versus detuned frequency of LO signal under 2.6 MHz EIT specturm shift. (c) The SNR as a function of S11 relative to a fixed LO and antenna power. (d) Results from the Rydberg-atom superheterodyne, plots of the beat-note intensity of the spectrum analyzer as functions of signal generator

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To demonstrate the comparable sensitivities of the integrated LO to other SSRs with a smaller applied LO power, we present the SNR as a function of S11 relative to a fixed LO and antenna power in the 925 to 946 MHz frequency range, as shown in Fig. 5(c). The practical operational bandwidth for this system is 26 MHz. There is a significant increase in sensitivity in the 933 to 959 MHz frequency range. Figure 5(c) illustrates that the SNR falls from 31.6 dB to 26.7 dB the increment of S11. It is noteworthy that the low S11 indicates minimal reflection loss in this system, highlighting the integrated LO’s ability to achieve equivalent sensitivities to other SSRs with reduced LO power.

As shown in Fig. 5(d), the beat-note intensity from the spectrum analyzer is a function of signal generator for the cases with resonator in yellow pentagrams and without in black dots. As indicated by the double-headed arrow, the two curves are shifted along the x-axis, suggesting an enhancement of 32 dB. This work identifies the intersections of the linear response curves and the noise floor of the spectrum analyzer as sensitivity. Ultimately, we find that the resonator provides a sensitivity of 213 \(n \text{V}/\text{cm}/\sqrt {\text{Hz}}\); 40 times more sensitivity than without resonator (8.13 \(\mu \text{V}/\text{cm}/\sqrt {\text{Hz}}\)).”

Table 1 summarizes the specific performance comparison with LO port integrated resonator in previous literatures [26, 27, 29, 30, 34]. Compared with the work in References [26, 27, 29], we provide a local oscillator port integrated resonator, which is more flexible for Rydberg atom-based superheterodyne. Compared with the work in References [26, 27, 29, 30, 34], the electrical size reaches a minimum of 0.088λ, which is critical for mobility in practical applications. Furthermore, the S11 decreases from −0.1 dB to −38.2 dB compared to the PPWG antenna radiation in Reference [34]. Furthermore, through the optimisation of experimental parameters, the sensitivity is successfully enhanced.

Table 1 Performance comparisons of Rydberg atom-based electric field measurement enhancement using resonators. Ref.: reference, \(f_{0}\): center frequency, ES: electrical size, EF: enhancement factor, S: sensitivity
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4 Conclusion

To summarize, LO port integrated resonator enables LO signal to be efficiently directed to the resonator through a PPWG. Results show that integrated resonator not only enhances mobility but also improves sensitivity by 32 dB. It is crucial for moving from laboratory to practical applications. Furthermore, the sensitivity is optimised through the optimisation of many parameters such as the LO signal amplitude, LO signal detuned frequency, and the power of the probe and coupling laser. Experiments have shown that sensitivity is 213 \(n \text{V}/\text{cm}/\sqrt {\text{Hz}}\) in the off-resonant region. The electrical size is 0.088λ and S11 reaches −38.2 dB. This work paves the way for Rydberg atom-based superheterodyne incorporated with LO port integrated resonator to improve mobility in practical scenarios.