The Rydberg atoms provide a platform for high-sensitivity^[1], wide-range^[2,3], and self-calibrated^[4] microwave sensing, which shows potential applications in microwave metrology, communications, and radar astronomy^[5,6]. However, due to the limitation of the electromagnetically induced transparency (EIT)^[7] linewidth, the instantaneous bandwidth of the Rydberg atomic system is less than 10 MHz^[8], typically. This finite bandwidth limits the data reception rates of the Rydberg atomic receiver and the application of the system in high-capacity communications^[9,10]. By optimizing the beam waist and the power of the lasers involved, the instantaneous bandwidth of the Rydberg atomic system can be improved^[11–13]. The lineshape of the microwave field resonant transition between distinct Rydberg states influences the reception bandwidth of the system. Applying an auxiliary microwave field or multi-frequency dressing microwave fields is an effective way to increase the response in the off-resonant and near-resonant systems^[14–20], thus increasing the operation bandwidth and data transfer rates.

Combining frequency comb technology with the Rydberg atoms, e.g.,  the microwave or optical frequency comb, enables the system to acquire more spectrum information^[21,22] and promotes the probability of capturing a fast time-varying signal. This is vital for cosmology detection, communications, cognitive radio, and radar applications. In addition, a continuous tunable waveguide-enhanced coupling atomic system^[3,23] helps to receive signals in different frequency bands.

Here, we achieve a tunable off-resonant Rydberg microwave frequency comb (MFC) spectrum to increase the off-resonant response of the Rydberg atoms. The demonstrated wideband continuous tunable MFC spectrum helps us recognize the spectrum pattern for an unknown frequency signal. This MFC method is essentially based on the coupling between multi-frequency microwave fields and the Rydberg atoms, and the obtained instantaneous response bandwidth is several times the bandwidth of the atomic heterodyne detection using a single local oscillator (LO) field. In addition, the meta-waveguide used in our system provides a broad operation range and strong field enhancement features compared with traditional waveguides. By combining the Rydberg MFC spectroscopy technique and a meta-waveguide-coupled Rydberg atomic system, instant microwave reception in a range of 36 MHz is achieved. A maximum detection range of 125 MHz for the MFC spectroscopy is achieved at an off-resonance frequency (2 GHz) from the Rydberg microwave transition frequency (10.78 GHz and 11.83 GHz).

As illustrated by Figs. 1(a) and 1(b), the room-temperature Rb85 Rydberg atoms are prepared and detected in a two-photon EIT scheme by two lasers^[24]. A probe laser of 780 nm, which is emitted through a single-frequency laser (Prelaser, FL-SF-780-CW), drives the transition 5P3/2↔5S1/2, and a 480-nm coupling laser that is generated from a narrow linewidth laser (Prelaser, FL-SF-480-CW) excites the atoms to the Rydberg state 58D5/2. The probe laser is locked to the saturated absorption spectrum of the Rb85 atoms, while the coupling laser is locked to the resonance peak of the two-photon EIT spectrum. These two beams are first focused by a pair of lenses (focal length f=200mm) and then overlapped in a 50-mm-long vapor cell. The power of the probe laser and coupling laser is 90 µW and 550 mW, with beam waists of 330 µm and 470 µm, respectively. The microwave field resonantly couples the atoms with two Rydberg transitions, 58D5/2↔57F7/2 and 59P3/2↔58D5/2, at 10.78 GHz and 11.83 GHz (calculated by the Python package, Alkali Rydberg calculator^[25]). However, to increase the tunability of the system, we set the operation frequency of the Rydberg MFC spectrum with a large red detuning Δ from the resonant transition frequencies of the Rydberg atoms. The EIT spectrum signal is obtained by measuring the transmission of the probe laser with a photodetector (Thorlabs, PDB450A-AC). Finally, the signal output from the photodetector is sent to an electric spectrum analyzer (ESA) (Ceyear, 4024F) to acquire the frequency spectrum of the output signal. A homemade spoof-surface-plasmon-polariton meta-waveguide^[26] is placed under the vapor to enhance the field intensity where the Rydberg atoms are excited by concentrating the microwave electric field in a small region^[26,27]. The signal microwave and LO fields are combined through a resistance power divider (RPD) and then input to the spoof-surface-plasmon-polariton^[28] meta-waveguide through a SubMiniature version A (SMA) connector.

By applying an LO microwave field that resonates with the Rydberg atoms, the sensitivity of the Rydberg atomic receiver is improved^[1]. Apart from the resonant working scheme, an off-resonant atomic heterodyne method has also been demonstrated in the Rydberg atomic system^[3,29]. Similarly, as shown in Fig. 1(a), an MFC field that is off-resonant with the Rydberg microwave transition is applied here. We replaced the single LO field in the off-resonant heterodyne detection scheme with a strong off-resonant MFC field (EMFC) at the input port of this meta-waveguide coupled system. When the microwave frequency is far from the resonance frequency of the system (10.78 GHz and 11.83 GHz), the energy levels of the Rydberg atoms are shifted by the beat signal between the MFC field (EMFC) and signal microwave field (ESig) through the AC-Stark shift^[2,3,30]. Thus, the response of the system is proportional to the AC-Stark shift, which is expressed as $$\begin{gathered} Eatom2≈(EMFC+ESig)2=(∑iNELOi+ESig)2 \\ ≈N|ELO|+2|ELO*ESig|(|EMFC|≫|ESig|), \end{gathered}$$(1)where Eatom is the electric field sensed by the atoms; EMFC and ESig represent the electric field of the MFC field and signal microwave field, respectively. The electric field for a specific comb line is marked as ELOi. N represents the total number of the comb lines. We assume that the electric field strength ELO is the same for all comb lines. In the experiment, the terms of the beat signal between each comb line ∑ij,i≠jNELOiELOj are eliminated as the beat frequency (>4MHz) is much larger than the instantaneous bandwidth (∼300kHz). The information of the microwave signal (amplitude, phase, and frequency) is determined from the output beat 2|ELO*ESig| between the signal and the MFC field from the EIT spectrum^[1,21,31]. A strong MFC LO field amplifies the output signal strength^[3], so we applied an MFC field with a much larger power (about 20 dB larger) than the resonant case (about −28dBm at the input port of the meta-waveguide). The intensity of the field sensed by the atoms is effectively enhanced by the meta-waveguide, which helps to reach the required strong LO field intensity. As shown in Fig. 1(c), the signal microwave field is generated through a microwave signal source (Rigol, DSG3136B). The MFC LO is generated from a vector signal source (Ceyear, 1465 F-V) using the multi-tone modulation module. Automatic data acquisition is achieved by controlling the signal sources and the ESA with a Python GUI (constructed by Python package PyQt5 and PyVISA). For testing the meta-waveguide-coupled atomic system performance, a free-space reception scheme is performed with a combination of an antenna and an LNA in Fig. 1(c) (dashed box).

The resonant sensitivity and off-resonant heterodyne response of the meta-waveguide coupled system are measured to show the capability to tune and work in a wide frequency range. As shown in Fig. 2(a), the power sensitivity of the system at 10.78 GHz is about −115dBm/Hz for the input port of the meta-waveguide (the losses of the cable and the RPD are corrected), and the linear dynamic range is about 75 dB. As shown in Fig. 2(b), the near-resonant response of the system is higher than the off-resonant response and is maximum at 10.78 GHz (marked by the red circle in the box). Compared with the resonant case, the off-resonant response of the system from 2 GHz to 8 GHz (marked by the dashed line) is reduced by ∼20dB but shows a smoother trend of change (marked by the dashed line). All of the responses are normalized to the resonant response at 10.78 GHz, and a response from 0.5 to 13.5 GHz is shown. To measure the instantaneous bandwidth of the system, the signal-to-noise ratio (SNR) of the output beat signal is recorded by scanning the frequency offset of the signal microwave field to the LO field. By setting the frequency of the LO field fLO to 10.78 GHz, we measure the resonant instantaneous bandwidth of the system. By setting fLO to 6 GHz, we use an off-resonant heterodyne method to measure the instantaneous bandwidth. As shown in Fig. 2(c), the system has an instantaneous bandwidth of about 300 kHz, both at 10.78 GHz and 6 GHz, limited by the EIT linewidth. This bandwidth could be increased further through the optimization of the beam waist.

We applied a single LO field to the meta-waveguide-coupled Rydberg atoms to test the system capability to detect the free-space signals. The free-space radio frequency signals are received by connecting an antenna and a low noise amplifier (LNA) to the input port of the meta-waveguide. We use an ESA to record all signals in the max-hold mode. As shown in Fig. 3(a), the FM radio signal at 107.4 MHz is retrieved both with and without an LNA (5 MHz–3 GHz, 30 dB gain). The signal is recorded in 1 minute. A suction cup omnidirectional antenna (50 MHz to 860 MHz) with a gain of 5 dBi is used to collect the FM radio signals. When the LO frequency is set to 2.44 GHz, the frequency spectrum of the WiFi signal in the lab is captured in 11 seconds at a resolution bandwidth of 30 kHz. The modulation bandwidth of the FM radio signal is smaller compared with the recorded frequency range (500 kHz), and the central frequency of the FM radio stays at 107.5 MHz. So we record the radio signal in a relatively long time to obtain a clear and stable signal spectrum. On the other hand, the WiFi signal changes rapidly and appears randomly in the frequency spectrum. A long-time average sampling in the max-hold mode will eliminate the detailed information in the output frequency spectrum, and thus the WiFi signal is captured in a short time. The signal frequency spectrum is shown by the orange line in Fig. 3(b). The system noise base is plotted with the blue line when the LO microwave field is turned off. Since the resolution bandwidth of the ESA is set to 30 kHz, the detailed irregularly modulated signals with the peak width at half height less than 30 kHz in the output spectrum are filtered by the ESA, and a series of peaks is retained in the output spectrum as a low-frequency envelope of the original signal.

The operation frequency of the Rydberg MFC spectroscopy can be agilely varied using the off-resonant atomic heterodyne method^[3,29] by tuning the central frequency fc of the MFC. To demonstrate the tunable off-resonant Rydberg MFC spectroscopy, a weak signal microwave field and MFC fields with different frequencies are combined by the RPD and then sent into the input port of the meta-waveguide. By scanning the weak (about −20dBm) signal microwave frequency, detecting the output beat signal, and recording the signal amplitude by an ESA^[30,32], the off-resonant Rydberg MFC spectrum with different central frequencies (fc=2GHz, 3 GHz, and 5.8 GHz) in a range of 36 MHz is plotted in Figs. 4(a)–4(c). As shown in Fig. 4(d), we set the central (0-order comb) frequency of the MFC fc as a reference (frequency zero point), and the frequency offset of the signal microwave from the central frequency of the MFC field fc is given by foffset=fMW−fc.(2)

Thus, the output (beat) signal frequency fb with its nearest comb line with a comb order n (n = 0, ±1, ±2, …) is given by fb=|foffset−nfr|(fb≤frep/2).(3)

In Figs. 4(a)–4(c), the MFCs used consist of 9 comb lines, and their repetition rates frep are the same (4 MHz). Their amplitudes output from the vector source are about 1 dBm (2 GHz), 4.5 dBm (3 GHz), and 1.3 dBm (5.8 GHz), respectively. The phases of the comb lines are initialized with a series of pseudo-random numbers to reduce the interference between individual comb lines. The amplitude of the output signal attenuates exponentially when the beat frequency fb increases and gets larger than the instantaneous bandwidth of the system, which is reflected by the response peaks in the spectrum. The nine peaks in Figs. 4(a)–4(c) have a similar variation trend due to the same pseudo-randomized phase setup (41°, 107°, 214°, 220°, 89°, 244°, 318°, 198°, and 322°). The phases of the comb lines are randomly initialized to reduce the destructive interference, which helps to increase the intensity of the MFC spectrum. We further experimentally studied on the phase setup of the MFC in the next section.

There is a strong coupling between the non-resonant microwave electric field due to the large polarizability of the Rydberg atoms^[33], and the interference between the individual comb lines is also intense, which is mediated by the Rydberg levels. The interference effects of the off-resonant Rydberg MFC spectrum versus the different phases of the comb lines and amplitude of the MFC are experimentally investigated when fc is 2 GHz. The profile of the output response peak is determined by the instantaneous bandwidth of the system, which is the same for different microwave frequencies. So we measure the signal amplitude near the center of each peak of the MFC spectrum to represent the total spectrum intensity and plot the results in Fig. 5. The output beat signal is measured with beat frequency fb=50kHz to get a maximum amplitude. Due to the finite modulation bandwidth of the vector signal source, the MFC span is limited to 125 MHz in Fig. 4(d). The flatness of the off-resonant Rydberg MFC spectrum is better compared with the resonant case^[21]. Considering this, the covering range can be easily extended if a wider MFC LO is used.

There are three phase configurations we used in Figs. 5(a)–5(c). In Fig. 5(a), the comb lines have the same phase (from the −4 to 4-order comb line, the phase of each comb line is 0°). In Fig. 5(b), a pseudo-randomized phase setup (41°, 107°, 214°, 220°, 89°, 244°, 318°, 198°, and 322°) is applied. In Fig. 5(c), an alternating phase configuration (30°, 0°, 90°, 0°, 240°, 0°, 90°, 0°, and 30°) is set. For the all-same-phase configuration in Fig. 5(a), the strong destructive interference suppresses the intensity of the MFC spectrum. A pseudo-randomized or alternating phase reduces the destructive interference between the comb lines, thus improving the response of the MFC spectrum. The MFC power should be near 1 dBm to get the optimal flatness of the MFC spectrum, as shown in Figs. 5(b) and 5(c), and is marked by red dashed line boxes. Increasing the LO power (from about −7 to 4 dBm) for the above two cases will improve the average spectrum response, but the flatness will decrease when the power is larger than 1 dBm. So there is a trade-off between the spectrum flatness and the intensity.

Our methods are not limited by the resonance linewidth of the Rydberg-Rydberg transitions, which allow for an extensive tuning range and a potentially large frequency span if a large bandwidth MFC is applied. However, determining the optimal phase and amplitude of the MFC LO field is a complex problem that will require more effort in the future to solve as the flatness and output spectrum intensity depend on multiple parameters, and complex interference exists between each comb line. More theoretical analysis considering the high-order response of the system using master equations or multi-mode Floquet^[1,3,30,34] calculations may help to solve this optimization problem. Deep-learning-based algorithms are also useful for solving this multi-parameter optimization problem^[35,36]. The microwave frequency comb method allows for the detection of the phase and frequency of the signal microwave field. This method improves the system response to the signal microwave field. However, it is important to note that the instant response range for the Rydberg MFC spectroscopy is not equal to the instantaneous bandwidth of the system, and the readout bandwidth of our method is still limited by the instantaneous bandwidth. Still, we can use this method to estimate the signal central frequency^[21]. The output signals can be processed through the deep-learning technique^[37–39] to get the pattern of the signal or classify the signal. Notice that an optical frequency comb method improves the readout bandwidth of the Rydberg electrometry by utilizing multiple probe laser fields^[22]. However, it is hard to recognize the phase and frequency of the microwave signal by this method. Combining advanced read-out methods with the Rydberg MFC technique may help to further improve the performance of the Rydberg atomic system in detecting wide-band signals^[22,40].

In conclusion, tunable off-resonant Rydberg MFC spectroscopy based on meta-waveguide-coupled Rydberg atoms is achieved in this work. In this way, we can set the central frequency of the spectrum arbitrarily from 0.5 GHz to 13.5 GHz. This overcomes the difficulty of Rydberg atoms only receiving resonant microwave signals. By optimizing the phase and power of the MFC field, the spectrum flatness is improved. The spectroscopy paves the way for agilely tunable wide-range microwave signal reception in the Rydberg atomic system. The MFC spectroscopy that covers several hundred of megahertz will be accomplished in the future by utilizing the relatively flat (from 2 GHz to 8 GHz) off-resonant response of the meta-waveguide- coupled Rydberg atoms.