1. Introduction

Over the last few decades, a wide variety of experimental quantum communications and processing devices have been demonstrated in laboratories, in communications and information related fields. Applications such as wireless quantum sensors [1, 2] are partially commercially available. Atom-based quantum techniques are emerging as a completely new and promising tool for advanced communications.

Accurate radio-frequency (RF) electromagnetic field sensing in free-space plays a fundamental role in wireless communication, and Rydberg atoms are remarkable quantum sensors for RF electric(E-) field measurements [3, 4]. An applied RF E-field induces strong ac-Stark coupling between Rydberg states, resulting in an Autler-Townes (AT) splitting [5] of a ladder-type electromagnetic-induced transparency (EIT) [6], which can convert the measurement of RF field into optical frequency determination [3]. Compared with conventional methods, this quantum-optical method has advantages of high sensitivity with a predicted shot noise limit of $<1\mu$ V $\cdot$cm${ }^{-1}\cdot$Hz${ }^{-1/2}$ [4, 7], high accuracy with an expected measurement uncertainty of $0.5\%$ [3, 8], ultra-broadband measurement covering from ∼100 MHz to sub-THz [9–12], and atom-based self-calibration to SI units. This method looks set to give rise to a new generation of RF E-field measurement standard [13].

In particular, some proof-of-concept works on wireless communication and remote sensing using Rydberg atoms have been presented recently [14–17]. When a ladder-type EIT is produced in a system where the upper level is a Rydberg state, a time varying RF E-field can be detected via its influence on the probe laser transmission. Thanks to the unique advantage of free-space RF field sensing, the quantum receiver has some great advantages compared with conventional electronics-based receivers, including but not limited to possibilities of weak signal and long-distance communication in free space or via a fiber link. All the main experiments on communication have been performed on the carrier of an optimized resonant frequency of atomic Rydberg states [14–17].

Nowadays, as wireless communication spectrum resources are becoming increasingly scarce, cognitive radio technology is being used to provide a solution for more efficient utilization of the radio spectrum [18]. As the very essence of cognitive radio, continuously tunable RF reception, which covers an entire frequency band, has become increasingly important for accessing underutilized spare sub-bands of the radio spectrum. In addition, broadband communication in higher frequency bands can provide the high speed, high capacity data transfer, essential to the fifth generation (5G) communication [19]. In a word, rather than retrieving the time-varying signals on a specific carrier frequency, we hope to make extensive use of the frequency band near the carrier frequency determined by the energy gap between the neighboring Rydberg states. This requires that Rydberg-atom-based quantum-optical radio communication features be available over the whole RF-carrier near the Rydberg resonance in question. If so, for example, different communication channels can work concurrently in different but close RF-carrier frequencies, which greatly enhances the communication capacity.

In this paper, we first study the spectral features of modulation signal transfer at different carrier frequency detunings relative to resonant frequencies of Rydberg states, such as how detection efficiency and linearity vary with modulation frequency and carrier power, as well as the dependence of the signal-to-noise ratio (SNR) as a function of various parameters including the RF detuning. After that we further qualitatively research the RF signal transfer properties over the carrier frequency domain by testing digital communication using a benchmark of pseudo-random binary sequence (PRBS) signal. The quality is usually characterized by the bit error rate (BER), which is defined as the ratio of the number of bit errors to the total number of transferred bits during one cycle. When the BER=0, the communication is said to be reliable. The experiment shows that the digital communication at a rate of 500 kbps exhibits reliable performance within the tunable bandwidth of 200 MHz at a 10.22 GHz carrier whereas the BER rises to 15% for an RF-detuning of $\pm 150$ MHz.

2. Theory and methods

The Rydberg atom EIT-based RF-receiver is shown schematically in Fig. 1 with the related energy levels of ${ }^{87}$Rb in the inset. It is similar to that of references [14–16] except for the (de-)coding part of the signal on the RF carrier and the differential detection. As shown in the inset, the long-lived ground state 5S${ }_{1/2}$ and high Rydberg state 59D${ }_{5/2}$ are coupled to a short-lived state 5P${ }_{3/2}$ by a weak probe laser at 780 nm and an intense coupling laser at 480 nm, respectively, forming a ladder-type EIT system. To minimize Doppler broadening, the probe and coupling laser beams are counter-propagating in the rubidium vapor [7]. The atomic vapor cell is a cube of side 8 mm operating at room-temperature (30±0.5${ }^{\circ }$C). The lasers with line widths ∼100 kHz are focused to the waist diameters (2w₀) of 120 μm and 200 μm, respectively. With the probe laser frequency locked to the resonant transition from 5S${ }_{1/2}$ (F=2) to 5P${ }_{3/2}$ (F’=3), we can observe an EIT signal by scanning the coupling laser. Extremely linearly polarized laser beams produced by Glan-Taylor prisms with an extinction ratio of 100,000:1 are employed to achieve a high-contrast EIT signal. In addition, a differential detection technique is employed to remove the Doppler-broadened background signal and to improve the SNR, rather than the usual configuration where elliptically polarized lasers are used [20]. In our setup, a pair of HWP and PBS is adopted to balance the transmitted and reflected laser powers, while another pair of HWP and BS used to collimate and co-polarize the orthogonal beam that serves as a reference signal for the balanced detector.

If we apply an RF field to couple another Rydberg state 60P${ }_{3/2}$ to 59D${ }_{5/2}$, a further spectral splitting will occur within the EIT peak. Due to the large electric dipole moment of Rydberg states, even a weak RF E-field can induce a strong coupling, leading to a remarkable ramification, known as Autler-Townes (AT) splitting [5], which makes it possible to use it as an RF receiver in wireless communication. The 10.22 GHz carrier is generated by a microwave signal generator (Agilent N5183A) amplitude-modulated externally by a kilohzertz RF generator (Agilent 33600A). The modulated signal at the carrier is sent towards the Rydberg atoms in the vapor cell by an X-band horn antenna with a far-field gain of 15.1 dBi.

 

Fig. 1 Experimental setup of the Rydberg-atom-based RF-receiver with differential detection. The coupling and probe lasers counter-propagate through the rubidium vapor cell, and form a ladder-type EIT. An RF E-field couples the two Rydberg states 59D_5/2 and 60P_3/2, resulting in an Autler-Townes splitting. Inset: Relevant energy levels of the rubidium atom. Note that a neighboring state 59D_3/2 lies about 50 MHz below 59D_5/2, which distorts the observed spectrum and it will be discussed in text later. AOM: acoustic optic modulator; BS: beam splitter; DM: dichroic mirror; GT: Glan-Taylor prism; HWP: half-wavelength plate; PBS: polarizing beam splitter.

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A spectral feature study of the probe laser for different RF detunings is helpful to find out the best RF carrier bandwidth of the Rydberg-atom-based receiver. When the applied RF field is frequency-detuned from the resonant transition between two Rydberg states, the observed spectroscopy of AT-splitting changes as well, becoming asymmetric as shown in the inset of Fig. 2, for example, at RF field detunings of $\pm 50$ MHz near a resonant frequency of 10.22 GHz for Rydberg transition from 59D${ }_{5/2}$ to 60P${ }_{3/2}$. The center frequency of the asymmetric AT-splitting separation, denoted by $f_{1/2}$, varies with RF-detuning ${\mathrm{\Delta }}_{RF}$ as well. The amount of variation of $f_{1/2}$ with respect to the resonant coupling laser frequency at zero RF-detuning, labeled as f₀, satisfies the relation $f_{1/2}-f_0=-\frac{1}{2}{\mathrm{\Delta }}_{RF}$, as shown in Fig. 2. This relation is supported by a four-level model simulation [8, 21]. In the following experiment, an AOM together with an external driven source will be used to tune the coupling laser frequency accurately to a specific RF detuning, and we shall monitor the probe beam intensity at $f_{1/2}$ to characterize the RF-field variation where the photodiode has the most sensitive response to the applied RF-field. This simple scheme provides a faster transfer rate than the method of encoding information into the AT-splitting width [16].

 

Fig. 2 The linear dependence of the frequency shift of the Autler-Townes splitting center, f_1/2 – f₀, on the RF detuning Δ_RF. A pair of typical asymmetric AT-splitting spectra is shown in the inset at RF detunings ±50 MHz relative to the energy difference between Rydberg states 59D_5/2 and 60P_3/2 (10.22 GHz).

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To well understand the optical transparency/absorption properties and dynamics of the atom-light system, we build up an optical Bloch equation under the rotating-wave approximation:

3. Results and analysis

Obviously, for a practical application of the Rydberg atom based RF receiver, the SNR of the optically detected signal is a vital evaluation parameter. We measured the optical signal at the center frequency $f_{1/2}$ versus the RF-field detuning ${\mathrm{\Delta }}_{RF}$ at different modulation frequencies after optimizing the coupling and probe beam intensities, and the corresponding SNR values are shown in Fig. 3. The probe Rabi frequency was ${\mathrm{\Omega }}_p=2\pi \times 25\pm 1$ MHz while coupling Rabi frequency was ${\mathrm{\Omega }}_c=2\pi \times 4.5\pm 0.5$ MHz. In the measurement, a wireless transmission of a kHz sinusoidal signals of peak-to-peak amplitude 2 V at a 10.22 GHz carrier of -10 dBm power is tested. The demodulated signal is interrogated by a spectrum analyzer (Keysight N9010B) with a resolution bandwidth of 100 Hz. Fig. 3(a) shows the SNR of the receiving signal over the RF detuning ranging from -150 MHz to +150 MHz for three different modulation frequencies $f_{mod}=1,10 \text{and} 100$ kHz. The SNR has a minimum value at RF detuning ${\mathrm{\Delta }}_{RF}=50$ MHz. This is due to the coupling of the neighboring state 59D${ }_{3/2}$, which lies about 50 MHz below 59D${ }_{5/2}$. On the whole, the SNR has acceptable values at $f_{mod}=1$ kHz. The SNR versus the RF-detuning is well consistent with the calculated data which are also shown in Fig. 3(a) in red line. The intrusion of the neighbor Rydberg states are not considered in the calculations. Actually the SNR depends on several parameters except for the RF detuning, in aspect of communications parameters, the frequency and magnitude of modulated signal, strength of carrier signal contributes the SNR as well. Optimization of the atomic density, Rabi frequency of probe and coupling lasers can also improve the SNR in aspect of quantum effects. In combination of these solutions, the degradation of the detection sensitivity (SNR) of the off-resonant RF field can be compensated to some extent.

 

Fig. 3 SNR of the differential demodulated signal on the balanced detector versus RF-detunings for different modulation frequencies f_mod (a) and its dependence on the modulation frequency for finer steps of f_mod (b). The SNR is larger than 10 dB in a carrier bandwidth up to 200 MHz for modulation frequencies less than 1 MHz. A calculation is also presented in (a) based on the steady solution of Eq. 1, where the intrusion of the neighbor Rydberg states is not included.

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It can also be seen from Fig. 3(b) where the SNR values are measured for various modulation frequencies at the resonant carrier. For the current setup, the demodulated SNR arrives at the most desirable value SNR=70 dB at 10 kHz modulation. Of course, many factors can affect the observed signal and its SNR, for example, photon-shot noise, the technical noise of the photodiode detector and the pre-amplifier, the coupling laser power and atomic density, etc. Fundamentally, the response of Rydberg atoms to an incident time-variant RF field is especially important since it determines the transfer rate in the communication. Physically, it is governed by the dynamic time for a steady EIT to build up again, characterized by the inverse of its dephasing rate [23, 24]. Many relaxation mechanisms contribute to the increasment of dephasing rate, such as collision between atoms at ground and Rydberg states. For the RF-field coupled states 59D${ }_{5/2}$ and 60P${ }_{3/2}$, the estimated dephasing rate is on the order of ∼100 kHz, by taking practical experimental paramentes into account. This could be an explanation of the observed maximum modulation frequency.

Care should be taken as to the nonlinearity of the photodiode gain response to the RF-field amplitude over its detuning range. Wider response and better linearity will lead to a higher SNR. The measurement results are presented in Fig. 4(a). One can see the photodiode has a good linear response for the RF-field varying from a weak field of ∼1 V/m to a strong one of ∼10 V/m, exhibiting a commendable performance in the applicable RF-field range. In this linear range, the best signal gain occurs at zero detuning, ${\mathrm{\Delta }}_{RF}=0$. This is consistent with the detuning for maximum SNR shown in Fig. 3. We can draw the EIT and AT spectrum at RF-resonance between the two Rydberg states 59D${ }_{5/2}$ and 60P${ }_{3/2}$, but at different incident RF field. It is shown in Fig. 4(b), where 0.18 V/m corresponding to a RF Rabi frequency of 5 MHz, 0.71 V/m to 20 MHz, 1.59 V/m to 45 MHz, and 3.54 V/m to 100 MHz, and 7.08 V/m to 200 MHz, which are all marked on the red line in Fig. 4(a). We can notice that the probe transmission at zero detuning of the coupling laser (when ${\mathrm{\Delta }}_c=0$) shows a maximum value at weak RF field illumination but its value decreases gradually with the RF-field increasing. When the Rabi frequency of RF field is enhanced up to 200 MHz (7.08 V/m), its EIT spectral transparency approaches to the lowest, corresponding to a low voltage on the photodiode as shown in Fig. 4(a).

 

Fig. 4 The response of the probe transmission to the applied RF-field over the RF-detuning range. The probe photodiode linearly responds to the RF-field in a wide dynamic range varying from weak field E ~ 1 V/m to strong end E ~ 10 V/m but has a maximum gain performance at zero detuning. Curves from top to bottom, respectively, corresponding to RF-detuning 0, -20 MHz, 20 MHz, -50 MHz, and 50 MHz with respect to the on-resonant frequency (10.22 GHz) in (a), which is well understood with the help of a theoretical simulation shown in (b). A careful selection of linear response range will give a pure spectrum as indicated by the red line in (c).

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This also indicates that choosing a moderate RF carrier-power as the working point is vital in the reliable Rydberg-atom-based RF-field communication, which is qualitatively described in a test shown in Fig. 4(c), where the black line represents the power spectrum when nonlinear region is included in capturing a 10 kHz signal over 10.22 GHz. It contains many nonlinear frequency modes. On the contrary, a careful selection for the working RF-field can single out all high frequency modes, providing a much better SNR in practical communications. The red line is obtained for the linear response region at zero detuning at RF-power of -10 dBm. Using a similar analysis can be performed for communication using an off-resonance RF carrier. With this linearity calibration, the powers of the carrier and modulation signal can be optimized for practical long-distance communication using specific radio propagation models [25].

Finally, we demonstrate a digital communication test using a PRBS signal transferred at different RF-carrier frequencies. The PRBS signal, recommended by the International Telecommunication Union (ITU), is commonly used in digital transmission testing [26]. Its transfer over the Rydberg-atom-based RF-receiver provides convincing support for its future practical application. Figures 5(a)-5(c) show three typical waveform transfers over different RF-detunings by the proposed off-resonance scheme for ${\mathrm{\Delta }}_{RF}=0,100$ MHz and 150 MHz, respectively. The transfer bit rate is fixed at 500 kbps. The red line waveforms represent the source signals which are digitally encoded for wireless communication, while the thin black lines are the optical signals detected using the Rydberg based RF-receiver. These optical signals are decoded back to the digital form shown in blue. Comparing them with the source signal, we can obtain the bit error rate (BER) for each case. The waveform transfer performs well at RF-detunings ${\mathrm{\Delta }}_{RF}=0 \text{and}$100 MHz, with no loss, but a BER of 15% exists at ${\mathrm{\Delta }}_{RF}=$150 MHz.

 

Fig. 5 A digital communication tested using a benchmark of PRBS signal transferring at a bit rate of 500 kbps. Three typical waveform transfers are shown in (a), (b) and (c), respectively, corresponding to RF-detuning 0, 100 MHz and 150 MHz with respect to the on-resonant frequency (10.22 GHz). The BER determined by comparing the decoded and source digital signals rises up with RF-detuning, up to 20% at ${\mathrm{\Delta }}_{RF}=$160 MHz in (d).

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This implies that Rydberg-atom-based RF-reception can provide a wideband carrier channel with a high transfer fidelity. This is borne out by a further measurement of the transfer BER over the whole carrier frequency domain as shown in Fig. 5(d). We can see that there is no loss of information over carrier RF-detuning from 0 to 100 MHz. Beyond ${\mathrm{\Delta }}_{RF}=$100 MHz, the BER increases linearly with the detuning depth, rising to over 20% at ${\mathrm{\Delta }}_{RF}=$160 MHz. Considering the negative detuning part, a bandwidth of 200 MHz with no loss can be achieved in our experiment. Once the transfer bit rate is increased to high values, for example, up to 1 Mbps, a larger BER is expected, a typical value of more than 30% being determined for 150 MHz off-resonant communication.

4. Conclusion

In summary, we have demonstrated broadband digital communication via highly-sensitive RF-reception based on the EIT of Rydberg atoms in a room-temperature atomic vapor. The RF-field is demodulated by detecting the optical power of a probe laser at the center frequency of RF-induced symmetric or asymmetric Autler-Townes splitting within an EIT window. Various factors affecting the feasibility of communication within a wide continuous RF-detuning are investigated. Besides the optimization of the coupling and probe laser powers [14, 15], attention should also be paid to the choice of a linear response of the probe density versus the RF field strength, the frequency range of the modulation signal as well as the RF-detuning depth itself. The digital communication tested using a standard PRBS signal transferred at a rate of 500 kbps shows a high fidelity for wireless communication within a tunable bandwidth of 200 MHz on a 10.22 GHz carrier. It is expected that the quality of communication also depends on the selection of Rydberg states, as the effective RF-detuning depth varies from state to state. This will inspire an attempt to employ many advanced communication techniques based on the Rydberg-atom RF receiver, such as the frequency-hopping spread spectrum (FHSS) and orthogonal frequency division multiplexing (OFDM) over the broadband RF-carrier.