A Rydberg atom-based amplitude-modulated receiver using the dual-tone microwave ﬁeld

We propose a Rydberg atom-based receiver for amplitude-modulation (AM) reception utilizing a dual-tone microwave ﬁeld. The pseudo-random binary sequence (PRBS) signal is encoded in the basic microwave ﬁeld (B-MW) at the frequency of 14.23 GHz. The signal can be decoded by the atomic receiver itself but more obvious with the introduction of an auxiliary microwave (A-MW) ﬁeld. The receiver’s amplitude variations corresponding to microwave ﬁeld are simulated by solving density matrices to give this mechanism theoretical support. An appropriate AM frequency is obtained by optimizing the signal-to-noise ratio, guaranteeing both large data transfer capacity (DTC) and high ﬁdelity of the receiver. The power of two MW ﬁelds, along with the B-MW ﬁeld frequency, is studied to acquire larger DTC and wider operating bandwidth. Finally, the readout of PRBS signals is performed by both the proposed and conventional mechanisms, and the comparison proves the obvious increment of DTC with the proposed scheme. This proof-of-principle demonstration exhibits the potential of the dual-tone scheme and oﬀers a novel pathway for Rydberg atom-based microwave communication, which is beneﬁcial for long-distance communication and weak signal perception outside the laboratory

Especially, optical readout methods based on electromagnetically induced transparency (EIT) and Aulter-Townes splitting (AT splitting) schemes of Rydberg atom-based receiver have unique communication advantages.Firstly, the Rydberg atom-based receiver reads modulated signals in real-time while do not require additional demodulation devices.The introduction of the atomic superheterodyne mechanism makes it possible to read out PM signals directly on the basis of the original AM and FM self-demodulation capabilities [8,10,11].Secondly, the frequency intervals between Rydberg levels covering from megahertz to terahertz bands allow the small-sized Rydberg receivers to overcome the Chu limit in the conventional dipole antenna, avoiding electromagnetic interference and miniaturizing the whole device [5,6].Thirdly, the atomic species, Rydberg levels, and spatial positions provide various channels for multiplexing in one atomic vapor cell [22][23][24][25][26]. Therefore, multiple channels are ready for transferring different information and signals.
Finally, the Rydberg atom-based receiver's robust dynamic range from nV/cm to V/cm scale has been confirmed, which refers to the strength ratio between the maximum and the minimum received signals without distortion [27,28].
The Rydberg atom-based receiver's data transfer capacity (DTC) is one of key parameters to determine the achievable communication rate for a channel.Various mechanisms have been proposed to increase the DTC of Rydberg atom-based receivers.The near photon-shot-noise channel capacity limit is realized by the phase-sensitive conversion of AM-encoded microwave signals into optical signals [1].The spatially distributed probe light beams are implemented as an array of atom-optical receivers, improving the DTC by increasing the number of channels [24].On the other hand, new mechanisms are also explored to extend the parallel operating bandwidth of Rydberg atom-based receivers.The signals in the MHz band are demodulated by a three-photon excitation scheme with an off-resonant heterodyne method [29].The Rydberg alternating current (AC) Stark's mechanism enables digital communication with operating carrier frequency continuously from 0.1 GHz to 5.0 GHz [30,31].Multiple resonant response profiles of a Rydberg atomic receiver are utilized to receive microwave with frequency ranging from 1.7 GHz to 116 GHz [32].A deep-learning algorithm is introduced to encode and decode the frequency-division multiplexed signals [33].However, the performance improvement of the Rydberg atom-based receivers still calls for development.
In this work, a Rydberg atom-based AM receiver utilizing a dual-tone microwave field is demonstrated.The basic microwave field (B-MW) field at a frequency of 14.23 GHz carries a pseudo-random binary sequence (PRBS) signal.The atomic receiver has selfdemodulation capability, which is enhanced since the introduction of an auxiliary microwave (A-MW).The performance of the receiver is simulated under different microwave fields to obtain the optimal operating conditions.The appropriate and applicable AM frequency, which indicates both large DTC and high fidelity of the receiver, is obtained.The power of the two microwave fields, together with the B-MW field frequency, is investigated to acquire appropriate DTC and operating bandwidth.Finally, the transferred PRBS signals between the proposed and conventional mechanisms are compared to verify the increment of DTC.This work builds a new avenue for Rydberg atom-based microwave communication, which brings it one step closer to practical application.Figure 1(b) shows the schematic diagram of the experimental setup.The probe beam with a wavelength of 780 nm from an external cavity diode laser (DL pro, Toptica) is first split into two beams by the combination of a half-wave plate (HWP1) and a polarization beam splitter (PBS1).One beam is used to lock probe laser frequency at the transition of 5S 1/2 (F = 3) -5P 3/2 (F = 4) by the saturation absorption spectroscopy (SAS) method.The other beam is employed to obtain the EIT spectroscopy and is injected into the center of the rubidium vapor cell.The coupling laser with 480 nm wavelength is from a frequencydoubled amplified diode laser (DLC TA-SHG pro, Toptica) and is split into two beams by the combination of HWP3 and PBS3.The reflected 480 nm beam is used to lock the frequency of the coupling laser by the EIT spectroscopy, and the transmitted beam counterpropagates and overlaps with the probe beam in the cell.Two broadband bi-convex spherical lenses with focal length of 150 mm are utilized to focus the beam and obtain higher energy densities in the interaction region.The power of probe and coupling lasers are 50 μW and 120 mW with beam diameters of 80 μm and 200 μm in the rubidium vapor cell, respectively.The corresponding Rabi frequencies are 29.7 × 2π MHz and 13.5 × 2π MHz.The cell has a length of 100 mm and 25 mm in diameter.The Rb vapor cell is at room temperature (∼298 K), and the Rydberg EIT spectra are slightly affected by the temperature fluctuations in the experimental environment.Additional absorption caused by increased temperature, or metal devices for thermal regulation will cause distortions and reading errors of microwave fields.Meanwhile, the receivers operating at room temperature are easy to integrate and adapt to practical needs.After that, probe laser passed the atomic vapor is filtered by a dichroic mirror (DM), collected by a photodiode detector (PD) and recorded by a spectrum analyzer (EXA signal analyzer, Keysight) and an oscilloscope (RTO2004, Rohde & Schwarz) simultaneously.The basic vector signal generator (B-SG) and auxiliary vector signal generator (A-SG) (SMB100A, Rohde & Schwarz) provide the B-MW and A-MW fields, respectively.A kilohertz PRBS signal, supplied by an arbitrary function generator (AFG) (3022C, Tektronix), is applied to the B-MW field by amplitude modulation.Then, these two microwave fields irradiate through two horn antennas (B-Horn and A-Horn) to the vapor cell at a distance of 50 cm to satisfy the far-field condition.The propagation direction of the microwave fields is perpendicular to both the probe and coupling beams.The probe, coupling beams, and microwave fields keep coaligned linear polarization.The unaligned polarization among laser and microwave fields will lead to an optical pumping effect and make magnetic sublevels with |m j | > 1/2 have a certain probability of population.Also, the co-aligned and linear polarization of laser and microwave fields estimates the residual EIT peaks during Aulter-Townes splitting process.The waveform of decoded PRBS signals is directly obtained on the oscilloscope by reading the output signal of the PD in real time.

Results and discussion
Applying the rotating-wave approximation, the Hamiltonian of the five-level system is given by [34][35][36]: where p , c , B-MW and A-MW are the Rabi frequencies of probe, coupling, B-MW and A-MW fields, respectively, and p , c , B-MW and A-MW are their frequency detunings.
Taking the Doppler effects into account, p and c are modified as p = p -2πν/λ p and c = c + 2πν/λ c [34][35][36].Here, ν is the velocity of the atoms, λ p and λ c are the wavelengths of corresponding field.Considering spontaneous radiation evolution, the atomic system satisfies the Lindblad equation [37,38]: where L is the Lindblad operator considering decay terms that can be expressed as [34][35][36]: Here, γ ij = ( i + j )/2 and i are the spontaneous decay rates of |i> state.The density matrix element ρ ii represents the atomic population in |i> state, while ρ ij stands for the coherent term between |i> and |j> states.The amplitude variation is deduced from the imaginary part of atomic susceptibility χ of the atoms with respect to the probe laser, which can be described as χ = (N|μ 21 | 2 )/( 0 p )ρ 21 .Here, N stands for atomic number density, 0 is the permittivity of free space, shows the reduced Plank's constant, μ 21 and ρ 21 are the dipole momentum and density matrix element between 5S 1/2 and 5P 3/2 states, respectively.In a steady state, the calculation of ρ 21 evolution can be extracted from the atomic density matrix.The vapor cell is kept at room temperature during the whole experiment, and atomic number density N stays at constant.So, the (N|μ 21 | 2 )/( 0 p ) is regarded as 1 during simulation process for simplify.The simulation of χ can be utilized to describe the atomic coherence caused EIT and the external electric field induced AT splitting spectra.
The typical EIT (dark blue line) and AT splitting (black line) spectra are shown first in Fig. 1(a).The amplitude at the resonant frequency (dash line) is the major parameter labeled as A. The EIT effect occurs when only the probe and coupling fields interact with atoms, and the A is marked as level "0" in this case.The AT splitting effect appears when a constant B-MW field is introduced and the A reduces to level "1".In particular, the A moves back and forth between "0" and "1" when the information is AM encoded in the B-MW field, and the modulated waveform can be reproduced by the optical readout method after simple NOT gate operation.The variation in A corresponds directly to the output of AM receiver.The further introduction of the A-MW field brings more obvious amplitude variation, which is directly reflected in the increment of SNR during information transmission.A quantitative simulation of amplitude variation is discussed below.The red dots and line indicate the dual-tone microwave field assisted receiver, while the blue dots and line stand for the single-tone case in the following Fig. 2(b) and (c).A quantified simulation result is presented in Fig. 2(b).The amplitude variation in the single-tone microwave field is theoretically simulated under the condition that p = 0.79 2 and c = 1.4 2 .The amplitude variations of the two receivers decrease as B-MW increases, but the receiver with a dualtone microwave field shows a superior amplitude variation.The best enhancement occurs at B-MW = 4.5 2 , which provides access to the performance improvement of the Rydberg atom-based AM receiver.The dependency of the A-MW on the amplitude variation is shown in Fig. 2(c).The amplitude variation increases with the increment of microwave strength in both cases, while the phenomenon is more obvious with the introduction of the A-MW field.Note that the amplitude variation of atomic receiver with dual-tone microwave field increases rapidly and approaches saturation when A-MW → 0.1 2 .
After elaborating the mechanism theoretically, the performances of receivers with single-tone and dual-tone microwave field are tested and compared experimentally.1000 kHz in dual-tone (red dots) and single-tone (blue dots) cases.The P B-MW is -10 dBm, and P A-MW is -15 dBm during the experiment.The SNR is obtained as the signal strength ratio at the corresponding f AM to the noise when no signal is loaded.The tendencies of SNR increases from 10 kHz to 1000 kHz because atoms hardly respond once the 1/f AM is less than the dynamic time for a steady EIT.The dynamic time is influenced by collisional relaxation, spontaneous emission, transit time broadening, etc. [39][40][41].It displays that the instantaneous bandwidth of the dual-tone microwave field assisted Rydberg atom-based receiver is around 220 kHz, larger than the single-tone microwave case.The introduction of the A-MW field significantly increases the SNR over the entire range.Especially, the maximum SNR of the readout signal increases by 3 dB with the involvement of the A-MW field when the f AM is 10 kHz.A large data rate is out of the responsible range of the receiver, which is limited by so-called instantaneous bandwidth.The abundant energy levels of Rydberg atoms and parallel use of multiple atomic species may give solutions to this problem.
The DTC of the readout signal is defined by the Shannon-Hartley theorem as [1,42].Here, S is the measured signal in volts, and N is the voltage noise spectral density.Figure 3(b) shows the variation of DTC versus f AM in the dual-tone microwave field (red dots) and the single-tone microwave field (blue dots) cases.The DTC mainly depends on the f AM and significantly increases when the value ranges from 0.1 kHz to 100 kHz.The SNR becomes the major constraint from 100 kHz to 1000 kHz in contrast, and DTC decreases rapidly.The calculated DTC increases by 6.22 dB compared to the single-tone microwave case.Although the maximum DTC of the Rydberg atom-based receiver is obtained, the SNR and fidelity of the readout signal fall into a lower value.For low modulation frequencies, the data capacity has a linear dependence on the modulation frequency.For faster modulations the DTC reaches a maximum, before decreasing due to the downhill SNR caused by a finite atom-switching time.Therefore, DTC has a maximum value for f AM variation, which is determined by the instantaneous bandwidth of the system and shows maximum in 100 kHz.For low modulation frequencies, the data capacity has a linear dependence on the modulation frequency.For faster modulations the DTC reaches a maximum, before decreasing due to the downhill SNR caused by a finite atom-switching time.Therefore, DTC has a maximum value for f AM variation, which is determined by the instantaneous bandwidth of the system and shows maximum in 100 kHz.Therefore, f AM Figure 4 shows the dependence of DTC on P B-MW , P A-MW , and B-MW of Rydberg atombased receiver.The red and blue dots represent the measurement results in dual-tone and single-tone cases.The DTC is extracted in Fig. 4(a) as P B-MW increases from -20 dBm to 25 dBm, setting P A-MW on the -15 dBm.The DTC shows the trend that increases first and then gradually saturates in both cases.The -2 dBm is the point where relative strength goes to reverse in the two cases.The receiver with a dual-tone microwave owns a larger DTC when the P B-MW in the range from -20 dBm to -2 dBm.The underlying reason is that B-MW continues to increase the atom population of 54P 3/2 state in both cases, leading to an increase in SNR and further in DTC.The saturation trend occurs owing to population inversion between 54P 3/2 and 54S 1/2 states and DTC no longer increases with P B-MW after reaching the threshold.The DTC inversion emerges when P B-MW is above -2 dBm.The major reason is the introduction of 54S 1/2 state leads to the increase of DTC but decreases the saturation threshold [36].The strength of basic microwave field larger than -2 dBm makes more atomic population at 53D 5/2 state, and the effect of auxiliary microwave field turns to saturation.However, this strength range is not the suitable microwave power range for communication [16], and also corresponds to a relatively larger Rabi frequency, which leads to the Stark frequency shift and makes resonant frequency change, which is unfavorable for EIT peak detection and communication.
Figure 4(b) shows the variation of DTC with P A-MW increasing from -30 dBm to 15 dBm while the P B-MW is kept at -10 dBm.A noticeable increase is observed for the P A-MW ranging from -30 dBm to -20 dBm, and a saturation tendency emerges when the P A-MW is above -20 dBm.The number of Rydberg atoms gradually reaches the limit of atomic population reversal between 54P 3/2 and 54S 1/2 states in Fig. 1  The B-MW field frequency determines the bandwidth of the Rydberg atom-based receiver, and its influence on the DTC is illustrated in Fig. 4(c).The DTC of the readout signal exhibits the maximum at resonant frequency of B-MW and gradually decreases as the frequency tunes to either blue or red detunings.The corresponding transmitted and readout signals with different B-MW field frequencies are shown in the Insets 1 and 2. Both near-resonant and off-resonant cases show higher SNR in dual-tone microwave field case.Note that the signal amplitude is more obvious, and the upper and lower edges are clearer in the case of dual-tone microwave field.The near-resonant B-MW field in the Inset 1 exhibits higher signal recovery and fidelity than the detuned case in the Inset 2.
The PRBS signal received from Rydberg atom-based receiver is compared with the original signal to evaluate the fidelity.The PRBS signal is produced in random and generated from an arbitrary waveform generator (AWG, Tektronix).The signal shows as cases of different modulation frequencies in practical application, so the width of transmitted signal is varied with time.The consistency between transmitted and received signals is kept, which shows the self-demodulation capability of atomic receivers and the potential for high-fidelity information transmission.The f AM is kept at 10 kHz, P B-MW is -10 dBm and P A-MW is -15 dBm.The input (black) and readout signals with dual-tone microwave field (red) and the single-tone microwave (blue) cases are recorded as shown in Fig. 5.The recovery rate and signal fidelity are defined as R rec = (f Df T )/f T and F sig = S T -(S D /S T )S D , respectively.The f T and f D represent the frequencies of transmitted and decoded signals, and S T and S D stand for their amplitudes, respectively [43,44].The R rec and F sig of the readout signal in the dual-tone microwave field case are 99.5% and 92.8%, while 98.9% and 80.9% in single-tone case.As a result, the enhanced features guarantees greater DTC and improved fidelity of the dual-tone microwave field assisted Rydberg atom-based receiver.

Conclusion
We present a Rydberg atom-based AM receiver utilizing a dual-tone microwave field in a 85 Rb atomic ensemble.The PRBS signal is encoded at the basic microwave field with the frequency of 14.23 GHz, then decoded by a Rydberg atom-based AM receiver.The introduction of auxiliary microwave field improves the SNR of received signals.The amplitude variations at the resonant frequency of EIT spectrum with different microwave fields involvement are investigated to provide theoretical support.The appropriate DTC of the proposed Rydberg atom-based AM receiver is about 62 kbit/s, and a 220 kHz instantaneous bandwidth is achieved.Finally, the DTC result of dual-tone microwave receiver is 6.22 dB larger than conventional single-tone microwave receiver.The dual-tone method improves Rydberg atom based receiver's response to microwave field variation, which is appropriate for not only AM scheme but also FM and PM schemes.This work takes a step closer to practical applications by opening up a new avenue for microwave communications based on Rydberg atoms.

Figure 1 (
Figure 1(a) shows the relevant energy levels diagram of 85 Rb.The atoms are excited from the ground state 5S 1/2 (F = 3) to the Rydberg state 53D 5/2 by a two-photon transition.The

Figure 3
Figure 3 (a) The SNR of received signals and (b) the DTC of Rydberg atom-based receiver versus the modulation frequency (f AM ) for dual-tone (red dots) and single-tone (blue dots) microwave field cases.The error represents the standard deviation of three measurements

Figure 4
Figure 4 (a) The DTC of the Rydberg atom-based receiver versus (a) the B-MW field power P B-MW , (b) the A-MW field power P A-MW , and (c) the B-MW field frequency B-MW .The error represents the standard deviation of three measurements.The transmitted and readout signals with and without the A-MW field for B-MW = 0 MHz in the Inset 1 and B-MW = +80 MHz in the Inset 2 (a).The Rydberg atom-based receiver's maximum DTC is about 62 kbit/s at P B-MW = -10 dBm and P A-MW = -15 dBm.

Figure 5
Figure 5 The transmitted (black) signal and received signals with dual-tone (red) and single-tone (blue) microwave field cases