Terahertz (THz) waves, spanning the spectral range of 0.1–10 THz, occupy a distinctive segment within the electromagnetic spectrum that bridges the gap between electronics and photonics [1]. Their exceptional properties have aroused widespread interest in more and more disciplines. For instance, THz waves can easily penetrate nonmetallic materials, do not harm living organisms, and produce characteristic fingerprint spectra for different substances. Therefore, THz waves have wide application prospects in areas such as chemical analysis [2,3], biomedical science [4–6], security screening [7], and communications [8]. In recent years, THz technology is being used in multitudinous fields; however, highly sensitive, room-temperature THz detectors are still rare. Therefore, THz detection technology is still one of the bottlenecks limiting the development of THz technology [9].

THz detectors can be broadly categorized as electronic detectors, photonic detectors, and thermal detectors. The detection mechanism of electronic detectors is mainly to detect the effect of electron movement or other electrical properties of the material induced by the THz electrical field. These electronic detectors include photoconductive antennas [10–12], Schottky diodes [13,14], glow discharge detectors [15–17], and field effect transistors [18–20]. Photonic detectors mainly include quantum well [21,22] and quantum dot [23] detectors and they have short response time and high detection sensitivity. Typically, their noise equivalent power (NEP) can be as small as 10−22W/Hz1/2. However, the strict operation conditions, such as low temperature and strong magnetic fields, limit their widespread application [23]. Thermal detectors such as bolometers [24,25] rely on THz wave to irradiate thermal-sensitive materials and cause changes in certain temperature-related measurable physical quantities. They have broad bandwidth and high sensitivity, but require a low operating temperature, which increases the operating cost and complexity of apparatus. Golay cells [26], pyroelectric detectors [27], and thermopile detectors [28] are operated at room temperature, but their sensitivities are much less than a bolometer. Usually, the response speed of thermal detectors is slower because they require the accumulation of heat to achieve significant physical effects. Therefore, high-speed and highly sensitive room-temperature THz detectors are urgently needed.

Quantum sensors, based on their highly coherent and well-controlled quantum structure, are able to measure weak electromagnetic signals with excellent sensitivity and precision [29]. The Rydberg atom measurement method is rooted in physics models of the atom-field interaction that is dependent only on invariable atomic parameters and fundamental constants [30]. This characteristic facilitates straightforward self-calibrated electric field measurements directly traceable to Planck’s constant [31]. Rydberg atoms are highly excited atoms with a high principal quantum number (n), high polarizability (∝n7), strong interactions (∝n4), and long lifetimes (∝n3). When an atom is excited to the Rydberg state by lasers, the absorption of the probe laser by the Rydberg atom is suppressed due to destructive interference, which results in a transparent window at resonance frequency, known as electromagnetically induced transparency (EIT) [32,33]. By converting the destructive interference into constructive interference, the electromagnetically induced absorption (EIA) effect occurs [34,35]. When an applied electric field is tuned in resonance or close to the transition frequency of the EIT/EIA spectral line, the original EIT/EIA signal peak will split into double peaks, which is known as Autler–Townes (AT) splitting [33]. Due to the proportional relationship between the bimodal interval and the external electromagnetic field intensity, THz wave can be detected by measuring the splitting interval. See Section 2 (Methods) for more details. The Rydberg atom detectors have been undertaken in the microwave and radio frequency (RF) field measurements, and the sensitivity can achieve the order of nV cm⁻¹ Hz^−1/2 [36–38]. In the THz frequency range, THz electric field measurements and real-time imaging based on Rydberg atoms have been preliminarily explored, and theoretical calculations have been performed based on a three-level or a four-level system [39–42]. However, since the transition energy gaps are large in the three- and four-level systems, lasers with shorter wavelengths and higher powers are needed [43], which increases the difficulty of the system. Compared to them, a five-level system of Rydberg atoms is an attractive method for THz detection, since it can use infrared lasers with lower power to obtain the desired Rydberg state. Additionally, infrared diode lasers are easier to use and maintain compared to the blue or ultraviolet lasers that are necessary for three- or four-level scheme [44]. Finally, five-level systems exhibit greater sensitivity in RF electric field detection compared to four-level systems [45]. Nonetheless, it is noteworthy that, to the best of our knowledge, a theoretical exploration of the THz wave detection with the five-level system of Rydberg atoms currently has not been done.

In this paper, the quantum coherence effect based on Rydberg atoms for detecting THz waves is investigated using our proposed five-level system of a cesium (Cs133) atom, the major factors affecting the detection effect are analyzed, and the sensitivity is numerically evaluated. At room temperature, its sensitivity is much higher than that of current THz wave detectors. This paper provides a theoretical basis for implementing an accurate detection of THz waves based on Rydberg atoms with SI traceability.

The schematic diagram of the proposed quantum THz detection system is shown in Fig. 1(a). Here, the Rydberg laser is overlapped with the dressing laser before being directed into the vapor cell. The probe laser counterpropagates with respect to the dressing and Rydberg lasers, passes through the vapor cell, and is detected by a photodiode (PD). The three lasers all pass through a vapor cell, which contains a Cs133 atom with natural isotopic abundances, while the THz wave is incident perpendicular to the three lasers. Figure 1(b) depicts the five-level model of Cs133 that is targeted by three infrared lasers and a THz field in Fig. 1(a). The three infrared lasers promote Cs133 atoms to the Rydberg state that is sensitive to the applied THz field with the frequency of 0.17 THz. The probe laser with a wavelength of 852 nm excites atoms to the 6P3/2 state, the dressing laser with a wavelength of 1470 nm takes the atoms from the 6P3/2 state to the 7S1/2 state, and the Rydberg laser with a wavelength of 806 nm is applied to tune the 7S1/2→19P3/2 transition. When the Rydberg laser scans near the resonance frequency, represented by ΔR in Fig. 1(b), the atomic absorption of the probe photon is inhibited due to coherent interference, which results in the transmission spectrum of the probe light exhibiting an EIT signal, as shown by the blue curve in Fig. 1(c). Once the atoms are excited to the 19P3/2 state, they can be used to sense a THz field with a frequency of 0.17 THz by coupling to the 18D5/2 state, which leads to AT splitting of the single EIT peak, as shown by the red curve in Fig. 1(c).

To simulate the THz wave detection by Rydberg atoms shown in Fig. 1, we calculate the power measured on the detector corresponding to the probe laser power by [46] P=P0exp(−2πLIm[χ]λp)=P0exp(−αL),(1)where P0 is the power of probe laser that was input into the vapor cell with the length of L, λp is the wavelength of the probe laser, Im[χ] is the imaginary part of susceptibility of the Cs133 vapor, and α=2πIm[χ]/λp is the Beer’s absorption coefficient for the probe laser. e−αL is the transmittance of the probe laser. The susceptibility for the probe laser is related to the density matrix element that is associated with the 7S1/2→19P3/2 transition by χ=−2nμp2ε0ℏΩpρ21,(2)where μp is the normalized transition-dipole moment of the probe laser, Ωp is the Rabi frequency of the probe laser, ε0 is the vacuum permittivity, ρ21 is the density matrix element between states 6S1/2 and 6P3/2, and n is the atomic density in the cell and is given by n=pkBT,(3)where T is the vapor temperature in Kelvin, kB is the Boltzmann constant, p is the saturated vapor pressure and is determined by [47] $$\begin{gathered} p=109.171−(3830/T)(solid), \\ p=109.717−(3999/T)(liquid). \end{gathered}$$(4)

The density matrix elements (ρij) are acquired by solving the Lindblad master equation using the QuTiP package [48] in Python software, written as ρ˙=−iℏ[H,ρ]+L(ρ),(5)where H is the Hamiltonian of the five-level system used in this paper, L(ρ) is the Lindblad operator that accounts for the decay processes in the atom, and ρ is the density matrix. The Hamiltonian isH=ℏ2[0Ωp000Ωp−2ΔpΩd000Ωd−2(Δd+Δp)ΩR000ΩR−2(ΔR+Δd+Δp)ΩTHz000ΩTHz−2(ΔTHz+ΔR+Δd+Δp)],(6)where Δp, Δd, ΔR, and ΔTHz are the single photon detunings of the probe laser, dressing laser, Rydberg laser, and THz electric field, respectively. Ωp, Ωd, ΩR, and ΩTHz are the Rabi frequencies associated with the probe laser, dressing laser, Rydberg laser, and THz electric field, respectively. The Lindblad operator is defined as L(ρ)=(γ2ρ22+γ4ρ44−12γ2ρ12−12γ3ρ13−12γ4ρ14−12γ5ρ15−12γ2ρ21−γ2ρ22+γ3ρ33+γ5ρ55−12(γ2+γ3)ρ23−12(γ2+γ4)ρ24−12(γ2+γ5)ρ25−12γ3ρ31−12(γ2+γ3)ρ32−γ3ρ33−12(γ3+γ4)ρ34−12(γ3+γ5)ρ35−12γ4ρ41−12(γ2+γ4)ρ42−12(γ3+γ4)ρ43−γ4ρ44−12(γ4+γ5)ρ45−12γ5ρ51−12(γ2+γ5)ρ52−12(γ3+γ5)ρ53−12(γ4+γ5)ρ54−γ5ρ55),(7)where γ2, γ3, γ4, and γ5 are the spontaneous decay rates of energy levels |2⟩, |3⟩, |4⟩, and |5⟩ that correspond to 6P3/2, 7S1/2, 19P3/2, and 18D5/2 in this calculation; we set γ2=5.234×2πMHz, γ3=3.3×2πMHz, γ4=0.01×2πMHz, and γ5=0.07×2πMHz. ρij are density matrix elements, and the density matrix is ρ=(ρ11ρ12ρ13ρ14ρ15ρ21ρ22ρ23ρ24ρ25ρ31ρ32ρ33ρ34ρ35ρ41ρ42ρ43ρ44ρ45ρ51ρ52ρ53ρ54ρ55).(8)We can substitute Eqs. (7) and (8) into Eq. (5), solve for the steady state where ρ˙=0, and combine with the total atomic population of the five levels being 1, tr(ρ)=1, to obtain the Bloch equation for the instantaneous steady state. Finally, the density matrix elements can be obtained by solving the Bloch equation.

In this section, we calculated the transmission spectra of probe light at different Rabi frequencies of the probe laser, dressing laser, Rydberg laser, and THz wave (Ωp, Ωd, ΩR, and ΩTHz), and found their impact on the EIT and EIA signals. Since both EIT and EIA arise from quantum coherence effects and involve transitions from excited states to the ground state [34], we obtained density matrix elements ρ11 and ρ14 by solving the density matrix, where ρ11 represents the atomic population of the ground state and ρ14 represents the coherence between the ground state and the Rydberg state. Finally, we provided a qualitative explanation for the conversion between EIT and EIA with ρ11 and ρ14.

The conversion process of the Cs133 five-level system from EIA to EIT with the increase of Ωp is depicted in Fig. 2(a). When the probe light with a relatively smaller Ωp is applied, most of the probe photons are absorbed by the electrons in the ground state of Cs133 atoms, which causes ρ11 to linearly decrease and an increase in ρ14, where ρ14 is less than 0, as shown in Fig. 2(b). The amplitude of the EIA signal decreases with increasing Ωp, and the process is represented by the shift from the blue curve to the green curve in Fig. 2(a). The Ωp of 3.6×2πMHz is the critical value of the transition from EIA to EIT. At this point, the visibility of the EIA-AT splitting curve reaches its minimum, leading to a deterioration of the detection ability for THz waves. When Ωp is approximately equal to 3.7×2πMHz, ρ14 increases to 0, which is indicated by gray lines in Fig. 2(b), and ρ11 is relatively small; therefore, the absorption of probe photons by Cs133 atoms is weak, which results in the manifestation of EIT. As Ωp increases, the amplitudes of EIT signals grow and the AT splitting peaks are more easily observed, so it is beneficial for improving the sensitivity of THz wave detection. During the process of increasing Ωp, the splitting interval (Δf) remains a constant, as indicated by the red, purple, and brown curves in Fig. 2(a). The reason is that the Δf is only proportional to the intensity of the incident THz electrical field.

The results presented in Figs. 3(a) and 3(b) demonstrate the impact of Ωd on the EIA and EIT signals and the conversion between EIT and EIA. Figures 3(c) and 3(d) are employed to analyze the variation of ρ11 and ρ14 with respect to Ωd. The Ωd has a significant impact on the transmittance of the probe light. In Fig. 3(a), the Ωd is small, and the transmittance of the probe light approaches zero. To see the curves clearly, the vertical axis is represented by logarithmic coordinates. However, it is difficult to measure such a small transmittance in actual testing, which makes it almost impossible to detect the THz wave. In Fig. 3(b), the transmittance is sufficiently large and the splitting interval is easily observed, which can be used for the detection of the THz waves. In Fig. 3(c), when Ωd is less than 3×2πMHz, ρ11 decreases with the increasing Ωd, since more ground state atoms are excited to the 7S1/2 state by the dressing light and probe light. When Ωd is larger than 5×2πMHz, it creates two competing excitation pathways with opposite amplitudes, 6P3/2↔7S1/2, which causes the absorption of the probe photons to be strongly suppressed [44] and an increase of ρ11, while ρ14 decreases with the increase of Ωd, as shown in Fig. 3(d). Furthermore, when Ωd ranges from 22×2π to 25×2πMHz, ρ14 is greater than 0, while ρ11 is relatively small. This indicates fewer atoms in the ground state compared to the Rydberg state, resulting in more transmitted probe laser, leading to EIT. However, in Fig. 3(d) when Ωd is approximately 25×2πMHz, ρ14 equals 0 as indicated by the olive dashed line, and ρ11 has increased to a higher value in the blue line, which means a relatively large population of atoms are in the ground state, the absorption of probe photons will be enhanced, and ultimately the EIA appears. As shown in Fig. 3(b), the amplitudes of EIT signals (blue, orange, and green curves) are greater than those of the EIA signals (red and purple curves), which means the EIT signals are more sensitive than the EIA signals in the THz detection with the current condition. Furthermore, with the increase of Ωd, both the baseline and amplitude of the EIT and EIA signals generally increase, while Δf remains constant due to the fixed THz electrical field. In the conversion region from EIT to EIA (Ωd≈25×2πMHz), the absolute peak value of the EIT or EIA signal is smaller, which reduces the observability of the AT splitting peaks and the sensitivity for THz wave detection, which will be discussed in detail in the next subsection. Because of the issues mentioned above, it is advisable to avoid this region in THz detection.

Figure 4 shows the impact of the Rabi frequencies of Rydberg light on the EIT/EIA signals, ρ11 and ρ14 in the case of Δp=Δd=ΔTHz=0, Ωp=4×2πMHz, and Ωd=20×2πMHz. Figure 4(a) displays the transmission spectra of the probe light in the absence of the THz electric field. When ΩR is less than 2×2πMHz, ρ11 decreases with the increase of ΩR, as shown by the red dashed curve in Fig. 4(c), and the amplitude of EIT signal decreases, as shown in the inset of Fig. 4(a). When ΩR exceeds 2×2πMHz, ρ11 rises with the increase of ΩR, as shown by the red dashed curve in Fig. 4(c), and the absorption of probe photons is intensified. As a consequence, when ΩR>3.2×2πMHz, ρ11 is large enough to lead to the transition from EIT (blue, orange, and green curves) to EIA (red, purple, and brown curves), which results in an increase in the peak value of EIA. In the presence of a THz electric field, Fig. 4(b) reveals that the original single peaks of EIT (orange, green, and red curves) and EIA (purple and brown curves) split into double peaks, and that the amplitude of EIT decreases and the amplitude of EIA increases with an increase in ΩR, while Δf remains constant. In Fig. 4(c), under the influence of the THz electric field, Cs133 atoms are excited to the higher level, and the excitation probability decreases, so the ρ11 (blue curve) with a THz electric field is greater than that without a THz electric field (red dashed curve) when ΩR<3.0×2πMHz. Simultaneously, due to the AT splitting altering resonance points and conditions, the effective ΩR is reduced [45]. Consequently, the interaction strength between atoms in the Rydberg state decreases [43], ultimately leading to a decrease in ρ11 with increasing ΩR, as shown by the blue curve in Fig. 4(c). In Fig. 4(d), ρ14 increases with the increase of ΩR, as shown by the olive dashed line, but under the influence of the THz field, when ρ14 is negative, it corresponds to EIA; when ρ14 is positive, it corresponds to EIT. However, as ΩR increases, ρ14 and ρ11 still maintain opposite trends.

The Rabi frequency of the detected THz wave also affects the EIT-AT signal. Figure 5(a) shows the EIT-AT signals at different THz electric fields. When the THz electric field is zero, the transmission spectrum of the probe light exhibits a single EIT peak that is depicted by the blue curve. With the application of a weak THz electric field (ΩTHz=1×2πMHz), the AT splitting occurs, but the splitting interval is challenging to distinguish, which is represented by the orange curve. When ΩTHz exceeds 1×2πMHz, the amplitudes of the EIT-AT curves decrease with the increase of ΩTHz, which are depicted by the green and red curves. In addition, the AT splitting interval linearly increases with ΩTHz, as illustrated in Fig. 5(b). The black squares represent the calculated results of Δf, and the red solid line represents the fitting curve by Δf=ΩTHz2π,R2=0.99834,(9)where R2 is the coefficient of determination used to evaluate the fitting results, and its value is very close to 1, indicating a highly linear relationship between the splitting intervals Δf and the Rabi frequency of the THz electric field ΩTHz. However, the calculation results deviate slightly from the linear relationship in the region of ΩTHz<1×2πMHz. This is primarily due to the nonlinear relationship between the AT splitting and THz field intensity under the influence of a weak THz electric field [39].

In the investigation of THz wave detection utilizing Rydberg atoms, the absolute peak of EIT or EIA signal is one of the important factors determining the observability of the splitting intervals. Generally, when the same THz electric field is detected, the higher the absolute peak of EIT signal, the higher the visibility of AT splitting, which indicates the system has higher sensitivity. The absolute peaks of EIT and EIA signals are defined as $$\begin{gathered} AEIT=Amax−Amin, \\ AEIA=Amin−Amax, \end{gathered}$$(10)where Amax and Amin are the maximum and minimum amplitudes of the EIT or EIA signal, respectively. A comprehensive study was conducted to investigate the impact of Ωp, Ωd, and ΩR on the absolute peaks of EIT (AEIT) and EIA (AEIA), and a three-dimensional scatter plot was drawn, as depicted in Fig. 6(a). The color of the spheres represents AEIA (red to white spheres) and AEIT (white to blue spheres). The maximum of |AEIA| is larger than the maximum of |AEIT|, so the EIA signal has the potential for a higher observability of the AT splitting. However, the dynamic range of AEIA (0 to −0.45) is greater than that of AEIT (0 to 0.13), which indicates that EIA is more sensitive to the changes of the lasers’ Rabi frequencies; that is, the EIA lacks robustness compared to the EIT in the THz detection. As mentioned in Fig. 3(b), the amplitude of EIT increases with the increase of Ωd; however, an increase in Ωd also leads to a decrease in the absolute peak value, resulting in a decrease in the detection sensitivity, as shown in Fig. 6(a). For a more straightforward comparison, Figs. 6(b), 6(c), and 6(d) present two-dimensional filled contour color plots with color fillings consistent with Fig. 6(a). The white region between the two dashed lines corresponds to the transitional area from EIA to EIT, which is indicated by white spheres in Fig. 6(a), and the black lines represent an absolute peak value equal to zero. In Figs. 6(b) and 6(d), the black line has a larger slope, and the white region in Fig. 6(d) is notably broader than in the other two figures, which indicates that Ωd is insensitive to the interconversion between EIA and EIT. In Fig. 6(c), the black line has a smaller slope, so it is more conducive to adjusting the ratio of Ωp and ΩR to achieve the transformation of EIT and EIA in the five-level system. The EIT effect emerges when Ωp and ΩR have a larger ratio. Consequently, to achieve optimal sensitivity for THz wave detection using a five-level system of Cs133 atoms, the Rabi frequencies of the three lasers should be maintained at a stronger probe light, dressing light, and weaker Rydberg light.

The detection of THz waves based on the quantum coherence effect of the Rydberg atom has high sensitivity. Therefore, we theoretically evaluated that the detection sensitivity of the proposed five-level system used quantum projection noise-limited sensitivity (S). The S can be calculated by [38] S=2ℏ4μTHzNτt,(11)where μTHz=174ea0 is the transition dipole moment caused by THz wave [49], e is the electron charge, a0 is the Bohr radius, ℏ is the reduced Planck’s constant, τ is the coherence time that depends on factors such as the atomic lifetime, t is the measurement time, and N is the number of atoms participating in the measurement. N is given by N=nAL,(12)where n is the atomic density in the cell, which is expressed by Eq. (3) in Section 2; however, only about 1/400 atoms participated in the measurement [50], A is the overlap area of laser spots and THz spots in the vapor cell, and L is the length of the vapor cell.

Equation (11) indicates that the quantum projection noise-limited sensitivity is decided by μTHz, τ, t, and N. Here, μTHz and τ are determined by the quantum system and the measurement time t was set as 1 s in the calculations. Therefore, S is only affected by N in a fixed quantum system. The number of atoms participating in the measurement is determined by the vapor temperature and the length of the vapor cell when the overlapped area is fixed, as shown in Eq. (12). Hence, we investigated the effect of the temperature and the length of the vapor cell on the S of the THz detection quantum system, and the results are shown in Fig. 7. In Fig. 7(a), the number of Cs133 atoms within the vapor cell increases with the increase of the vapor temperature. When the temperature reaches the Cs133 melting point of 28.5°C, the slope of the curve sharply increases, which is marked by the red color; this means a significant increase in the number of atoms due to the phase transition. With the increasing temperature, the atomic concentration in the vapor cell increases, which leads to an improved sensitivity, and S decreases, as depicted in the insets of Fig. 7(b). Additionally, we highlight the variation of S during the phase transition of Cs133 atoms, and the region between 28°C and 29°C is marked by the black square in Fig. 7(b), which is zoomed in and displayed in Fig. 7(b). The slopes of the curves near to the melting point temperature, represented by the green color, are significantly greater than those of the orange line. In the process of measuring the THz electric field using Rydberg atoms, the interaction length between Rydberg atoms and various lasers and THz waves is another crucial factor influencing the sensitivity, and the calculated relationships of S and cell length are illustrated in Fig. 7(c). As the length of the vapor cell increases, the interaction lengths between the lasers, THz waves, and Cs133 atoms also increase, and the number of Cs133 atoms participating in the measurement rises, which augments the sensitivity. By the Cs133 five-level system proposed in this study, the quantum projection noise-limited sensitivity is as small as 10−9 V m⁻¹ Hz^−1/2. However, in practical experiments, the sensitivity may be affected by other factors, such as the laser linewidth, laser power, and detector noise.

In this paper, we have proposed a quantum THz detection system, which applies three infrared lasers (Ωp, Ωd, ΩR) to excite Cs133 atoms to the Rydberg state that is sensitive to an electromagnetic wave with a frequency of 0.17 THz (ΩTHz). Concurrently, we have comprehensively analyzed the primary influencing factors in THz wave detection by solving the Lindblad master equation. The results revealed that Ωd exerts a significant effect on the amplitude of EIT, and at larger Ωp and Ωd, and smaller ΩR, the absolute peak exhibits its maximum value. At this time, the visibility of AT splitting is maximum, thus achieving better detection sensitivity for THz electric fields. Additionally, our calculations indicated that the application of different proportions of Ωp, Ωd, ΩR in the five-level system can realize the transition between EIT and EIA, and ρ11 and ρ14 play a crucial role in the conversion between EIT and EIA. Specifically, Ωd has the least influence on this transition, and EIT signals appear when there is a relatively larger ratio of Ωp and ΩR. In the critical region where the EIA and EIT transitions occur, the visibility of the AT splitting interval is minimal, which significantly affects the sensitivity of the THz electric field detection. Furthermore, we numerically evaluated the sensitivity of the five-level system used in this paper for detecting THz waves by S, and analyzed the influence of the temperature and vapor cell length on the sensitivity. Theoretically, the S can reach the order of 10−9 V m⁻¹ Hz^−1/2, which is a great improvement compared to other room temperature THz wave detectors, such as Golay cells, pyroelectric detectors, and Schottky diodes. Conclusively, this paper provides a theoretical foundation for the highly sensitive detection of THz waves utilizing the Rydberg atomic five-level system. Furthermore, it serves as a significant catalyst in advancing the realm of THz science and technology.

Acknowledgment. We acknowledge discussions with Yan Peng and Ruijie Peng from University of Shanghai for Science and Technology, Haitao Tu from South China Normal University, Haogong Liu from Xi’an Polytechnic University, and Dong Li from Microsystems and Terahertz Research Center, China Academy of Engineering Physics. We gratefully acknowledge HZWTECH for providing computation facilities.