Detuning radio-frequency electrometry using Rydberg atoms in a room-temperature vapor cell

Traceable radio-frequency electric field measurements with high sensitivity have been demonstrated from the centimeter wave to the millimeter wave using room-temperature Rydberg atoms. Here, we investigate the splitting spectra of electromagnetically induced transparency induced by a radio-frequency electric field, which is detuned from the resonant frequency of transitions between two Rydberg states, 47D5/2 ↔ 48P3/2. By varying the detuning of the radio-frequency electric field, we measure the separation between the two peaks, and in particular their relative height. The resulting measured resonant transition frequency between two Rydberg states is found to exhibit a visible change when the power of the radio-frequency electric field is varied, thus causing uncertainty in the traceable radio-frequency electric field measurement. Further, the effect of detuning of the probe light on the radio-frequency electric field measurement is presented.


Introduction
Rydberg atoms are highly excited atoms with a large principal quantum number. Compared with the ground state atoms, Rydberg atoms have a large orbital radius and large dipole moment [1], and are very sensitive to the external electric field [2,3]. Electromagnetically induced transparency (EIT) is a phenomenon where the transmission of a weak probe field increases in an absorbing medium when a strong coupling field is used [4]. The effect of EIT is widely used in nonlinearity enhancement [5,6], electric dipole moment measurements [7,8], optical quantum storage [9,10] and other aspects.
Rydberg EIT has been used to measure radio-frequency electric-field (RF E-field) strengths over a large range of frequencies (1 GHz to 500 GHz) by using a room-temperature vapor cell [11][12][13][14][15][16]. An RF E-field applied to the Rydberg atoms results in the Autler-Townes (AT) splitting of the Rydberg EIT transmission spectrum. Using this approach, RF E-fields can be measured with a high sensitivity, approximately 30 µV cm −1 Hz −1/2 [12]. Millimeter wave E-field spatial distributions have recently been demonstrated with a high resolution Laser Physics Detuning radio-frequency electrometry using Rydberg atoms in a room-temperature vapor cell Linjie Zhang 1,2,4 , Yue Jia 1,2 , Mingyong Jing 1,2 , Liping Guo 3 , Hao Zhang 1,2 , Liantuan Xiao 1,2 and Suotang Jia 1,2 of approximately λ/650 [14,15], where λ is the wavelength of the millimeter E-field. The polarization of RF E-fields was also identified; the resolution of polarization reached 0.5° [13]. As a core component of RF E-field electrometry, the effects of the size of the vapor cell on the accuracy of the measurements was investigated [16]. Simons et al demonstrated the improved measurement sensitivity of a weak detuning RF E-field strength compared to an on-resonant RF electric field [17].
In this work, we investigated the EIT-AT spectrum of a ladder four-level system involving Rydberg states in a roomtemperature vapor under the condition of the detuning RF E-field and a probe light. The splitting separation and the relative height ratio of two peaks of the AT spectra are presented respectively as a function of the frequency detuning and the power of the RF E-field. It is shown that the power of the RF E-field will cause the frequency shift of the 47D 5/2 -48P 3/2 Rydberg transition. The model of a four-level system theory is used and provides a good qualitative explanation for the experimental observation. Moreover, the effect of the EIT-AT splitting on the detuning of the probe light is demonstrated. Figure 1(a) is the four-level atomic energy level scheme in our experiments. In a typical three-level EIT system, a 'dark state' is created by a probe field (Ω p ) and a coupling field (Ω c ), which prevent the resonant absorption of the probe laser. An applied RF field that is resonant with the two adjacent Rydberg states breaks down the three-level system and causes the probe photons to be absorbed again. The EIT spectrum will split into two peaks: this is called AT splitting. The AT splitting (Δf ) is proportional to the RF E-field strength E and is described as

Experimental approach
where ℘ MW is the transition dipole moment of the adjacent Rydberg states and is Planck's constant, and Ω M is the Rabi frequency of the Rydberg state transition induced by the RF E-field [18]. Figure 1(b) shows the experimental setup. We use a cylindrical vapor cell with 50 mm length and 20 mm diameter containing cesium ( 133 Cs) atom vapor. The probe laser drives the 6S 1/2 (F = 4) → 6P 3/2 (F′ = 5) transition and is generated with an external cavity diode laser. To observe an EIT signal, we apply a counter-propagating coupling laser (~510 nm) with a scanning frequency near the transition between 6P 3/2 and 47D 5/2 . The coupling laser (Toptica TA-SHG110) is supplied by a frequency doubled amplified diode laser at 1020 nm. The Rabi frequency of the probe (coupling) laser is Ω p = 8.08 MHz × 2π (Ω c = 2.05 MHz × 2π). Both lasers have the same linear polarization and are locked to a high-finesse Fabry-Pérot cavity made by the ultra-low-expansion glasses. Each laser has a spectral bandwidth less than 1 kHz according to the cavity transmission linewidth. It is necessary to shift the laser frequency to the resonance transition by the doublepassed acoustic-optical modulator (AOM). The RF E-field at about 6.946 GHz, which couples resonantly to the trans ition of 47D 5/2 ↔ 48P 3/2 , is led into the vapor cell by a horn antenna. The frequency of the coupling laser is scanned by an AOM so that the precise frequency of the splitting interval can be obtained. Meanwhile, the laser power is stabilized by a proportion-integral-differential feedback loop system by controlling the AOM diffraction efficiency to reduce the intensity noise.

Experimental results and discussion
The EIT signal and AT splitting spectrum of the probe transmission induced by an RF E-field are shown in figure 2(a).  (1), the splitting separation can be given by where Δ M = f − f 0 is the frequency detuning of the RF E-field, f is the measured E-field frequency, f 0 is the resonance frequency of the 47D 5/2 ↔ 48P 3/2 transition. Here, Δf 0 is the splitting interval induced by the resonant RF E-field and is equal to the Rabi frequency of the RF resonance transition in our experimental configuration. The RF E-field with the resonance frequency has the minimum AT splitting. Figure 3(a) shows the dependence of the splitting interval Δf on the absolute radio frequency f. The simulation curves using equation (2)  The detuning RF E-field would cause larger splitting and improve the measurement sensitivity than that of the resonance frequency [17]. Moreover, the splitting measurement can offer a method to obtain the resonance frequency of the Rydberg transition through equation (2). However, the simulation results of the resonance frequency for the different RF E-field strength in figure 3(a) is shown in figure 3(b). It is noted that the resonance frequency shifts with the variation in RF E-field strength, which is similar to the effect of the molecule photo-association [19]. The RF E-field induced the frequency shift of the resonance transition between Rydberg states. Here, the induced frequency shift is smaller than 0.3 MHz according to the mean values in figure 3(b). Considering the error of the fitting results, the maximum frequency shift is about 1.2 MHz.
To investigate the frequency shift with the power of detuning RF E-fields in detail, we illustrate the relation of the height Laser Phys. 29 (2019) 035701 ratio of two AT splitting peaks to the detuning of the E-field frequency. The radio frequency is varied in a small range from 6.9455 GHz to 6.9485 GHz. In figure 4(a), we show the height ratio of right peak H 2 to left peak H 1 of AT splitting curves for the different radio frequencies with the selected output power: −13 dbm, −15 dbm, −17 dbm, −19 dbm and −21 dbm. Figure 4(a) shows the slope of the height ratio is dependent on the applied RF E-field power. The smaller RF power produces the bigger slope. The curves corresponding to the variation in RF power intersect at a resonant frequency when the height ratio of the AT splitting peaks is equal to 1. It is noted that the frequency shift is about 1.1MHz under the different RF E-field power. Figure 4(b) shows the theoretical simulation curves for the four-level ladder system, which is similar to [20]. Here, the frequency shift induced by the RF E-field is not considered. The Hamiltonian matrix of the fourlevel ladder system is given: where Ω p , Ω c , Ω M are the Rabi frequency corresponding to the probe laser, coupling laser and the RF E-field, respectively, ϕ is the initial phase of the RF E-field. Here, Δ p , Δ c and Δ M are the detuning associated with the probe, coupling and RF E-field corresponding to the transition of states, respectively. The master equations are given aṡ where Γ kj = δ jk γ j , the subscripts i, j , k correspond to the states, Γ ik indicates the spontaneous radiation attenuation  from |i > to |k > , and the parameter γ j is the decay rate for the various states. Here, ρ kj is the coherence between states k and j . Solving the probability density equations can obtain ρ 21 , to obtain the dispersion and absorption spectra of Rydberg EIT including the RF E-field. The imaginary part of the ρ 21 can be calculated as where Laser Phys. 29 (2019) 035701 We select reasonable experimental parameters to fit the height ratio of the splitting peaks by using equation (5). Here, we set Δ p = 0, γ 12 = 2π × (5.  figure 4(b), all curves cross at the resonance frequency (Δ M = 0) in which the relative height ratio equals 1. It is reasonable to conclude that the RF E-field causes the resonant frequency shift and the height ratio of AT splitting can offer the reference to calibrate the resonance frequency for different RF E-field strengths. For the frequency shift introduced by the splitting fitting in figure 3 and height ratio in figure 4, the introduced error of the E-field strength using equation (2) is smaller than 1% for the weak field measurement. Furthermore, we investigate the effect of the detuning of the probe field on the RF E-field measurement. The RF E-field  strength is about 31 mV cm −1 according to the calculation using the splitting of the resonant frequency. The frequency of the probe laser varies to the resonance frequency f p of transition 6S 1/2 , F = 4 ↔ 6P 3/2 , F′ = 5, f p ± Γ and f p ± 2Γ, where Γ is the natural linewidth of 6P 3/2 . Figure 5 shows the AT splitting measurement versus the detuning of the probe laser and the RF E-fields. The minimum splitting interval indicates the resonance for both the probe laser and the RF field. The detuning of the probe laser will produce a measurement error smaller than 6.3% in our experiments.
When the RF field is off-resonant with the Rydberg states, the splitting interval shows the asymmetric characteristic for the red detuning and blue detuning of the probe laser. While the probe laser is tuned on the blue side, the measurement deviation of the RF E-field strength is bigger than that which is measured when the probe laser is red detuning. Red detuning of the probe laser will produce more measurement deviation for the blue detuning RF E-field. The asymmetric characteristic is attributed to the asymmetric shift of hyperfine levels under the stray magnetic field around the vapor cell [21,22].
In conclusion, we investigated the effect of the detuning frequency of adjacent Rydberg state transitions on the RF E-field metrology in a cascade four-level system in a roomtemperature vapor cell. We demonstrated the dependence of both the splitting intervals and the height ratio of two splitting peaks on the RF E-field frequency. The frequency shifts of Rydberg transition induced by the RF E-field are investigated. Furthermore, we showed the effect of probe detuning on the RF E-field measurement. Red detuning of the probe laser will produce the asymmetric measurement deviation in the detuning RF E-field metrology.