Microwave Electrometry with Multi-Photon Coherence in Rydberg Atoms

A scheme for the measurement of a microwave (MW) electric field is proposed via multi-photon coherence in Rydberg atoms. It is based on the three-photon electromagnetically induced absorption (TPEIA) spectrum. In this process, the multi-photon produces a narrow absorption peak, which has a larger magnitude than the electromagnetically induced transparency (EIT) peak under the same conditions. The TPEIA peak is sensitive to MW fields, and can be used to measure MW electric field strength. We found that the magnitude of TPEIA peaks shows a linear relationship with the MW field strength. The simulation results show that the minimum detectable strength of the MW fields is about 1/10 of that based on an common EIT effect, and the probe sensitivity could be improved by about four times. Furthermore, the MW sensing based on three-photon coherence seems to be robust against the changes in the control field and shows a broad tunability, and the scheme may be useful for designing novel MW sensing devices.

It is know that an EIT system has good coherence and its probe spectrum is widely used for MW measurement.Recently, three-photon coherence attracts researchers' interest.For example, observation of three-photon electromagnetically induced absorption (TPEIA) [17] in atomic systems, constructive interference in the three-photon absorption [15], [16], demonstration of three-photon coherence condition [18], and its extention to Rydberg atoms [19].Three-photon coherence has important applications, such as overcoming residual Doppler shifts [20], miniaturizing of MW field sensors and receivers [21] and its dependence on the AT splitting with two coupling lasers [22].While, few reaserch involves MW measurement based on three-photon coherence.
In this paper, we propose a scheme to measure MW electric fields based on three-photon coherence in Rydberg atoms.A probe and a control fields counter-propagate through the atomic system [23].The 87 Rb atoms are excited from the ground states to the Rydberg states, and the absorption spectrum of three-photon transition shows a single absorption peak around the resonant frequency.Due to three-photon coherence, a strong TPEIA peak appears under the three-photon resonance condition.It is interesting to find that the TPEIA peak changes linearly with the MW electric field strength.This may be used to detect the MW electric field.The simulation results show the sensitivity is enhanced by about 4 times and its minimum detectable strength of the MW electric field can be increased by more than one order of magnitude, compared with a common EIT scheme.Also, the scheme shows a wide tunability.This may help to design novel MW sensing devices.

II. MODEL AND BASIC EQUATION
Fig. 1(a) shows a four-level ladder-type atomic system.The relevant atomic energy levels of 87 Rb are 5S 1/2 (|1 ) , 5P 3/2 (|2 ), 53D 5/2 (|3 ), and 54P 3/2 (|4 ).A probe laser Ω p with a wavelength of λ p ∼ 780 nm and a coupling laser Ω c with λ c ∼ 480 nm counter-propagate through the atoms and drive |1 ↔ |2 and |2 ↔ |3 transition, respectively.A MW field drives the Rydberg transition of the states |3 to |4 .A similar system has been used in intracavity EIT [26], THz field measurement [27], nonlinear optical effects [28], and so on.In the interaction picture and after the rotating wave approximation, the Hamiltonian of the system can be written as where H is the interaction Hamiltonian, the Rabi frequency is denote the detunings of the corresponding fields, respectively.µ ij (i, j = 1, 2, 3, 4) is the transition dipole moment from state |i to state |j .The dynamic evolution of the system can be described using the density-matrix method as follows [30]: where L (ρ) denotes the decoherence processes.The time evolution of density matrix elements can be written as with ρ ij =ρ * ji and the closure relation j ρ jj = 1, (i,j=1,2,3,4).Here, is the decay from the states |i to |j , and γ ij = (Γ i +Γ j )/2, Γ i is the population decay rate.We consider ρ and the susceptibility of the Rydberg atoms is then Doppler-averaged: where N is Rydberg atom density.µ 12 is the dipole moment of transition |1 ↔ |2 , ε 0 is the dielectric constant of vacuum, u = 2k B T /m is the most probable velocity of the atoms, k B is the Boltzmann constant, T is the temperature of system, m is the mass of the atom.Then, we obtain the three-photon coherence term in ρ 21 [31], [32], and the threephoton coherence part of the atomic susceptibility is where

III. RESULTS AND DISCUSSION
We first consider the TPEIA spectrum changing with the MW field in a Doppler-free scheme, where the probe field counter-propagates with the control field (see Fig. 2).The 87 Rb atomic density is N = 10 −7 cm −3 and spontaneous decay Γ 2 = 6γ = 2π × 6MHz in the following discussion [4].And the following discussion is scaled by γ for simplicity.Fig. 2(a) shows that the TPEIA signal has one absorption peak through the three-photon process when the lasers interact with atoms resonantly.The basic property of TPEIA agree well with the previous study [17], [33], and here we pay attention to the variation of TPEIA with MW fields.For a weak MW field, the TPEIA peak increases with the strength of MW field, as shown in Fig. 2(a), the absorption peak becomes strong with increase of the MW field strength, and the peak linewidth becomes a little broad due to homogenous broadening effect.Thanks to the three-photon coherence, the population transfers from the ground state to the Rydberg state, and the peak value of absorption spectrum can be improved in the range of weak MW field.Fig. 2(b) shows the variations of magnitude of TPEIA peak as a function of the MW field strength.The TPEIA peak becomes strong by increasing MW field, and the linewidth remains narrow.While, the magnitude of TPEIA peak changes nonlinearly with the MW field strength, which may be not suitable for the linear measurement of MW electric field.Generally, most of the experiments are performed in a room temperature vapour.So, the Doppler effect is obvious due to mismatch of the coupling-field wavelength, where the probe and control fields counter-propagate through the atomic vapor.The variations of the absorption spectrum are shown in Fig. 3(a).The TPEIA peak becomes strong with increase of the MW field.However, when the strength of the MW field is further increases, the TPEIA signal is suppressed and two transmission windows far away from resonance (see Fig.

3(b))
. We pay more attention to the enhanced TPEIA peak and explore its application in precise measurement.
It is interesting to find that the magnitude of TPEIA peak varies linearly with the MW field, as shown in Fig. 4(a).The numerical results show that the curve slop based on three-photon coherence is about 4. Fig. 4(b) shows the linear measurement of MW field based on the common EIT method, where the frequency splitting of EIT peaks changes linearly with the MW electric field strength.The slope of measurement curve based on EIT is about 1 from simulation.The comparison of Fig. 4(a) and Fig. 4(b) shows that the curve slope based on TPEIA is about 4 times larger than that of EIT method.The MW electric field strength could be estimated from changes in the magnitude of TPEIA peaks.It is known that, the larger curve slope under the same condition results in the better detection sensitivity.This indicates that the probe sensitivity could be improved by about a factor of 4 due to the three-photon coherence.
It is important to detect the minimum strength of the MW field for precise measurement.Fig. 5(a) shows the minimum detectable strength MW min for the three-photon resonance case.According to the Rayleigh criterion [35], the the corresponding spectrum resolution is about 0.02γ, which means the minimum detectable strength of the MW field is about 0.02γ based on the EIT scheme, as shown in Fig. 5(b).Our simulation results show that the minimum detectable strength of the MW field is about 0.002γ for the TPEIA spectrum, which is about 1/10 that based on an common EIT effect (see Fig. 5(a)).This indicates that the minimum detectable strength could be improved by 10 times due to the threephoton coherence.The above discussions deal with the weak MW field.In the dressed-state picture, the probe and control transition consist of a classical Rydberg EIT scheme.There is an EIT window around the resonant frequency, and it can be understood in the EIT theory [42].The coupling fields Ω c dresses the states |2 and |3 , and two new eigenstates appear, i.e., |+ and |− .With coupling of the probe field, two transition channels appear, |1 → |+ and |1 → |− .When the MW field drives the Rydberg transition |3 → |4 , the Rydberg-EIT is disturbed and an enhanced absorption peak builds up, which referred to be as TPEIA.6(a) shows the effect of large MW field on the magnitude of the Doppler-averaged TPEIA spectrum.For example, when the MW field strength Ω m ≤ 2.8γ, the TPEIA peak becomes strong with increase of the MW field, as shown Fig. 6(a).While, the TPEIA peak decreases with the further increase of the MW field.So, the TPEIA peak reaches a maximum at Ω m = 2.8γ under the given condition.This is because the two dressed states induced by the MW field are well separated, and the AT splitting of Rydberg EIT is dominant over TPEIA peak in the regime of the large MW field [19].The constructive interference for three-photon coherence gradually weakens.When the TPEIA peak decreases with Ω m , the peak-to-peak distance of the two transmission peaks increases with Ω m , as shown in Fig. 3(a).This effect can be well-understood in the dressed-state picture [34].The effect of the control field on the intensity of the absorption spectrum is shown in Fig. 6(b).The magnitude of TPEIA peak increases when the control field becomes strong.The strong control field induces the good multi-photon coherence and contributes to the large TPEIA peak.Of course, if the control field is too large, the Rydberg EIT evolves into Aulter-Tonwns splitting, and the inter-path interference weakens, resulting in decrease of TPEIA peak.In addition, the numerical results show that the linewidth of the absorption spectrum is about 1.15γ.The linewidth of TPEIA peak is a little broader than that of Doppler-free scheme.This is due to the residual Doppler effect as a result of the temperature of the atomic vapor.The above discussions are based on resonant interaction of the control field.Fig. 7 shows the effect of the control field detuning on the TPEIA spectrum.The figure shows that the absorption window shifts with the control field detuning ∆ c .While the width and peak value of the TPEIA spectrum basically remain unchanged.This means that the scheme has a broad detection range and some tunability.

IV. CONCLUSION
In summary, we theoretically investigate TPEIA spectrum of Rydberg atoms and propose to use three-photon coherence to detect the weak MW electric field.Due to the multi-photon coherence, there is constructive interference in the TPEIA at the resonant frequency.It is interesting to find that the magnitude of TPEIA peaks change linearly with the MW field, which can be used to detect the MW electric field.The numerical results show the sensitivity based on TPEIA is about 4 times larger than that of the EIT scheme.Its minimum detectable is about one order of magnitude smaller than that of the EIT scheme.Moreover, the MW measurement based on TPEIA shows a broad detection range and some tunability.The proposed scheme may help to design novel MW-sensing devices.
ij = 0.The coherence term ρ 21 can be obtained by solving the above equation.With consideration of the residual Doppler effect, the frequency detunings of the control and probe fields are modified as δ

Fig. 3 .Fig. 4 .
Fig. 3. (a) The Doppler-averaged TPEIA spectrum and (b) variation of the TPEIA spectrum with MW field.The other parameters are the same as in Fig. 2(a).

Fig. 5 .
Fig. 5. (a) Doppler-averaged TPEIA spectrum and (b) transmission spectrum of EIT.The other parameters are the same as in Fig. 2(a).

Fig. 6 .Fig. 7 .
Fig. 6.(a) The effect of large MW field on the intensity of the Doppleraveraged TPEIA spectrum and (b) peak intensity as a function of the control field strength with Ωm = 2.8γ.The other parameters are the same as in Fig. 2(a).