1. Introduction

When four-level Rydberg atoms interact with local light and microwave (MW) fields, the electromagnetically induced transparency (EIT) spectrum induced by probe and control (coupling) lasers will be split because the strong local MW field induces Autler-Townes (AT) splitting of the EIT peak [1]. The peak-to-peak separation is sensitive to the electric (E) field strength of local MW. And MW E-field strength detection can thus be changed into frequency measurement of local probe laser, the detection sensitivity is thus enhanced substantially [2,3,4,5]. Compared with traditional dipole antennas [6], EIT allows the advent of new quantum technologies and devices [7]. EIT spectrum of Rydberg atoms has enhanced capabilities and novel applications in MW sensing and metrology, including sub-wavelength imaging [8], laser stabilization [9], electronic communication [10], quantum gates [11,12], MW polarization direction measurement [2], atomic receivers [13,14], etc.

One particular interest in Rydberg-EIT-based communications is detecting emerging fields, including both analog amplitude and frequency modulation [15,16] and pulsed MW field detection and measurements [17]. Besides, in order to realize high-bandwidth receivers and devices, it is important to investigate time-dependent behaviors and transients in Rydberg EIT, because temporal EIT behaviors in atom-field interaction determine the fundamental bandwidth limits of EIT-based sensors and receivers for communications and pulse detection [17].

In this paper, we investigate the time-dependent response of four-level cold Rydberg atomic system with local light and MW fields in EIT theoretically. Our results show that the local control laser acts as a switch. When the strength of the control laser is weaker than the spontaneous decay rate from the excited state 5P_3/2 to the ground state 5S_1/2, population transition from state 53D_5/2 to state 5S_1/2 is forbidden. The transient absorption for the probe laser will monotonically decrease to its steady state. And the steady-state time window of the coupled atomic system is dependent on both the atomic medium and the local control laser. In contrast, when the control field is stronger than the spontaneous decay rate, the transition channel from state 53D_5/2 to state 5S_1/2 is open. Part populations of state 53D_5/2 can be stimulated back to state 5S_1/2 through a two-photon resonant transition, which leads to that the transient absorption for the probe laser will be oscillatorily damped to its steady-state value. And the oscillation frequency is dependent on both the control field and the spontaneous decay rate between states 5S_1/2 and 5P_3/2. Transient populations of state 5S_1/2 will be enhanced and transient negative absorption for the probe laser appears. The steady-state time window is reduced and just depends on the atomic medium. When a third MW field is applied, the time-dependent EIT transmission spectrum can be split at the resonant frequency because of the AT splitting of level 53D_5/2. The stable splitting distance shows a linear dependence on the continuous MW-field strength. Furthermore, the simulation results show that the frequency splitting of the time-dependent EIT spectra is sensitive to the E-field strength of local MW pulse, and can be used to sense pulsed MW fields.

2. Model and basic equation

In the present work, states $|1 \rangle $, $|2 \rangle $, $|3 \rangle $ and $|4 \rangle $ represent the states 5S_1/2, 5P_3/2, 53D_5/2 and 54P_3/2 of ⁸⁷Rb, respectively. A probe laser couples states $|1 \rangle $ and $|2 \rangle $ with a Rabi frequency of ${\Omega _p}$. A control laser is tuned to the upper transition $|2 \rangle \leftrightarrow |3 \rangle $ with a Rabi frequency of ${\Omega _c}$. The probe and control lasers are both linearly polarized. The third driving field, a MW field supplied by horn antenna is set to excite the atomic transition between the two Rydberg states $|3 \rangle $ and $|4 \rangle $ with a Rabi frequency of ${\Omega _m}$. More details can be found in Fig. 1. In the simulation, the probe and control lasers are continuous lights, while the MW field could be either a continuous or a pulsed field.

 

Fig. 1. Schematic setup for experiment and energy level diagram. Four-level cold Rb atoms are confined in a vapor cell. A probe laser couples atomic transition $|1 \rangle - |2 \rangle $ with Rabi frequency ${\Omega _p}$. A control laser drives $|2 \rangle - |3 \rangle $ transition with Rabi frequency ${\Omega _c}$, and a MW field drives $|3 \rangle - |4 \rangle $ transition with Rabi frequency ${\Omega _m}$. M₁, probe laser high reflection and control laser high transmission lens. M₂, probe laser high transmission and control laser high reflection lens. PD, photodiode.

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In the interaction picture and after the rotating wave approximation, the final expression of the Hamiltonian for the atomic system can be given by

Here $\hbar $ is the Plank constant. The Rabi frequencies of the probe, control and MW fields are ${\Omega _p} = {\mu _{12}}{E_p}/\hbar $, ${\Omega _c} = {\mu _{23}}{E_c}/\hbar $ and ${\Omega _m} = {\mu _{34}}{E_m}/\hbar $, with ${\mu _{12}}$, ${\mu _{23}}$ and ${\mu _{34}}$ being the atomic transition dipole moments between levels $|1 \rangle $ and $|2 \rangle $, levels $|2 \rangle $ and $|3 \rangle $, and levels $|3 \rangle $ and $|4 \rangle $, respectively. ${E_p}$, ${E_c}$ and ${E_m}$ stand for the amplitudes of the electronic component of the probe, control and MW fields, respectively. The detunings of the three fields are given by ${\Delta _p} = {\omega _p} - ({\omega _2} - {\omega _1})$, ${\Delta _c} = {\omega _c} - ({\omega _3} - {\omega _2})$ and ${\Delta _m} = {\omega _m} - ({\omega _3} - {\omega _4})$, respectively. ${\omega _p}$, ${\omega _c}$ and ${\omega _m}$ are the frequencies of the probe, control and MW fields, respectively. And $\hbar {\omega _i}$ ($i = 1,$ $2,$ $3$ and $4$) is the eigenvalue of state $|i \rangle $.

To account for decay processes, the Lindblad equation is used to describe the dynamics of our system which is governed by the following master equation for the four-level density matrix $\rho $ [18,19]:

Here ${\gamma _{jk}}$ ($j = 1,$ $2$ and $3$ and $k = 2,$ $3$ and $4$, $j \ne k$) denotes the spontaneous decay rate from level $|j \rangle $ to level $|k \rangle $. ${\gamma _{jk}} = ({{\gamma_j} + {\gamma_k}} )/2$, with ${\gamma _j}$ (${\gamma _k}$) being the decay rate of level $|j \rangle $ ($|k \rangle $). Level $|1 \rangle $ is the ground state and we set ${\gamma _1} \equiv 0$ throughout the paper. The optical Bloch equation for each density matrix of the coupled four-level system can thus be expressed as

In the simulation, we consider a closed system requiring ${\rho _{11}} + {\rho _{22}} + {\rho _{33}} + {\rho _{44}} = 1.$ The probe laser (or the probe and control lasers) is (are) switched on firstly, and the switching time can be set as $t ={-} {t_0}$µs. When the system has reached its steady state, the control laser (or the MW field) is switched on, and the switching time is set as $t = 0$. Then the initial conditions for Eqs. (4) can be considered as ${\rho _{11}}(t ={-} {t_0}) = 1$ and ${\rho _{jk}}(t ={-} {t_0}) = 0$ ($j = 1 - 4$, $k = 2 - 4$). The temporal solutions of Eqs. (4) can be readily obtained numerically, and the time-dependent atomic susceptibility $\chi (t)$ can thus be written as [4]

Here l is spatial length of the atomic medium, and c is the speed of free-space light.

3. Results and discussion

Figure 2 shows the numerical results of the time-dependent atomic susceptibility Im [χ(t)] versus evolution time t for different control field strength ${\Omega _c}$. In the simulation, the probe laser is switched on at the beginning of the time evolution (at a time $t ={-} 3.0$µs). When the atomic medium has reached its steady state, the control laser is switched on (at $t = 0$). It can be assumed that the present four-level system is reduced into a standard three level ladder-type configuration. The detunings of the probe and control fields are ${\Delta _p} = {\Delta _c} = 0$, i.e. the two lasers are both resonant. Here we adopt a weak probe field with a strength of ${\Omega _p} = 2\pi \times 0.01$MHz. And the decay rates of levels $|2 \rangle $, $|3 \rangle $ and $|4 \rangle $ are chosen as ${\gamma _2} = 2\pi \times 6.1$MHz, ${\gamma _3} = 2\pi \times 1.0$kHz and ${\gamma _4} = 2\pi \times 0.5$kHz, respectively.

 

Fig. 2. Atomic susceptibility Im [χ(t)] versus evolution time t. The parameters are chosen as ${\gamma _2} = 2\pi \times 6.1$MHz, ${\gamma _3} = 2\pi \times 1$kHz and ${\gamma _4} = 2\pi \times 0.5$kHz, ${\omega _p} = 2\pi \times 3.84 \times {10^8}$MHz, ${\Omega _p} = 2\pi \times 0.01$MHz, ${\Delta _p} = {\Delta _c} = 0$ and $l = 5$cm. The strengths of the control field are ${\Omega _c} = 2\pi \times 2$, $2\pi \times 5$ and $2\pi \times 10$MHz, respectively.

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${\mathop{\rm Im}\nolimits} [\chi ]$ exhibits absorption properties when quantum coherence is induced by external fields in atomic medium. In the simulation, all the atoms populate on level $|1 \rangle $ at the beginning of the time evolution, and the initial conditions are chosen as ${\rho _{11}}(t ={-} 3\mu s) = 1.0$ and ${\rho _{jk}}(t ={-} 3\mu s) = 0$ ($j = 1 - 4$, $k = 2 - 4$). Therefore, the transient absorption for the probe laser is ${\mathop{\rm Im}\nolimits} [\chi (t ={-} 3\mu s)] = 0$ at the beginning, see Fig. 2. As time increases, the atomic medium will exhibit its absorption property for the probe laser because the probe field induces polarization of the atomic medium, and a few atoms can be coherently excited from level $|1 \rangle $ onto level $|2 \rangle $ by absorbing one photon of the probe field, even though most of the atoms still populate on level $|1 \rangle $. It leads to that ${\mathop{\rm Im}\nolimits} [\chi (t)]$ increases as time evolves. When the atomic medium reaches a new steady state after the application of the probe laser, ${\mathop{\rm Im}\nolimits} [\chi (t)]$ reaches its maximum and keeps stable. As shown in Fig. 2, ${\mathop{\rm Im}\nolimits} [\chi (t)]$ reaches the maximum in the vicinity of $t ={-} 2.6$µs, and shows no further variation with time evolution until the switching on of the control laser at $t = 0$.

When the local control laser is weak, for example, when ${\Omega _c} = 2\pi \times 2.0$MHz, the transient medium absorption for the probe field ${\mathop{\rm Im}\nolimits} [\chi (t)]$ monotonically decreases to its steady state with time evolutions after the application of the control field, see the dotted red curve in Fig. 2. And the steady-state time window ${T_w}$ (the time for the atomic medium to reach its steady state after the application of the control laser) lasts about $2.0$µs. Nevertheless, when the local control field becomes strong, for instance, when ${\Omega _c} = 2\pi \times 5.0$MHz and $2\pi \times 10.0$MHz, ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will be oscillatorily damped to the steady-state value. The steady-state time window ${T_w}$ is reduced to $0.66$µs and keeps stable with a further increase of the control field, see the dashed blue and solid gray curves in Fig. 2. Besides, there appears transient negative absorption (or gain) for the probe laser. If we consider the initial conditions to be similar as those in Ref. [21], the obtained three ${\mathop{\rm Im}\nolimits} [\chi (t)]$ evolution curves will fit the three curves in Fig. 2 well.

In order to disclose the underlying mechanism of the dependence of the transient absorption for the probe laser ${\mathop{\rm Im}\nolimits} [\chi (t)]$ on the control field strength ${\Omega _c}$, we simplify Eqs. (4) to the first order of ${\Omega _p}$ as following

We can obtain the analytical solutions of Eqs. (7) with the Laplace transform, which can be expressed as

Here we consider the resonant case with ${\Delta _p} = {\Delta _c} = 0$ and ${\Omega _1} = \sqrt {{\gamma _{12}}^2 - {\Omega _c}^2} $. ${\gamma _{13}}$ is neglected because ${\gamma _{13}} < < {\gamma _{12}}$ and ${\gamma _{13}} < < {\Omega _c}$. For simplicity, the population localization ratio of state $|3 \rangle $ is set to $0$ at the beginning of the time evolution. ${\rho _{12}}(t)$ and ${\rho _{13}}(t)$ describe the temporal one- and two-photon coherence between levels $|1 \rangle $ and $|2 \rangle $, and levels $|1 \rangle $ and $|3 \rangle $, respectively. Here we consider a weak probe laser, i.e. in the bare-state picture one can see most of the atoms still populate on level $|1 \rangle $ after the application of the probe field. Therefore, we can use ${\rho _{12}}(t)$ and ${\rho _{13}}(t)$ depicting the temporal populations of levels $|2 \rangle $ and $|3 \rangle $, respectively.

As ${\Omega _c}$ increases, the varying of ${\mathop{\rm Im}\nolimits} [\chi (t)]$ with time evolutions can be divided into three regions:

Region I, $0 < {\Omega _c} < {\gamma _{12}}$. In this region, the one-photon coherence ${\rho _{12}}(t)$ between levels $|1 \rangle $ and $|2 \rangle $ is imaginary at the beginning of the time evolution (see Eq. 8(a)), the dispersive response for the probe laser vanishes. In the bare-state picture one can see a few atoms can be coherently excited from level $|1 \rangle $ onto level $|2 \rangle $ through a one-photon resonant transition. When the control laser is switched on, populations on level $|2 \rangle $ will take a Rabi oscillation between levels $|2 \rangle $ and $|3 \rangle $ and be excited onto level $|3 \rangle $ with a frequency ${\Omega _c}$ [21,22]. However, in region I ${\Omega _c} < \frac{{{\gamma _2}}}{2} = {\gamma _{12}}$, the population oscillation frequency ${\Omega _c}$ between levels $|2 \rangle $ and $|3 \rangle $ is less than half of the decay rate of level $|2 \rangle $. Besides, ${\gamma _3} < < {\gamma _2}$ and ${\gamma _3} < < {\Omega _c}$, the atomic stimulated absorption between levels $|2 \rangle $ and $|3 \rangle $ is allowed, while the atomic stimulated emission is forbidden. E.g. populations on level $|2 \rangle $ can be excited onto level $|3 \rangle $, while populations on level $|3 \rangle $ can not be simulated back to level $|2 \rangle $ (and then back to level $|1 \rangle $), which leads to the temporal two-photon coherence ${\rho _{13}}(t)$ between levels $|1 \rangle $ and $|3 \rangle $ approaches to its steady-state value monotonically, see the dotted red and solid black curves in Fig. 3(b). The stable population localization ratio of level $|2 \rangle $ will be ${\rho _{22}} = 0$ when the atomic system reaches its steady state. The excited populations are stabilized at level $|3 \rangle $ at last. The transient absorption for the probe laser ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will monotonically approach to the steady-state valve with reducing populations of level $|2 \rangle $ after the application of the control field, see the dotted red and solid black curves in Fig. 3 (a). And the steady-state time window ${T_w}$ is dependent on both the external control field (${\Omega _c}$) and the atomic medium (${\gamma _{12}}$), a weaker ${\Omega _c}$ will lead to a longer ${T_w}$.

 

Fig. 3. Results obtained with Eqs. (8). The strengths of the control field are ${\Omega _c} = 2\pi \times 2$, $2\pi \times 3$, $2\pi \times 5$ and $2\pi \times 10$MHz, respectively. The other parameters are the same as in Fig. 2.

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From Eq. 8(b) one can find that the steady-state value of ${\rho _{13}}(t)$ is $- {\Omega _p}/{\Omega _c}$. The stable population localization ratio of state $|3 \rangle $ is inversely proportional to the control field strength ${\Omega _c}$, see Fig. 3(b).

Region Π, ${\Omega _c} = {\gamma _{12}}$, the analytical solutions of Eqs. (7) can be given by

In this region, ${\Omega _c} = \frac{{{\gamma _2}}}{2}$, populations on level $|3 \rangle $ can not be simulated back to level $|2 \rangle $ too, which leads to that the transient absorption for the probe laser ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will monotonically approach to its steady state after the switching on of the control laser. ${\Omega _c} = {\gamma _{12}}$, the steady-state time window is dependent on ${\gamma _{12}}$, i.e. on the atomic medium only. The time window is shorter than that in Region I.

Region III, ${\Omega _c} > {\gamma _{12}}$. In this region ${\Omega _1} = \sqrt {{\gamma _{12}}^2 - {\Omega _c}^2} $ is imaginary and we set ${\Omega _2} = \sqrt {{\Omega _c}^2 - {\gamma _{12}}^2} $, then Eqs. (8) can be rewritten as

The population oscillation frequency ${\Omega _c}$ between levels $|2 \rangle $ and $|3 \rangle $ is larger than half of the decay rate of level $|2 \rangle $ in this region. Therefore, populations on level $|3 \rangle $ can be simulated back to level $|2 \rangle $, and then back to level $|1 \rangle $. The two-photon coherence ${\rho _{13}}(t)$ between levels $|1 \rangle $ and $|3 \rangle $ will be oscillatorily damped to its steady-state value with time evolutions, see Eq. (10b) and the dashed blue and solid gray curves in Fig. 3(b). ${\rho _{13}}(t)$ leads to a damped oscillatory population transition between levels $|3 \rangle $ and $|1 \rangle $ as time evolves, and the transition frequency is $\sqrt {{\Omega _c}^2 - {\gamma _{12}}^2} /2$. This contributes that the transient absorption for the probe field ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will be oscillatorily damped to its steady-state value with a frequency of $\sqrt {{\Omega _c}^2 - {\gamma _{12}}^2} /2$. For instance, when the control field strength is ${\Omega _c} = 2\pi \times 5.0$MHz, the oscillation period for ${\mathop{\rm Im}\nolimits} [\chi (t)]$ lasts about $0.505$µs, and when ${\Omega _c} = 2\pi \times 10.0$MHz, the oscillation period reduces to $0.210$µs, see the dashed blue and solid gray curves in Fig. 3(a). Atoms populate on level $|3 \rangle $ can be stimulated back to level $|1 \rangle $ through a two-photon coherent transition, which leads to enhanced transient populations of level $|1 \rangle $ and transient negative absorption for the probe field. ${\Omega _c} > {\gamma _{12}}$, the steady-state time window of the atomic system is only dependent on the atomic medium (${\gamma _{12}}$) and keeps stable with a further increase of the control laser.

In order to further confirm the above explain, we display the time-dependent populations of levels $|1 \rangle $, $|2 \rangle $ and $|3 \rangle $. The probe laser couples the atomic transition between levels $|1 \rangle $ and $|2 \rangle $. When we consider a weak probe field with a strength of ${\Omega _p} = 2\pi \times 0.01$MHz, the population localization ratio of state $|2 \rangle $ is less than $0.1\%$ when the control laser is switched on at $t = 0$, which is difficult to be distinguished from the background noise. Weak probe field approximations can not be considered in the present scheme. Therefore, we can improve transient population localization ratios of sates $|2 \rangle $ and $|3 \rangle $ through enhancing probe laser strength in the simulation.

The transient populations of levels $|1 \rangle $, $|2 \rangle $ and $|3 \rangle $, and the transient absorption for the probe laser ${\mathop{\rm Im}\nolimits} [\chi (t)]$ varying with evolution time t are shown in Fig. 4. The strength of the probe field is enhanced into ${\Omega _p} = 2\pi \times 2.0$MHz. The control field strength is ${\Omega _c} = 2\pi \times 2.0$MHz for the upper two panels and ${\Omega _c} = 2\pi \times 10$MHz for the lower two panels, respectively. When ${\Omega _c}$ is weaker than the spontaneous decay rate ${\gamma _{12}} = 2\pi \times 3.05$MHz from level $|2 \rangle $ to level $|1 \rangle $, part populations of level $|1 \rangle $ can be excited onto level $|2 \rangle $ by absorbing one photon of the probe field. While atoms populate on level $|2 \rangle $ will take a Rabi oscillation after the application of the control laser and are immediately excited onto level $|3 \rangle $ because the one-photon coherence between levels $|2 \rangle $ and $|3 \rangle $ is imaginary [22], see Fig. 4(a1). This part of excited atoms are stabilized at level $|3 \rangle $ because the population oscillation frequency ${\Omega _c}$ between levels $|2 \rangle $ and $|3 \rangle $ is less than half of the decay rate of level $|2 \rangle $. When the atomic medium reaches its steady state, the stable population localization ratios of the three levels are ${\rho _{11}}(t \ge {T_w}) = 0.204$, ${\rho _{22}}(t \ge {T_w}) = 0.0$ and ${\rho _{33}}(t \ge {T_w}) = 0.796$, respectively. The monotone decreasing populations of level $|2 \rangle $ as time increases after the application of the control laser leads to that the transient absorption for the probe field ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will be monotonically damped to its steady-state value, see the solid blue curve in Fig. 4(b1).

 

Fig. 4. Transient populations of levels $|1 \rangle $, $|2 \rangle $ and $|3 \rangle $ (Figs. (a1) and (a2)) and absorption for the probe field Im [χ(t)] (Figs. (b1) and (b2)) as functions of evolution time t. The parameters are the same as in Fig. 2 except that the probe field strength is ${\Omega _p} = 2\pi \times 2$MHz. The control field strength is ${\Omega _c} = 2\pi \times 2$MHz for the upper two panels and ${\Omega _c} = 2\pi \times 10$MHz for the lower two panels, respectively. The solid gray curves in Figs. (b1) and (b2) are the dotted red and solid gray curves in Fig. 2.

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When the control field strength ${\Omega _c}$ becomes stronger than the spontaneous decay rate ${\gamma _{12}}$, the obtained results become complicated. About $23.1\%$ populations of state $|1 \rangle $ are excited onto state $|2 \rangle $ through a one-photon resonant transition when the probe laser is switched on at $t ={-} 3$µs, see the solid black and dashed blue curves in Fig. 4(a2). The control laser is turned on at $t = 0$, and most populations of level $|2 \rangle $ are immediately excited onto the higher level $|3 \rangle $ through absorbing one-photon of the strong control laser, rather than being stimulated back to the meta-stable state $|1 \rangle $ through the two side-band transition channels, $|{2d} \rangle \to |1 \rangle $ and $|{3d} \rangle \to |1 \rangle $ [21], see the dotted red curve in Fig. 4(a2). Meanwhile, about $10\%$ populations of state $|1 \rangle $ are excited onto state $|3 \rangle $ through a two-photon resonant transition. As time evolves, $75\%$ populations of state $|3 \rangle $ are excited back to state $|1 \rangle $ when 0.068µs$\le t \le $0.176µs, accompanying emitting one photon of the probe field and one photon of the control field. This part of returned populations, which include the “pre-excited” and “re-excited” populations of state $|1 \rangle $ before and after the application of the control laser at $t = 0$ lead to the enhanced transient population localization ratio of level $|1 \rangle $, and the transient negative absorption for the probe laser locating in the vicinity of $t = 0.114$µs in Fig. 4(b2).

With the weak probe field approximation one can find that the stable population localization ratio of level $|3 \rangle $ is inversely proportional to the control field strength ${\Omega _c}$. This conclusion is still valid here. ${\rho _{33}}(t \ge {T_w})$ for ${\Omega _c} = 2\pi \times 2.0$MHz is larger than that obtained with ${\Omega _c} = 2\pi \times 10$MHz, see the dotted red curves in Figs. 4(a1) and 4(a2).

The transient absorption for the probe field ${\mathop{\rm Im}\nolimits} [\chi (t)]$ (see the left panels) and the medium EIT transmission spectrum $T(t)$ (see the right panels) as functions of evolution time t and probe field detuning ${\Delta _p}$ are shown in Fig. 5. The probe and control lasers are switched on at $t ={-} 8.0$µs and left on. The probe laser strength is ${\Omega _p} = 2\pi \times 0.01$MHz. One main focus of the present work is on investigating time-dependent medium EIT transmission spectrum, therefore, the control laser strength is chosen as ${\Omega _c} = 2\pi \times 2.0$MHz. A control field with a strength stronger than the spontaneous decay rate ${\gamma _{12}} = 2\pi \times 3.05$MHz from level $|2 \rangle $ to level $|1 \rangle $ will contribute transient negative absorption for the probe laser, which is unfavourable to the detection of the transmission spectrum, see Eq. (6). The probe laser absorption for the atomic medium at probe frequency reduces to the steady-state valve (${\mathop{\rm Im}\nolimits} [\chi (t)] = 0$) as time increases, which leads to that the transmission spectrum for the probe laser can be found at ${\Delta _p} = 0$ around $t ={-} 7.0$µs, see the upper two panels of Fig. 5. It takes about $1\sim 2$µs for the atomic system to reach steady-state EIT [23].

 

Fig. 5. Transient absorption for the probe field Im [χ(t)] (Figs. (a) and (c)) and common logarithm of medium transmission spectrum $T(t)$ (Figs. (b) and (d)) as functions of evolution time t and probe field detuning ${\Delta _p}$ without (the upper panels) and with (the lower panels) a continuous MW field. The parameters are the same as in Fig. 2 except that the control field strength is ${\Omega _c} = 2\pi \times 2$MHz. The MW field strength is ${\Omega _m} = 2\pi \times 0.75$MHz.

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When a third continuous MW field with a frequency of ${\omega _m} = 14.233$GHz and a strength of ${\Omega _m} = 2\pi \times 0.75$MHz is switched on at $t = 0$, this field will couple the atomic transition between the Rydberg states $|3 \rangle $ and $|4 \rangle $ with a strength of ${\Omega _m}$. The transient absorption for the probe field ${\mathop{\rm Im}\nolimits} [\chi (t)]$ changes after the application of the MW field (at about $t = 0.5$µs, see Fig. 5(c)). As time increases, ${\mathop{\rm Im}\nolimits} [\chi (t)]$ oscillatorily decreases to the steady-state value locating at ${\Delta _p} ={\pm} {\Omega _m}$. The local MW field induces AT splitting of level $|3 \rangle $. The “dark state” created by the probe and control fields will become into a “bright state”, and the probe photons can be absorbed again on resonance [24]. The EIT spectrum locating at the resonant interaction ${\Delta _p} = 0$ will thus be split into two, see Fig. 5(d).

The stable frequency separation $\Delta f$ between the two EIT spectra is dependent on the local MW field strength ${\Omega _m}$. See Fig. 6, when ${\Omega _m} = 2\pi \times 0.25$, $2\pi \times 0.5$, $2\pi \times 1.0$ and $2\pi \times 2.0$MHz, the stable frequency splitting is $\Delta f = 2\pi \times 0.5$, $2\pi \times 1.0$, $2\pi \times 2.0$ and $2\pi \times 4.0$MHz, respectively. There shows a linear dependence between the stable spectrum-to-spectrum separation and the MW E-field strength, which can be used to sense continuous MW fields [25,26,27].

 

Fig. 6. Common logarithm of medium transmission spectrum $T(t)$ as functions of evolution time t and probe field detuning ${\Delta _p}$. The parameters are the same as in Fig. 2 except that the control field strength is ${\Omega _c} = 2\pi \times 2$MHz. The MW field strength is $2\pi \times 0.25$MHz for (a), $2\pi \times 0.5$MHz for (b), $2\pi \times 1.0$MHz for (c) and $2\pi \times 2.0$MHz for (d).

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The atomic medium needs some time to re-polarize after the application of the local MW field, which leads to the transient EIT signal locating in the vicinity of ${\Delta _p} = 0$ after the application of the MW field at $t = 0$, as can be found in Fig. 6. The response speed for the atomic medium to re-polarize depends on the local MW E-field strength, a stronger field leads to a shorter response time and a faster response speed. Therefore, the duration time for the transient EIT transmission signal locating at ${\Delta _p} = 0$ is reducing with increasing strength of the local MW field.

Besides sensing continuous MW fields, more importantly, the time-dependent EIT transmission signals of the coupled atomic system can also be used to detect pulsed MW fields. In the present work, the time dependent electric field of the local MW pulse is defined as ${\Omega _m}(t) = {\Omega _{m0}}\sin {({\pi t/{T_m}} )^2}\sin ({{\omega_m}t} )$ [28]. Here ${\Omega _{m0}}$ is the peak strength, ${T_m} = 16.0$µs is the duration and ${\omega _m} = 14.233$GHz is the frequency of the MW pulse. The probe and control fields are turned on at $t ={-} 16$µs and left on. The pulsed MW field is switched on at $t = 0$, the system has reached the steady state then. For simplicity, we just consider the influence of the wave-packet envelope on the medium EIT transmission spectrum [29,30,31].

When a pulsed MW field with a peak strength of ${\Omega _{m0}} = 2\pi \times 1.5$MHz is turned on at $t = 0$, the local MW E-filed gradually increases to its peak strength. Compared with a continuous MW E-field with a strength of ${\Omega _m} = 2\pi \times 1.5$MHz, the atomic medium needs a longer response time to re-polarize, which leads to that the splitting of the transient probe field absorption ${\mathop{\rm Im}\nolimits} [\chi (t)]$ and the EIT transmission spectrum $T(t)$ starts around $t = 1.7$µs after the application of the local pulsed MW field, see Fig. 7. The frequency splitting between the transmission spectra reaches its maximum in the vicinity of $t = 9$µs, as can be found in Fig. 7(b). Compared with the peak strength of the MW pulse which locates at $t = 8$µs, there shows a time delay of $1$µs. The largest splitting between the EIT spectra is sensitive to the peak strength of the MW pulse ${\Omega _{m0}}$. For instance, when ${\Omega _{m0}} = 2\pi \times 0.25$, $2\pi \times 0.5$, $2\pi \times 1.0$ and $2\pi \times 2.0$MHz, the largest frequency splitting is $2\pi \times 0.5$, $2\pi \times 1.0$, $2\pi \times 2.0$ and $2\pi \times 4.0$MHz, respectively, as can be found in Fig. 8. The local exciting MW pulse terminates at a time $t = 16$µs, and the AT splitting of level $|3 \rangle $ disappears. The probe photons can be absorbed again on resonance when $t \ge 16$µs. Therefore, it is possible to sense the strength of a pulsed MW field utilizing medium time-dependent EIT transmission spectra.

 

Fig. 7. Same as in Fig. 5 except a third local pulsed MW field with a duration of ${T_m} = 16.0$µs, a central frequency of ${\omega _m} = 14.233$GHz and a peak strength of ${\Omega _{m0}} = 2\pi \times 1.5$MHz is switched on at $t = 0$.

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Fig. 8. Same as in Fig. 6 except that the continuous MW field is changed into a pulsed MW field.

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4. Conclusion

In conclusion, we investigated the time-dependent medium EIT transmission signals of one coupled atomic system theoretically in the present work. Our results show that the local control laser acts as a switch. When the control laser strength ${\Omega _c}$ is weaker than the spontaneous decay rate ${\gamma _{12}}$ from level $|2 \rangle $ to level $|1 \rangle $, populations on level $|3 \rangle $ can not be simulated back to level $|1 \rangle $. The transient absorption for the probe laser ${\mathop{\rm Im}\nolimits} [\chi (t)]$ monotonically decreases to its steady state. And the steady-state time window ${T_w}$ of the atomic system is dependent on both the atomic medium and the local control laser. In contrast, when ${\Omega _c} > {\gamma _{12}}$, the transition channel from level $|3 \rangle $ to level $|1 \rangle $ is open. Part populations of level $|3 \rangle $ will be stimulated back to level $|1 \rangle $ through a two-photon resonant transition, which leads to that ${\mathop{\rm Im}\nolimits} [\chi (t)]$ will be oscillatorily damped to the steady-state value. The oscillation frequency is dependent on both ${\Omega _c}$ and ${\gamma _{12}}$, a stronger control field will lead to a higher oscillation frequency. Transient populations of level $|1 \rangle $ will be enhanced and transient negative absorption for the probe laser appears. When a third MW field is applied, the AT splitting of level $|3 \rangle $ will give rise to frequency splitting of the time-dependent EIT transmission spectrum, and the stable splitting distance shows a linear dependence on the continuous MW E-field strength. Furthermore, our simulation results show that the medium time-dependent EIT spectra are sensitive to the E-field strength of local MW pulse, and can be used to sense pulsed MW fields.