When a two-level system is driven by a strong coupling field of a resonant frequency, the populations of the states undergo coherent Rabi oscillations, as a result, there appears sidebands known as Autler–Townes (AT) doublet.^[1] The AT doublet has been demonstrated in both atomic^[2] and molecular systems,^[3–5] as well in quantum dots^[6–8] and in superconducting qubits.^[9,10] One important application of the AT effect is to use it to measure the absolute value of the molecular transition dipole moment^[3] and its internuclear distance dependence.^[11] On the other hand, the effects of inhomogeneous Doppler line broadening on the AT splitting have been investigated.^[12,13]

Traditionally, AT doublet was observed in three-level system, where the states are probed through a transition to or from a third level by probe absorption or fluorescence excitation spectroscopies. In this paper, we shall study the AT effect of high-lying state in a cascade four-level system. Our method is based on the generalized n-photon resonant -wave mixing with phase conjugation geometry, which is a general method to study the highly energy-state through monitoring the transition between the first excited state and the ground state .^[14] Since the transition is usually very strong, this spectroscopy can be very efficient. Previous, we have used six-wave mixing (SWM) to study the doubly excited autoionizing Rydberg states of Ba. Here, we shall used phase conjugate SWM to probe the AT doublet of high-lying state in a Doppler-broadened cascade four-level system. The critical role of Doppler broadening on the observation of AT splitting is investigated.

Let us consider a cascade four-level system (Fig. 1), where the states between and , and , and and are coupled by dipolar transitions with resonant frequencies , , and , and dipole moment matrix elements μ₁, μ₂, and μ₃, respectively. In the phase conjugate SWM, beams 3 and have the same frequency ω₃ and a small angle exists between them. Beam 1 with frequency ω₁ propagates along the opposite direction of beam 3, while beam 2 with frequency ω₂ can propagate either along the direction of beam 3 or along the opposite direction.

Fig. 1.

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	Fig. 1. Energy-level diagram for resonant SWM in a cascade four-level system.

Assuming that so that ω_i drives the transition from to , the simultaneous interactions of atoms with beams 1, 2 to 3 will induce atomic coherence between and through resonant three-photon transition. This three-photon coherence is then probed by beams and 2, and as a result an SWM signal of frequency ω₁ is generated almost opposite to the direction of beam .

Let the detunings be represented by , so after a canonical transformation the effective Hamiltonian is the following

The nonlinear polarization responsible for the SWM signal is proportional to the off-diagonal density matrix element ρ₁₀. We shall consider the solution in the first and second order in E₁ and E₂, respectively, while to all order in E₃. This can be done by solving the following equations through perturbation

First, we have . On the other hand, since beam 3 is a strong field, which couple the states and , we have to solve the following equations

Now, we consider the special case of SWM with phase conjugation geometry. Equation (6) contains the contribution from FWM, i.e.,

By subtracting this term, we obtain

We then consider the phase-matching condition. Let the complex incident laser fields be written as ( ) while , where and are the wave vectors of beams i and , respectively, then ( ) and , with ( ) and . In the following discussions we assume that , then the corresponding off-diagonal density matrix element for SWM is given by

Let us now consider the Doppler effect. The nonlinear polarization responsible for the SWM signal is given by averaging over the velocity distribution function , i.e.,

Here is obtained from Eq. (9) by using the Doppler-shift frequency detunings, i.e., ( ) and , N is the density of the atoms, while

In this section, we shall study how the AT effect affects the SWM spectrum in a Doppler-broadened system. For comparison, we start with the homogeneously broadened system. The resonant condition can obtain by solving the pole structure in Eq. (9), i.e.,

When we scan with , the SWM spectrum is given by

It shows that the resonances occur at and when . On the other hand, if we scan with , then the SWM spectrum becomes

Now, we investigate the AT splitting in the SWM spectrum in a Doppler-broadened system. We assume that beam 2 propagates along the direction of beam 3, so , and , while is approximated as because the angle between beams 3 and is typically very small. We express in Eq. (10) as a function of v explicitly, then

Therefore, we have

in Eq. (18) has four poles, i.e., , , and , and the integral in Eq. (20) consists mainly of the contributions of atoms with velocities , , and , which vary as we scan the incident laser frequencies . Here, v₁, v₂, and are the real parts of , , and , respectively. Considering the case that v₁, v₂, , then, since is a pole of second order, the resonance of appears when or .

We neglect the relaxation rates and define

Then, for the resonant condition , we have

On the other hand, for we have

Now, we study the spectra of scanning and separately. First, let us consider the spectrum of with . According to Eqs. (21) and (22), the resonances are given by and , respectively. This spectrum is the same as that in the homogeneously broadened system. Next, we consider the spectrum of with . Unlike the spectrum of , it is found that the resonance conditions become and , which depends on the ratios between the magnitudes of the wave vectors.

In this section we shall present numerical results of SWM spectra in a cascade four-level system with parameters . Let us start with the homogeneously broadened case. Figures 2(a) and 2(b) present the SWM intensity versus and , respectively, with , (solid curve), 5 (dashed curve), and 10 (dash–dotted curve), while for panel (a) and for panel (b). When is scanned, we observe three peaks with AT splitting appears at [see Fig. 2(b)]. By contrast, there is only single peak at when we scan [Fig. 2(a)]. To understand these spectra, we consider the limit , , for simplicity. For the SWM spectrum of , we have

Physically, for the spectrum of , the peaks at and at correspond to the one-photon transition between states and and two-photon transition from to the dressed states, respectively, leading to the fact that the signal of the former is much larger than that of the latter.

Fig. 2.

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	Fig. 2. (color online) SWM intensity versus (a) and (b) in a homogeneously broadened system with parameters and (solid curve), 5 (dashed curve), and 10 (dash–dotted curve), while for panel (a) and for panel (b).

Now, let us consider the SWM spectra in a Doppler-broadened system with . Figures 3(a) and 3(b) present the SWM intensity versus and , respectively, when , and (solid curve), 5 (dashed curve), and 10 (dash–dotted curve), while for panel (a) and for panel (b). First of all, these spectra are Doppler-free. Moreover, the SWM spectra of scanning [Fig. 3(b)] are similar to those in the homogeneously broadened case [Fig. 2(b)] in which there exists three peaks with AT splitting appears at . By contrast, unlike the corresponding case in the homogeneously broadened system the SWM spectra of [Fig. 3(a)] exhibits three peaks also with AT splitting appears at

Physically, the peak at is derived from the condition , while the AT splitting from the condition . Since there are of the same order, leading to the simultaneous appearance of three peaks in both the SWM spectra of and .

Fig. 3.

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	Fig. 3. (color online) SWM intensity versus (a) and (b) in a Doppler-broadened system with when and (solid curve), 5 (dashed curve), and 10 (dash–dotted curve), while for panel (a) and for panel (b).

Now, we consider another Doppler-broadened case with = 0.4. figures 4(a) and 4(b) present the SWM intensity versus and , respectively, when and (solid curve), 50 (dashed curve), and 100 (dash–dotted curve), while for panel (a) and for panel (b). Unlike the previous cases, the SWM spectrum is extremely broad. Physically, the broadening is due the polarization interference between atoms of different velocities, as we shall discuss in the next section. This can be verified through artificially ignoring the polarization interference by calculating

The dotted curves in Figs. 4(a) and 4(b) present the results with , which show again very narrow peaks, indicating that the broadening of the SWM spectrum is due to the polarization interference.

Fig. 4.

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	Fig. 4. (color online) SWM intensity versus (a) and (b) in a Doppler-broadened system with when and (solid curve), 50 (dashed curve), and 100 (dash–dotted curve), while for panel (a) and for panel (b). The dotted curves in panels (a) and (b) are the results for when the polarization interference are ignored.

Finally, let us consider the case with and . Figures 5(a) and 5(b) presents the SWM intensity versus and , respectively, when and (solid curve), 10 (dashed curve), and 20 (dash–dotted curve), while for panel (a) and for panel (b). Here, we observe a quite different SWM spectrum with only two narrow peaks with the central peak disappeared. These spectra are similar to the conventional AT spectrum in the three-level system.

Fig. 5.

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	Fig. 5. (color online) SWM intensity versus (a) and (b) in a Doppler-broadened system with and when and (solid curve), 10 (dashed curve), and 20 (dash–dotted curve), while for panel (a) and for panel (b).

In this section we shall study how the polarization interference affects the phase conjugate SWM spectroscopy in a Doppler-broadened system. Let us compare the polarizations corresponding to the linear absorption (L), FWM and SWM in a Doppler-broadened system, which, from the perturbation theory, are given by

We start with the linear absorption, where atoms with velocity v interact the field G₁ at time . Since , we have in Eq. (23) given by , i.e., atoms with velocity v will acquire a relative phase in a time interval τ, causing dephasing between atoms of different velocities. More specifically, after integrating over the velocity distribution function we obtain a term . In other words, atoms excited during the time interval will contribute to the linear absorption, leading to the Doppler broadening of the line shape.

Now, we turn to the FWM with incident beams given by and , where and . As is well known, FWM is intrinsically Doppler-free when the incident beams have narrow linewidth, because if beam 1 with frequency ω₁ is within the Doppler profile of the transition from to , then only atoms in a specific velocity group will contribute to the signal. However, the polarization interference can cause the FWM spectrum extremely broad. Specifically, atoms with velocity v interact with the fields G₁, G₂, and at times , , and , so the atomic polarizations acquire phases , , and , respectively, at each interaction. As a result, we have . Physically, there is dephasing between atoms of different velocities after the first field G₁. However, the subsequent interactions with fields G₂ and can induce rephasing process, and then, cause constructive interference between atoms of different velocities if . In this case, the FWM signal originates mainly from the moments when the complete rephasing occurs, i.e., . In other words, we can obtain a Doppler free FWM spectrum when . Otherwise, the randomization of the phases of the atomic polarizations will cause additional broadening of the spectrum line.^[15–17]

Finally, we turn to the SWM. Atoms with velocity v interact with the fields G₁, G₂, G₃, , and at times , , , , and so the atomic polarizations acquire phases , , , , and , respectively, at each interaction. As a result, we have . The complete rephasing occurs when , which can be satisfied if . In other words, the SWM spectrum is Doppler free only when , otherwise, the SWM spectrum becomes extremely broad.

Now, we investigate how the polarization interference affects the phase conjugate SWM spectrum. According to Eq. (8) the contribution to SWM signal consists of two components, i.e., an SWM term shown in Eq. (6) and an FWM term shown in Eq. (7), corresponding to the AT splitting and the central peak in the SWM spectrum, respectively. The polarization interference can have different effects on these two terms according to the values of ζ₂ and ζ₃. Let us first consider the case when . In this case the polarizations of both the FWM term and the SWM term can be rephasing completely, leading to the appearance of three resonant peaks with Doppler-free linewidth in the SWM spectrum, as shown in Fig. 3. Now let consider the case of . If simultaneously, then the phases of the polarizations of both the FWM term and the SWM term are randomized, so we obtain a extremely broadening of the SWM spectrum [see Fig. 4]. On the other hand, if then since the SWM term has narrow linewidth while the FWM term is so broad that its peak value is greatly reduced, as a result, the SWM spectrum exhibits only two narrow peaks corresponding to the AT doublet.

In this paper, we propose an AT spectroscopy based on phase conjugate SWM to detect AT doublet of high-lying state in a Doppler broadened cascade four-level system. This is important because it can be used to measure the dipole moment between the transition of highly excited states. It is worth mentioning that AT splitting have been observed previously by multi-wave mixing. For example, AT splitting of FWM^[18] and SWM^[19] have been demonstrated in electromagnetically induced transparency atomic media. However, in most previous works AT doublet was observed in low-lying state. By contrast, our work is concentrated on the AT doublet of high-lying states. The advantages of our method are the following. Compared to the method of detecting fluorescence from excited states, our scheme is still realizable even when state or has long radiative lifetimes so that direct detection of the fluorescence is difficult. This is because transitions between these states are driven by lasers, and so long as to is a strongly coupled transition, the wave-mixing signal can be emitted via this strong transition.

We also discussed the effect of Doppler broadening on the SWM spectrum from a time-domain viewpoint. Although this subject has been discussed in Refs. [15] and [16], here we concentrate on the physics underlying. Specifically, due the atomic motion, the atomic polarizations acquire different phases for atoms with different velocities as time evolves. The Doppler free SWM spectrum can be obtained only when the atomic polarization can be rephasing again at certain time after the interactions of all the incident fields.