Optical coherence arising from atom-field interaction in the field of quantum optics results in considerable interesting and counterintuitive phenomena, such as coherent population trapping,^[1] enhancement of Kerr nonlinearity,^[2] lasing without inversion,^[3] laser cooling of atoms,^[4] electromagnetically induced absorption.^[5] Typically, as a significant quantum interference effect with wide applications in quantum information processing and quantum nonlinear optics, electromagnetically induced transparency (EIT),^[6–7] where a weak probe field can travel through a quantum optical medium without attenuation when another strong field is simultaneously applied to the system, has been extensively studied in various physical systems.^[8–12] One of the simplest configurations is a Λ-type three-level structure involving two lower states coupled to a single excited state by two classical fields, as seen in the first EIT observation.^[13] Beyond steady-state analysis, time-dependent EIT is studied via the Schrödinger–Maxwell formalism.^[14] In practice, EIT has also been an important foundation for many useful devices. For example, a low-loss all-optical microdisk switch based on EIT and quantum Zeno effect^[15] has been demonstrated.^[16]

Previous study showed that two nearly resonant modes sharing a common reservoir can yield a quantum destructive interference between the modes, which is so-called Fano interference.^[17] Using the analysis of decaying-dressed states in the three-level system, Anisimov and colleagues demonstrated that the nature of EIT dip is the result of suppression of absorption induced by Fano interference and EIT can occur even at very small control-field amplitude in terms of the system’s decay rates.^[18] However, as the control intensity increases, two dressed states try to separate and Fano interference is greatly weakened until it disappears completely. And in the absence of Fano interference, a dip in the probe absorption line is just due to a doublet structure, which is known as Autler–Townes splitting (ATS).^[19] Because a dip presented in the absorption profile displays a seemingly identical feature for EIT and ATS, to discriminate EIT from ATS is a very important issue in quantum optical theory and experiment.^[20–23]

A significant breakthrough on objective test of EIT and ATS has been proposed by Anisimov and colleagues again, based on the aforementioned decaying-dressedstate formalism and Akaike’s information criterion.^[24] Subsequently, a corresponding experimental observation is carried out in a coherently driven cold Λ-type atomic system, which shows an excellent agreement with that proposal.^[25] Moreover, an investigation on dressed-state realization of the transition from EIT to ATS is demonstrated in a three-level superconducting quantum circuit,^[26] which may be beneficial to analysis of some relevant phenomena described by dressed-state model.

In this paper, we investigate EIT and ATS in a five-level system which can be decomposed into a four-level Y-type subsystem and a three-level Λ-type subsystem. Here in line with the previous EIT and ATS criteria proposed in the three-level system, we show an intersection of EIT with ATS in the four-level subsystem. By setting the appropriate traveling frequency of the strong driving and signal fields, we further show that an EIT dip is sandwiched between two ATS dips in the five-level system, which can be clearly described by the combination of dressed-state theory and Fano interference.

Let us consider a closed five-level atomic system interacting with four semi-classical laser fields, as depicted in Fig. 1. A weak probe field with Rabi frequency Ω _p drives the transition while three strong control, driving and signal fields with Rabi frequencies Ω _c , Ω _d and Ω _s couple the transitions , , and , respectively. For such a matter-field interaction configuration, the system’s Hamiltonian is given by (ħ = 1)

Fig. 1

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	Fig. 1 Schematic diagram of a five-level atomic system interacting with four classical optical fields. A weak probe field with Rabi frequency Ω p drives the transition, at the same time, three strong control, driving and signal fields with Rabi frequencies Ω c , Ω d and Ω s couple the transitions , , , respectively.

The time evolution of the atomic system can be described by a standard density matrix formalism

In the above equations we have inserted phenomenologically the relaxation rate for the level as well as the dephasing rate γ _ij for the transition and i, j ∈ {1,2,3,4,5}). The remaining equations are calculated by .

In the weak probe-field regime, we just need to solve these equations to first-order perturbation expansion in Ω _p and give steady-state expression of the off-diagonal density matrix element used to describe the system’s linear response to the probe field. Assuming that the system population is initially in the ground state (i.e. ), we have

It is well known that optical linear absorption is quantified by the imaginary part of the first-order matrix element , which can be employed to characterize ATS and EIT.

On the basis of Eq. (5) and previous three-level EIT and ATS criterion obtained in Refs. [18, 24], we now study the detailed characters of linear response to the probe field by the manner of numerical analysis and then show the mixture of EIT and ATS in the five-level system.

First, dropping the control field Ω _c , this five-level system reduces to a four-level Y-type system, which can be regarded as being consisted of two ladder-shaped (LS3) and (LS4) subsystems. It is clear from Refs. [18, 24] that for the LS3 (LS4) subsystem EIT occurs in the weak field condition while ATS appears in the strong field limit . Figure 2 plots the response profiles of linear absorption Im as a function of the probe detuning Δ _p for the Y-type system (red solid line) and its subsystems (black dashed-dotted line for LS3 and blue dashed line for LS4). The parameters given in Fig. 2 ensure that LS4 and LS3 subsystems are in the EIT and ATS regimes respectively, except for the subgraph Fig. 2(c) sketched in the both EIT cases. For the same decay rates γ₃₁ = γ₄₁ in the subsystems LS3 and LS4, we find that ATS displays stronger suppression than EIT on the probe-field absorption, as seen in Fig. 2(a). Besides, in the light of the almost identical red and blue lines, we point out that in the Y-type system EIT effect is disturbed and concealed by ATS. In Fig. 2(b), we see the EIT dip becomes deep as γ₄₁ decreases, and as a result the four-level absorption profile, where a narrow steep valley at Δ _p = 0 mainly comes from the EIT effect and the remaining section of the transparency window and two peaks characterize ATS, displays an intersection of EIT with ATS. For the close decay rates γ₃₁ (γ₄₁) and γ₂₁, it is exhibited from Fig. 2(c) that the four-level system can restrain the probe-field absorption more efficiently in contrast with the three-level system, which is of especial significance in multi-wave mixing where EIT can be used to reduce the linear absorption of generated field and enhance the nonlinear susceptibility.^[27–28] By setting the nonzero detuning of the driving field, two separate ATS and EIT dips for the Y-type system are shown in Fig. 2(d).

Fig. 2

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Fig. 2 (Color online) Imaginary part of the matrix element for the Y-type system (red solid line) and its subsystems (black dashed-dotted line for LS3 and blue dashed line for LS4) as a function of the probe field detuning Δ p . The parameters are as follows:(a) Δ d = Δ s = 0, Ω d = 2γ21, Ω s = 0.8γ21, γ31 = γ41 = 0.1γ21, (b) Δ d = Δ s = 0, Ω d = 2γ21, Ω s = 0.8γ21, γ31 = 100γ41 = 0.1γ21, (c) Δ d = Δ s = 0, Ω d = Ω s = 0.7γ21, γ31 = γ41 = 0.2γ21, (d) Δ s = 0, Δ d = γ21, Ω d = 2γ21, Ω s = 0.8γ21, γ31 = 100γ41 = 0.1γ21.

In terms of the above study, we next discuss the response spectrum of the five-level system, which can be deemed to be a composite combination of the mentioned four-level Y-type subsystem (FYTS) and a three-level Λ-type subsystem (TΛTS).

Figure 3 shows the absorption curves Im versus the probe detuning Δ _p for the five-level system (red solid line) and two subsystems (black dashed-dotted line for FYTS and blue dashed line for TΛTS). It is obvious that three dips presented in the five-level system result from two dips in the FYTS and one dip in the TΛTS, as demonstrated in Fig. 3(a) where FYTS is in the ATS regime and TΛTS is in the EIT situation. More concisely, an EIT dip is sandwiched between two ATS dips, i.e., a mixture of multi-dip structure from EIT and ATS in the absorption line, which can be clearly explained by the combination of dressed-state theory and Fano interference. For example, three absorption peaks in the black profile are the result of occurrence of three dressed states (one primary and two secondary) induced by driving and signal fields and the gaps between two adjacent peaks lead to two dips. Meanwhile, the dip in the blue line is due to the interference of decaying-dressed states arising from the hybrid of dressed states and reservoirs and in essence such interference can be attributed to Fano interference. In Fig. 3(b), we also show three ATS dips for the five-level system in which the divided two subsystems are both the ATS regimes.

Fig. 3

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	Fig. 3 (Color online) Imaginary part of the matrix element for the five-level system (red solid line) and two subsystems (black dashed-dotted line for FYTS and blue dashed line for TΛTS) as a function of the probe field detuning Δ p . (a) Ω c = 0.8γ21, (b) Ω c = 2.5γ21. The other parameters are as follows: Δ c = 0, Δ s = −Δ d = γ21, Ω d = Ω s = 2γ21, γ31 = γ41 = 100γ51 = 0.1γ21.

In summary, we have studied EIT and ATS in a five-level atomic system resolved into a four-level Y-type subsystem and a three-level Λ-type subsystem. The general solution of the off-diagonal steady-state matrix element responsible for the linear absorption of probe field has been obtained via the standard density matrix approach. Based on the three-level criterion on EIT and ATS, we have demonstrated an intersection of EIT with ATS in the four-level system divided further into two three-level systems. Subsequently, we have shown that an EIT dip can be sandwiched between two ATS dips (i.e., a mixture of multi-dip structure from EIT and ATS) in the five-level system. This mixture can be elucidated well by the combination of dressed-state theory and Fano interference. Our investigation on multi-dip structure of EIT and ATS may be useful in quantum nonlinear optics, for instance four-wave mixing.