Quantum optics studies the interactions of light and matter, no matter whether in the semiclassical method or quantum theory. An optical cavity can be used to enhance the interaction strength of matter and light, modify the rate of spontaneous radiation, and realize the reversible exchange of energy in the cavity.^[1–3] It has also been used in diverse areas of research and applications such as nonlinear optical devices, quantum entanglement, and optical frequency metrology.^[4–6] Extensive application of optical cavities and the studies on quantum optics and quantum electrodynamics have developed into a new research field of cavity quantum electrodynamics (cavity-QED). The interaction of two-level atoms and an optical cavity is the basic physical situation about cavity-QED first introduced by Jaynes and Cummings.^[7] It is interesting to extend studies of cavity-QED to the interactions of cavity modes and multilevel atoms,^[8,9] such as the intracavity electromagnetically induced transparency (EIT) in three-level or four-level atoms confined in an optical cavity, Fano-type interference in a three-level V-type atom–cavity system, and optical bistability in four-level atoms coupled with an optical cavity.^[10–16]

Cavity-QED with atomic systems has been an active area of research during the last few decades. Especially the optical switch based on cavity-EIT is still a hot topic today.^[17–23] Many other interesting research projects are also based on cavity-QED. For example, the internal states of an atom represent one qubit and the quantum state of the field inside the cavity represents the other in recent cavity-QED to realize implementation of a quantum phase gate,^[24–27] and also the quantum networks.^[28]

Here, we report a theoretical analysis of some single atoms trapped in an optical cavity, which has a closed four-level loop with three free-space lasers and an applied probe laser driving the cavity mode. This system displays varying EIT splitting and cavity normal modes shifting. We analyze the intensity ratio and phase shift of the output intracavity probe field by means of input-output theory to study how these applied free-space fields affect the double-EIT peaks and the shifting of the normal modes splitting sidebands. As a result, we find a way to realize the gradual transfer between the single-EIT peak and double-EIT peaks. We believe this work will be useful for other researchers to investigate light controlling light in a cavity and furthermore to realize much subtler all-optical switch and quantum phase gate via coupled cavity–atom systems.

This paper is organized as follows. In Section 2, we introduce our model and discuss the transmission and reflection theoretically. In Section 3, we present our numerical calculation results and make a more specific analysis. The conclusions are drawn in Section 4.

The cavity–atom system is shown in Fig. 1. Some single four-level atoms are located in a single mode cavity. A probe laser injected externally from the left side of the cavity mirror drives the cavity mode and has a frequency detuning from the atomic transition – by . The cavity mode is coupled to the atomic transition – with frequency detuning . The classical control lasers injected into the cavity perpendicularly to the propagation of the probe laser drive the atomic transitions – and – with Rabi frequencies , and frequency detunings , , respectively, as depicted in Fig. 1. Levels and are coupled by a microwave laser injected by a microwave horn with Rabi frequency and frequency detuning . Two detectors measure the transmitted and reflected intracavity probe signals, respectively. The level configuration is also shown in Fig. 1. We choose () transitions of Rb. Three lower states , , and correspond to 5SRb F = 1 (m = 1), F = 1 (m = −1), and F = 2 (m = −1), respectively.

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of the cavity–atom system and the level structure of atoms. The probe laser propagates along the axis of the cavity and the control lasers are injected into the cavity from outside.

The Hamiltonian for this coupled cavity–atom system under three-photon resonance condition () reads^[16]

For simplicity, we consider an asymmetric Fabry–Perot cavity with field decay rates and through the left and right mirrors, respectively. The operator () is the field operator of an electromagnetic mode impinging from the left (right) side, while () describes the outgoing field to the left (right) side. We can obtain^[29]

In this section, we present some numerical results to analyze the influence of the coherent control lasers on the cavity–atom system. We calculate the optical intensity ratio , and the optical phase shift , of the itracavity probe field versus probe field detuning . For simplicity, we assume that the cavity mode is resonance with the transition – and all results are based on full-resonance condition (). We show the results in Figs. 2–5. There are four eigenstates i.e., two dressed intracavity dark states (central peaks) and two normal modes (broader sidebands).^[15] When , this cavity–atom system will present the typical intracavity single EIT, namely, a dark state formed with states and (). However, this dark state can be manipulated by to create double EIT (dressed dark states). The dressed dark states consist of coherent superposition of the state and the intracavity dark state. The two normal modes come from the vacuum Rabi splitting. They are the combination of the intracavity bright state and the excited state . When there are no control lasers, the splitting of the two normal modes sidebands is . While in our system, the splitting is . The second term in the above formula comes from the nonzero closed interaction phase and the third control laser . It is different from Ref. [15], we care about how and another factor φ affect these dressed intracavity dark states and the coupled cavity–atom normal modes.

Fig. 2.

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	Fig. 2. (color online) (a) The intensity ratio of the transmitted probe field versus the probe frequency detuning in different closed interaction phases. Panels (b) and (c) show the results of φ = 0, , π and , , , respectively.

Fig. 3.

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	Fig. 3. (color online) The intensity ratio of the transmitted probe field versus the probe frequency detuning for (a) φ = 0, (b) , (c) . The parameters used here are , N = 2, , , , , and , , .

Fig. 4.

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	Fig. 4. (color online) Reflected spectra of intracavity probe field (a) with different closed interaction phases when and (b) with different control Rabi frequencies when .

Fig. 5.

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	Fig. 5. (color online) The phase shift of (a), (c), (e) the transmitted probe field and (b), (d), (f) the reflected probe field versus probe frequency detuning : (a), (b) , ; (c), (d) , ; (e), (f) , . The other parameters are , N = 2, , , φ = 0, , and π.

Figure 2 shows the transmission spectra in different closed interaction phases φ when and the other parameters are , N = 2, , , and . From Fig. 2(a), we can see that the transmission spectra are symmetrical about plane . This is shown more obviously in Figs. 2(b) and 2(c). When the closed interaction phase is changed discretely from 0 to π in Fig. 2(b), we find that the EIT peak changes from one in the position of negative frequency detuning to the symmetrical splitting double-EIT peaks and then to one peak again while at positive frequency detuning. However, things are reversed when φ is altered from π to 2π discontinuously in Fig. 2(c). Indeed, this phenomenon is not only presented in the intracavity dark states, but also for the splitting normal modes. As we can see, the position of the normal mode sidebands will also change with varying closed interaction phase φ, and is also the plane of symmetry. Therefore, it is only necessary for us to explore the transmitted probe field with φ changing from 0 to π. Here we obtain the eigenvalues of double-EIT as

Figure 3 depicts the transmitted spectra of the intracavity probe field with varying from 0 to discretely at φ = 0, , and π, respectively. As we can see, when (the black dotted lines in Fig. 3), the shape of transmission is always symmetrical about line no matter what the closed interaction phase φ is. This can be explained by a simple four-level atom–cavity model. We all know that this system can degenerate to a three-level -type cavity-EIT configuration when ..^[11] We have also verified it in our analysis formulas. When , the central EIT peak is split into two peaks symmetrically and the peaks separation is approximately equal to .^[15] However, when we apply another coherent control laser , namely, and , it will destroy the manipulation of on the coherent superposition of cavity-EIT (intracavity dark state) and the atomic state and also on the two normal modes. For example, in Fig. 3(a), the spectrum features four symmetrical peaks when (black dotted line), with increasing to 0.5, the symmetrical splitting of the dark state is broken. When is equal to (), the double EIT peaks will become a single peak at the position of frequency detuning , and the principle is similar to the situation in Fig. 3(c) except that the single peak will be at the position of frequency detuning . In Fig. 3(b), the central peaks are a little different from those in Figs. 3(a) and 3(c). As it is shown to us, whatever the control Rabi frequency is, there will always be two peaks symmetrical about line . This can be explained by Eqs. (11) and (12) with .

As we have mentioned before, the coherent control laser will also influence the normal modes. In Fig. 2(b), we can see that the normal modes shift with varying φ and the two normal modes are symmetrical about line only when . In Figs. 3(a)–3(c), we can also find the shifting of the normal modes with varying intensity of control Rabi frequency . The peak shifting of the normal modes approximately satisfies the following expression:

We also analyze the reflected intracavity probe field, as shown in Fig. 4, on which the influence of the control laser is the same as the transmitted intracavity probe field. For simplicity but to be clear, we only show the reflected intensity ratio with different φ when and with different control Rabi frequencies when . The parameters used in Figs. 4(a) and 4(b) are the same as those used in Figs. 2 and 3, respectively.

Besides the intensity ratio of the intracavity probe field, we care about the phase shift of the intracavity probe field as well. Therefore, in Figs. 5 and 6, we show how the closed interaction phase φ and the strength of the coherent control laser affect the angular dispersions of the transmitted and reflected intracavity probe fields. As depicted, the phase shift of both transmitted and reflected intracavity probe fields can be controlled by adjusting the closed interaction phase φ. Especially for the reflected spectra in Fig. 5(b), they present sharp abnormal angular dispersions at the EIT peaks, which means that we can realize large cross phase modulation (XPM) () with low intensity control laser.^[31] Moreover, the reflected spectra have sharper slopes comparing with the transmitted spectra from Figs. 5(a) and 5(b). To explain the difference between the reflected and transmitted dispersions, we draw the phase curves at three situations: , , , where the strength of the probe field injected into the cavity increases with increasing. As shown in Figs. 5(a), 5(c), and 5(e), the dispersions of the transmitted spectra at EIT resonance keep almost the same at the three situations. This shows that the intracavity field distribution keeps the same with different . However, the phase of the reflected spectra is decided by the difference between the distribution of intracavity field and that of atomic absorption (Eq. (10)) and on the other hand, the larger abnormal dispersion is obtained at EIT windows^[31] when the interaction strength is enlarged. So the dispersion of the reflected field increases with increasing and even has a very sharp slope in contrast to the transmitted field. According to these analyses, our system with large is advantageous to obtaining a large phase shift of the reflected spectra.

Fig. 6.

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	Fig. 6. (color online) The phase shift of the reflected probe field versus probe frequency detuning for (a) φ = 0 and (b) . The parameters are , N = 2, , , , , , , .

When we make the closed interaction phase φ = 0, the asymmetry on the phase shift of the double-EIT peaks is derived from free space control laser (control Rabi frequency ) as shown in Fig. 6(a). While , the angular dispersion at double EIT peaks is always symmetrical as shown in Fig. 6(b). That is reasonable, since the real part of the susceptibility can be simplified as 15 namely, there is always when .

It is clear that the evolution of the double intracavity EIT can be manipulated flexibly by either the intensity or the closed interaction phase of the control lasers.

We study a cavity–atom system which consisted of some single four-level atoms coupled with an asymmetrical cavity. By applying the third coherent control laser , this system contains a closed interaction phase φ that can modify the transmitted and reflected intracavity probe fields, namely, besides the strength of the control Rabi frequency, there is also another factor that can influence the EIT peaks and the normal modes. We specifically analyze the changing of this kind of double-EIT peaks and the shifting of the cavity normal modes depending on the closed interaction phase and strength of the control lasers. Therefore, it enables us to control the cavity EIT to realize a gradual transfer between single intracavity EIT and double intracavity EIT via changing either the signal intensity or the closed interaction phase. The large XPM () in our system could be useful for quantum phase gate and all-optical temporal differentiation.^[32]