1. Introduction

It is crucial to control optical absorption, especially to realize extreme absorption for the development of photonic devices [1–3], and in the process of optical quantum information, e.g., quantum communication and quantum computation [4,5]. Electromagnetically induced transparency (EIT) [6], which happens when the destructive interference of two transition paths occurs, has been used for manipulating photon absorption in optical storage, quantum phase gate, optical switching, and optical probe [7–10]. Generally, the EIT system is with a strong coupling laser and the strong nonlinearity, however, the optical absorption efficiency of which is less than $100\%$. Recently, studies show that the coherent perfect absorption (CPA) (the absorption efficiency is $100\%$) has captured significant attention in light-matter interaction, and can be realized in solid-state Fabry-Perot devices [11,12], metamaterials [13,14] and cavity quantum electrodynamics (CQED) [15,16]. Besides the manipulation on photon absorption, CPA has potential applications in photon detection and optical sensor [17,18]. The physical origin behind CPA in CQED system is the interference between transmitted and reflected fields at the cavity interfaces. And to achieve CPA, the interaction between input fields and media must satisfy time-reversal symmetry of radiation [19].

Previous studies of CPA are in the linear classical regime or two-level nonlinear quantum regime [20,21], where the CPA mode is generally non-tunable. For example, in a two-level atom-cavity system [21], dual-frequency CPA modes at $\Delta _p=\pm 4.5\Gamma$ are obtained for a given input intensity $|a_{in}|^2\approx 55$ ($\Delta _p$ is the frequency detuning of probe field, and $\Gamma$ is the decay rate of atomic excited level). The frequency of CPA is variable only with some certain intensities of the probe fields. For practical applications, the frequency of CPA should be tunable for a given signal. Therefore, the susceptibility of the media in a proposed scheme must be controllable to manipulate the interference at the interfaces. One of the methods is introducing EIT-type quantum interference which is induced by a strong coherent laser coupling the excited state of a two-level atom with another hyperfine level of ground state of the atom.

Here, we propose a scheme where CPA modes are manipulated in a three-level atom-cavity nonlinear quantum regime. In the scheme, some cold atoms are trapped in a single-mode optical cavity which is driven by two coherent probe lasers from two ends. A control laser is coupled to the atoms causing the destructive interference between two transition paths. And the control laser is used to manipulate the interaction between input fields and atoms. With the EIT-type quantum interference induced by the control laser, CPA mode does not occur at resonant frequency of the probe fields, instead, four CPA modes occur at non-resonant frequencies for a given input intensity of the probe fields. In addition, at a given frequency of the probe fields, CPA can be transferred to coherent non-perfect absorption (CNPA) by the control laser, and vise versa. The scheme provides the potential application of controllable perfect absorber in complicated system.

In Sec. 2, we present the theoretical model, and we obtain the solution of output fields by master equation and input-output relation. We also analyze the CPA condition and the manipulation on it by the control laser. In Sec. 3, we firstly discuss the intensity and frequency of input probe fields for CPA. And then, we discuss the output-input relation with different relative phases of input probe fields. Finally, we discuss the manipulation on CPA and CNPA by the control laser. And the conclusion is presented in Sec. 4.

2. Theoretical model and analysis

The schematic diagram of the system is shown in Fig. 1. We consider a system consisting of some cold three-level atoms coupled to a two-sided optical cavity. Two coherent probe lasers $a_{in,l}$ and $a_{in,r}$ are injected into the cavity from two interfaces, and drive the atomic transition $|1\rangle \rightarrow |3\rangle$ with a frequency detuning $\Delta _p=\omega _p-\omega _{31}$. $\Delta _{ac}=\omega _c-\omega _{31}$ is the frequency detuning between the cavity mode and the atomic transition $|1\rangle \rightarrow |3\rangle$. Two detectors D$_1$ and D$_2$ are applied to measure the output fields $a_{out,l}$ and $a_{out,r}$ from two interfaces of the cavity. A control laser is coupled to the atomic transition $|2\rangle \rightarrow |3\rangle$ with a frequency detuning $\Delta _1=\omega _{1c}-\omega _{32}$. Here $\omega _c$ is the frequency of the cavity mode, $\omega _p$ and $\omega _{1c}$ are frequencies of the probe and control lasers, and $\omega _{mn}$ ($m,n=1,2,3$) is the frequency of corresponding atomic transition.

 

Fig. 1. (a) Schematic diagram of a two-sided cavity filling with some (b) three-level atoms.

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For accessible implementation, the atoms can be trapped in a vapor cell placed inside the optical cavity, and the hyperfine energy levels 5$S_{1/2}$ $F=1$, 5$S_{1/2}$ $F=2$ and 5$P_{3/2}$ of $^{87}$Rb can be selected as levels $|1\rangle$, $|2\rangle$, and $|3\rangle$, respectively. Correspondingly, a laser of 780 nm can be used as the probe and control lasers. At the same time, a Doppler-free configuration can be implemented by injecting the control laser to the cavity through two beam splitters (BS) with a small angle to the cavity axis.

The Hamiltonian of the system is [1],

In semiclassical approximation, we treat the expectation values of field operators as the corresponding fields, e.g., $\langle a\rangle =\alpha$ and $\langle a^{\dagger }\rangle =\alpha ^*$ [22]. According to the Liouville equation of motion with the decay process [23],

The intracavity field can be solved from the following differential equation,

According to Eq. (6), the input fields must be identical when this system is acted as a perfect absorber, i.e., $a_{in,l}=a_{in,r}$ and $\varphi =0$. Therefore, the intracavity field intensity under the condition of CPA ($\langle a_{out,l}\rangle =\langle a_{out,r}\rangle =0$) can be derived as,

The cavity-atom detuning $\Delta _{ac}$ is also obtained as a function of $\Delta _p$, $\Omega _1$ and $\kappa$, which is not presented here for its complicated form. At the same time, according to Eqs. (6) and (8), the relation between the intensity and frequency of the probe fields under the CPA condition can be derived as the following simple form, $|\alpha _{in}|^2=T |\alpha |^2$, where $T=\kappa \tau$ is the transmission coefficient of the cavity. It is clear that the formula is a quartic equation about $\Delta _p$, therefore, varying system parameters to match the CPA condition can lead to the manipulation on single-, dual-, three- and four-frequency CPA.

When $\Omega _1=0$, the scheme can be acted as a two-level CPA system [21]. While $\Omega _1\neq 0$, the coupling between the atomic levels $|2\rangle$ and $|3\rangle$ forms a destructive quantum interference between two absorption channels of the transition $|1\rangle \rightarrow |3\rangle$. And this will induce an interference control on CPA condition. In the following, we will study the manipulation on CPA from the decay rates ($\kappa$) of the cavity mirrors and the Rabi frequency ($\Omega _1$) of the control laser. Furthermore, we also study the output intensity controlled by the frequency, the intensity, and the phase of the input probe fields.

3. Numerical results and discussion

Figure 2 shows the frequency range of CPA. The parameters in this section are, $\gamma _{12}=0.001\Gamma$, $g\sqrt {N}=10\Gamma$, and $T=0.01$. And the cavity-atom detuning $\Delta _{ac}$ is obtained by CPA condition $\langle a_{out,l}\rangle =\langle a_{out,r}\rangle =0$. In Fig. 2(a), the output intensity can be zero at the resonant frequency only when the input probe fields are absent, which indicates that CPA cannot occur at the resonant frequency for EIT system. This is caused by zero absorption of the atomic media at the resonant frequency, which causes $n^{''}=0$ ($n^{''}$ is the imaginary part of refractive index). Together with the quartic equation of $|\alpha _{in}|^2$ versus $\Delta _p$, it accounts for the phenomenon shown in Fig. 2, where four-frequency CPA occurs at a specific input intensity in the three-level cavity-EIT system rather than the dual-frequency CPA in the two-level atom-cavity system [21]. While at non-resonant frequencies, the needed input intensity achieving CPA is stronger for smaller decay rate $\kappa$, which also forms the wider frequency range of CPA. This indicates that a cavity with the higher finesse leads to the stronger nonlinear excitation regime of CQED.

 

Fig. 2. The input intensity under CPA condition for (a) $\Omega _1=\Gamma$ with $\kappa =0.5\Gamma$ (dashed red line), $\kappa =\Gamma$ (solid green line), and $\kappa =1.5\Gamma$ (dot-dashed blue line), and (b) $\kappa =\Gamma$ with $\Omega _1=0.5\Gamma$ (dashed red line), $\Omega _1=\Gamma$ (solid green line), and $\Omega _1=1.5\Gamma$ (dot-dashed blue line).

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In Fig. 2(b), the frequency range of CPA has little difference for different $\Omega _1$. However, for a specific frequency of the probe fields, the intensity to realize CPA is usually different for different $\Omega _1$. For example, when $|\Delta _p|=\pm 5\Gamma$, the needed input intensities are $\left |\alpha _{in}\right |^2=125.5$ for $\Omega _1=0.5\Gamma$, $\left |\alpha _{in}\right |^2=152.9$ for $\Omega _1=\Gamma$, and $\left |\alpha _{in}\right |^2=165.8$ for $\Omega _1=1.5\Gamma$, respectively. This is on account of the manipulation on polariton states by the control laser. In the system, the first excited polariton states contain two bright polariton states $|\Psi _+\rangle =\frac {1}{\sqrt {2}}(\frac {1}{\sqrt {N}}\sum _{j=1}^{N}|1,\ldots 3_{j},\ldots 1\rangle |0_c\rangle +|1,\ldots 1,\ldots 1\rangle |1_c\rangle )$ and $|\Psi _-\rangle =\frac {1}{\sqrt {2}}(\frac {1}{\sqrt {N}}\sum _{j=1}^{N}|1,\ldots 3_{j},\ldots 1\rangle |0_c\rangle -|1,\ldots 1,\ldots 1\rangle |1_c\rangle )$, and a dark polariton state $|\Psi _d\rangle =\frac {1}{\sqrt {g^2N+\Omega _1^2}}(g\sum _{j=1}^{N}|1,\ldots 2_{j},\ldots 1\rangle |1_c\rangle -\Omega _1|1,\ldots 1,\ldots 1\rangle |0_c\rangle )$. Here $|1_c\rangle$ and $|0_c\rangle$ are one-photon and zero-photon states of the cavity mode. With $\Omega _1 \ll g\sqrt {N}$, the control laser can be treated perturbatively, and then the bright polariton state $|\Psi _+\rangle$ will be split into two dressed polariton states $|\Phi _+\rangle =\frac {1}{\sqrt {2}}(|\Psi _+\rangle +|2\rangle )$ and $|\Phi _-\rangle =\frac {1}{\sqrt {2}}(|\Psi _+\rangle -|2\rangle )$. The destructive interference between two excitation paths induces cavity EIT at the frequency of polariton resonance. Therefore, input probe fields are not absorbed by atoms. At other frequencies, the EIT-type quantum interference makes the output intensities from two interfaces dependent on the relative phase $\varphi$ and intensities of the probe fields.

In the following, we compare the difference of output intensities from two frequencies of the probe fields for $\varphi =0$, $\varphi =\pi /2$, and $\varphi =\pi$ in Fig. 3. The other parameters are $\Omega _1=\Gamma$, $\kappa =\Gamma$, $\Delta _{ac}=-6.9\Gamma$ for Fig. 3(a), and $\Delta _{ac}=-6.5\Gamma$ for Fig. 3(b). When $\varphi =2n\pi$, two output intensities $|\alpha _{out,r}|^2$ and $|\alpha _{out,l}|^2$ are identical as shown by the dot-dashed blue lines, and CPA is manifested at a given frequency and intensity of the probe fields because of the fully destructive interference between the transmission and the reflection of two input probe fields at two ends of the cavity. When $\varphi =(2n+1)\pi$, two output intensities $|\alpha _{out,r}|^2$ and $|\alpha _{out,l}|^2$ are identical and equal to the input intensity $|\alpha _{in}|^2$ as shown by the dotted black lines, the reason of which is that no light can be coupled into the cavity for the destructively interference of two input probe fields [21]. For other values of $\varphi$, two output intensities $|\alpha _{out,r}|^2$ and $|\alpha _{out,l}|^2$ are generally different, and CPA does not occur (see the dashed red and solid green lines).

 

Fig. 3. The output intensity versus input intensity for (a) $\Delta _p=7.2 \Gamma$ and (b) $\Delta _p=7 \Gamma$ with $\varphi =0$ (dot-dashed blue lines), $\varphi =\pi /2$ (dashed red lines for $|\alpha _{out,r}|^2$ and solid green lines $|\alpha _{out,l}|^2$), and $\varphi =\pi$ (dotted black lines).

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As discussed in Sec. 2, CPA occurs when $\varphi =0$ as shown by the dot-dashed blue lines in Fig. 3. However, for $\Delta _p$ above the threshold value of CPA frequency, the CQED system is driven in a linear excitation regime and CPA can be observed only with an extremely weak intensity of the probe fields. For example, in Fig. 3(a), $|\alpha _{out}|^2$ is linearly dependent on $|\alpha _{in}|^2$, and CPA occurs for $|\alpha _{in}|^2 \ll 5$ at $\Delta _p=7.2\Gamma$. It is clearly presented from Fig. 2 that the frequency range of CPA at these parameters is $\Delta _p=-7.1\Gamma \sim 7.1\Gamma$. Therefore, when $|\Delta _p|$ is decreased from the threshold value of CPA frequency, e.g., $\Delta _p=-7\Gamma$, $|\alpha _{out}|^2$ is nonlinearly dependent on $|\alpha _{in}|^2$ and the needed intensity for CPA is above $|\alpha _{in}|^2=15$ as shown in Fig. 3(b).

In spite of that, the CPA at a given frequency of the probe fields is adjustable. Although it indicates in Fig. 2(b) that the variation of $\Omega _1$ has little effect on the frequency range of CPA, changing $\Omega _1$ to control single-path absorption can realize the transfer between CNPA and CPA. In Fig. 4, the manipulation on output intensity by control laser is depicted. The parameters are $\kappa =\Gamma$ and $\varphi =0$. For $\Delta _p=7.2\Gamma$, CPA does not occur when $\Omega _1=0.5\Gamma$, or CPA occurs with $|\alpha _{in}|^2<1$ when $\Omega _1=\Gamma$ (the dashed red and solid green lines shown in Fig. 4(a)). However, when $\Omega _1=1.5\Gamma$, CPA can be clearly manifested at the frequency with $|\alpha _{in}|^2\approx 14$ (the dot-dashed blue line shown in Fig. 4(a)). In three-level EIT system, the manipulation on single-path absorption by $\Omega _1$ will also disturb nonlinear CPA condition. For example, in Fig. 4(b), when $\Omega _1=0.5\Gamma$, the input intensity for CPA is $|\alpha _{in}|^2=6.3$. When $\Omega _1=\Gamma$, the needed intensity is $|\alpha _{in}|^2=15.9$. When $\Omega _1$ is increased to $\Omega _1=1.5\Gamma$, the needed intensity is increased to $|\alpha _{in}|^2=30.3$ which is not manifested with dot-dashed blue line in Fig. 4(b).

 

Fig. 4. The output intensity as the function of $|\alpha _{in}|^2$ with $\Omega _1=0.5\Gamma$ ($\Delta _{ac}=-7.2\Gamma$ for (a) and $\Delta _{ac}=-6.9\Gamma$ for (b)) (dashed red lines), $\Omega _1=\Gamma$ ($\Delta _{ac}=-6.9\Gamma$ for (a) and $\Delta _{ac}=-6.5\Gamma$ for (b)) (solid green lines), and $\Omega _1=1.5\Gamma$ ($\Delta _{ac}=-6.4\Gamma$ for (a) and $\Delta _{ac}=-6\Gamma$ for (b)) (dot-dashed blue lines).

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This indicates that for three- or four-level atom-cavity system, the coherent perfect absorption is realizable and controllable by EIT-type quantum interference.

4. Conclusion

In conclusion, we have analyzed the interference control on CPA condition in a three-level atom-cavity system. Generally, the decay rates of the cavity mirrors can be as small as possible to widen the frequency range of CPA. While for an input probe laser with a given frequency, the needed input intensity for CPA is larger as the decay rates are smaller, which forms the CPA with a stronger excitation regime. However, the needed intensity of CPA is controllable by the control laser, which provides the application of CPA with weak fields in a low-loss cavity. Besides, the transfer between a CPA mode and a CNPA mode is realized by the control laser. In addition, with EIT-type quantum interference induced by the control laser, four CPA modes instead of two CPA modes are obtained for a given input probe laser, which has potential application in the study of wide-band perfect absorber.