Tunable interaction-free all-optical switching in a five-level atom-cavity system

Tiantian Liu; Gongwei Lin; Fengxue Zhou; Li Deng; Shangqing Gong; Yueping Niu

doi:10.3788/COL201715.092702

Abstract

A scheme is proposed for tunable all-optical switching based on the double-dark states in a five-level atom-cavity system. In the scheme, the output signal light of the reflection and the transmission channels can be switched on or off by manipulating the control field. When the control light is coupled to the atom-cavity system, the input signal light is reflected by the cavity. Thus, there is no direct coupling between the control light and the signal light. Furthermore, the position of the double-dark states can be changed by adjusting the coherent field, and, thus, the switching in our scheme is tunable. By presenting the numerical calculations of the switching efficiency, we show that this type of the interaction-free all-optical switching can be realized with high switching efficiency.

With the development of optical communications and quantum information network, the all-optical switching, in which a signal light beam can be fully controlled by a control light field, has attracted a lot of attention in recent years^[1–14]. Particularly, with the optical nonlinearities, schemes of interaction-free all-optical switching have been proposed and demonstrated^[15–19]. In these interaction-free schemes, it is possible to eliminate the direct coupling between the signal light and the control light. Then, the signal photon loss in the absorbing medium is suppressed. However, these schemes^[15–19] are limited by the achievable optical nonlinearities for low light intensities.

Recently, Zou et al.^[20] proposed a new method for interaction-free all-optical switching based on three-level atoms confined in an optical cavity. They showed that with the quantum interference effect^[8,21–23], a weak control light can be used to suppress the normal-mode excitation from the signal light. Thus, the interaction-free all-optical switching would be operated at low light intensities. In this Letter, we present an alterable scheme for interaction-free all-optical switching based on the quantum interference effect^[8,21–23]. In our scheme, we use the double-dark states^[24–26] in the five-level atom-cavity system to realize the optical switching. As the dark state does not contain the excited states, the switching in our scheme is not affected by the spontaneous emission and can be realized with higher switching efficiencies, compared to the previous scheme^[20]. What is more, the frequency position of the double-dark states can be changed by adjusting the coherent field; thus, the switching in our system is tunable.

We consider a composite atom-cavity system that consists of a single mode cavity containing $N$ five-level atoms, as shown in Fig. 1. Each atom contains three ground states $| 1 〉$ , $| 3 〉$ , $| 4 〉$ and two excited states $| 2 〉$ , $| 5 〉$ . The cavity mode and a pump laser resonantly couple the atomic transitions $| 1 〉 \to | 2 〉$ and $| 2 〉 \to | 3 〉$ with the coupling strengths $g$ and $Ω_{1}$ , respectively. The twofold levels, labeled 3 and 4, are coupled by a coherent field with the Rabi frequency $Ω_{2}$ , which can be a microwave or quasi-static field^[26]. These four states coupled by the cavity mode and two coherent fields constitute a four-level electromagnetic induced transparence (EIT) system, which contains the double-dark state structure^[25]. In addition, the cavity mode is driven by a signal field $ξ_{p}$ with frequency detuning $Δ_{p} = ω_{p} - ω_{a}$ ; here, $ω_{p}$ is the frequency of the signal field, and $ω_{a}$ is the eigenfrequency of the cavity mode. A free-space control laser $Ω_{3}$ drives the atomic transition $| 4 〉 \to | 5 〉$ with frequency detuning $Δ = ω_{3} - ω_{54}$ , where $ω_{3}$ is the frequency of the control laser, and $ω_{54}$ is the transition frequency between states $| 5 〉$ and $| 4 〉$ . The Hamiltonian of the system can be written as $H = \sum_{k = 1}^{N} Δ | 5 〉_{k} 〈 5 | + \sum_{k = 1}^{N} (g a^{†} σ_{12}^{(k)} + Ω_{1} σ_{23}^{(k)} + Ω_{2} σ_{34}^{(k)} + Ω_{3} σ_{54}^{(k)} + H.c.) + i (ξ_{p} a^{†} e^{- i Δ_{p} t} - ξ_{p}^{*} a e^{i Δ_{p} t}),$ (1)where $a$ ( $a^{†}$ ) is the annihilation (creation) operator of the cavity mode, and $σ_{l m}^{(k)} = | l 〉_{k} 〈 m | (l, m = 1 - 5)$ is the atomic transition operator for the $k$ th atom. $ξ_{p}$ is related to the laser power P by $ξ_{p} = \sqrt{(2 κ P / ℏ ω_{p})}$ , where $κ$ denotes the cavity decay rate. Define the collective operator of the atoms $S_{p}^{†} = \frac{1}{\sqrt{N}} \sum_{k = 1}^{N} | p 〉_{k} 〈 1 |$ with $p = 2, 3, 4, 5$ . Then, the Hamiltonian $H$ can be represented by $H = Δ S_{5}^{†} S_{5} + g \sqrt{N} (a^{†} S_{2} + S_{2}^{†} a) + Ω_{1} (S_{2}^{†} S_{3} + S_{3}^{†} S_{2}) + Ω_{2} (S_{3}^{†} S_{4} + S_{4}^{†} S_{3}) + Ω_{3} (S_{4}^{†} S_{5} + S_{5}^{†} S_{4}) + i (ξ_{p} a^{†} e^{- i Δ_{p} t} - ξ_{p}^{*} a e^{i Δ_{p} t}) .$ (2)

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Figure 1.(a) Schematic diagram for the tunable interaction-free all-optical switching in a five-level atom-cavity system. (b) The schematic diagram of the input–output channels. (c) The detailed output states of the all-optical switching. When there is no control light, there is output signal light from the transmission channel, but no light from the reflection channel; when the control light is on, the signal light cannot be coupled into the cavity, the output is switched to the reflection channel, and there is no output light from the transmission channel. There is no direct coupling between the signal light and the control light.

Considering photon losses from the cavity and the decays of the atoms, the dynamic equations of the system are given in a rotating frame by the quantum Langevin equations, where $γ_{i} (i = 2, 5)$ denotes the decay rate of the atom. Here, we have ignored the decoherence rates between the ground states. $a_{in} (t)$ and $ζ_{i} (t)$ with zero mean values denote the input vacuum noises associated with the cavity mode and the atomic system, respectively. They obey the nonvanishing commutation relations^[27] $〈 a_{in} (t) a_{in}^{†} (t^{'}) 〉 = δ (t - t^{'})$ and $〈 ζ_{i} (t) ζ_{i}^{†} (t^{'}) 〉 = δ (t - t^{'})$ . Under the mean-field approximation $〈 Q c 〉 = 〈 Q 〉 〈 c 〉$ ^[28], the mean value equations are given by

In this section, we show the tunable interaction-free all-optical switching based on the five-level atom-cavity system. Without loss of generality, we assume that $γ_{2} = γ_{5} = γ$ . Then, the steady-state solution of the system is given by $〈 a 〉 = \frac{ξ_{p}}{i Δ_{p} + κ + \frac{g^{2} N}{i Δ_{p} + \frac{γ}{2} + \frac{Ω_{1}^{2}}{i Δ_{p} + \frac{Ω_{2}^{2}}{i Δ_{p} + \frac{Ω_{3}^{2}}{i (Δ_{p} - Δ) + \frac{γ}{2}}}}}} .$ (5)

In this system, the input signal light is coupled into the cavity and results in two output channels: the reflected signal light from the cavity and the transmitted signal light though the cavity. The reflected signal field from the cavity is given by^[20] $a_{R} = \sqrt{2 κ} 〈 a 〉 - ξ_{p}$ , and the transmitted signal field through the cavity is given by $a_{T} = \sqrt{2 κ} 〈 a 〉$ . The schematic diagram of the input–output channels are showed in Fig. 1(b). The states of the two output channels are controlled by the free-space control light $Ω_{3}$ . Figure 1(c) shows the detailed output states.

In Fig. 2, we plot the reflected signal light intensity $I_{R} = a_{R}^{*} a_{R}$ and the transmitted signal light intensity $I_{T} = a_{T}^{*} a_{T}$ normalized by the input signal intensity $I_{in} = {| ξ_{p} |}^{2}$ versus the signal laser detuning $Δ_{p}$ . From the solid blue line in Fig. 2(a), one can observe that when the control laser is absent ( $Ω_{3} = 0$ ), there are two deep dips in the middle of the reflected signal spectrum. Correspondingly, there are two high peaks in the middle of the transmitted signal spectrum, as shown in Fig. 2(b) (solid blue line). The two dips/peaks represent the double-dark states in frequency $\pm d$ with $d = \sqrt{\frac{g^{2} N + Ω_{1}^{2} + Ω_{2}^{2} - \sqrt{{(g^{2} N + Ω_{1}^{2} + Ω_{2}^{2})}^{2} - 4 g^{2} N Ω_{2}^{2}}}{2}}$ . The double-dark states do not contain the excited states and are nearly unaffected by the spontaneous emission. The two lower dips/peaks beside the double-dark states in Fig. 2 are the bright states, which contain the excited states with large spontaneous emission.

Figure 2.(Color online) (a) Reflected signal intensity ( $I_{R} / I_{in}$ ) and (b) the transmitted signal intensity ( $I_{T} / I_{in}$ ) as functions of the frequency detuning of the signal field $Δ_{p}$ . Solid blue line, $Ω_{3} = 0$ MHz; dashed red line, $Ω_{3} = 2 MHz$ . The rest of the parameters are $g \sqrt{N} = 16 MHz$ , $ξ_{p} = 2 MHz$ , $γ = 1 MHz$ , $κ = 2 MHz$ , $Ω_{1} = 16 MHz$ , and $Ω_{2} = 10 MHz$ .

The dashed red lines in Figs. 2(a) and 2(b) plot the reflection and transmission spectrum when the control laser with frequency detuning $Δ = - d$ is present ( $Ω_{3} \neq 0$ ). The spectrum show that at dark-state resonance ( $Δ_{p} = - d$ ), as the control laser field induces destructive interference, the transmitted signal light is suppressed, while the reflected signal light is maximized. Therefore, the control laser can be used to switch the signal light. So, the coupled atom-cavity system can be used to perform an all-optical switching controlled by a weak control light $Ω_{3}$ . When the control laser is coupled to the system, the signal light cannot be coupled into the cavity and, then, is reflected from the cavity. Therefore, the all-optical switching is performed without the direct coupling between the control light and the signal light, and, thus, it is interaction free.

We plot the resonant frequency $d$ of the dark state versus the coherent field $Ω_{2}$ in Fig. 3. From Fig. 3, we can see that the resonant frequency $d$ of the dark state is a monotonically increasing function of $Ω_{2}$ and is saturated at a moderate $Ω_{2}$ value. That is to say, the position of the double-dark states can be manipulated by adjusting the intensity of the coherent field. With a large $g \sqrt{N}$ value ( $g \sqrt{N} = 100 MHz$ ), we can get a large tunable range, as shown in Fig. 3(b). As shown in the experiment, $g \sqrt{N}$ would reach to 190 MHz even in the low quality cavity with many atoms^[29]. Hence, the optical switching in our scheme would be tunable in a large range.

Figure 3.Resonant frequency of the dark state as function of the coherent intensity $Ω_{2}$ . Other parameters are (a) $g \sqrt{N} = 16 MHz$ and $Ω_{1} = 16 MHz$ and (b) $g \sqrt{N} = 100 MHz$ and $Ω_{1} = 100 MHz$ .

Next, we analyze the efficiencies of the interaction-free all-optical switching. From Fig. 2, one can see that when the control light is present, the signal light is reflected from the cavity (the reflection channel is opened), while the transmitted light through the cavity is suppressed (the transmission channel is closed). When the control light is absent, the transmission channel is opened, and the reflection channel is closed. Following the method in Ref. [20], one can obtain the switching efficiency of the reflection output channel and the transmission output channel: $η_{R / T} = \frac{I (o) - I (c)}{I_{in}}$ . Here, $I (o)$ is the signal output intensity when the switching is opened, $I (c)$ is the signal output intensity when the switching is closed $[I (o) > I (c)]$ , and $I_{in}$ is the input signal intensity.

Figures 4–6 show the switching efficiencies $η_{R}$ and $η_{T}$ versus the parameters $g \sqrt{N}$ , $Ω_{3}$ , $Ω_{1}$ , and $Ω_{2}$ , separately. From Fig. 4, one can see that $η_{R}$ and $η_{T}$ are monotonically increasing functions of $g \sqrt{N}$ and are saturated at moderate $g \sqrt{N}$ values. This means that the high efficiency depends on a moderately large coupling coefficient $g \sqrt{N}$ . Figure 5 shows that the interaction-free all-optical switching in the cavity-atom system can be done with a relatively weak control laser. The switching efficiency (especially $η_{T}$ ) increases rapidly with the increasing control field, but saturates at $Ω_{3}$ values smaller than other parameters, like $g \sqrt{N}$ and $Ω_{2}$ . This is similar to the figures in the three-level atom-cavity system^[20]. Figure 6 plots $η_{R}$ and $η_{T}$ versus the pump field $Ω_{1}$ and the coherent field $Ω_{2}$ . It shows that the pump field $Ω_{1}$ and the coherent field $Ω_{2}$ have little impact on the switching efficiency. From Figs. 4–6, one can find that the switching efficiency decreases with the increasing of the atom decay rate $γ$ .

Figure 4.(Color online) Switching efficiencies of (a) the reflected output field $η_{R}$ and (b) the transmitted output field $η_{T}$ versus the atom-cavity coupling $g \sqrt{N}$ with $γ = 1 MHz$ (black solid lines), $γ = 2 MHz$ (red dashed lines), and $γ = 3 MHz$ (blue dotted lines), respectively. Other parameters are $ξ_{p} = 2 MHz$ , $κ = 3 MHz$ , $Ω_{1} = 16 MHz$ , $Ω_{2} = 10 MHz$ , $Ω_{3} = 2 MHz$ , and $Δ_{p} = Δ = - d$ .

Figure 5.(Color online) Switching efficiencies of (a) the reflected output field $η_{R}$ and (b) the transmitted output field $η_{T}$ versus the control field $Ω_{3}$ with $γ = 1 MHz$ (black solid lines), $γ = 2 MHz$ (red dashed lines), and $γ = 3 MHz$ (blue dotted lines), respectively. Other parameters are $Δ_{p} = Δ = - d MHz$ , $ξ_{p} = 2 MHz$ , $κ = 3 MHz$ , $g \sqrt{N} = 16 MHz$ , $Ω_{2} = 10 MHz$ , and $Ω_{1} = 16 MHz$ .

Figure 6.(Color online) Switching efficiencies of (a) the reflected output field $η_{R}$ and (b) the transmitted output field $η_{T}$ versus the pump field $Ω_{1}$ with $γ = 1 MHz$ (black solid lines), $γ = 2 MHz$ (red dashed lines), and $γ = 3 MHz$ (blue dotted lines), respectively. Other parameters are $Δ_{p} = Δ = - d MHz$ , $ξ_{p} = 2 MHz$ , $κ = 3 MHz$ , $g \sqrt{N} = 16 MHz$ , $Ω_{2} = 10 MHz$ , and $Ω_{3} = 2 MHz$ . (c) $η_{R}$ and (d) $η_{T}$ versus the coherent field $Ω_{2}$ with $γ = 1 MHz$ (black solid lines), $γ = 2 MHz$ (red dashed lines), and $γ = 3 MHz$ (blue dotted lines), respectively. Here, $Ω_{1} = 16 MHz$ , and other parameters are the same as (a) and (b).

Finally, we address the experimental feasibility of the present scheme. The atomic configuration can be chosen from the hyperfine states of the Rb atoms ( $γ = 3 MHz$ ). The atoms are confined in a 5 cm cavity with a finesse of 150 ( $κ = 10 MHz$ ) with $g \sqrt{N} = 50 MHz$ ( $N \approx 10^{4}$ atoms). We choose the parameters $Ω_{3} = 1.5 MHz$ , $Ω_{2} = 6 MHz$ , and $Ω_{1} = 5 MHz$ , and the switching efficiency is derived to be $η_{R} = 0.86$ and $η_{T} = 0.98$ . Compared with the switching efficiency in Ref. [20] with $η_{R} = 0.83$ and $η_{T} = 0.56$ under the same parameters of the cavity and the intensity of the control field, there is little difference between the reflection efficiency $η_{R}$ , but the transmission efficiency $η_{T}$ is greatly enhanced in our scheme. This is because in Ref. [20], the switching is realized based on the bright states, which contain the excited states. Thus, the transmitted light through the cavity is greatly influenced by the spontaneous emission. However, in our scheme, the switching is achieved based on the double-dark states, which do not contain the excited states and are nearly unaffected by the spontaneous emission. Therefore, the switching efficiency of the transmission channel is much higher than that in Ref. [20].

In conclusion, we demonstrate a tunable interaction-free all-optical switching in a five-level atom-cavity system. Compared with the interaction-free all-optical switching in the three-level atom-cavity system^[20], we utilize the double-dark states rather than the bright states to realize the all-optical switching; thus, the switching efficiency is greatly improved. What is more, the position of the double-dark states can be changed by adjusting the coherent interaction, so the switching in our system is tunable.

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