1. Introduction

Electromagnetically induced transparency (EIT) has been widespread concerned because of its wonderful feature and promising applications [1], such as laser sideband cooling, light velocity controlling, quantum memory, and so on. Besides in the three-level atom systems, EITs have also been found in other systems such as four-wave-mixing schemes [2, 3], optomechanics [4–6], metamaterials [7, 8], plasma [9], and even vacuum [10]. It is well known that vacuum possesses numerous virtual photons due to vacuum fluctuation, which is the reason of Casimir force [11] and Lamb shift [12, 13]. More important, the vacuum fluctuation can be modified by artificial electromagnetic environments, and the vacuum-related quantum phenomena can also be controlled by these man-made structures.

High-finesse optical cavity has been well developed and widely used in QED systems [14–17]. It is also used in EIT experiment to strengthen the coupling between atoms and the driving field [18, 19]. In 2011, an extreme case of EIT was realized in the experiment, which is called as “Vacuum-Induced Transparency” (VIT) because the driving field is replaced by the vacuum cavity mode, there was no real photon initially in the cavity [20–22]. Although VIT has promising applications on photon switch or transistor [23], it is hard to be achieved in the conventional cavity and is difficult to be widely used up to other optical or quantum devices, because of the mediocre coupling coefficient between atoms and cavity.

In recent decades, the development of metamaterials provides new tools to control light or photons, even vacuum. Such materials can possess negative effective permittivity ($\epsilon$-negative, EN), negative effective permeability ($\mu$-negative, MN), or negative refractive index (left-hand materials, LHM) at interesting frequencies after subtle design. Until now metamaterials had been achieved from microwave [24] to visible light [25–29]. Due to their unique properties, metamaterial have promising applications on quantum optics, such as quantum interference enhancement [30–32], long-lived entanglement [33], perfect lens [34], superradiance [35], and Casimir force [36–38]. In 2003, a structure which pairing an EN slab and an MN slab was proposed by Alu and Engheta [39]. It was found that highly local of the fields appear near the interface, because such structure can support both the tunnel modes and surface plasmonic modes simultaneously. If a dipole is placed near the interface, the high local field provides a strong coupling between dipole and field. Therefore various harmonic systems, i.e. classical oscillator [40–42] and two-level atom [43], are considered to be sandwiched between MN and EN slabs, and the spectra of Rabi splitting were observed.

Taking advantage of the high locality of the surface mode near the interface between MN and EN slabs, in this work, we consider a thin layer of three-lever atoms doped very close to such interface. The combo structure made by the matched MN and EN slabs acts as a high Q-cavity which resonantly couples to one of the atomic dipole moments, and greatly alter the response of another atomic dipole to the weak probe field. Compared to the standing wave mode in the ordinary cavity, the surface modes in our scheme lead to the huge mode density and provide an extremely strong coupling to the atomic dipole. We focus on the case that the field in the combo structure is in the vacuum state, and discuss the phenomenon of VIT. As the atom-field coupling in our combo structure is determined by the location of atoms, various VIT phenomena can be observed if the atomic layer locates in different positions. For comparison, it is hard to observe such significant VIT phenomena in conventional micro-cavity because the atom-filed coupling coefficient in conventional micro-cavity is a few orders less than that in our scheme.

Our paper is organized as follows. In Sec. II, we introduce the cavity-based EIT model and extend it to VIT naturally. In Sec. III, We discussed the calculation of the cooperation parameter, which is the key parameter related to VIT, by Green tensor concerning the matching MN and EM slabs. In Sec. IV, the transmission spectra are calculated, and reveal the VIT phenomena within such metamaterials structure. The relative discussions and an application of single-photon switch are presented in Sec V. We draw the conclusion in Sec. VI.

2. Model and theory

We start from the usual EIT system, which is a $\Lambda$-type three-level atom interacting with a weak probe field and the coherent driving field simultaneously. The atomic scheme and relative parameters are shown in the inset of Fig. 1. By ignoring the dissipation, the Hamiltonian of the system is given

 

Fig. 1 A single layer of $\Lambda$-type three-level atoms located near the interface between a $\mu$-negative slab and a $\epsilon$-negative slab. The level scheme of an atom is shown in the dashed circle. The directions of polarization are indicated by double-headed arrows.

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The response of atom to probe field is embodied by the density matrix ${\rho }_{ab}$. In the case of the weak probe field, it is reasonable to assume a vanishing atomic excitation. And then, in the steady state, one is able to deduce the analytical results of density matrix according to the master equation [1, 44]. If we extend the single atom to a bulk atomic gas with total number N and volume V, the linear susceptibility of the atomic gas can be expressed as

Where ${\mu }_{ab}$ is the electric dipole moment of atomic transition $|a〉\leftrightarrow |b〉$ and$N/V$ is the atom number density. Here$\Delta ={\omega }_{ab}-{\omega }_p$ is the detuning between the transition $|a〉\leftrightarrow |b〉$and the probe fields, ${\Delta }_c={\omega }_{ac}-{\omega }_c$ is the detuning between the transition $|a〉\leftrightarrow |c〉$and the coherent driving fields, while $\delta =\Delta -{\Delta }_c={\omega }_{cb}-\left({\omega }_p-{\omega }_c\right)$ is the two-photon detuning. The decoherence rates are defined as ${\gamma }_{ab}={\Gamma }_{ab}+{\Gamma }_{ac}+{\gamma }_a$ and ${\gamma }_{bc}={\gamma }_c$ in which ${\Gamma }_{ab}$ (${\Gamma }_{ac}$) is the spontaneous emission rates from state $|a〉$ to states $|b〉$ ($|c〉$). ${\gamma }_a$and ${\gamma }_c$ are the dephasing rates of states $|a〉$ and $|c〉$, respectively.

The decoherence rate${\gamma }_{bc}$is a key quantity of EIT. If ${\gamma }_{bc}\ne 0$, the susceptibility should have a nonzero imaginary value at resonance$\Delta \text{=}{\Delta }_c=\delta \text{=}0$, i.e. $\chi \propto i{\gamma }_{bc}/{\left|{\Omega }_c\right|}^2$when ${\Omega }_c>>{\gamma }_{bc},{\gamma }_{ab}$. As ${\gamma }_{bc}$ cannot be eliminated exactly in experimentally, the EIT can only be improved by enhancing the Rabi frequency of the coherent driving field ${\Omega }_c$.

The Rabi frequency ${\Omega }_c$ relates to both the coupling coefficient $g$ ($\text{2}g$ is the vacuum Rabi frequency) and the intensity of the driving field. For the weak driving filed which can be described by photon number $n_c$, the Rabi frequency ${\Omega }_c$can be rewritten as

For a high $g$, ${\Omega }_c$ could be significant even if the coherent driving field is absent, i.e.$n_c=0$, and has the value of ${\Omega }_c=2g$. When $2g$ is larger than the decoherent rate ${\gamma }_{bc}$, the EIT phenomena are still apparent in absent of driving field. Such a phenomenon is named as vacuum induced transparent (VIT).

The effective way to enhance the coupling coefficient $g$ is placing the atom into a cavity. Therefore several cavity-based VIT or vacuum Rabi splitting were analyzed, such as cavity-atoms system [10, 20], semiconductor quantum dots in photonic crystal [45], and semiconductor circuit [46].

For the usual cavity-based VIT scheme, the transition $|a〉\leftrightarrow |c〉$ is resonant with the cavity mode in order to enhance g. Meanwhile, the transition $|a〉\leftrightarrow |b〉$, which corresponds to the probe field, is far detuning from the cavity mode. To describe the cavity-based VIT, a quantity named as the single-atom cooperation parameter is defined as

As mentioned before ${\gamma }_{ab}$and ${\gamma }_{bc}$ include several channels. However, they can be simplified in the QED system. For${\gamma }_{bc}$, its prominent element is the cavity losses $\kappa$ at the cavity frequency or equivalent${\omega }_{ac}$, i.e. ${\gamma }_{bc}\approx \kappa$. Meanwhile, for ${\gamma }_{ab}$, its dominant contribution comes from the decay rate from $|a〉$to$|b〉$, i.e.${\gamma }_{ab}\approx {\Gamma }_{ab}$. Thus the single-atom cooperativity parameter is approximated as [20, 23]

Correspondingly, the susceptibility $\chi$ in Eq. (2) can be rewritten as the function of $\eta$, which is (for derivation, see Appendix) [13]

Here $\tilde{\Delta }={2\Delta /\Gamma }_{ab}$ and $\tilde{\delta }=2\delta /\kappa$are the normalized probe-atom detuning and the two-photon detuning, respectively [20].

In the conventional cavity-based VIT system, the single-atom cooperativity parameter can reach $7.2\pm 0.5$ theoretically, and achieve only about 3.4 in experiment [20], which is limited by the dipole momentum and standing-wave cavity mode.

Here we propose a new kind of cavity possessing larger coupling coefficient g but smaller cavity loss $\kappa$, so that it is helpful to improve the VIT phenomena. Such special cavity is a double-layer structure, which is combined with a $\mu$-negative(MN) slab and a $\epsilon$-negative(EN) slab, shown in Fig. 1. The interface between two single-negative slabs is defined as the x-y plane at z = 0.

The permeabilities and permittivities of the EN and the MN slabs can be reasonably described by Drude models as

Where ${\omega }_e$ and ${\omega }_m$ are the bulk plasma frequencies. ${\gamma }_e$ and ${\gamma }_m$ refer to the dissipations of the materials. In this paper, these parameters are set as ${\omega }_e={\omega }_m=1.732{\omega }_0$ (${\omega }_0=3.0\times {10}^{15}Hz$), ${\gamma }_e={\gamma }_m=1.67\times {10}^{-8}{\omega }_0=5\times {10}^7{s}^{-1}$. In addition, the thicknesses of slabs are set to be $d_1\text{=}{d}_2={\lambda }_0=2\pi c/{\omega }_0$.

Such two-layer structure exhibits the unique properties at the frequency ${\omega }_0=3.0\times {10}^{15}Hz$. According to ref [42], at such frequency ${\epsilon }_1({\omega }_0)={\mu }_2({\omega }_0)=2$and ${\mu }_1({\omega }_0)={\epsilon }_2({\omega }_0)=-2$, the structure supports a tunneling mode and various surface plasmonic modes. For both kinds of modes, the electric fields are concentrated at the interface between EN and MN slabs, which acts as a high-Q cavity. This Q factor does not relate to the Q factor of each material, but from this special combination of EN and MN slabs. The tunneling mode responds to propagating wave out of the structure, and contributes to the cavity loss $\kappa$. Meanwhile, the surface plasmon modes can coherently interact with the atom, and provide a huge g when the atom is located at the interface.

To show the tunneling mode, we consider the propagating wave incident from left to the combo of EN and MN slabs without three-level atom, and plot the transmittance spectrum in Fig. 2.

 

Fig. 2 The transmittance spectrum of the combo of EN and MN slabs for the normal incident. Relative parameters are presented in the context.

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From Fig. 2, the transmission spectrum shows a serial of transparent windows. Especially at $\omega \approx {\omega }_0\text{=}3\times {10}^{15}{s}^{-1}$, the transparent peak is ultra narrow which refers to the tunneling mode. It is a perfect cavity mode with low loss and high Q-factor. In addition, around $\omega \text{=}8.2\times {10}^{15}{s}^{-1}$ (the probe window we marked), as both the permittivity and the permeability of the slabs become positive and act as dielectrics, there is a wide transparent band.

To show the property of the surface plasmon modes at ${\omega }_0$, we choose three surface modes with propagating constant $K_{\left|\right|}=K_0,10K_0,100K_0$ and plot their normalized electric field distributions in Fig. 3.

 

Fig. 3 The normalized electric field intensity distribution of the surface mode in the combo of EN and MN slabs at $\omega \text{=}{\omega }_0\text{=}3\times {10}^{15}{s}^{-1}$. Three modes with propagating constants $K_{\left|\right|}=K_0,10K_0,100K_0$ are shown.

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From Fig. 3, the normalized electric field of three surface modes concentrates at the interface z = 0 and decays exponentially away from the interface. The FWHM (full width at half maximum) of the mode distribution decreases with the increasing of propagating constant $K_{\parallel }$. As the surface modes ($K_{\parallel }>K_0={\omega }_{\text{0}}/c$) can exist for arbitrary propagating constant $K_{\parallel }$ [43], the combo structure provides a huge coupling coefficient g when an atom with transition frequency ${\omega }_0$ is located at the interface. Furthermore, the coupling coefficient g would decrease exponentially when the atom is away from the interface. As a result, the combo of EN and MN provide a perfect platform to realize the controllable VIT.

3. Cooperativity parameter

Based on the unique property of the combo structure mentioned above, the atomic level scheme is chosen so that the transition $|a〉\leftrightarrow |c〉$ is resonant with the surface mode and the tunneling mode, i.e. ${\omega }_{ac}={\omega }_0=3.0\times {10}^{15}{s}^{-1}$and its dipole momentum ${\mu }_{a\text{c}}=6\times {10}^{-29}C\cdot m$ is along the z-axis direction. Therefore it achieves to high g and lower $\kappa$; Meanwhile, the transition $|a〉\leftrightarrow |b〉$possesses the frequency at the center of the wider transparent window, i.e. ${\omega }_{ab}=8.2\times {10}^{15}{s}^{-1}$and the dipole momentum ${\mu }_{a\text{b}}=6\times {10}^{-30}C\cdot m$ in the x-axis. so the probe field can transmit through the combo and interact with the dipole ${\mu }_{a\text{b}}$. We consider a thin layer of three-level atoms doped at $z_0$.

According to Eq. (5), the single-atom cooperativity parameter can be determined by the value of $4g^2$,${\Gamma }_{ab}$, and $\kappa$. As mentioned before, $4g^2$ is the coupling intensity between the vacuum mode and atomic transition $|a〉\leftrightarrow |c〉$. It can be calculated by the classical electromagnetic Green tensor $\text{G}(z_0,z_0,\omega )$ [32, 47–50], and has the expression as

The integral over $K_{\parallel }$ in Eq. (9) is from 0 to infinite but mainly originate from the surface modes (the part meet that $K_{\parallel }>{\omega }_0/c$). So all quantities are the function of frequency $\omega \text{=}{\omega }_{ac}={\omega }_0=3.0\times {10}^{15}{s}^{-1}$.

_{${\Gamma }_{ab}$}is the spontaneous decay rate of transition $|a〉\leftrightarrow |b〉$, and is mainly related to the electromagnetic modes at ${\omega }_{ab}=8.2\times {10}^{15}{s}^{-1}$. Similar to Eq. (9), using Green tensor in a different polarization, it has the detail expressions as

Where $K_1=\sqrt{{\epsilon }_1{\mu }_1}\cdot \omega /c$ is wave number in MN slab, $K_{\parallel }$ is the $x-y$ plane component of the wave vector, and $K_{1z}^{}$ satisfying$K_{1z}^2=K_1^2-K_{\parallel }^2$ is the z-component of the wavevector. $r_{ij}^{TE}$ and $r_{ij}^{TM}$ are the reflection coefficients at the interface between the ith and jth slab when the TE and TM polarized field incident from ith slab to jth slab, respectively ($i\ne j$;$i,j\in \left\{0,1,2\right\}$ is marked in Fig. 1). $R_{12}^q$ ($q\in \left\{\text{TE,TM}\right\}$) are the reflection of the MN slab when the TE or TM polarized field incident from EN slab to MN slab, which is defined

The cavity loss $\kappa$ can be measured by the FWHM of the tunnel mode centered at ${\omega }_{ac}={\omega }_0=3.0\times {10}^{15}{s}^{-1}$. From Fig. 2, we get the FWHM of tunnel mode is $5\times {10}^7{s}^{-1}$. So in this paper, the cavity loss is $\kappa =5\times {10}^7{s}^{-1}$and it is independent of the atom.

As $4g^2$ related to the surface mode directly, it is sensitive to the position of atom film. To achieve a proper cooperativity parameter, we plot $4g^2$, ${\Gamma }_{ab}$ and $\eta$ as a function of the position of atom film in Fig. 4.

 

Fig. 4 Cooperativity parameter $\eta$ (black solid line), the coupling of control field 2g (blue dash line), and the decay rate of atomic transition$|a〉\to |b〉$ ${\Gamma }_{ab}$ (red dot line) change with the position of atoms.

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For the decay rate of atomic transition$|a〉\to |b〉$, as the combo are dielectric at ${\omega }_{ab}=8.2\times {10}^{15}{s}^{-1}$, ${\Gamma }_{ab}$ is nearly the constant for different positions. However, 2g increase exponentially when atoms tend to the interface $z=0$. For example, $\eta$ is 10⁻¹⁰ for $z_0={\lambda }_0$, $\eta$ reaches 1 for z₀ = $0.02{\lambda }_0$, and $\eta$ is about 10⁷ for z₀ = $0.0001{\lambda }_0$. As a result, the VIT phenomena would be easily achieved in such structure theoretically.

4. Result

To discuss the input-output problem, a thin atomic layer is set near the surface at $z_0$. The atomic areal density is $N/\sigma$, and then the atom layer possesses the resonant optical depth of the ensemble $\aleph ={\left|{\mu }_{ab}\right|}^2{\omega }_{ab}N/2c\hslash {\epsilon }_0{\Gamma }_{ab}\sigma$ with a length L along the probe beam [20]. Thus the transmission amplitude of probe field is $t=\exp \left(i{K}_{ab}L\chi /2\right)$, where the influence of the structure on the transmission is ignored because the frequency of probe field is set at the center of the transparent window. The transmission amplitude is only affected by the atomic susceptibility $\chi$. $\chi$ is related directly to Eq. (6) and should be modified correspondingly for the atomic layer as follows (for derivation, see Appendix),

When both the probe field and the cavity mode are resonant with or very close to their respective atomic transitions, i.e. $\Delta =\delta =0$, as well as the atoms are close to the interface between EN slab and MN slabs enough, the phenomenon of VIT can be found for weak probe light. Recently, several experiments discussed the laser-cooled atoms optically trapped inside a high-finesse optical cavity, for example$\aleph =0.\text{4}$ or $\aleph =0.\text{5}$for a double-pass [20], and $\aleph =0.\text{9}$ in a later experiment [23]. Therefore, in our scheme, we assume a low areal density of the atomic layer as $2\times {10}^{12}{m}^{-2}$ and the optical depth$\aleph =0.2\text{5}$.

When the atom layer is in free space, the transmission spectrum shows a Lorentz dip near resonance and $T(\Delta =0)=0.37$, shown in the red dashed curve in Fig. 6. However, when the atom layer is doped in the combo structure, the result changes. For example, when the position of atoms is $z_0=0.01{\lambda }_0$, the cooperative parameter reaches$\eta =8.4$, which is higher than the normal cavity [20, 23]. The susceptibility at resonance is $\chi (\Delta =0)=0.43i$ which become nearly zero, shown in Fig. 5 and then the transmission probability ($T={\left|t\right|}^2$) at resonance changes to $T(\Delta =0)=0.9$, shown in the blue solid curve in Fig. 6. The VIT phenomena happen. Because the dissipation of metamaterials brings a large $\kappa =5\times {10}^7{s}^{-1}$, which also limits the coherence, the transparent window doesn’t get 1. The FWHM of the VIT peak is about $3.3\times {10}^8{s}^{-1}$, which is narrower than the vacuum Rabi frequency ${\Omega }_c{|}_{n_c=0}=2g=1.4\times {10}^9{s}^{-1}$, and larger than the cavity loss $\kappa$.

 

Fig. 5 The real part (blue solid line) and imaginary part (red dash line) of susceptibility of the atom layer in the combo structure. The atom layer is located at $z_0=0.01{\lambda }_0$ which leads to $2g=1.4\times {10}^9{s}^{-1}$ and $\eta =8.4$. Other parameters are ${\Delta }_c=0$, ${\Gamma }_{ab}=4.7\times {10}^9{s}^{-1}$, $\kappa =\text{2}\pi \times \text{8}\times {10}^{\text{6}}Hz$, $\aleph =0.25$.

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Fig. 6 The transmission spectrum of the atomic layer in the combo structure (Blue solid curve). The atom layer is located at $z_0=0.01{\lambda }_0$ which leads to $2g=1.4\times {10}^9{s}^{-1}$ and $\eta =8.4$. Other parameters are ${\Delta }_c=0$, ${\Gamma }_{ab}=4.7\times {10}^9{s}^{-1}$, $\kappa =\text{2}\pi \times \text{8}\times {10}^{\text{6}}Hz$, $\aleph =0.25$. For comparison, the transmission spectrum of the atom layer in free space is also plotted in the red dashed curve.

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In a real device, if the atom areal density is improved to $5\times {10}^{12}{m}^{-2}$, which means there is one atom in each circle with a radius of a half wavelength, the transmittance of atom layer in free space is less than 0.1, but the transmittance of atom layer in our combo structure is still around 0.8.

5. Discussion

One of the advantages of our structure is the coupling between atomic transition and the cavity mode could be controlled by the position of atoms. With the controllable coupling, different phenomena could be found.

For the usual EIT system, when the coherent driving field is very weak, a sharp transmission window possesses a tiny linewidth. In this case, the sharp transmission peak is based on quantum interference primarily. If we further decrease the power of the driving field, the FWHM of this narrow peak will decrease and stop at a value which equals to the decoherence rate ${\gamma }_{bc}\approx \kappa$. This quantum interference phenomenon is called coherent population trapping (CPT) [51] and be used to measure the dephasing time experimentally [52]. In our VIT case, if the atoms are located at $z_0=0.033{\lambda }_0$ the cooperative parameter $\eta$ is about $0.31$, which is smaller than 1, the interaction is in the weak-coupling region. The transmission spectrum is plotted in Fig. 7. A small peak can be found in the bottom of the transmission dip, and the FWHM of the peak is equal to the cavity loss $\kappa$. So Fig. 7 is a CPT dominating case in our VIT system [53].

 

Fig. 7 The transmission spectrum of the atom layer in the combo structure. The atom layer is located at $z_0=0.033{\lambda }_0$, which leads to$2g=2.3\times {10}^8{s}^{-1}$and $\eta =0.24$. Other parameters are the same with those in Fig. 5. It shows the phenomena of Coherent population trapping. The FWHM of the small peak is $2\pi \times 8\times {10}^{6}Hz$, which is almost equal to the cavity loss.

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Different with CPT, in which the peak width is controlled by nature of the quantum interference [54], there is another phenomenon called Autler-Townes splitting (ATS) [55–58], which shows two distinguished resonances in the spectrum. And in some situations, there is a crossover from EIT to ATS [59, 60]. As a special case of EIT, VIT also combines two different physical processes, CPT which only modifies the atom states, and ATS which is dominated by an external field [61].

To reach ATS region, the coupling 2g should much larger than the decoherence ${\gamma }_{bc}\approx \kappa$. We set the atom extremely close to the interface of two slabs, $z_0=0.001{\lambda }_0$, and the cooperative parameter $\eta =8.3\times {10}^3$, which is never possible in a normal cavity. The transmission spectrum splits into two independent Lorenz dips and separated by ${\Omega }_c{|}_{n_c=0}=2g=2\pi \times 7.07\times {10}^{9}Hz$ (vacuum Rabi frequency), see Fig. 8. This is the signatures of Autler-Townes splitting. In this case, the cavity loss $\kappa$ is much less than the coherent coupling 2g, so the dressed states dominate the ATS phenomenon [62]. As the density of atoms is low, the lowest value of the transmission dips is far from 0. If we further increase the density of atoms, the transmission dips should close to 0.

 

Fig. 8 The transmission spectrum of the atom layer in the combo structure. The atom layer is located at $z_0=0.001{\lambda }_0$, which leads to${\Omega }_c=2g=2\pi \times 7.0\times {10}^{9}Hz$and $\eta =8.3\times {10}^3$. Other parameters are the same as those in Fig. 5. Two discrete dips demonstrate the Autler-Townes splitting, and the splitting value is equal to ${\Omega }_c=2g$.

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In addition, the ATS phenomenon in our structure can be applied to the single photon switcher. As the combo structure with matching MN and EN slabs is transparency not only for the probe field near ${\omega }_{ab}$ but also for the tunnel mode at ${\omega }_{ac}$, the coupling ${\Omega }_c$ can be improved by the photons through tunnel mode. Therefore one or a few photons in the tunneling mode can block the transmission of the probe field [63–66].

According to ${\Omega }_c=2g\sqrt{n_c+1}$, by inputting one photon at ${\omega }_{ac}$ through tunnel mode ($n_c=1$), the splitting of two transmission dips can change from $2g$ to $\text{2}\sqrt{\text{2}}g$ when 2g is much larger than $\kappa$. We increase the areal density of atom into $\text{8}\times {10}^{12}{m}^{-2}$, so that $\aleph =\text{1}$, and keep other parameters unchanged. The transmission spectrum is plotted in Fig. 9. The blue solid curve refers to the case of $n_c=1$, while the red dashed curve refers to the case of $n_c=0$. It is clear that the splitting of two dips for $n_c=1$ is enhanced to $\text{2}\pi \times 9.9GHz$, which is larger than $\text{2}\pi \times 7.0GHz$ for $n_c=0$. If we set the frequency of the probe field at $\Delta \text{=}\pm \text{2}\pi \times \text{3}\text{.5GHz}$, the transmission will change from 0.02 to 0.96 for the case of $n_c=1$. It behaves as a single photon switch. When the number of photon in the tunnel mode increases ($n_c>1$), the splitting of two transmission dips will also increase with the Rabi frequency ${\Omega }_c$, and then the signal to noise ratio of the switch will be improved.

 

Fig. 9 The transmission spectrum of the atom layer in the combo structure. The blue solid curve refers to the case of $n_c=1$, while the red dashed curve refers to the case of $n_c=\text{0}$. The areal density of atom is $N/\sigma \text{=8}\times {10}^{12}{m}^{-2}$, so that$\aleph =\text{1}$. The atomic layer is located at $z_0=0.001{\lambda }_0$.

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The loss of metamaterials is the main problem with such kinds of structure, because most of metamaterials use metal structure to achieve negative permeability and permittivity. But in recent years, the all-dielectric metamaterials base on Mie resonances give a wonderful solution for low loss [67, 68]. At the wavelength of the infrared quantum dot (1530nm), perfect reflectors with a 99.7% reflectance have been made from a hexagonal lattice of silicon cylinder resonators as the single-negative metamaterials [69]. And a simulation work shows almost zero absorption reflector at visible wavelengths by a single-negative metasurface made from an array of TiO₂ nanoparticle arranged in square lattice embedded in air [70]. We notice that the losses in TiO₂ structures could be as low as ${\text{10}}^{\text{-6}}$ level (the imaginary part of the refractive index) [71], which is close to what we use here. In addition, the result of VIT with different losses when the atom is located at $z_0=0.0001{\lambda }_0$ is shown in Fig. 10. The splitting is still distinguishable (green curve) when the dissipative is${\gamma }_{e,m}=1.67\times {10}^{-5}{\omega }_0$ which is as large as real experiments [72]. The single-atom cooperativity parameters can reach 830 (two orders better than the ordinary cavity) at the dissipative ${\gamma }_{e,m}=1.67\times {10}^{-\text{6}}{\omega }_0$.

 

Fig. 10 VIT transmission spectrum under different losses ($z_0=0.0001{\lambda }_0$). For ${\gamma }_{e,m}=1.67\times {10}^{-4.5}{\omega }_0$, cavity loss $\kappa$ is larger than the decay rate of the atom $\Gamma$, so what we can see is only a transmission dip around $\Delta \text{=0}$. The transmission at $\Delta \text{=0}$ increases when ${\gamma }_{e,m}=1.67\times {10}^{-5}{\omega }_0$. As the dissipations decrease to $1.67\times {10}^{-5.5}{\omega }_0$ or less, the splitting becomes significant and VIT phenomena can be distinguished clearly.

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For the location of the three-level system, it depends on the kinds of system. For quantum dot [16, 73] or quantum well, their position could be controlled by the time of chemical vapor deposition. In meta-atoms and radio frequency metamaterials systems [74], it is much easier since the wavelength is on the scale of a few millimeters. So the meta-atoms made by antennas could be moved by translation stage or by hand directly. In the visible region, sub-wavelength structures in transmission mode have been fabricated by GaN-based integrated-resonant unit elements [75]. In the future, as microfabrication technology advances, the single-negative metamaterial structure with the novel properties and such low loss may be produced at the infrared and visible range.

6. Conclusion

The EIT properties of a three-lever atom located near the interface of pairing single-negative metamaterial slabs are studied. Due to the particular distribution of electromagnetic field mode in the structure, it could be looked as a perfect single mode cavity, and it is much easier to achieve strong coupling than the traditional cavity. Because of the expression of Rabi frequency, ${\Omega }_c=2g\sqrt{n_c+1}$, the transparency of a three-lever atomic layer can be induced by the vacuum mode when the coupling is strong enough. Moreover, by adjusting the position of the atomic layer, we can control the coupling between atoms and the vacuum mode, and that’s why CPT and ATS could be observed. The coupling can be improved by move the atomic layer closer to the interface, and it has little impact on the non-resonant channels. Therefore, we give a demonstration of the optical switch controlled by a single photon. Furthermore, ultrastrong coupling [76, 77] could be achieved in this system easily and it is a possible way to reach deep strong coupling [78, 79].

With the development of microfabrication, the precise location of atom or meta-atom in artificial structures is possible in the experiment, and various kinds of low loss metamaterials have been come up or made [67, 80–83]. Besides, single-negative metamaterials in the visible frequency have been designed in recent years [25, 28, 29, 70]. In addition, this model is also applicable to the microwave frequencies, which is easy to implement experimentally [74]. Based on these technological advances and application needs, this VIT structure has promising applications on the integrated devices in quantum memory and optical switch.