1. Introduction

The phenomenon of electromagnetically induced transparency (EIT), a typical quantum interference effect, has been extensively investigated mainly in atomic gases [1]. Usually, EIT is observed in a three-level atomic system with a Λ configuration exposed to a strong controlling field and a weak probe field, where two ground states are linked to a common excited state. Using EIT, the absorption of the probe field can be largely suppressed even it is tuned to a strong one-photon resonance. Except for the large suppression of optical absorption, EIT may also result in substantial alternations of many other physical properties, including the drastic change of dispersion that can significantly reduce group-velocity and hence result in slow (subluminal) light.

In addition to the single EIT mentioned above, a double EIT has also been observed in a four-level atomic system exposed to three laser fields with a N-type level configuration [1]. In this case, two strong laser fields, named as the controlling and assisted fields, respectively, determine the quantum interference property of the probe field. The double EIT is very useful for optical and quantum information processing and manipulation since the information can be stored and retrieved in multiple channels separately.

Many important applications of EIT have been proposed, including lasing without population inversion [2], enhanced nonlinear optical effect [1, 3], quantum computation and telecommunications [4, 5], quantum memory [6], all-optical switching [7], ultraslow optical solitons [8, 9]. However, the requirement of low-temperature environment and the large size for atomic EIT severely hamper the implementation of EIT in chip-scale applications.

Recently, much attention has been paid to the study of the classical analogues of three-level atomic EIT in various systems, such as coupled resonators [10, 11], electric circuits [12], plasmonic structures [13–16], and so on. Particularly, the plasmonic analogue of EIT in metamaterial structures [17,18] have attracted growing interest because of their effective medium characters. EIT metamaterials can not only work in the terahertz regime but also can be used to design chip-scale optical devices. Important applications of EIT metamaterials have been proposed, including optical buffers, highly sensitive sensors, ultrafast switches, and electromagnetically induced absorbers, etc [19–21]. However, to authors’ knowledge a classical analogue of EIT in four-level atomic system with N-type level configuration has not been realized up to now.

On the other hand, the control of light speed is highly desirable for many practical applications. Although subluminal light has been demonstrated in atomic systems based on EIT [1] and a superluminal light (or called as fast light [22]) has been reported by using the active Raman gain schemes [23–26], a practical system supporting both subluminal and superluminal light propagations are still rare. Recently, some theoretical proposals showed the possibility of changing light propagation from subluminal to superluminal in atomic systems [27]. However, in these proposals, either coupling of the dipole transition forbidden levels are required or some very special level structures are used, which make them hard to be realized practically.

In this article, we propose a scheme to design a new type of optical metamaterial that can mimic the functionality of a four-level N-type atomic system with EIT. Different from the most EIT metamaterials realized up to date where only a single EIT is possible and can only be controlled passively by varying the geometrical parameters of resonator structures [28–34], in our scheme a transition from a single EIT to a double EIT of terahertz radiation may be easily achieved by actively tuning the intensity of the infrared pump field or passively tuning the geometrical parameters of resonator structures. Additionally, in our EIT metamaterial the group velocity of a terahertz radiation can be changed from subluminal to superluminal by adjusting the pump field intensity. The scheme suggested here may be used to construct chip-scale slow and fast light devices and to realize rapidly responded switching of terahertz radiation at room temperature.

2. EIT metamaterial equivalent to N-type atomic system

The unit cell of present EIT metamaterial structure consists of a cut wire (CW) and two pairs of split-ring resonators (SRRs). As in Ref. [14], we assume Si islands (denoted by the red spots) are positioned in gaps of one SRR-pair, as shown in Fig. 1(a). As the resonances of CW and SRR are both determined primarily by their geometrical dimensions and the size of the subwavelength unit cell ensures that terahertz waves do not diffract at normal incidence, thus the EIT metamaterial array behaves as an effective medium. An incident terahertz beam is collimated on the metamaterial array with the electric field parallel to the CW. An infrared pump beam with a spot diameter being much larger than the terahertz beam diameter enables a uniform excitation aperture for the terahertz transmission. The aim of introducing such optical pump beam is to change the electrical conductivity of the Si islands and hence to actively control characters of the EIT metamaterial. Fig. 1(b) shows a possible experimental arrangement of the optical pump-terahertz probe measurement.

 

Fig. 1 (a) Schematic of the unit cell. The plasmonic system consists of a bright element (CW) and two dark elements (SRRs). The geometrical parameters are L = 85, w = 5, a = 29, b = 5, d₁ = 10, d₂ = 14, h = 495, P_x = 80 and P_y = 120μm, respectively. The right hand of panel (a) is microscopic image of the fabricated metamaterial. (b) Possible experimental arrangement of the optical pump-terahertz probe measurement. (c) Equivalent atomic model. Energy level structure and excitation scheme of the lifetime-broadened four-level N-type atomic system with upper energy level |4〉, |3〉 and lower energy levels |1〉, |2〉. Δ₂, Δ₃ and Δ₄ are detunings. Ω_p, Ω_c, and Ω_d are half Rabi frequencies of the probe, control, and assisted fields, respectively.

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The CW resonator array has a typical localized surface plasmon (LSP) resonance at a frequency within the terahertz regime, while the SRRs support an inductive-capacitive (LC) resonance at the same frequency. In the EIT metamaterial the CW is directly excited by the incident radiation along the CW. However, the SRRs are weakly coupled to the radiation due to the perpendicular orientation of the electric field. The near field coupling between the CW and SRRs excites the LC resonance in the SRRs. Thus, the CW and SRRs serve as the bright and dark modes, respectively, for the radiation excitation.

Now we adapt the widely used coupled Lorentz oscillator model to analyze the near field interaction between the two elements in an EIT metamaterial unit cell. Each cell can be regarded as a four-level atom involving a ground state |1〉 and three excited states |2〉, |3〉, and |4〉 with N-type level configuration (Fig. 1(c)). The LSP bright mode resonance in the CW corresponds to the dipole-allowed transition |1〉 → |3〉. The first SRR-pair simultaneously couples to the CW and the second SRR-pair corresponding to the transitions |3〉 → |2〉 and |2〉 → |4〉, respectively. The coupling between the CW and the second SRR-pair is very weak and hence can be neglected. A destructive interference may occur between the pathways |1〉 → |3〉 and |1〉 → |3〉 → |2〉 → |3〉, and between the pathways |1〉 → |3〉 and |1〉 → |3〉 → |2〉 → |4〉 → |2〉 → |3〉, which can result in the double EIT phenomenon.

It is instructive to first analyze the quantum destructive interference character in the four-level atomic EIT system shown in Fig. 1(c). A weak probe field of angular frequency ω_p and wavevector k_p, i.e., E_p(r, t) = e_pℰ_p(r, t)e^{i(k_p·r−ω_pt)} + c.c., drives the transition |1〉 → |3〉; a strong control field of angular frequency ω_c and wavevector k_c, i.e., E_c(r, t) = e_cℰ_c(r, t)e^{i(k_c·r−ω_ct)} + c.c., drives the transition |2〉 → |3〉; in addition, a strong assisted field of angular frequency ω_d and wavevector k_d, i.e., E_d (r, t) = e_d ℰ_d (r, t)e^{i(k_d·r−ω_dt)} +c.c., drives the transition |2〉 → |4〉. Here e_c, e_p, and e_d are, respectively, unit vectors denoting the polarization of the control, probe, and assisted laser fields, with ℰ_c, ℰ_p, and ℰ_d being their corresponding envelopes. Under electric-dipole and rotating-wave approximations, the equations of motion governing atomic dynamics read

Equation (1) can be solved by searching for the solutions a_j = â_je^−iλt with the eigenvalue λ. After substituting these solutions into Eq. (1), we obtain the equation

Now we turn to study the interference property of the metamaterial introduced above. Such metamaterial system can be described by the coupled Lorentz oscillator model as did in Refs. [13, 14]

The solutions of Eq. (5) can be searched in the form x_j(t) = x̃_j(ω)e^−iωt + c.c. (j = 1, 2, 3). It is easy to obtain

The propagation dynamics of the envelope E₀ of the terahertz radiation in the metamaterial can be described by Maxwell equation. Under the slowly-varying envelope approximation, the equation reads

In our numerical example, the system parameters are chosen as κ₁ = 0.1 THz², κ₂ = 0.05 THz², γ₁ = 0.05 THz, γ₂ = 0.025 THz, and γ₃ = 0.025 THz for the absence of the infrared pump field. When the infrared pump field is applied, the parameters γ₁, γ₃, κ₁, and κ₂ do not change significantly, while the damping rate γ₂ is increased obviously from 0.025 THz (corresponding to the pump power 0 mW) to 0.271 THz (corresponding to the pump power 1350 mW). Physically, the incensement of γ₂ is caused by the increased conductivity of the Si islands under photoexcitation [14].

Shown in Fig. 2 is the normalized susceptibility χ/χ₀ as a function of frequency and pump-field intensity, with χ₀ being the normalization constant. The real part Re(χ/χ₀) (blue dashed line) and imaginary part Im(χ/χ₀) (red solid line) of χ/χ₀ characterizes the dispersion and absorption properties of the terahertz radiation, respectively. Fig. 2(a) is for the case without the infrared pump field. We see that two transparency windows corresponding to the double EIT.

 

Fig. 2 The real part Re(χ/χ₀) (blue dashed line) and imaginary part (red solid line) of the normalized susceptibility χ/χ₀ as a function of frequency and the pump-field intensity. The power of the infrared pump field is taken as 0 (a), 75 mW (b), 320 mW (c), and 1350 mW (d), respectively.

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When increasing the pump-field power to 75 mW, one has γ₂ = 0.041 THz, the absorption of the terahertz radiation near the resonance becomes weaker than that in Fig. 2(a), as shown in Fig. 2(b). The separation between the two EIT transparency windows becomes also narrower. However, it becomes flatter in comparison with Fig. 2(a). The slope of the dispersion becomes zero when the power of the pump field arrives to 320 mW, which leads to γ₂ = 0.083 THz, as shown in Fig. 2(c).

As the pump field arrives the power 1350 mW, the absorption of the terahertz radiation near the resonance frequency is almost vanished, as shown in Fig. 2(d). In this situation, two EIT transparency windows are merged into one corresponding to the single EIT.

Besides the active control for realizing the transition from the double EIT to the single EIT (thus the crossover from the superluminal to the subluminal propagations, see in the next section) by changing the infrared pump field, we can also realize such transition by passively changing the geometrical parameters of resonator structures. Specifically, we can vary the distance d₂ between two SRR pairs, which is associated with the coupling coefficient κ₂. Decreasing d₂ can increase κ₂, corresponding to the increase of the half Rabi frequency Ω_d in the N-type atomic system.

Shown in Fig. 3 is the normalized susceptibility χ/χ₀ as a function of the frequency and κ₂. The other parameters are the same with those used in Fig. 2. Fig. 3(a) shows the case of κ₂ = 0. We see that only one EIT transparency window is opened, corresponding to a single-EIT. When increasing κ₂ [see Fig. 3(b)–(d)], a small absorption appears in the center of the EIT transparency window and the single EIT transparency window splits into two EIT transparency windows gradually. That is to say, the single EIT is transferred into the double EIT when κ₂ is increased.

 

Fig. 3 The real part Re(χ/χ₀) (blue dashed line) and imaginary part Im(χ/χ₀) (red solid line) of the normalized susceptibility χ/χ₀ as a function of frequency and the coupling coefficient κ₂. Coupling coefficient κ₂ is taken as 0 (a), 0.015 THz² (b), 0.035 THz² (c), and 0.045 THz² (d), respectively.

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3. Transition from subluminal and superluminal propagations of terahertz radiation

We now show that the EIT metameterial proposed here can be used to realize subluminal propagation, superluminal propagation, and their transition of terahertz radiation in a single metamaterial system. Note that the group velocity of the system can be expressed as V_g = c/n_g, where the group index n_g is defined by

If n_g > 1, the group velocity of the radiation is smaller than c, thus the propagation of radiation is subluminal. However, if the 0 < n_g < 1 or n_g < 0, the group velocity of the radiation is larger than c or it becomes negative, thus the propagation of radiation is superluminal. Particularly, a negative group velocity means that the peak of the emerging pulse occurs at an earlier time than the peak of the pulse at the entrance to the medium [22].

Fig. 4(a) and Fig. 4(b) show the group index n_g and the absorption coefficient $\alpha =\frac{{\omega }_0}{2c}\text{Im}\left(\chi \right)$ as functions of the power of the infrared pump field, respectively. From Fig. 4(a), we see that with the increase of the pump power the group index changes from −350 to nearly 50. The region n_g < 1 corresponds to the superluminal propagation and the region n_g > 1 corresponds to the subluminal propagation. We get n_g = 1 (i.e. the group velocity equals c) when the pump power is about 320 mW.

 

Fig. 4 (a) The group index n_g as a function of the pump power. (b) The absorption coefficient α as a function of the pump power. (c) The superluminal propagation (V_g = −3.3 c) of the terahertz radiation with pump power 300 mW. (d) The subluminal propagation (V_g = 0.02 c) of the terahertz radiation with pump power 1350 mW. The initial condition is E = E₀e^{−0.25(t/τ)2} with τ = 10⁻¹² s.

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From Fig. 4(b), we see that α decreases rapidly with the increase of the pump field power. Thus to avoid a large absorption while ensuring the superluminal propagation, a weak pump field is necessary. To this end, we can use a pump field with power a little less than 320 mW, say 300 mW. At this power, α = 1.50 cm⁻¹ while the group velocity is −3.3 c. On the other hand, the incident radiation propagates only a very short distance (the thickness of a metamaterial structure is several hundred micrometers) to pass the medium. Thus, the small absorption accompanied by the superluminal propagation will not lead to a serious attenuation of the terahertz radiation. When the power of the pump field is increased to 1350 mW, the absorption is decreased to α = 0.78 cm⁻¹. Thus, the group velocity varied from −3.3 c to 0.02 c can be realized by increasing the pump power from 300 mW to 1350 mW with only a small absorption.

Fig. 4(c) (Fig. 4(d)) shows the superluminal (subluminal) propagation with V_g = −3.3 c (V_g = 0.02 c) of the terahertz radiation for pump power 300 mW (1350 mW). When plotting the figure, the parameters of the system are chosen as the same as used in Fig. 2. The propagation length is chosen as 0.05 cm. The initial condition is chosen to be E = E₀e^{−0.25(t/τ)2} with τ = 10⁻¹² s. As expected, an evident time advancement −0.5 × 10⁻¹² s is observed in Fig. 4(c), and an evident time delay 83.3 × 10⁻¹² is obtained in Fig. 4(d). However, in both the superluminal and subluminal propagations, the terahertz pulse have slight attenuations due to the existence of small absorption in the system.

4. Summary

We have proposed a scheme for designing a new type of optical metamaterial that can mimic the EIT characters of four-state atomic systems of N-type energy-level configuration. We have demonstrated that in such metamaterial a transition from a single EIT to a double EIT of terahertz radiation may be easily realized by actively tuning the intensity of the infrared pump field or passively tuning the geometrical parameters of resonator structures. In addition, the group velocity of the terahertz radiation can be varied from subluminal to superluminal by changing the pump field intensity. The scheme suggested here may be used to construct chip-scale slow and fast light devices and to realize rapidly responded switching of terahertz radiation at room temperature.