## Abstract

Two independent significant developments have challenged our understanding of light-matter interaction, one, involves the artificially structured materials known as metamaterials, and the other, relates to the coherent control of quantum systems via the quantum interference route. We theoretically demonstrate that one can engineer the electromagnetic response of composite metamaterials using coherent quantum interference effects. In particular, we predict that these composite materials can show a variety of effects ranging from dramatic reduction of losses to switchable ultraslow-to-superluminal pulse propagation. We propose parametric control of the metamaterials by active tuning of the capacitance of the structures, which is most efficiently engineered by embedding the metamaterial structures within a coherent atomic/molecular medium. This leads to dramatic frequency dependent features, such as significantly reduced dissipation accompanied by enhanced filling fraction. For a Split-ring resonator medium with magnetic properties, the associated splitting of the negative permeability band can be exploited for narrow band switching applications at near infrared frequencies involving just a single layer of such composite metamaterials.

©2008 Optical Society of America

The previous decade has seen the emergence of a new class of electromagnetic materials, called *metamaterials* [1] that are essentially composite materials, usually structured on length scales much smaller than the wavelength of radiation. Metamaterials can exhibit a host of novel properties such as negative dielectric permittivity (*ε*) [2], negative magnetic permeability (*µ*) [3] and negative index of refraction, [4] as well as large positive magnitudes of the above material parameters. Such extraordinary properties of the metamaterials arise primarily due to a clever design of the underlying structure leading to an electromagnetic resonance at a desired frequency. Depending on the specific geometry, the structures are usually independently driven by either the electric field or the magnetic field of the radiation resulting in enhanced dielectric or magnetic response, respectively. Effects ranging from electromagnetic invisibility [5] and negative refraction [6] to imaging with resolution beyond the classical diffraction limit (Super-lens) [7] are based on these properties. Most metamaterials have carefully structured metallic inclusions: as electromagnetic waves propagate through them, they drive currents in loops which is accompanied by dynamic accumulation of charges within the structure leading to an L-C resonance, that is responsible for all the novel features of metamaterials. Initial proof-of-principle experiments on metamaterials and the investigation of the above effects were performed with microwaves [4, 6] and radiowaves [8]. In the past few years, however, metamaterials with such enhanced properties have been demonstrated across the electromagnetic spectrum, from zero frequency [9] to optical frequencies [10].

Metamaterials for future applications will be required to be highly transparent (low-loss), tunable, reliable, and perhaps even reconfigurable. One of the outstanding challenges has been to make the metamaterials less absorptive which is crucial to most applications. Although much attention has been paid to the use of low loss materials in the construction of the metamaterial, the very resonant nature that gives rise to its unique properties also renders it intrinsically dissipative and dispersive [1]. On the other hand, we have well known counterintuitive phenomena in the area of *coherent control*, where highly lossy resonant atomic/molecular media can be rendered literally transparent [11]. This is achieved by careful engineering of the quantum pathways for the desired transition such that an exact cancellation of the transition probability occurs over the frequency range of interest. This effect is a macroscopic manifestation of an intrinsically quantum mechanical effect, and is widely known as *electromagnetically induced transparency* [12, 13] (EIT). The essential requirement for EIT is a strong coherent control radiation field that drives the atoms or molecules into coherence across the appropriate quantum levels. The creation of quantum interference [14] in atomic media results in myriad counter-intuitive effects like ultra-slow light [15], superluminal light [16], making an otherwise opaque medium transparent [12, 13] and demonstration of ultra-high index of refraction [17]. The quantum interference effects arise from a careful creation of competing pathways for the electron excitation within the atom, that can be maneuvered to interfere either constructively or destructively [18]. Note that coherent control intrinsically arises due to the quantum nature of the atoms while the properties of metamaterials can be derived from purely classical considerations. Here we present an approach that combines the central ideas of coherent control with those of metamaterials resulting in a new class of actively controllable, low-loss composite metamaterials.

We now show that the properties of a coherent atomic or molecular medium can be utilized to manipulate the metamaterial properties by embedding the metamaterial structural units in the coherent medium. The capacitance of the metamaterial units depends on the dielectric permittivity of the embedding coherent medium, which can be actively manipulated by a control field applied at an appropriate frequency where the metamaterial is transparent due to a pass band (see Fig. 1). Each metamaterial unit can be modeled as a resonant L-C circuit and its resonance frequency can be accurately tuned via its capacitance. The properties of the metamaterial can be most effectively controlled via the dielectric properties of the *resonant* atomic medium driven to coherence. Here we demonstrate two magnetic metamaterial designs that operate in two distinct frequency regimes and offer enormous control over radiation: from complete tunability of the metamaterial parameters and drastic reduction of losses, with the possibilities of ultra-slow and superluminal light to extremely narrow band metamaterial switches.

First, we present coherent control of the ubiquitous metallic Split-ring resonator (SRR) medium [3] at mid-infrared frequencies. For incoming radiation with the magnetic field along the axis of the SRR, the induced currents in the SRR have a resonance at a frequency *ω _{m}* given by
${\omega}_{m}=1/\sqrt{LC}$, where

*L*is the net inductance and

*C*is the net capacitance of the SRR units. The effective magnetic permeability of the SRR medium is reasonably described by [19]

where *f _{m}* is the filling fraction, Γ

*governs the dissipation rate, and are both related to the structural parameters of the SRR [19]. The capacitance of the SRR arises principally from the capacitive gaps in the loop [Fig. 1(a)]. Now, we consider the SRR to be embedded in a frequency dispersive (resonant) dielectric medium. The capacitance strongly depends on the frequency, thus drastically changing the magnetic response of the SRR.*

_{m}A coherent atomic medium with a Λ-level structure [see Fig. 1(a)] when driven to EIT, offers a dielectric permittivity [14]

to the probe radiation at frequency *ω* close to the atomic resonance frequency *ω*
_{1}, where Ω* _{c}* is the Rabi frequency of the control field and

*γ*

_{13}is the decay rate between the atomic levels |1〈 and |3〉 while

*κ*is the oscillator strength that can be controlled via the atomic number density. The capacitance of the SRR, when such a coherent atomic medium is placed within the capacitive gaps, is approximately given by

*C*(

*ω*)=

*ε*, where

_{0}ε(ω)A/d*A*and

*d*are the area and width of the capacitive gaps in the SRR, respectively. The resonant interaction of the magnetically driven SRR with the driven dielectric material within the capacitive gaps gives rise to entirely new resonant features. Further, each new resonance has an effective width Γ(

*ω*) and a renormalized filling fraction

*f*(

*ω*) that differ significantly from the bare SRR medium.

Consider the dispersion of *µ*
_{eff} shown in Fig. 2(a) for the composite medium. The embedding dielectric medium is assumed to have a Lorentz dispersion centered close to the magnetic resonance frequency of the bare SRR structure (*ω*
_{1}=73.6THz,*ω _{m}*=74.9THz,

*γ*

_{13}=0.24GHz and Γ

_{m}=4.8THz). This Lorentz dispersion corresponds to Ω

*→0 in Eq. (2). Such a dispersion can be readily realized by either using resonant quantum dots or choosing a Raman transition (see for example [20]) with a strong pump field such that the Raman resonant probe frequency lies in the vicinity of the magnetic resonance frequency. Strong modulation of*

_{c}*µ*

_{eff}is obtained when the dielectric resonance frequency gets close to the bare SRR resonance frequency. The large dispersion, with the consequent large group indices obtainable shows the possibility of a new class of media for ultra-slow or superluminal light, wherein the magnetic interaction with the medium determines the passage of light.

Similarly in Fig. 2(c), we show the behaviour of *µ*
_{eff} for SRR with an embedding medium exhibiting EIT, where the control field at a lower frequency propagates in the metamaterial due to a passband. For the parameters of Fig. 2(c), the single broad magnetic resonance peak has split into three peaks, where all the peaks have significantly narrower widths indicating drastically reduced dissipation. The spacing of the peaks is mainly determined by the strength of the control field, while the extent of modulation in *µ*
_{eff} depends on the proximity of the EIT line center with the bare SRR resonance frequency. Thus, complete control over the dispersion and dissipation of the metamaterial can be attained via the control field.

The most well-known counterintuitive aspect of EIT, namely zero dissipation at the EIT line center, manifests here as drastically reduced dissipation (Γ_{eff}) in the metamaterial [Fig. 2(d)]. Such reduction of the dissipation occurs primarily due to vanishing currents in the SRR loops and is also reflected in the reduction of the filling fraction [Fig. 2(d)]. It appears as if the EIT aided effective magnetic medium mimics vacuum at these frequencies, where *µ*
_{eff}→1 around the EIT line center. To further validate these effects, we present the results of photonic bandstructure calculations based on the transfer matrix method [21] in Fig. 2(b) for two-dimensional SRR with the cross-sections shown in Fig. 1(a) with *a*=600nm, *b*=312nm, *L*=144nm, *d*= 24nm, and *D*=24nm. The band dispersions [Fig. 2(b)] agree reasonably with those predicted by the approximate analytic formulae [Eq. (1) & Eq. (2)] although the exact locations of the computed bandgaps, due to *µ*
_{eff}<0 are shifted.

Since EIT media are very popular at optical frequencies, we consider SRRs made of silver (shown in Fig. 3) with resonance frequency of about 272 THz. The dielectric permittivity of silver used in the simulations is obtained from Ref. [22]. Coherent control in this structure is realized by immersing the SRR metamaterial in metastable helium gas [23] that exhibits EIT at ~277 THz (*λ*=1083 nm) [24], in the vicinity of the metamaterial resonance. The control laser frequency ~425 THz (*λ*=706 nm), used to obtain EIT, lies within a (higher) pass band of this metamaterial. Infusion of metastable helium gas into the metamaterial results in a resonant response, and the formation of a corresponding bandgap of about 50 GHz near 277 THz [see Fig. 4(a)]. Switching on the control field (Ω* _{c}*=10

_{γ21}=24 GHz) further splits this bandgap, creating two nearly dispersion-free bands near the EIT line center [see Fig. 4(b)].

The bandwidths of the gaps and the dispersion of the allowed bands are all controllable by the amplitude and the frequency of the control radiation. For example, a blue-shifted control laser would require red-shifting the probe laser in order to maintain the two-photon resonance condition, thus providing the possibility to align the EIT resonance at any desired location in/around the bandgap. Wider bandgaps would be obtained as the EIT line center gets closer to the bare SRR resonance frequency. In Fig. 4(c), we show the reflectance and the transmittance of light incident on a slab of SRR medium composed of just one unit cell layer, with and without the EIT control field. Note the sharp switching of reflectance and transmittance at specific frequencies that are entirely governed by the control field parameters. In particular, around 277 THz, the transmittance drops dramatically from nearly 95% to a few percent within about 10 GHz bandwidth indicating its potential use for an extremely narrow-band switching applications. Recently *non-resonant* rescaling of capacitance resulting in the tuning of the resonance frequency by up to 20% has been demonstrated at about 1THz [25]. We would like to note the following: first, we have assumed that the EIT effect is itself not hampered by the metamaterial. This assumption is well justified as we have chosen the EIT control field to lie within a propagating band and well within the first Brillouin zone, hence the control field would be uniform across the medium. The introduction of the metastable atomic gas within the metamaterial structure will not affect the atomic decay rates significantly. Second, we have not attempted to homogenize the structure that is less than a quarter of the free space wavelength. We have calculated and presented only the photonic bandstructure and transmission/reflection properties of this metamaterial in Fig. 4.

Before we conclude, we discuss the effects of inhomogeneous broadening caused by imperfect fabrication of the metamaterial units. We do require the nanostructures to have high quality (and low surface roughness). Present day FIB based nanotechnology does have the technical capability to meet such requirements. However, the thickness of the capacitive gaps is extremely critical and could be slightly different in different units resulting in a distribution of the values of the resonant frequency. This is a particularly important issue for the nanostructures discussed here. We have analyzed this by including inhomogeneous broadening due to the imperfect manufacturing by assuming a Gaussian distribution for the resonant frequency *ω _{m}*, with a width comparable to the bare SRR linewidth (Γ

*), and found that all the above narrow band effects continue to remain nearly unaltered. This robust response we believe is due to the enhanced frequency dependent filling fraction that remains independent of ω*

_{m}*, this can be easily seen by substituting Eq. (2) in Eq. (1) and identifying the renormalized filling fraction. This effect far out-weighs the effects of inhomogeneous broadening and the narrow peak due to EIT remains unaffected. We have also undertaken similar calculations involving cut wire metamaterials that rely on plasmonic resonances for their dielectric or magnetic response, these can also be coherently controlled (whose details will be published separately elsewhere). Collections of silver or gold nanorods can be accurately fabricated with little size dispersion and such metamaterials will hence be even less prone to the effects of inhomogeneous broadening.*

_{m}We have demonstrated here a new paradigm of parametric control of metamaterials. The metamaterial response is completely transformed by introducing coherently controllable resonant media into the design. Our results provide a new mechanism to reducing dissipation inherent to metamaterials, which has been the main hindrance in developing applications. Despite the control scheme being inherently dispersive, we stress that it is entirely tunable and can be used effectively for a variety of narrow band applications. Our results bring together two powerful, yet disparate ideas to manipulate on-demand the optical properties of such composite materials.

## Acknowledgment

The authors thank Srihari Kesavamurthy for discussions, and Sir John Pendry for critical comments, particularly pointing out the issue of inhomogeneous broadening. SAR thanks the Department of Science and Technology (India) for funding.

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