Investigation On Negative Refractive Index Based On The Spontaneously Generated Coherence

The development of left-handed materials with negative refractive indices, which have both negative dielectric permittivity and negative magnetic permeability in a certain frequency region, has encountered a lot of difficulties. In the 30 years after the concept of left-handed material was put forward, few people has paid attention to it because no one can find real left-handed materials in the world, although they have many novel and interesting properties, such as negative group velocity, inverse Doppler shift, abnormal Cerenkov radiation, and perfect imaging. It is not until the beginning of the 21st century that this situation has a favorable turn.In 1999, Pendry, one British scientist, proposed a scheme to simultaneously realize negative dielectric permittivity and negative magnetic permeability, which makes it possible to realize left- handed materials in experiment.Pendry first designed a micro-structure unit, metallic split ring resonators (SRRS), which can produce strong magnetic response. One SRRS is composed of two concentric ring resonators with small openings. When a magnetic field vertical to the SRRS is applied, according to the Faraday's law of electromagnetic induction, we know that the current and then the inductance will be induced. In addition, due to the space between the inner and outer rings, the capacitance will also exist. Then one can achieve LC resonance related to the geometry size and shape of SRRS, and the magnetic permeability of an array of periodic SRRS can be expressed asPendry also presented a method of realizing negative dielectric permittivity. He found that an array of periodic conductive wires is similar to the plasma as far as its electromagnetic response is concerned. Plasma is a neutral system composed of a large number of charged particles moving almost freely, and its permittivity is given by where is the eigen-frequency of a plasma. Clearly, whenω<ωp, it will have negative values. Usually the eigen-frequency of metallic plasma is in the UV band, so one can achieve negative permittivity in the visible optical band. To facilitate the macroscopic management, one need to realize negative permittivity in the microwave band. This requires much lower eigen-frequency of plasma. An array of periodic metallic wires has very low effective electron density n_e and very high effective electron quality m_eff, thus has very low eigen-frequency of plasma, This allows us to achieve negative permittivity with reasonable values in the microwave band.Based on this scheme, several groups in the world have observed the special electromagnetic properties belonged to only left-handed materials, which provides important experimental support for the relevant theories.In 2003, a group in Turkey's Bilkent University realized in experiment the two-dimensional photonic crystal based negative refraction phenomenon using a square lattice structure composed of the sapphire coshes. As a comparison, they repeated their experiment with the plate composed of ordinary polystyrene ball to replace the photonic crystal. Once again, the experimental results agreed well with numerical simulations by showing the negative refraction phenomenon.In the above, we have briefly introduced two different methods for realizing left-handed materials in the microwave band with the classical electromagnetism. In the following, we will introduce a third method for realizing left-handed materials in the optical band, which is based on the electromagnetically induced transparency theory in quantum optics. The basic idea is to utilize quantum coherence effect and local field effect in dense atomic gases to control the permittivity and magnetic permeability of two different transitions so that the two quantities simultaneously have negative values in the same frequency region.This method was first separately proposed by Oktel et al. and Shen et al. where they consider a simple three-levelΛsystem. The energy space between the lowest and the middle levels is the same as that between the middle and the highest levels, and the lowest and middle levels have the same parity while the highest one is different from them in parity. Thus, the system has one magnetic-dipole transition and one electric-dipole transition at an optical frequency. It is well known that the coupling coefficient for the electric field isΛ 1 /α(α=137 is the fine structure constant) times larger than that for the magnetic field when an optical wave interacts with an atomic system, so one must find a proper way to make the density matrix element on the magnetic -dipole transition much larger than that on the electric-dipole transition so that the permittivity and the magnetic permeability on the two transitions are of the same order. It has been verified that the technique of quantum coherence based on electromagnetically induced transparency can well fulfill this task. To ensure the magnetic permeability and the permittivity negative simultaneously, the atomic density must be large enough so that the local field effect arising from dipole-dipole interaction has to be taken into account.In 2006,Thommen et al further considered how to realize the left-handed material with a four-level atomic system via the technique of quantum coherence. Compared with the three-level system, the four-level system is much more feasible for performing corresponding experiments due to the freedom in energy level choice.In this thesis, we further develop the technique of quantum coherence of realizing left-handed materials with negative refractive index, i.e., simultaneously consider the spontaneously generated coherence (SGC) and the laser induced coherence. Here SGC refers to the quantum interference between two partially overlapping decay channels when one atom goes from a doublet of closely- lying levels to a common level or vice versa. Consequently, the two closely-lying levels become quantum correlated via the partially indistinguishable spontaneous emission process. In the recent two decades, people have carried out a lot of work on SGC, and found that SGC can be utilized to control the optical properties of relevant media, such as emission, absorption, and dispersion.Now, we begin to treat in detail a dense atomic gas composed of the–type four-level atoms with SGC as shown in Fig. 1.For dense atomic gas, we should take into account the local field effect arising from dipole- dipole interaction between atoms because there are many atoms in a region of one wavelength. In fact, the local field effect can greatly reduce the atomic gas density required by the realization of negative refractive index so that it can have reasonable values. That is, the local field effect is very helpful for relevant experimental study.First, with the dipole approximation and the rotating-wave approximation, we start from the interaction Hamiltonian and the master equation of density operator to achieve a set of equations for density matrix elements. Then, via both analytical expressions in the limit of weak probe and full numerical calculations, we investigate in detail the effect of SGC, local field effect, and the external coherent field on permittivity and magnetic permeability. We find that, when the atomic density is large enough, the real parts of permittivity and magnetic permeability can have negative values simultaneously in a certain frequency region, which means that the left-handed material with negative refractive index can be realized (see Fig.2).Compared with those atomic systems without SGC, the required atomic density for realizing negative refractive index can be reduced about one order in our atomic system. In particular, for our atomic system, the negative refractive index may be accompanied by little absorption in case relevant parameters have appropriate values.We also briefly discussed how to effectively simulate SGC in the dressed state representation with real atoms, since it is difficult or impossible to find real atoms with SGC in the world.Our research has further developed the coherent atomic gas method for realizing left-handed materials. We have considered SGC for the first time in the research of left-handed materials, which then led to several new conclusions, and also added new freedom for controlling negative refractive index and other optical properties.