Theoretical Investigations On Negative Refractive Index And Quantum Phase Gate With Atomic Coherence

This thesis for doctorate is mainly to investigate the negative refraction based on spontaneously generated coherence (SGC) and electromagnetically induced transparency (EIT) and to achieve efficient XPM through coherently enhanced Kerr nonlinearity in atomic system with quantum interference effect. My thesis contains three parts as followed.Electromagnetically induced negative refraction in an atomic system with spontaneously generated coherenceIn 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.We investigate a dense gas of four-level atoms with spontaneously generated coherence (SGC) as shown in Fig.1. In this system, the two lower 1 and 2 have the same parity, as well as the two closely-lying upper levels 3 and 4 are opposite in parity to the two lower levels. 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. By the full numerical calculations, we investigate in detail the effect of SGC, local field effect, and the external coherent field on the dielectric permittivity and the magnetic permeability.At first, we investigated the permittivity and the magetic permeability as a function of the probe detuning in the dense atomic system when the SGC exist or not. From Fig.2 we can see that when SGC is maximal, the real parts ofεrandμrare simultaneously negative in a small frequency band aroundΔp= 0. That is, the SGC effect greatly enhances the magnetic response of a dense atomic gas so that one can realize negative refractive index at much lower atomic densities.Then, we study the effect of external coherent field on the achieving LHM in the case of maximal SGC. It can be found that EIT resulted from laser induced atomic coherence is quire important for realizing negative refraction. Thus the coupling strength should be much stronger than the coherence decay rate on the magnetic transition, so that the EIT effect is strong enough and the magnetic permeability has negative values in the center of the EIT window. On the other hand, the coupling strength should not be infinitely large, otherwise the electric permittivity may become positive, which is surely not favorable for achieving negative refraction. In addition, we have discussed the influence of local field effect on the electric permittivity and the magnetic permeability. It is well known that, the local field effect, whose strength directly depends on the atomic density, plays a significant role in achieving large enough magnetization and the negative refraction. By the numerical calculation, the saturating atomic density is about 10²⁴ m^-3.Finally, we have investigated that the SGC influence the negative refractive index as two incoherent fields interact with transitions｜2〉←→｜3〉and｜2〉←→｜4〉, respectively. Fig.3(a) and Fig.3(b) show that there is a more remarkable SGC when two weak incoherent pumps are applied, which lead to the negative refraction accompanying with little absorption aroundΔ_p= 0.We know that it is hard or impossible to find a real atomic system with near-degenerate levels having non-orthogonal dipoles, so our considered scheme is feasible to carry out the negative refraction based on SGC in the partially dressed state picture of a coherently driven field. Moreover, we also can engineer suitable energy schemes to realize negative refraction in semiconductor quantum-well structures, in which quantum coherence similar to SGC appears.Coherent control of negative refraction based on local-field enhancement and dynamically induced chiralityIn this part, we theoretically investigate the optical properties of a multi-level close-loop atomic system, and discuss how to achieve the negative refraction using the chirality based on electromagnetically induced coherence. Using the semi-classical theory of atom-field interaction, we study in detail the complex refractive index by the numerical calculation.This atomic configuration as shown in Fig.4, which is composed of two basic EIT system which are coupled by the coherence termρ₁₂: one is in the Ladder configuration (see Fig.4(b)), while the other is the Lambda configuration (see Fig.4(c)). In the Ladder system, the probe magnetic transition｜1〉←→｜3〉can be finished through either the direct coupling of B por the cross coupling ofρ₁₂·E_p·Ω_c. Similarly, the probe electric transition｜2〉←→｜4〉in the Lambda system also have two paths: the direct coupling one bridged by E_p and the cross coupling one bridged byρ₂₁·B_p·Ω_c.Firstly, we discuss dynamically induced chirality for achieving negative refraction by changing the steady population at levels｜1〉and｜2〉. As we gradually decreaseρ₁₁, the transparency window in the Im ( n ) curve becomes narrower and the negative-valued part of the Re ( n ) curve tends to the zero line in the Fig.5(a) and Fig.5(b). In this case, the electric susceptibilityχ_e is enhanced while the magnetic susceptibilityχ_m and the two chirality coefficientsξ_EH andξHE are weakened. Conversely, if we increaseρ₁₁starting from the balanced case ofρ₁₁ = 0.5, the negative-valued part of the Re ( n ) curve once again tends to the zero line, but the transparency window becomes wider instead of narrower as shown in the Fig.5(c) and Fig.5(d). In this case, the enhanced susceptibility is the magnetic oneχ_m but not the electric oneχ_e. Thus we may conclude that: (a) the direct coupling coefficientχ_m in the Ladder EIT system is always negligible for the probe response; (b) the direct coupling coefficientχ_e in the Lambda EIT system answers for the probe transparency; (c) the cross coupling coefficientsξ_EH andξHE are critical for achieving the negative refraction around a transparency window.In the next, we examine the complex refractive index with the different values of the two dephasingsγ₁₃ andγ₂₃. We know that the electric susceptibilityχ_e in the Lambda EIT system, which is very sensitive toγ₂₃, so the probe absorption becomes more and more severe as gradually increasingγ₂₃. Thus one can suppress the residual absorption in the transparency window by increasing the driving Rabi frequencyΩ_c.In addition, the other magnetic dephasingγ₁₃ how to influence the negative refraction. By the numerical calculations, we can see thatγ₁₃ is not as important asγ₂₃becauseγ₁₃affects n only throughξ_HE whileγ₂₃ affects n through bothχ_e andξ_EH. In other words, the Lambda EIT system dealing with the probe electric transition plays a more important role than the Ladder EIT system dealing with the probe magnetic transition.Finally we discuss the dependence of the complex refractive index n on the driving phaseΦ, one most important characteristic of the close-loop atomic system. We note from Fig. 6 that there exist a series of mode hopping atΦ(?) 2 kÏ€+ 0.5Ï€andΦ(?) 2 kÏ€+ 1.5Ï€. If the driving phase is increased fromΦ(?) 2 kÏ€+ 0.5Ï€toΦ(?) 2 kÏ€+ 1.5Ï€(fromΦ(?) 2 kÏ€+ 1.5Ï€toΦ(?) 2 kÏ€+ 2.5Ï€), the refractive index Re ( n ) atΔ_E =-0.4MHzwill change from negative to positive values (from positive to negative values). The refractive index Re ( n ) atΔ_E = 0.8MHz, however, varies in the reverse direction for the same driving phase modulation. That is, we can manipulate the refractive index Re ( n ) at a fixed frequency via the periodic phase modulation. To well suppress the probe absorption as attaining the negative refraction, we should carefully set the relative phaseΦ= kÏ€+ 0.5Ï€.In this part, we perform semiclassical analysis on the coherent light-atom interaction in a five-level tripod system with inherent symmetry. By adjusting a microwave field without destroying the symmetry, we can achieve efficient XPM through the enhanced cross-Kerr nonlinearities in two distinct schemes: one is based on EIT while the other uses coherent Raman gain. In both schemes, the slowed down probe and signal pulses always have matched group velocities due to the symmetric light-atom interaction. When XPM is to produce a conditional phase shift of the order ofÏ€, two-qubit phase gates become feasible to realize at the resonance frequencies. Then, using two-qubit phase gates, one-qubit Hadamard gates, and optical delay lines to generate three-photon GHZ states.In the five-level configuration as shown in Fig.7, the four transitions｜0〉←→｜2〉,｜1〉←→｜4〉,｜2〉←→｜4〉, and｜3〉←→｜4〉interact with a microwave field E_d, a probe field E_p ofσ⁺ polarization, a coupling field Ec ofÏ€polarization, and a signal field E_s ofσ^- polarization, respectively, which could be realized in ⁸⁷Rb atoms. In the following, we will consider two different schemes for achieving greatly enhanced Kerr nonlinearities: one is based on the EIT phenomenon while the other utilizes the coherent Raman gain. In this EIT scheme, the microwave and coupling fields are required to be much stronger than the probe and signal fields (Ω_c,dΩ_p,s) so that we can setρ₁₁ =ρ₂₂ = 0.5. According to the master equation of the density operator, we can deduce the analytical formula of the first-order linear and third-order nonlinear susceptibilities about the weak fields when this atomic system is steady. In Fig. 8, we first plot the linear absorption and dispersion spectra of the probe field using the expression of the linear susceptibility. As we can see, one EIT window is produced in the positive detuning region while another EIT window exists in the negative detuning region. Due to the intrinsic symmetry of our considered system, two EIT windows are expected to respectively appear in the positive and negative detuning regions of the signal spectra (not shown). In these EIT windows accompanied by steep normal dispersion, it is possible to remarkably reduce group velocities of the probe and signal pulses as well as to greatly suppress the probe and signal absorption. We further plot the real and imaginary parts ofχ_p³ around the two EIT windows in Fig. 9. It is found that has a negative peak at the center of the left EIT window while a positive peak at the center of the right EIT window. It is clear that, within the EIT windows, Re[χ_p³] is dramatically enhanced while both Im[χ_p³] and Im[χ_p³]are greatly suppressed at a specific point. The same remarks hold true for Re[χ_p³], Im[χ_p³], and Im[χ_p³] as a result of the inherent symmetry of the five-level tripod system.In this Raman gain scheme, the microwave (coupling) field is assumed to be much weaker (stronger) than the probe and signal fields (Ω_c>>Ω_{p ,s}>>Ω_d) so that we can set c₀ = 1 with c₀ being the probability amplitude at level｜0〉. Using the dynamic equations for atomic probability amplitude, we could deduce the steady-state solutions of the probe and signal linear susceptibilities as well as the probe and signal nonlinear susceptibilities. According to the expression of the linear susceptibility, we plot the real and imaginary parts ofχ_p¹ as a function of the probe detuningΔ_p in Fig.10. As we can see, there exists a narrow gain peak at the probe resonance accompanied by very steep normal dispersion, which is resulted from the three-photon hyper-Raman process. Fig. 11 shows that the nonlinear susceptibilityχ_p³ behaves very similar to the linear susceptibilityχ_p¹ except that its amplitude is about two-order smaller. As for the signal susceptibilitiesχ_s¹ andχ_s³, they should show the same spectral features as their probe counterparts due to the inherent symmetry of our considered system.Finally, we propose here a new method to generate three-photon GHZ states using both linear and nonlinear optical elements. As shown in Fig. 12, the schematic diagram of the setup is composed of five half-wave plates to act as single qubit Hadamard gates, two Kerr media to act as double qubit QPGs, and two EIT media (or optical fiber loops) to act as optical delay lines.In conclusion, we have investigated a four-level atomic system with SCG for achieving the left-handed property of negative refractive index. In particular, with appropriate parameters, we can obtain negligible absorption at certain frequencies of negative refraction. We also have studied a close-loop atomic system for achieving negative refraction with little absorption in the optical regime. It could be better understood if we decompose the multi-level atomic system into two coupled three-level EIT systems. It is worth nothing that we also can periodically tune the refractive index between positive and negative values by modulating the driving phase. Finally, we can achieve efficient XPM through the enhanced cross-Kerr nonlinearities in two distinct schemes: one is based on EIT and the other uses coherent Raman gain by adjusting a microwave field in a five-level system. We may further attain a quantum phase gate and generate the three-photon GHZ state when a condition phase shift of order ofÏ€is viable in the presence of XPM.