利用原子相干效应实现光频负折射率和偏振量子相位门的理论研究
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摘要
光与物质作用导致的原子相干效应从激光诞生之后,一直是光学领域的热门研究课题,它是量子光学中最重要的基本物理现象之一。目前,针对各种新的材料和介质,人们还在不断深入研究与原子相干效应有关的各种现象,如:相干粒子数俘获、电磁感应光透明、光群速度减慢、光学非线性系数增强和相干诱导光子带隙结构等。在本论文中,我们主要研究如何利用相干原子气体方法在光学波段实现负折射率材料,以及在一个具有对称结构的五能级原子系统中,产生大的Kerr非线性系数,进而实现量子相位门和三光子GHZ态。
本论文共分五个部分,具体内容如下:
第一部分:介绍几个基本的量子相干概念及由量子相干引起的与本论文研究内容有关的几种物理效应。
第二部分:介绍左手材料和手性材料的基本性质与发展历程,非线性极化率以及与量子信息相关的几个基本概念。
第三部分:研究一个存在自发辐射相干效应(SGC)的四能级原子系统,利用自发辐射相干效应对介质的介电常数和磁导率进行有效调控。经过理论推导和数值模拟发现,控制自发辐射相干效应和洛伦兹局域场效应以及外加驱动场强度等参数,可在一定频率范围内同时得到负的介电常数和负的磁导率,进而实现左手材料。我们还讨论了如何在缀饰态表象下模拟自发辐射相干效应,以便避开SGC效应存在的苛刻条件。
第四部分:研究一个具有闭合环路的高密度多能级原子系统,进一步发展相干原子气体方法,即利用电磁感应手性和强的局域场效应实现负折射率。理论分析发现,通过控制低能级粒子数布居、洛伦兹局域场效应(原子密度)、退相干速率以及驱动场强度等参数,可在光频波段实现零吸收的负折射率。由于该原子系统具有一个闭合环路,因此,相对位相对复折射率的影响非常显著。当频率一定时,可以通过周期调制相对位相来控制折射率的变化。
第五部分:研究一个具有对称结构的五能级Tripod型原子系统,利用电磁感应透明机制和拉曼增益机制,在探测跃迁共振频率附近获得大的交叉Kerr非线性系数。理论分析发现,在EIT机制中,XPM可以在两个不同频率处无吸收获得;而在拉曼增益机制中,XPM伴随增益只能在一个频率处获得。在这两种机制中,探测场和信号场之间的对称相互作用不会因微波场的调制而破坏。另外,当XPM足够大可以产生一个π量级的条件相移时,就可以用其实现双量子比特相位门;于是,我们就可以利用双量子比特相位门,单比特Hadamard门和光延迟线实现三光子GHZ态。
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 coherence
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.
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 10~(24) 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 chirality
In 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 termρ_(12): 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 ofρ_(12)·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 byρ_(21)·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 decreaseρ_(11), 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 increaseρ_(11)starting from the balanced case ofρ_(11) = 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 dephasingsγ_(13) andγ_(23). We know that the electric susceptibilityχ_e in the Lambda EIT system, which is very sensitive toγ_(23), so the probe absorption becomes more and more severe as gradually increasingγ_(23). Thus one can suppress the residual absorption in the transparency window by increasing the driving Rabi frequencyΩ_c.
In addition, the other magnetic dephasingγ_(13) how to influence the negative refraction. By the numerical calculations, we can see thatγ_(13) is not as important asγ_(23)becauseγ_(13)affects n only throughξ_(HE) whileγ_(23) 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 kπ+ 0.5πandΦ(?) 2 kπ+ 1.5π. If the driving phase is increased fromΦ(?) 2 kπ+ 0.5πtoΦ(?) 2 kπ+ 1.5π(fromΦ(?) 2 kπ+ 1.5πtoΦ(?) 2 kπ+ 2.5π), 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= kπ+ 0.5π.
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 ofπ, 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 ofπpolarization, and a signal field E_s ofσ~- polarization, respectively, which could be realized in ~(87)Rb 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,d)Ω_(p,s)) so that we can setρ_(11) =ρ_(22) = 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_p~(3) 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_p~(3)] is dramatically enhanced while both Im[χ_p~(3)] and Im[χ_p~(3)]are greatly suppressed at a specific point. The same remarks hold true for Re[χ_p~(3)], Im[χ_p~(3)], and Im[χ_p~(3)] 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 c_0 = 1 with c_0 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_p~(1) 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_p~(3) behaves very similar to the linear susceptibilityχ_p~(1) except that its amplitude is about two-order smaller. As for the signal susceptibilitiesχ_s~(1) andχ_s~(3), 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 ofπis viable in the presence of XPM.
本论文共分五个部分,具体内容如下:
第一部分:介绍几个基本的量子相干概念及由量子相干引起的与本论文研究内容有关的几种物理效应。
第二部分:介绍左手材料和手性材料的基本性质与发展历程,非线性极化率以及与量子信息相关的几个基本概念。
第三部分:研究一个存在自发辐射相干效应(SGC)的四能级原子系统,利用自发辐射相干效应对介质的介电常数和磁导率进行有效调控。经过理论推导和数值模拟发现,控制自发辐射相干效应和洛伦兹局域场效应以及外加驱动场强度等参数,可在一定频率范围内同时得到负的介电常数和负的磁导率,进而实现左手材料。我们还讨论了如何在缀饰态表象下模拟自发辐射相干效应,以便避开SGC效应存在的苛刻条件。
第四部分:研究一个具有闭合环路的高密度多能级原子系统,进一步发展相干原子气体方法,即利用电磁感应手性和强的局域场效应实现负折射率。理论分析发现,通过控制低能级粒子数布居、洛伦兹局域场效应(原子密度)、退相干速率以及驱动场强度等参数,可在光频波段实现零吸收的负折射率。由于该原子系统具有一个闭合环路,因此,相对位相对复折射率的影响非常显著。当频率一定时,可以通过周期调制相对位相来控制折射率的变化。
第五部分:研究一个具有对称结构的五能级Tripod型原子系统,利用电磁感应透明机制和拉曼增益机制,在探测跃迁共振频率附近获得大的交叉Kerr非线性系数。理论分析发现,在EIT机制中,XPM可以在两个不同频率处无吸收获得;而在拉曼增益机制中,XPM伴随增益只能在一个频率处获得。在这两种机制中,探测场和信号场之间的对称相互作用不会因微波场的调制而破坏。另外,当XPM足够大可以产生一个π量级的条件相移时,就可以用其实现双量子比特相位门;于是,我们就可以利用双量子比特相位门,单比特Hadamard门和光延迟线实现三光子GHZ态。
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 coherence
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.
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 10~(24) 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 chirality
In 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 termρ_(12): 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 ofρ_(12)·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 byρ_(21)·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 decreaseρ_(11), 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 increaseρ_(11)starting from the balanced case ofρ_(11) = 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 dephasingsγ_(13) andγ_(23). We know that the electric susceptibilityχ_e in the Lambda EIT system, which is very sensitive toγ_(23), so the probe absorption becomes more and more severe as gradually increasingγ_(23). Thus one can suppress the residual absorption in the transparency window by increasing the driving Rabi frequencyΩ_c.
In addition, the other magnetic dephasingγ_(13) how to influence the negative refraction. By the numerical calculations, we can see thatγ_(13) is not as important asγ_(23)becauseγ_(13)affects n only throughξ_(HE) whileγ_(23) 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 kπ+ 0.5πandΦ(?) 2 kπ+ 1.5π. If the driving phase is increased fromΦ(?) 2 kπ+ 0.5πtoΦ(?) 2 kπ+ 1.5π(fromΦ(?) 2 kπ+ 1.5πtoΦ(?) 2 kπ+ 2.5π), 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= kπ+ 0.5π.
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 ofπ, 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 ofπpolarization, and a signal field E_s ofσ~- polarization, respectively, which could be realized in ~(87)Rb 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,d)Ω_(p,s)) so that we can setρ_(11) =ρ_(22) = 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_p~(3) 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_p~(3)] is dramatically enhanced while both Im[χ_p~(3)] and Im[χ_p~(3)]are greatly suppressed at a specific point. The same remarks hold true for Re[χ_p~(3)], Im[χ_p~(3)], and Im[χ_p~(3)] 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 c_0 = 1 with c_0 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_p~(1) 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_p~(3) behaves very similar to the linear susceptibilityχ_p~(1) except that its amplitude is about two-order smaller. As for the signal susceptibilitiesχ_s~(1) andχ_s~(3), 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 ofπis viable in the presence of XPM.
引文
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