| Quantum interference effects based on the interaction between light and matter can effectively change the polarization, absorption, dispersion and refractive index of the medium. It's one of the hottest areas of the researches on quantum optics and optical physics in recent years. In nonlinear optics, the refractive index change by the light intensity is called optical Kerr effect. It has wide researches and applications on cross-phase modulation (XPM), self-phase modulation (SPM), resulting in single photon optical soliton and the enhancement of the refractive index. To achieve giant Kerr nonlinearity without optical resonant absorption with a weak pulse is very important to all-optical switches and optical quantum logic operations. We know that the free electron in a quantum well has a series of separated levels which are similar to atom, so we can get desired levels in artificial quantum structures by changing the parameters of quantum well. While considering the practical application, semiconductor has large electric dipole moment and nonlinear optical coefficients, so extending the study on nonlinear optics in semiconductor structure is very attractive.In this thesis, we first design a semiconductor quantum well structure to achieve a simple three-level system, shown in Fig. 1. We investigate an AlGaAs/GaAs double quantum well with a continuum. The ground subband of the shallow well and the first excited subband of the right deep well are coupled by tunneling. Then two new subbands |2> and |3> are created and tunneling to a continuum. The split between subbands |2> and |3> is proportional to the strength of the coupling and inversely proportional to the thickness of the thin potential barrier. The thinner the barrier, the stronger the coupling, the bigger the split. Since the two resonance transitions from the ground state | 1> to subbands |2> and |3> are coupled by the thin barrier to the continuum, the optical absorption shows Fano-type interference. Following the standard processes, the system dynamics can be described by equations of motion for the probability amplitudes of the states in a rotating frame. We solve the coupled amplitude equations in steady state in the nondepletion approximation and the weak field approximation, and get the expression of the polarizability. Then we draw the real part and imaginary part of the linear polarization after normalized, rating to the probe energy in Fig. 2:We can see from Fig. 2 that the imaginary part of the linear polarization reduces to zero at the resonant frequency, showing zero absorption. At the same point the refractive index varies a lot. That is to say, we get a high dispersion at the resonant frequency, which is as the same as in an atom.And then, we designed a double semiconductor quantum well structure based on the quantum tunneling in Fig. 3 to study linear and nonlinear absorption and refractive index. In this structure, the two excited subbands of the shallow well and the deep well are coupled by the mid barrier, and then two new subbands |2> and |3> are formed and tunneling to a continuum through the right barrier. The two resonance transitions from the ground state | 1> of the deep well to subbands |2> and |3> are coupled by the thin barrier to the continuum, the optical absorption shows Fano-type interference.As the coupling and probe field are both weak, we obtained the expression of the first-order and third-order polarizability in steady state in the nondepletion approximation, by choosing the appropriate parameters we obtain the enhanced Kerr nonlinearity with zero absorption, shown in Fig. 4. The parameters are as follows:Δ0 =3.50meV,Δs =1.0meV,Γ2 =1.30meV,Γ3 =1.92meV,Γ4 =0.12meV,q1 =2.46, q 4 =?1.35。 In this figure, Im(χ'(1 ) ), Re(χ'( 3) ) and Im(χ'( 3) ) respectively account for the linear absorption, two-photon absorption, and XPM. At the point A the probe energy is 222.648meV, where the linear absorption and two-photon absorption are respectively 5.881E-5meV-1 and -0.001meV-3, while the corresponding real part of third-order polarization is -5.9meV-3. In other words, we achieve a large Kerr nonlinear while linear and two-photon absorptions are vanished.And we follow to investigate the effects of parameters. We find that signal detuning effects a little; and the split of levels |2> and |3> should be as small as possible, but it can not be reduced without limit; the most effected parameters are dephasing rate which is determined by the environment temperature, and relaxation rate of level |4> which can be controlled by modifying the width and the height of the right barrier. from 0meV to 3.0meV. As you see that the absolute value of the real part of third-order polarizability becomes smaller. Therefore, if we want to obtain a large Kerr nonlinearity, the relaxation rate of level |4> should be as small as we can control.In Fig. 6 (a), Fig. 6 (b), Fig. 6 (c) and Fig. 6 (d) the dephasing rate varies from 0meV to 3.0meV. As you see that the absolute value of the real part of third-order polarizability becomes smaller. So, if we want to achieve a large Kerr nonlinearity, we can control the temperature below 10K in experiments. The smaller dephasing rate, the stronger the quantum tunneling effect, the stronger the Kerr nonlinearity.We know that in the semiconductor quantum well structure has a large dipole matrix element, which makes the required field strength pump small enough. So our semiconductor quantum well structure can achieve enhancement Kerr nonlinearity under the conditions of weak pumping field. In nonlinear optics, the total refractive index contains the linear and nonlinear parts. After the effective length, the phase shift of probe field is proportional to the real part of the nonlinear refractive index. So long as we get a high enhancement Kerr nonlinear with vanishing linear and nonlinear absorptions in semiconductor quantum well, we'll be able to achieve the cross-phase modulation under weak field conditions, which will have great practical value on classical and quantum information processing.
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