含驱动项光学频率上转换系统的非经典效应
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摘要
由于辐射场的非经典态(如压缩态和纠缠态)，在光通信、精密光学测量和量子信息处理中的潜在的应用，吸引了人们广泛的研究兴趣。长期以来，光学参量下转换被认为是重要的压缩态和纠缠态的产生源，并对其各种性质进行了很好的研究。然而，对于频率上转换过程，似乎很少人注意其压缩和纠缠效应的产生。大量的努力集中在开发新的非线性材料，拓广频率变换范围，提高功率转换效率等。一些工作对单个上转换过程的量子涨落，信息转移和时间演化等进行了研究。但大多数这些模型没有考虑驱动项对耦合场动力学行为的影响。Prants首次研究了含驱动项上转换系统的时间演化。但是仅导出了参量共振条件下的演化算符。因此，有必要对这一系统开展更深入的研究，如探讨失谐情形下，该系统的动力学及其非经典效应等。
本文首先对光频率转换技术的发展状况进行了一个简短的回顾，对含驱动项频率上转换系统的量子动力学模型做了详细的介绍。接着通过选用适当的含时幺正算符，对信号和闲置光子数算符的线性组合进行相似变换，构造出含驱动项的频率上转换系统的含时不变量，得到了相应的辅助微分方程。采用Lewis-Riesenfeld量子不变量理论，求出了系统薛定谔方程的显示解析解,并借此分析了初态、失谐频率和驱动项对输出光场的强度、量子起伏的影响。发现输出场的量子起伏与驱动项无关，但是明显依赖于初始状态和相对失谐参量Γ。当系统最初处于双模相干态，输出场在任意失谐下都不出现压缩。当信号模初始处于单模相干压缩态，闲置模初始处于单模相干或真空态时，输出信号场的最大压缩和输入场是相同的，但输出闲置场的最大压缩，随失谐量的增加而减小。因此只有当参量共振发生时，系统的时间演化才能有效地将一束低频压缩光束转换为一束高频压缩光束。另外，研究发现，上转换系统输出闲置场和信号场是纠缠的。在无驱动项和参量共振条件下，当信号场初始处于单模粒子数态、闲置场初始处于真空态时，系统两输出模式的纠缠度随相互作用时间作周期性的变化。当双模光子数相等时输出态纠缠度最大。最后,值得一提的是，我们的解还能描述一个级联准相位匹配上转换过程的量子动力学行为。因此，本文的研究结果对于非线性光学和量子光学的研究有着较重要的意义。
The study of squeezed and entangled states of the Radiation field has attractedmuch interest due to their potential applications in optical communication, preciseoptical measurement and quantum information processing. Optical parametricdown-conversion has long been considered as an important source of the squeezed andentangled state of light and has been well investigated. However, for the frequencyup-conversion process, little attention seems to have been paid to the generation ofsqueezing and entanglement. A lot of effort has been focused on developing newnonlinear materials, extending the frequency conversion range, improving the powertransfer efficiency, and so on. Some work has been done on the quantum fluctuation,information transfer and time evolution in the single up-conversion. Most of thesemodels did not considered the influence of the driving term on the dynamic behaviorsof the coupled fields. The evolution of an up-conversion model with a driving termwas first studied by Prants. However, only an evolution operator under parameterresonance was derived. Therefore, it is necessary to carry out a further research onthis system, such as to explore the dynamics and non-classical effects of the systemunder frequency detuning.
In this Master's thesis, firstly, the research development of nonlinear opticalfrequency conversion technique is reviewed briefly, the quantum dynamical model offrequency up-conversion with a driving term, is introduced in detail. Then employinga similarity transformation of the linear combination of the signal and idler photonnumber operators, by a time-dependent unitary operator, we construct an invariant andderive corresponding auxiliary differential equations, for the parametricup-conversion system with a driving term. Using the Lewis-Riesenfeld quantuminvariant theory, an explicit analytical solution for the time evolution operator of thesystem is obtained. Then it is used to investigate mainly the influence of the initialsqueezed state, the frequency detuning, and the driving term, on the quantumfluctuations of the coupled fields. It is found that the quantum fluctuations of theoutput beams are independent of the driving term, but obviously dependent on theinitial states and the relative detuning parameterΓ. When the system is initially in atwo-mode coherent state, no squeezing occurs in the output fields for arbitrarydetuning. When the signal mode starts in a single-mode squeezed coherent state, the idler mode starts in a single-mode coherent or vacuum state, the maximum squeezingin the output signal field is the same as its input for arbitrary detuning, but themaximum squeezing for the output idler field decreases as the detuning increases.Therefore, only when a parameter resonance occurs, can the time evolution for thesystem effectively convert a low-frequency squeezed light into a high-frequencysqueezed light. In addition, we also find that the idler field, produced by the generalup-conversion, is entangled with the output signal field. In the case of no driving termand no detuning, the signal field starts in a single-mode number state, and the idlerfield starts in a vacuum state, the evolution of the usual simple frequencyup-conversion makes the entanglement degree between the two modes varyperiodically with interaction time. The output state is maximally entangled when thetwo-mode output photons are equal to each other. At last, it is worth mentioning thatour solution can also describe the quantum dynamical behavior of a cascadedquasi-phase-matching up-conversion. Therefore, our results could be useful for thestudies in the nonlinear optics and quantum optics.
本文首先对光频率转换技术的发展状况进行了一个简短的回顾，对含驱动项频率上转换系统的量子动力学模型做了详细的介绍。接着通过选用适当的含时幺正算符，对信号和闲置光子数算符的线性组合进行相似变换，构造出含驱动项的频率上转换系统的含时不变量，得到了相应的辅助微分方程。采用Lewis-Riesenfeld量子不变量理论，求出了系统薛定谔方程的显示解析解,并借此分析了初态、失谐频率和驱动项对输出光场的强度、量子起伏的影响。发现输出场的量子起伏与驱动项无关，但是明显依赖于初始状态和相对失谐参量Γ。当系统最初处于双模相干态，输出场在任意失谐下都不出现压缩。当信号模初始处于单模相干压缩态，闲置模初始处于单模相干或真空态时，输出信号场的最大压缩和输入场是相同的，但输出闲置场的最大压缩，随失谐量的增加而减小。因此只有当参量共振发生时，系统的时间演化才能有效地将一束低频压缩光束转换为一束高频压缩光束。另外，研究发现，上转换系统输出闲置场和信号场是纠缠的。在无驱动项和参量共振条件下，当信号场初始处于单模粒子数态、闲置场初始处于真空态时，系统两输出模式的纠缠度随相互作用时间作周期性的变化。当双模光子数相等时输出态纠缠度最大。最后,值得一提的是，我们的解还能描述一个级联准相位匹配上转换过程的量子动力学行为。因此，本文的研究结果对于非线性光学和量子光学的研究有着较重要的意义。
The study of squeezed and entangled states of the Radiation field has attractedmuch interest due to their potential applications in optical communication, preciseoptical measurement and quantum information processing. Optical parametricdown-conversion has long been considered as an important source of the squeezed andentangled state of light and has been well investigated. However, for the frequencyup-conversion process, little attention seems to have been paid to the generation ofsqueezing and entanglement. A lot of effort has been focused on developing newnonlinear materials, extending the frequency conversion range, improving the powertransfer efficiency, and so on. Some work has been done on the quantum fluctuation,information transfer and time evolution in the single up-conversion. Most of thesemodels did not considered the influence of the driving term on the dynamic behaviorsof the coupled fields. The evolution of an up-conversion model with a driving termwas first studied by Prants. However, only an evolution operator under parameterresonance was derived. Therefore, it is necessary to carry out a further research onthis system, such as to explore the dynamics and non-classical effects of the systemunder frequency detuning.
In this Master's thesis, firstly, the research development of nonlinear opticalfrequency conversion technique is reviewed briefly, the quantum dynamical model offrequency up-conversion with a driving term, is introduced in detail. Then employinga similarity transformation of the linear combination of the signal and idler photonnumber operators, by a time-dependent unitary operator, we construct an invariant andderive corresponding auxiliary differential equations, for the parametricup-conversion system with a driving term. Using the Lewis-Riesenfeld quantuminvariant theory, an explicit analytical solution for the time evolution operator of thesystem is obtained. Then it is used to investigate mainly the influence of the initialsqueezed state, the frequency detuning, and the driving term, on the quantumfluctuations of the coupled fields. It is found that the quantum fluctuations of theoutput beams are independent of the driving term, but obviously dependent on theinitial states and the relative detuning parameterΓ. When the system is initially in atwo-mode coherent state, no squeezing occurs in the output fields for arbitrarydetuning. When the signal mode starts in a single-mode squeezed coherent state, the idler mode starts in a single-mode coherent or vacuum state, the maximum squeezingin the output signal field is the same as its input for arbitrary detuning, but themaximum squeezing for the output idler field decreases as the detuning increases.Therefore, only when a parameter resonance occurs, can the time evolution for thesystem effectively convert a low-frequency squeezed light into a high-frequencysqueezed light. In addition, we also find that the idler field, produced by the generalup-conversion, is entangled with the output signal field. In the case of no driving termand no detuning, the signal field starts in a single-mode number state, and the idlerfield starts in a vacuum state, the evolution of the usual simple frequencyup-conversion makes the entanglement degree between the two modes varyperiodically with interaction time. The output state is maximally entangled when thetwo-mode output photons are equal to each other. At last, it is worth mentioning thatour solution can also describe the quantum dynamical behavior of a cascadedquasi-phase-matching up-conversion. Therefore, our results could be useful for thestudies in the nonlinear optics and quantum optics.
引文
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