双光子跃迁调控与量子聚焦
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摘要
量子控制作为一个全新的学科领域正在蓬勃发展,作为近代科学前沿中最为活跃的边缘学科领域之一,它涉及量子力学、光学、化学、控制论、信息科学等众多领域。随着激光、观测、纳米等技术的迅速发展,进行实验研究的量子控制手段也在不断的改进和增加,这必将促进量子控制论的发展,促进与量子力学系统相关联的化学反应、生物细胞、量子信息、纳米材料等领域的量子控制研究的发展。
随着超快脉冲的迅速发展,激光脉冲整形技术取得了突破性的进展,人们可以获得任意复杂的光脉冲波形,实现更为精细的量子控制。激光与物质相互作用的研究吸引了大批的科学研究者。不同的激发机制产生不同的现象或效应。瞬时频率随时间线性变化的啁啾脉冲光场作用于原子会产生一些奇特的量子效应,在量子相干控制研究中有着很大的优越性,因而在理论和实验中经常被采用。啁啾过程的扫频效应会实现多个共振激发过程的发生,从而导致多通道的量子干涉效应或量子衍射效应的产生。同样控制光场的相位也可以实现对干涉和衍射过程的控制。而在光波的干涉和衍射过程中,干涉相消或干涉相长是由光波的相位关系决定的,只要相位相同的光场与原子相互作用就可以实现干涉相长的过程,从而利用较弱的光场来实现较大的激发效率。
本文利用二阶微扰理论得到了激发态布居几率的解析表示,研究了可控脉冲光场作用下的双光子作用过程。选择最佳相位函数对脉冲整形,可实现双光子跃迁几率的增强。所得结果对利用光谱学来探究物质的结构等领域的理论和实验有一定的参考价值。通过深入的研究发现双光子跃迁几率的振幅实际上是光场在晶体中频域的夫琅禾费衍射过程,并且这个衍射的结果取决于狭缝的宽度。基于成熟的整形脉冲技术对光场进行裁剪或对其相位调制,可以实现类似于多缝干涉的压缩结果,实现量子聚焦。因而这一设计思想对于量子控制、信息处理等具有很大的实际意义。该工作的意义还在于可以指导实验,同时为教学提供演示。本文由如下三个部分构成。
第一部分微扰理论
在这一部分中介绍了相互作用基本理论。在电偶极近似下,得到了原子和辐射场整个系统的哈密顿(?)。在此基础上,通过求解薛定谔方程,采用迭代法并利用与时间有关的微扰理论讨论(?)各次幂的项对跃迁速率的贡献,得到激发态几率振幅表达式和辐射跃迁几率的一般表达式。介绍了双光子跃迁原理,并通过二阶微扰理论讨论双光子过程的跃迁几率,最终得到激发态的布居几率的演化,此结果是我们后面章节研究的理论基础,对于后期工作也有重大意义。
第二部分脉冲整形及双光子跃迁调控
在这一部分中基于二阶微扰理论,分析了整形脉冲作用于原子系统的双光子过程。深入研究了在远离共振和近共振两种情况下双光子跃迁几率的演化规律。结果表明,远离共振时,利用整形光脉冲作用于原子系统,会使得某些频率对应处的跃迁几率消失,光可以无损耗的通过介质：在近共振时,选择最佳相位函数对脉冲整形,可实现双光子跃迁几率的增强。证明了在相干量子调制等一些物理过程的研究中啁啾效应有着很大的优越性。所得结果对利用光谱学来探究物质的结构等领域的理论和实验研究有一定的参考价值。
第三部分量子聚焦效应
在这一部分中利用目前使用最多的激光相干量子控制法,对啁啾脉冲光场作用下双光子跃迁几率的演化规律进行更加深入的研究。利用二阶微扰理论得到了啁啾脉冲作用下激发态布居几率的解析表示,通过分析可以看出,双光子跃迁几率的振幅实际上是光场在晶体中频域的夫琅禾费衍射过程。这个衍射的结果取决于狭缝的宽度,当狭缝(泵浦光场的频率宽度)较窄时,激发态布居几率的演化规律只是一个单缝衍射的过程。当狭缝较宽时,激发态布居几率类似两个直边衍射相加。基于成熟的整形脉冲技术对光场进行裁剪或对其相位调制,实现类似于多缝干涉的压缩结果。经压缩后跃迁几率增强,实现量子聚焦的目的。因而这一设计思想对于量子控制、信息处理等具有很大的实际意义。
Quantum control is booming as a brand new discipline, it is one of the most active element in the modern scientific front edge discipline, which involved in quantum mechanics, optics, chemical, cybernetics, information science, etc. With the rapid development of laser, observation and nanometer technology, quantum control methods carried out experimental research are also improving and increasing constantly, which will promote the development of quantum cybernetics, promote the development of chemical reaction, biological cells, quantum information, nano-materials and other fields'quantum control research associated with quantum mechanics system.
With the rapid development of the ultrafast pulse, laser pulse shaping technology have made breakthrough progress.people can get any complex optical pulse waveform to realize more elaborate quantum control. The research of interaction between laser and material attracts a lot of scientific researchers. Different stimulated mechanisms produce different phenomena or effects.The chirped pulse light field who's instantaneous frequency vary linearly with time produce some peculiar quantum effects when interact with atom,which have more prodigious advantages in some coherent quantum control physical process researches,thus the chirped pulse light field was often used in theory and experiment.The chirp sweep frequency effect can implement multiple resonance stimulate processes,thus causing the multi-channel quantum interference effect or quantum diffraction effect. At the same time, the interference and diffraction effect can also be controlled by modulating the optical field phase.Constructive interference or destructive interference is decided by the phases relationship of light waves, as long as having the same phase optical field interacted with atom, constructive interference can be achieved, thus the weaker light field can realize the large stimulated efficiency.
Based on the second order perturbation theory, we get the analytic expression of the excited state population probability and study the two-photon processes under the controllable chirp pulse. The bigger two-photon transition probability can be obtained by choosing the best shaping function under the resonance case.The results have certain reference value by using spectroscopy to explore material structure and other areas in theory and experiment. Through studying in-depth we find that the amplitude of the two-photon transition probability is actually the Fraunhofer diffraction process in frequency domain when the light goes through crystal and the results depend on the width of the slit. Based on the mature shaped pulse technology, we can realize the compression of the multiple-slit interference by cropping the optical field or modulating the phase.After compressing the the transition probability is greatly enhanced and quantum focused effect can be relized. Consequently the design idea is of great practical significance for quantum control、information processing and so on. Also the significance of this work is guiding the experiment in the lab environment and providing demonstration for teaching. This paper is composed of the following three parts.
The first part is perturbation theory
In this part, it mainly introduces the fundamental theories of the interaction. Under electric dipole approximation, we get Hamiltonian H, of the whole system between atoms and radiation field. Through solving the Schrodinger equation, using iterative method and the perturbation theory related with time, we discuss the every term of HI contributes to the transition probability and get the expression of the excited state probability amplitude and transition probability. It introduces two-photon process and discusses transition probability of the two-photon processes based on the second-order perturbation theory, and finally we gain the evolution of population probability of the excited state, which is the basic theory for later chapters and also has significant meaning for research work.
The second part is that pulse shaping and two-photon transition control
In this part, based on the second-order perturbation theory,the two-photon processes in the atom system excited by a shaped pulse are analyzed.The evolution of the two-photon transition probability is studied detailedly under two cases of nonresonance and resonance.It's shown that the transition probabilities disappeared at some corresponding frequencies under the nonresonance case, and thus the laser pulse can get through the medium without loss.Meanwhile,the bigger two-photon transition probability is obtained by choosing the best shaping function under the resonance case.It's proved that chirped effect has great advantages in some coherent quantum modulation of physical process research, The results has certain reference value when using spectroscopy to explore material structure and other areas in theory and experiment.
The third part is quantum focused effect
In this part, the evolution of the two-photon transition rules under the chirp pulse light field are analyzed using the current laser coherent quantum control method. Based on the second order perturbation theory, we get the analytic expression of the population probability of the excited state. It's shown that the amplitude of the two-photon transition probability is actually Fraunhofer diffraction process in frequency domain when the light goes through crystal.The results of the diffraction depend on the width of the slit. It is just a single slit diffraction when the slit (the frequency width of the pump field) is narrower.When the slit is wider, the evolution of the population probability of the excited state is similar to the adding of two straight edge diffraction together. Based on the mature shaped pulse technology, we can realize the compression of the multiple-slit interference by cropping the optical field or modulating the phase. After compressing the the transition probability is greatly enhanced and the quantum focused effect can be relized. Consequently the design idea is of great practical significance for quantum control、information processing and so on.
随着超快脉冲的迅速发展,激光脉冲整形技术取得了突破性的进展,人们可以获得任意复杂的光脉冲波形,实现更为精细的量子控制。激光与物质相互作用的研究吸引了大批的科学研究者。不同的激发机制产生不同的现象或效应。瞬时频率随时间线性变化的啁啾脉冲光场作用于原子会产生一些奇特的量子效应,在量子相干控制研究中有着很大的优越性,因而在理论和实验中经常被采用。啁啾过程的扫频效应会实现多个共振激发过程的发生,从而导致多通道的量子干涉效应或量子衍射效应的产生。同样控制光场的相位也可以实现对干涉和衍射过程的控制。而在光波的干涉和衍射过程中,干涉相消或干涉相长是由光波的相位关系决定的,只要相位相同的光场与原子相互作用就可以实现干涉相长的过程,从而利用较弱的光场来实现较大的激发效率。
本文利用二阶微扰理论得到了激发态布居几率的解析表示,研究了可控脉冲光场作用下的双光子作用过程。选择最佳相位函数对脉冲整形,可实现双光子跃迁几率的增强。所得结果对利用光谱学来探究物质的结构等领域的理论和实验有一定的参考价值。通过深入的研究发现双光子跃迁几率的振幅实际上是光场在晶体中频域的夫琅禾费衍射过程,并且这个衍射的结果取决于狭缝的宽度。基于成熟的整形脉冲技术对光场进行裁剪或对其相位调制,可以实现类似于多缝干涉的压缩结果,实现量子聚焦。因而这一设计思想对于量子控制、信息处理等具有很大的实际意义。该工作的意义还在于可以指导实验,同时为教学提供演示。本文由如下三个部分构成。
第一部分微扰理论
在这一部分中介绍了相互作用基本理论。在电偶极近似下,得到了原子和辐射场整个系统的哈密顿(?)。在此基础上,通过求解薛定谔方程,采用迭代法并利用与时间有关的微扰理论讨论(?)各次幂的项对跃迁速率的贡献,得到激发态几率振幅表达式和辐射跃迁几率的一般表达式。介绍了双光子跃迁原理,并通过二阶微扰理论讨论双光子过程的跃迁几率,最终得到激发态的布居几率的演化,此结果是我们后面章节研究的理论基础,对于后期工作也有重大意义。
第二部分脉冲整形及双光子跃迁调控
在这一部分中基于二阶微扰理论,分析了整形脉冲作用于原子系统的双光子过程。深入研究了在远离共振和近共振两种情况下双光子跃迁几率的演化规律。结果表明,远离共振时,利用整形光脉冲作用于原子系统,会使得某些频率对应处的跃迁几率消失,光可以无损耗的通过介质：在近共振时,选择最佳相位函数对脉冲整形,可实现双光子跃迁几率的增强。证明了在相干量子调制等一些物理过程的研究中啁啾效应有着很大的优越性。所得结果对利用光谱学来探究物质的结构等领域的理论和实验研究有一定的参考价值。
第三部分量子聚焦效应
在这一部分中利用目前使用最多的激光相干量子控制法,对啁啾脉冲光场作用下双光子跃迁几率的演化规律进行更加深入的研究。利用二阶微扰理论得到了啁啾脉冲作用下激发态布居几率的解析表示,通过分析可以看出,双光子跃迁几率的振幅实际上是光场在晶体中频域的夫琅禾费衍射过程。这个衍射的结果取决于狭缝的宽度,当狭缝(泵浦光场的频率宽度)较窄时,激发态布居几率的演化规律只是一个单缝衍射的过程。当狭缝较宽时,激发态布居几率类似两个直边衍射相加。基于成熟的整形脉冲技术对光场进行裁剪或对其相位调制,实现类似于多缝干涉的压缩结果。经压缩后跃迁几率增强,实现量子聚焦的目的。因而这一设计思想对于量子控制、信息处理等具有很大的实际意义。
Quantum control is booming as a brand new discipline, it is one of the most active element in the modern scientific front edge discipline, which involved in quantum mechanics, optics, chemical, cybernetics, information science, etc. With the rapid development of laser, observation and nanometer technology, quantum control methods carried out experimental research are also improving and increasing constantly, which will promote the development of quantum cybernetics, promote the development of chemical reaction, biological cells, quantum information, nano-materials and other fields'quantum control research associated with quantum mechanics system.
With the rapid development of the ultrafast pulse, laser pulse shaping technology have made breakthrough progress.people can get any complex optical pulse waveform to realize more elaborate quantum control. The research of interaction between laser and material attracts a lot of scientific researchers. Different stimulated mechanisms produce different phenomena or effects.The chirped pulse light field who's instantaneous frequency vary linearly with time produce some peculiar quantum effects when interact with atom,which have more prodigious advantages in some coherent quantum control physical process researches,thus the chirped pulse light field was often used in theory and experiment.The chirp sweep frequency effect can implement multiple resonance stimulate processes,thus causing the multi-channel quantum interference effect or quantum diffraction effect. At the same time, the interference and diffraction effect can also be controlled by modulating the optical field phase.Constructive interference or destructive interference is decided by the phases relationship of light waves, as long as having the same phase optical field interacted with atom, constructive interference can be achieved, thus the weaker light field can realize the large stimulated efficiency.
Based on the second order perturbation theory, we get the analytic expression of the excited state population probability and study the two-photon processes under the controllable chirp pulse. The bigger two-photon transition probability can be obtained by choosing the best shaping function under the resonance case.The results have certain reference value by using spectroscopy to explore material structure and other areas in theory and experiment. Through studying in-depth we find that the amplitude of the two-photon transition probability is actually the Fraunhofer diffraction process in frequency domain when the light goes through crystal and the results depend on the width of the slit. Based on the mature shaped pulse technology, we can realize the compression of the multiple-slit interference by cropping the optical field or modulating the phase.After compressing the the transition probability is greatly enhanced and quantum focused effect can be relized. Consequently the design idea is of great practical significance for quantum control、information processing and so on. Also the significance of this work is guiding the experiment in the lab environment and providing demonstration for teaching. This paper is composed of the following three parts.
The first part is perturbation theory
In this part, it mainly introduces the fundamental theories of the interaction. Under electric dipole approximation, we get Hamiltonian H, of the whole system between atoms and radiation field. Through solving the Schrodinger equation, using iterative method and the perturbation theory related with time, we discuss the every term of HI contributes to the transition probability and get the expression of the excited state probability amplitude and transition probability. It introduces two-photon process and discusses transition probability of the two-photon processes based on the second-order perturbation theory, and finally we gain the evolution of population probability of the excited state, which is the basic theory for later chapters and also has significant meaning for research work.
The second part is that pulse shaping and two-photon transition control
In this part, based on the second-order perturbation theory,the two-photon processes in the atom system excited by a shaped pulse are analyzed.The evolution of the two-photon transition probability is studied detailedly under two cases of nonresonance and resonance.It's shown that the transition probabilities disappeared at some corresponding frequencies under the nonresonance case, and thus the laser pulse can get through the medium without loss.Meanwhile,the bigger two-photon transition probability is obtained by choosing the best shaping function under the resonance case.It's proved that chirped effect has great advantages in some coherent quantum modulation of physical process research, The results has certain reference value when using spectroscopy to explore material structure and other areas in theory and experiment.
The third part is quantum focused effect
In this part, the evolution of the two-photon transition rules under the chirp pulse light field are analyzed using the current laser coherent quantum control method. Based on the second order perturbation theory, we get the analytic expression of the population probability of the excited state. It's shown that the amplitude of the two-photon transition probability is actually Fraunhofer diffraction process in frequency domain when the light goes through crystal.The results of the diffraction depend on the width of the slit. It is just a single slit diffraction when the slit (the frequency width of the pump field) is narrower.When the slit is wider, the evolution of the population probability of the excited state is similar to the adding of two straight edge diffraction together. Based on the mature shaped pulse technology, we can realize the compression of the multiple-slit interference by cropping the optical field or modulating the phase. After compressing the the transition probability is greatly enhanced and the quantum focused effect can be relized. Consequently the design idea is of great practical significance for quantum control、information processing and so on.
引文
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[38]Tamborenea P I, Metiu H. Quantum control of electrons in semiconductor nanostructures using spatially uniform electric fields [C]//In:Proceedings of the 40th IEEE conference on Decision and Control, Orlando, Florida USA, 2001,281-286.
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[40]Sudzius M, Loser F, Lyssenko V G, et al. Coherent control of Bloch oscillations-form breathing mode quantum beats to spatial oscillations [C]//Quantum Electronics Conference, IQEC,1998,12.
[41]Bardeen C J, Wilson K R, Yakovlev V V, et al. Quantum control of molecular motion and multiphoton processes in I2 by use of ultrashort chirped pulses [C]//Quantum Electronics and Laser Science Conference, QELS,1997,164-165.
[42]程代展.量子控制一个全新的学科领域[J].控制与决策,2002,17(5)：513-531.
[43]Paz J P. Protecting the quantum word [J]. Nature,2001,412:869-870.
[44]唐志祥,张登玉.量子相干与量子消相干[J].激光杂志,2003,24：35-38.
[45]Doherty A, Doyle J, Mabuchi H, et al. Robust control in the quantum domain. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia,2000,949-954.
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[1]Sarkar C, Bhattacharya R, Bhattacharyya S.S, et al. Population transfer to excited vibrational levels of H2 molecule by stimulated hyper-Raman passage with chirped laser pulses[J]. The Journal Of Chemical Physics,2007,127(10):1-8.
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[19]郑淮斌,仇旭,李永放。绝热条件下二能级系统相干控制动态分析[J].陕西师范大学学报：自然科学版,2007,35(4)：28-32.
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[1]李永放,任立庆, 马瑞琼, 仇旭, 刘娟, 樊荣, 伏振兴。波函数的时域衍射[J]。中国科学G辑,2009,39(4)：600-605。
[2]Nirit Dudovich, Dan Oron, Yaron Silberberg. Coherent Transient Enhancement of Optically Induced Resonant Transitions [J]. Phys. Rev. Lett,2002,88:123004 (1-4).
[3]Jerome Degert, Wendel Wohlleben, Beatrice Chatel, Marcus Motzkus, Bertrand Girard. Realization of a Time-Domain Fresnel Lens with Coherent Control [J]. Phys. Rev. Lett,2002,89:203003 (1-4).
[4]S.E.Harris. Chirp and compress:Toward single-Cycle Biphotons [J].Phys. Rev. Lett. 2007,98,063602.
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[6]Barak Dayan, Avi Pe'er, Asher A. Friesem, and Yaron Silberberg. Two Photon Absorption and Coherent Control with Broadband Down-Converted Light [J]. Phys. Rev. Lett.2004,93:023005(1-4).
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[12]李永放,李百宏,任立庆,等.和频过程中的绝热与透热过程.中国科学：物理学力学天文学,2010,40：1468-1475。
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[38]Tamborenea P I, Metiu H. Quantum control of electrons in semiconductor nanostructures using spatially uniform electric fields [C]//In:Proceedings of the 40th IEEE conference on Decision and Control, Orlando, Florida USA, 2001,281-286.
[39]Haycock D L, Cheong K I, Jessen P S. Quantum state control via tunneling in an optical potential [C]//Quantum Electronics and Laser Science Conference, QELS,1999,80-81.
[40]Sudzius M, Loser F, Lyssenko V G, et al. Coherent control of Bloch oscillations-form breathing mode quantum beats to spatial oscillations [C]//Quantum Electronics Conference, IQEC,1998,12.
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[1]过巳吉.非线性光学[M].空军导弹学院：西北电讯工程学院出版社,1986：262-264.
[2]喀兴林.高等量子力学[M].第二版.北京：高等教育出版社,2004：480-484.
[3]周世勋.量子力学教程[M].北京：高等教育出版社,2006：153-160
[4]过巳吉.非线性光学[M].空军导弹学院：西北电讯工程学院出版社,1986：280-285.石顺祥,陈国夫,赵卫,等.非线性光学[M].西安：西安电子科技大学出版社,2003：
[5]石顺祥,陈国夫,赵卫,等.非线性光学[M].西安：西安电子科技大学出版社,2003：241-246.
[6]Glauber R J.The Quantum Theory of Optical Coherence [J].Phys.Rev.1963, 130:2529-2539.
[7]过巳吉.非线性光学[M].空军导弹学院：西北电讯工程学院出版社,1986：294-296.
[8]谭维翰.非线性与量子光学[M].第二版.北京：科学出版社,2000：92-94.
[9]曾谨言.量子力学(卷Ⅱ)[M].北京：科学出版社,2005：224-227.
[10]钱梅珍,崔一平,杨正名.激光物理[M],第二版,北京：电子工业出版社,2001：105-111.
[11]Meshulach D, Silberberg Y. Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse [J]. Physical Review A,1999,60 (2):1287-1291.
[1]Sarkar C, Bhattacharya R, Bhattacharyya S.S, et al. Population transfer to excited vibrational levels of H2 molecule by stimulated hyper-Raman passage with chirped laser pulses[J]. The Journal Of Chemical Physics,2007,127(10):1-8.
[2]Sohn J.Y, Ahn Y.H, Park D.J. Tunable terahertz generation using femtosecond pulse shaping [J]. Applied Physics Letters,2002,81(1):13-15.
[3]Dudovich N, Oron D, Silberberg Y. Coherent Transient Enhancement of Optically Induced Resonant Transitions [J]. Physical Review Letters,2002,88:123004(1-4).
[4]Dudovich N, Dayan B, Gallagher Faeder S.M, et al. Transform-Limited Pulse Are Not Optimal for Resonant Multiphoton Transitions [J]. Physical Review Letters, 2001,86:47-50.
[5]Meshulach D, Silberberg Y. Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse [J].Physical Review A,1999,60 (2):1287-1291.
[6]Oron D, Dudovich N, Yelin D, et al. Narrow-Band Coherent Anti-Stokes Raman signals from Broad-Band pulses [J].Physical Review Letters,2002,88:063004(1-4).
[7]Oron D, Dudovich N, Silberberg Y. Single-Pulse Phase Contrast Nonlinear Raman Spectroscopy [J].Physical Review Letters,2002,89:273001(1-4).
[8]Oron D, Dudovich N, Yelin D, et al. Quantum control of coherent anti-Stokes Raman processes [J].Physical Review (A),2002,65:8-12.
[9]Wollenhaupt M, Prakelt A, Sape-Tudoran C, et al. Femtosecond strong-field quantum control with sinusoidally phase-modulated pulses [J]. Physical Review (A).2006,73:1-15.
[10]付振兴,李永放,蔡晓燕.啁啾场作用于二能级原子跃迁的时域动态变化分析[J].陕西师范大学学报：自然科学版,2008,36(5)：20-26.
[11]李永放,任立庆,马瑞琼等.利用相位可控光场实现量子态波函数时域演化的量子控制[J].物理学报,2010,59：1671-1675.
[12]姚启钧.光学教程[M].第3版,北京：高等教育出版社,2002：532-533.
[13]Meshulach D, Silberberg Y. Coherent quantum control of two photon transitions by a femtosecond laser pulse[J].Nature,1998,396:239-242.
[14]Byron F.W, Fuller R.W. Mathematics of Classical and Quantum Physics物理学中的数学方法[M].蔡维译.北京：科学出版社,1982：384-398.
[15]Djotyan G P, Bakos J S, Sorlei Z S, et al; Coherent control of atomic quantum states by single frequency-chirped laser pulses [J]. Physical Review A,2004, (70):063406 (1-7)
[16]伏振兴,李永放,刘碧蕊.非绝热条件下二能级系统中的粒子布居儿率振[J].长江大学学报：自然科学版,2008,5(2)：12-14.
[17]李永放,任立庆,马瑞琼等.波函数的时域衍射[J].中国科学G辑,2009,39(4)：600-605.
[18]XIAO MING, ZHUANG QI, GE YANG. Quantum control using laser [J].Chinese Science Bulletin,1999,44(19):2017(in Chinese).
[19]郑淮斌,仇旭,李永放。绝热条件下二能级系统相干控制动态分析[J].陕西师范大学学报：自然科学版,2007,35(4)：28-32.
[20]吕乃光.傅里叶光学[M].第2版,北京：机械工业.
[1]李永放,任立庆, 马瑞琼, 仇旭, 刘娟, 樊荣, 伏振兴。波函数的时域衍射[J]。中国科学G辑,2009,39(4)：600-605。
[2]Nirit Dudovich, Dan Oron, Yaron Silberberg. Coherent Transient Enhancement of Optically Induced Resonant Transitions [J]. Phys. Rev. Lett,2002,88:123004 (1-4).
[3]Jerome Degert, Wendel Wohlleben, Beatrice Chatel, Marcus Motzkus, Bertrand Girard. Realization of a Time-Domain Fresnel Lens with Coherent Control [J]. Phys. Rev. Lett,2002,89:203003 (1-4).
[4]S.E.Harris. Chirp and compress:Toward single-Cycle Biphotons [J].Phys. Rev. Lett. 2007,98,063602.
[5]S.Sensarm, G.Y.Yin, S.E.Harris. Generation and Compression of Chirped Biphotons [J]. Phys. Rev. Lett.2010,104,253602.
[6]Barak Dayan, Avi Pe'er, Asher A. Friesem, and Yaron Silberberg. Two Photon Absorption and Coherent Control with Broadband Down-Converted Light [J]. Phys. Rev. Lett.2004,93:023005(1-4).
[7]Dan Oron, Nirit Dudovich, and Yaron Silberberg, Single-Pulse Phase-Contrast Nonlinear Raman Spectroscopy, Phys. Rev. Lett.2002,89,273001-4.
[8]Nirit Dudovich, Barak Dayan, Sarah M. Gallagher Faeder, and Yaron Silberberg, Transform-Limited Pulses Are Not Optimal for Resonant Multiphoton Transitions, Phys. Rev. Lett.2001,86,47-50.
[9]Nirit Dudovich, Dan Oron, Yaron Silberberg. Quantum Control of the Angular Momentum Distribution in Multiphoton Absorption Processes [J]. Phys.Rev.Lett, 2004,92:103003(1-4).
[10]Broers B, Noordam L D. Diffraction and focusing of spectral energy in multiphoton processes[J].Phys. Rev. A,1992,46:2749-2756.
[11]Rickes T, Yatsenko L P, Steuerward S, Halfmann T, Shore B W, Vitanov N V, Bergmann K. Efficient adiabatic population transfer by two-photon excitation assisted by a laser-induced Stark shift [J]. Chem. Phys.,2000,113:534-546.
[12]李永放,李百宏,任立庆,等.和频过程中的绝热与透热过程.中国科学：物理学力学天文学,2010,40：1468-1475。