量子光学中与经典Fresnel变换对应的若干新幺正算符
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摘要
经典光学(Fourier光学)的一个重要组成部分是Fresnel衍射(Fresnel变换)及关于衍射的Collins公式。我国学者范洪义等人首先用相干态表象研究了Fresnel衍射的量子对应,利用有序算符内的积分(IWOP)技术他们建立了一个量子力学Fresnel算符,以实现量子光学中的Fresnel变换,它与经典Collins公式相对应。自量子的Fresnel变换提出以来,它被广泛的应用于讨论经典光学与量子光学理论之间的关系,及与其他光学变换的关系。本文用量子光学的观点重新审视若干经典光学变换,采用的是压缩相干态表象和相干纠缠态表象,提出与经典Fresnel变换对应的新的量子光学中的若干幺正算符,如单模和双模广义Fresnel算符,Fresnel-Hadamard互补算符,并给出其物理意义。这些新幺正算符的经典对应给经典光学提供了新变换的可能性。这篇论文研究的出发点是基于如下的考虑：范洪义等人给出的Fresnel算符是通过相干态在相空间的一个代表点运动到另一点sz-rz*而导出的,这里ss*-rr*=1。根据相空间的直观分析,一个相干态对应于图形上面积为h／2的小圆。量子Fresnel变换表明相空间的一个小圆移动到另一个小圆。ss*-rr*=1保证了该变换是辛变换,同时也保证了刘维定理(相体积不变)的满足。考虑到相干态是压缩相干态中的一个特殊态矢,以及压缩相干态在量子光学和量子信息等领域的广泛应用,我们从压缩相干态在相空间的代表点(一个椭圆)的运动及用IWOP技术提出了广义Fresnel算符。鉴于量子纠缠的理论也渗透到量子光学中,我们又基于相干纠缠态表象,提出了一个Fresnel-Hadamard互补算符,它对一个光分束器的两个输出光场和分别起到了Hadamard变换和Fresnel变换的作用(该光分束器的两个输入光场光场分别为a1与a2)。
论文具体安排如下：
第一章,首先介绍本文工作的研究背景,然后介绍经典Fourier光学中的Fresnel衍射公式,以及在经典框架内介绍了衍射的Collins公式的推导过程。
第二章,为了建立起从经典到量子之间的”桥梁”,我们介绍了一些背景知识。首先介绍一些常见表象如坐标、动量、粒子数、及相干态等表象。然后介绍真正起到上述“桥梁”作用的有序算符内的积分(IWOP)技术的提出背景和内涵。
第三章,我们介绍了范等人从量子光学的相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为Fresnel算符。从相空间的直观分析,Fresnel算符对应于相干态在相空间的代表点(一个为面积h／2的圆)的运动变换。我们还简单介绍范等人从量子光学的双模相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为双模Fresnel算符。
第四章,我们从量子光学的相干压缩态表象及用IWOP技术提出了与Fresnel变换对应的量子算符,称之为广义Fresnel算符。从相空间的直观分析,广义Fresnel算符对应于压缩相干态在相空间的代表点(一个椭圆)的运动。
第五章,我们基于双模相干压缩态表象介绍了与双模Fresnel算符对应的双模广义Fresnel算符。
第六章,基于相干纠缠态表象,我们发现了一个Fresnel-Hadamard互补算符,它对双模光场a1与a2经过一个光分束器的输出光场和分别起到了Hadamard变换和Fresnel变换的作用。以上讨论表明,从量子光学的新表象和有序算符内的积分(IWOP)技术,可以找到新的光学变换。
最后,我们给出一些总结和展望。
One of the important parts in classical optics (Fourier optics) is the Fresnel diffrac-tion and its Collins formula. Scholars of China Fan Hong-yi et.al firstly studied the quantum correspondence of Fresnel diffraction by using the coherent state. By using the Integral technique within ordered product (IWOP) of operators they constructed a quantum Fresnel operator to realize the Fresnel transformation in quantum optics and it corresponds to the classical Collins formula. Since the introduction of the quantum Fresnel transformation, it has been widely applied to the discussion of the relation be-tween the classical optics and the quantum optics and the relations with other optical transformations. In this paper will use the view of quantum optics to resurvey some classical optical transformation by using the squeezed coherent state and coherent en-tangled state representation to put forward some new unitary operators (e.g. the single mode and two mode generalized Fresnel operators, the Fresnel-Hadamard complimen-tary operator) in quantum optics corresponding to the classical Fresnel transformation and give their physical meanings. The classical correspondence of these new unitary operators may provide probability for some new transformations in classical optics. The starting point of our work is based on the below considering:the Fresnel operator given by Fan Hong-yi et.al is derived from a coherent's moment in phase space from point to point sz-rz*, where ss*- rr*= 1. According to the intuitional analysis, a coherent state is graphically corresponding to a small round. The quantum Fresnel transformation indicates a small round moving to another small round. The condition ss*-rr*=1 guarantees the transformation is symplectic and also guar-antees satisfying the Liouvil theorem (the invariability of phase volume). According to that the coherent state is a special case of squeezed coherent state and thinking that the squeezed coherent state has been got widely used in quantum optics and quantum information, the generalized Fresnel operator we introduce in this paper is just based on the moment of squeezed coherent state(graphically corresponding to a ellipse in phase space) and by using the IWOP technique. Whereas the filtering of the theory of quantum entanglement to quantum optics, based on the coherent entangled state, we introduce a Fresnel-Hadamard complementary operator, which can play the roles of Hadamard transformation and Fresnel transformation respectively for the two output fields of and of a beamsplitter (The two input fields of the beamsplitter is a1 and a2).
My Ph.D dissertation is arranged as following:
In chapter one, we introduce some background knowledge of our work, the Fres-nel diffraction formula in classical Fourier optics and the deriving process to obtain the Collins formula in classical frame.
In chapter two, in order to bridge the classics to quantum, we introduce some background knowledge. Firstly, we introduce some basic representation like the co-ordinate, momentum, particle number and the coherent state representations, then we introduce the background and meaning of the Integral technique within ordered product (IWOP) of operators which really has the above bridge function.
In chapter three, we introduce the quantum operator corresponding to the classi-cal Fresnel transformation, i.e. the Fresnel operator constructed by Fan et.al, which is based on the coherent state representation in quantum optics and the IWOP technique. According to the intuitional analysis in phase space, the Fresnel operator corresponds to the moving transformation of the representing point(a round with h/2 area) of co-herent state in phase space. we also introduce the quantum operator corresponding to the classical Fresnel transformation, i.e. the two-mode Fresnel operator constructed by Fan et.al, which is based on the two-mode coherent state representation in quantum optics and the IWOP technique.
In chapter four, based on the squeezed coherent state representation in quantum optics and the IWOP technique, the quantum operator corresponding to the classical Fresnel transformation, i.e. the generalized Fresnel operator is introduced. According to the intuitional analysis in phase space, the generalized Fresnel operator corresponds to the moving transformation of the representing point(an ellipse with h/2 area) of squeezed coherent state in phase space.
In chapter five, based on the two-mode squeezed coherent state representation we introduced the two-mode generalized Fresnel operator corresponding to the two-mode Fresnel operator.
In chapter six, based on the coherent entangled state, we find a Fresnel-Hadamard complementary operator. For the optical fields and considered as two out-put fields after the two mode optical fields a1 and a2 passing through a beamsplitter, the Fresnel-Hadamard complementary operator can play the roles of Hadamard trans-formation and Fresnel transformation respectively. Above discussion indicates that from the new representation in quantum optics and using the Integral technique within ordered product (IWOP) of operators we can find new optical transformations.
Finally, we give some conclusions and expectations.
论文具体安排如下：
第一章,首先介绍本文工作的研究背景,然后介绍经典Fourier光学中的Fresnel衍射公式,以及在经典框架内介绍了衍射的Collins公式的推导过程。
第二章,为了建立起从经典到量子之间的”桥梁”,我们介绍了一些背景知识。首先介绍一些常见表象如坐标、动量、粒子数、及相干态等表象。然后介绍真正起到上述“桥梁”作用的有序算符内的积分(IWOP)技术的提出背景和内涵。
第三章,我们介绍了范等人从量子光学的相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为Fresnel算符。从相空间的直观分析,Fresnel算符对应于相干态在相空间的代表点(一个为面积h／2的圆)的运动变换。我们还简单介绍范等人从量子光学的双模相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为双模Fresnel算符。
第四章,我们从量子光学的相干压缩态表象及用IWOP技术提出了与Fresnel变换对应的量子算符,称之为广义Fresnel算符。从相空间的直观分析,广义Fresnel算符对应于压缩相干态在相空间的代表点(一个椭圆)的运动。
第五章,我们基于双模相干压缩态表象介绍了与双模Fresnel算符对应的双模广义Fresnel算符。
第六章,基于相干纠缠态表象,我们发现了一个Fresnel-Hadamard互补算符,它对双模光场a1与a2经过一个光分束器的输出光场和分别起到了Hadamard变换和Fresnel变换的作用。以上讨论表明,从量子光学的新表象和有序算符内的积分(IWOP)技术,可以找到新的光学变换。
最后,我们给出一些总结和展望。
One of the important parts in classical optics (Fourier optics) is the Fresnel diffrac-tion and its Collins formula. Scholars of China Fan Hong-yi et.al firstly studied the quantum correspondence of Fresnel diffraction by using the coherent state. By using the Integral technique within ordered product (IWOP) of operators they constructed a quantum Fresnel operator to realize the Fresnel transformation in quantum optics and it corresponds to the classical Collins formula. Since the introduction of the quantum Fresnel transformation, it has been widely applied to the discussion of the relation be-tween the classical optics and the quantum optics and the relations with other optical transformations. In this paper will use the view of quantum optics to resurvey some classical optical transformation by using the squeezed coherent state and coherent en-tangled state representation to put forward some new unitary operators (e.g. the single mode and two mode generalized Fresnel operators, the Fresnel-Hadamard complimen-tary operator) in quantum optics corresponding to the classical Fresnel transformation and give their physical meanings. The classical correspondence of these new unitary operators may provide probability for some new transformations in classical optics. The starting point of our work is based on the below considering:the Fresnel operator given by Fan Hong-yi et.al is derived from a coherent's moment in phase space from point to point sz-rz*, where ss*- rr*= 1. According to the intuitional analysis, a coherent state is graphically corresponding to a small round. The quantum Fresnel transformation indicates a small round moving to another small round. The condition ss*-rr*=1 guarantees the transformation is symplectic and also guar-antees satisfying the Liouvil theorem (the invariability of phase volume). According to that the coherent state is a special case of squeezed coherent state and thinking that the squeezed coherent state has been got widely used in quantum optics and quantum information, the generalized Fresnel operator we introduce in this paper is just based on the moment of squeezed coherent state(graphically corresponding to a ellipse in phase space) and by using the IWOP technique. Whereas the filtering of the theory of quantum entanglement to quantum optics, based on the coherent entangled state, we introduce a Fresnel-Hadamard complementary operator, which can play the roles of Hadamard transformation and Fresnel transformation respectively for the two output fields of and of a beamsplitter (The two input fields of the beamsplitter is a1 and a2).
My Ph.D dissertation is arranged as following:
In chapter one, we introduce some background knowledge of our work, the Fres-nel diffraction formula in classical Fourier optics and the deriving process to obtain the Collins formula in classical frame.
In chapter two, in order to bridge the classics to quantum, we introduce some background knowledge. Firstly, we introduce some basic representation like the co-ordinate, momentum, particle number and the coherent state representations, then we introduce the background and meaning of the Integral technique within ordered product (IWOP) of operators which really has the above bridge function.
In chapter three, we introduce the quantum operator corresponding to the classi-cal Fresnel transformation, i.e. the Fresnel operator constructed by Fan et.al, which is based on the coherent state representation in quantum optics and the IWOP technique. According to the intuitional analysis in phase space, the Fresnel operator corresponds to the moving transformation of the representing point(a round with h/2 area) of co-herent state in phase space. we also introduce the quantum operator corresponding to the classical Fresnel transformation, i.e. the two-mode Fresnel operator constructed by Fan et.al, which is based on the two-mode coherent state representation in quantum optics and the IWOP technique.
In chapter four, based on the squeezed coherent state representation in quantum optics and the IWOP technique, the quantum operator corresponding to the classical Fresnel transformation, i.e. the generalized Fresnel operator is introduced. According to the intuitional analysis in phase space, the generalized Fresnel operator corresponds to the moving transformation of the representing point(an ellipse with h/2 area) of squeezed coherent state in phase space.
In chapter five, based on the two-mode squeezed coherent state representation we introduced the two-mode generalized Fresnel operator corresponding to the two-mode Fresnel operator.
In chapter six, based on the coherent entangled state, we find a Fresnel-Hadamard complementary operator. For the optical fields and considered as two out-put fields after the two mode optical fields a1 and a2 passing through a beamsplitter, the Fresnel-Hadamard complementary operator can play the roles of Hadamard trans-formation and Fresnel transformation respectively. Above discussion indicates that from the new representation in quantum optics and using the Integral technique within ordered product (IWOP) of operators we can find new optical transformations.
Finally, we give some conclusions and expectations.
引文
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[109]Hong yi Fan and Yue Fan. Epr entangled states and complex fractional fourier transformation. Euro. Phys. J. D,21:233,2002.
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[115]D. Dragoman. Classical versus complex fractional fourier transformation. J. Opt. Soc. Am. A,26:274,2009.
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[123]Hong yi Fan and Hai liang Lu. New two-mode coherent-entangled state and its application.J. Phys. A,37:10993,2004.
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