Nonlinear Squeezed States and Multiplication Rule of Nonlinear Squeezing Operators Gained via Nonlinear Coherent State Representation and IWOP Technique

Using the nonlinear coherent state representation we derive nonlinear squeezed states and the multiplication rule of nonlinear squeezing operators. We find that the symplectic matrices multiplication rule in nonlinear coherent state projection operator representation maps into the multiplication rule of successive nonlinear squeezing operators.The technique of integral within an ordered product of operators plays an essential role in deriving the multiplication rule.     

Phase Properties of Nonlinear Coherent States

Using the Pegg-Barnett fornalism we study the phase probability distributions and the squeezing effects ofmeasured phase operators in the nonlinear coherent states introduced by R.L. de Matos Filho and W. Vogel to describethe center-of mass motion of a trapped ion and the q-coherent states. Moreover, we have obtained the completenessrelation of nonlinear coherent states and proved that the q-Fock state |n>q introduced in many papers is, in fact, theusual Fock state.     

Nonlinear two-mode squeezing obtained by analysing two-mode exponential quadrature operators in entangled state representation

By analysing the properties of two-mode quadratures in an entangled state representation (ESR) we derive from ESR some complicated exponential quadrature operators for nonlinear two-mode squeezing, which directly leads to wave function of the nonlinear squeezed state in ESR.     

一种新的偶奇非线性相干态中测量相位算符的压缩效应

利用Bamett和pegg提出的测量相位算符讨论了一种新的偶奇非线性相干态的压缩效应。结果表明，在这种偶奇非线性相干态中测量相位算符不存在CN和SN类压缩，只存在CS类压缩。利用数值计算方法研究了CS类压缩效应，结果表明与通常的偶奇相干态的不同的是，偶非线性相干态在大区域内出现压缩效应而奇非线性相干态在小区域内才出现压缩效应。     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group

We find that the coherent state projection operator representation of symplectic transformation constitutes a loyal group representation of symplectic group. The result of successively applying squeezing operators on number state can be easily derived.     

Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group

We find that the coherent state projection operator representation of symplectic transformation constitutesa loyal group representation of symplectic group. The result of successively applying squeezing operators on numberstate can be easily derived.     

The P-representation on Nonlinear Coherent State Basis

Using the technique of integration within an ordered product of nonlinear Bose operators and by composing the nonlinear coherent state's over-completeness relation, we construct the corresponding P-representation theory. The generalized P-representation for some nonlinear Bose operators can be established.     

Collins diffraction formula studied in quantum optics

We find that the Collins diffraction formula in cylindrical coordinates is just the transformation matrix element of a three-parameter two-mode squeezing operator in the deduced entangled state representation. This is a new tie connecting the unitary transform in quantum optics to the generalized Hankel transform in Fourier optics. The group multiplication rule of the squeezing operators maps to the Collins formula related to two successive Hankel transforms.     

Production of the Superposition of Nonlinear Coherent States and Entangled Nonlinear Coherent States

In this paper, by using the parity operator as well as the nonlinear displacement-type operator, we define new operators which by the action of them on the vacuum state of the radiation field, superposition of two nonlinear coherent states and two-mode entangled nonlinear coherent states are generated. Also, we show that via the generalization of the presented method, the superposition of more than two nonlinear coherent states and n-mode entangled nonlinear coherent states can be generated.     

Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis

We find that the coherent state projection operator representation of the two-mode squeezing operator constitutes a loyal group representation of symplectic group, which is a remarkable property of the coherent state. As a consequence, the resultant effect of successively applying two-mode squeezing operators are equivalent to a single squeezing in the two-mode Fock space. Generalization of this property to the 2n-mode case is also discussed.     

Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law

We find that classical similarity transformations in the coherent state representation projects onto the similarity transformation operators (STO), these operators constitute a loyal representation of symplectic group. Remarkably, the multiplication rule of the STOs naturally leads to the quantum optical generalized ABCD law, which is the quantum mechanical correspondence of the classical optical ABCD law. Throughout the whole derivation, the technique of integration within an ordered product (IWOP) of operators is employed.     

Symplectic Group Representation of the Two-Mode Squeezing Operator in the Coherent State Basis

We find that the coherent state projection operator representation of the two-mode squeezing operator constitutes a loyal group representation of symplectic group, which is a remarkable property of the coherent state. As a consequence, the resultant effect of successively applying two-mode squeezing operators are equivalent to a single squeezing in the two-mode Fock space. Generalization of this property to the 2n-mode case is also discussed.     

多光子非线性过程中的振幅平方压缩

本文研究了多光子非线性光学过程中的振幅平方压缩,证明了K光子么正算符Sk(z)(K>2)对非真空相干态的作用在—级近似下会产生振幅平方压缩.以及Sk(z)(K>2,K≠4)对真空态的作用不会产生振幅平方压缩.发现4个光子么正算符S4(z)对真空态的作用可以产生振幅平方压缩,这表明振幅平方压缩是一种独立于二阶压缩、反聚束以及亚泊松统计分布的新的光场非经典效应.     

非线性相干态光场与二能级原子相互作用的量子特性

借助于数值计算方法, 研究了非线性相干态光场与一个二能级原子相互作用过程中原子的布局反转、 光子的反聚束效应以及光场振幅平方压缩的时间演化特性, 并讨论了光场参量α、原子的初始状态以及非线性相 干态Lamb-Dicke参数η对该相互作用系统量子特性的影响.     

Three-Mode Cyclic Squeezed States Constructed via EPR Entangled State

We construct the three-mode cyclic squeezed states and analyze its squeezing property by using the technique of integration within an ordered product of operators and the natural representation of the two-mode squeezing operator in the Einstein-Podolsky-Rosen entangled state basis.     

压缩真空场与原子非线性作用系统中原子的量子特性

研究了附加克尔介质的压缩真空场与二能级原子依赖强度耦合相互作用系统中原子的反转特性,并采用密度算符间的距离研究了该模型中原子量子态的演化规律.详细地讨论了克尔非线性作用的强度以及初始压缩真空场的压缩度对原子反转和原子量子态演化的影响.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.