New Development Of Beam Splitter Operator Theory And Its Applications

Beam splitter is a basic and important linear element,which plays an important role for splitting,transformation and entanglement in experiments for quantum optics and quantum information.For a long time,the researchers have some understanding of the performance of entanglement,but the analysis is not deep enough because of the lack of advanced mathematical theory.In this thesis,we shall take use of the IWOP technique(integration within an ordered product)of operators to construct quantum mechanics entanglement representation and the operator theory of beam splitter,including representation,normally ordering and Weyl ordering theories,such that some new properties of beam splitter can further be excavated.In addition,the method can also be directly extended to discuss the function of multi-cascaded beam splitters and Mach-Zehnder interferometer.Based on these,some applications of beam splitter are also considered about preparing new quantum state of light and quantum teleportation.The specific research contents are as follows.(1)Using the IWOP technique and completeness of representation,we obtained the integral expressions of single-and two-variable Hermite polynomials.Based on this,we further derived the mother functions for single-variable and two-variable even-Hermit and odd-Hermit polynomials.This method seems more concise and direct.In addition,some applications such as deriving new formula and constructing new two-mode quantum states are discussed.It is shown that two-variable Hermite polynomials excitation can used to produce entangled coherent states.In addition,by using the IWOP technique,Weyl expansion of operator is derived.Then,three new ordering formulas of operators are presented,which have been applied to calculate Q-P and P-Q ordering of some operators.As other applications of Weyl expansion,the formula of photon counting is also easily obtained.In particular,noticing the Weyl ordering of displacement operator is itself,Weyl ordering invariance under similarity transformations is conveniently proved,instead of using of Wigner operator.(2)Beam splitter is a basic linear element in quantum optics,which plays an important role in preparation of entangled states and quantum measurement.On the basis of the transformation relation of operators at input ports and output ports,we derived the natural representations in different representations.Using the natural expression rather SU(2)Lie algebra relation,as well as the IWOP technique,we have conveniently and concisely derived the normally ordering form,exponential expression of beam splitter operator.Many forms of decompositions for beam splitter operator can also directly obtained by virtue of its natural representation and the orthogonality of coordinate states.Furthermore,it can be used to achieve entangled state representation of quantum mechanics and their corresponding Schmidt decomposition.Furthermore,the case has been extended to the case of multi-cascaded beam-splitters.As applications,two-cascaded beam-splitters are used to prepare an entanglement state representation and its Schmidt decomposition,and qubit states by conditional measurement.These studies provide an effective way for the quantum teleportation with continuous variable,and the preparation of multi-mode entanglement state representation and qubit states.In addition,a general method is shown for obtaining the total operator and the corresponding normally ordering form as well as Schmidt decomposition of the linear systems consisting of beam-splitters.(3)Based on the traditional Kimble-Braunstein scheme of quantum teleportation and the orthogonal property of entangled state representation,we derived the mean density operator of the output state,which means that we constructed the relation between the output density operator and the characteristics functions of the input state to be teleported and the entangled resources.The characteristics function of the output state is also derived which is a concise form relating these two characteristic functions above.Then we further obtained a new formula for calculating the quantum teleportation fidelity for the coherent state input and any two-mode entangled resource.It is shown that the fidelity of teleportation can be easily calculated when the Q-function of the normally ordering form of entangled resource is known.This way is convenient for getting the fidelity of teleportation.As its applications,some Gaussian and non-Gaussian entangled states are examined to teleport the coherent state.(4)Based on an asymmetrical beam splitter and conditional measurement,we have proposed a scheme for generating non-Gaussian states in a realistic case.Photon subtraction,photon addition and quantum catalysis can be seen as three special cases of the scheme.Then we studied the nonclassical properties of such quantum states,by deriving analytical expressions for photon number,squeezing effect,Mandel Q parameter and Wigner function.It is shown that these properties not only depend on squeezing parameter and measurement,but also on the transmissivity of the ABS.Photon addition and quantum catalysis present a higher probability in the regions of low and high transmissivity,respectively.Although both single-photon subtraction and addition share a clear negative volume,the latter can be considered as a better choice for producing nonclassical states than the two others,if both success probability and negative volume in the small squeezing region are taken into consideration.Quantum catalysis is better in the large squeezing region and with low transmissivity.These results provide some reference for generating such non-Gaussian states.