Study On Operator Sorting And Quantum Shearing Proble

At present,the preparations,properties and applications of non-Gaussian quantum states are one of the hot topics in quantum optics.The reason is that the traditional Gaussian quantum states have obvious disadvantage in quantum information processing,such as the inability to complete general quantum computation,and the inability to achieve the optimal entanglement distillation which determines the success or failure of quantum communication.Non-Gaussian quantum states can not only improve the entanglement performance effectively,but also provide a new physical carrier for quantum information processing such as long-distance quantum key distribution and quantum computing.Therefore,the preparation and manipulation of non-Gaussian quantum states have attracted great attention and become an important research topic in quantum optics.To this end,we propose a quantum scissor scheme by using beam splitters and conditional measurements for preparing non-Gaussian quantum states and investigating their quantum properties.The main research work of this thesis includes the following four aspects:1.Based on the IWOP technique(integration within an ordered product)of operators,the ordered arrangement forms of a series of power operators are derived.We introduce a differential method for calculating the operator ordering and use it to present the normal and anti-normal ordering products of combination of coordinate and momentum operators.Compared to the existed ways,our method is neater and simpler in deriving the above operator ordering.As their important byproducts,the mutual transformation formulas between the normal ordering and anti-normal ordering are also obtained,which have good universality.Using the mutual transformation formulas,the generating functions of even and odd bivariate Hermite polynomials which are widely used in mathematical physics is obtained.Furthermore,we arrange the power operator of the product of the photon creation operator and the photon annihilation operator in their normally and anti-normally ordered product forms by using special functions and the mutual transformation formulas between normal and anti-normal orderings of operators.Besides,the Q-and P-ordered forms of the power operator of the product of coordinate operator and momentum operator are also obtained by the analogy method.Based on the ordered arrangement of these power operators,the P representation of chaotic light field and the differential relation between Hermite polynomials and Laguerre polynomials are obtained.2.As we know,beam splitter is one of the basic linear devices in quantum optics,which plays an important role in the preparation of non-Gaussian quantum states.On the basis of the transformation relation of operators at input and output ports,we derive the natural expression in coherent state representation.Using the natural expression and the IWOP technique,we derive the normally ordering form and exponential expression of beam splitter operator.Furthermore,the case is extended to 2-cascaded beam splitters.In addition,the entanglement characteristics of the output state of beam splitter are studied.Finally,the quantum catalysis of beam splitter and the quantum scissors of 2-cascaded beam splitters are briefly discussed.3.A quantum scissor scheme is designed by means of beam splitter and condition measurements.Via inputting and detecting single photon states,quantum scissor operation is equivalent to a mixed superposition of three pure state projection operators,which means that the output states are always truncated for any input state.We theoretically prepare a class of non-Gaussian quantum states via thermal state truncation and investigate their statistical properties using average photon number,gain intensity and signal to noise ratio.It is shown that the intensity gain and signal to noise ratio greater than one can be achieved by modulating the thermal parameter and the transmissivity,which realizes the signal amplification and enhancement.Besides,quantum scissor operation can generate the highly non-classical quantum state by investigating the negativity volume of Wigner function.4.We propose a scheme which allows quantum state truncation via the combined action of a displacement operation and two beam splitters and obtain the equivalent operator of displacement-based quantum scissors.We take the thermal state,coherent state and squeezed vacuum state as the input states and derive the analytical expressions of the output states.Their statistical properties are analyzed in detail by average photon number and signal to noise ratio,and the nonclassicality of the output states is studied via the negativity of Wigner function.The results show that the displacement-based quantum scissors can effectively enhance the success probability and the nonclassicality of the output states.