Beam splitter (BS) 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 between operators at input ports and output ports, we derive the natural representations in different representations. Using the natural expression rather than SU(2) Lie algebra relation, as well as the technique of integration within ordered product (IWOP) of operator, we can conveniently and concisely derive the normally ordering form and exponential expression of BS operator. Many forms of decompositions for BS operator can also be directly obtained by its natural representation and the orthogonality of coordinate states. Furthermore, the entangled state representation and corresponding Schmidt decomposition can be conveniently obtained.
Beam splitter (BS) 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 between operators at input ports and output ports, we derive the natural representations in different representations. Using the natural expression rather than SU(2) Lie algebra relation, as well as the technique of integration within ordered product (IWOP) of operator, we can conveniently and concisely derive the normally ordering form and exponential expression of BS operator. Many forms of decompositions for BS operator can also be directly obtained by its natural representation and the orthogonality of coordinate states. Furthermore, the entangled state representation and corresponding Schmidt decomposition can be conveniently obtained.
Beam splitter,as a kind of linear optics instruments,has many applications such as in quantum optics and quantum information,including the preparation of nonclassical quantum states and entangled state representation.In Heisenberg picture,on the one hand,the relation of input-output of beam splitter can be easily obtained.Especially for the multicascaded beam-splitters,the input-output relation can also be directly obtained by the input-output relation of single beam splitter.On the other hand,we often need to calculate the probabilities of detecting photon number in many cases,thus we need to turn into Schrdinger picture for simplifying our calculation.Based on the equivalence between both pictures,the relation between transformation matrixes connecting these two pictures is derived.That is to say, the transform matrix corresponding to the Schrdinger picture can be obtained by transposing the transform matrix in Heisenberg picture.This concise relation constructs a bridge connecting two pictures and simplifies our calculation in the Schrdinger picture rather than step by step.Using the relation between transform matrixes of both pictures and combining the technique of integration within ordered product of operator,we further consider the coordination representation,normally ordering form and exponential expression of single beam-splitter.Then we further examine the coordination representation,normally ordering form and exponential expression of two-cascaded beam-splitters.As a generalization,the method is extended to the case of multi-cascaded beam-splitters.These investigations provide an effective way to prepare multi-mode entangled states and qubit states.In addition,a general method is shown of obtaining the total operator and its normally ordering form as well as Schmidt decomposition of the linear systems consisting of beam-splitters.As applications,2-cascaded beam-splitters is used to generate a new quantum mechanics representation and prepare the qubit states with the help of conditional measurement.The Schmidt decomposition of three-mode entangled state representation can be directly obtained by the coordination representation of 2-cascaded beam-splitters,which shows the property of entanglement.In addition,based on this representation we can clearly see that when the input states of first beam splitter are two coordinate states,the output states cannot be entangled.This implies that although the coordinate states are nonclassical,the entangled state can not be prepared either.The new proposed quantum mechanics representation will be further used to investigate the optical transformations,including wavelet transformation,Fourier transform,fractional Fourier transform,et al.Therelevant discussion will be our aim in the future research.
Abstract A beam splitter operator is a very important linear device in the field of quantum optics and quantum information. It can not only be used to prepare complete representations of quantum mechanics, entangled state representation, but it can also be used to simulate the dissipative environment of quantum systems. In this paper, by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator, we present the coherent state representation of the operator and the corresponding normal ordering form. Based on this, we consider the applications of the coherent state representation of the beam splitter operator, such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables. Furthermore, we extend our investigation to two single and two-mode cascaded beam splitter operators, giving the corresponding coherent state representation and its normal ordering form. In addition, the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated, and the fidelity is discussed. The above results provide good theoretical value in the fields of quantum optics and quantum information.
Using the technique of integration within an ordered product of operators we construct a generalized two-mode entangled state, which can be generated by an asymmetrical beam splitter (BS). Some important properties of this state, such as orthogonality and Schmidt decomposition, are also discussed by deriving the expression of BS operator in coordinate representation. As its applications, to conjugate state, obtain operator identities, generate new squeezing operators (squeezed state) are also presented. It is shown that the fidelity of quantum teleportation can be enhanced under certain case by using the asymmetrical new squeezed state as entangled resource.
In this work, we present a comprehensive analytical study of a lossless quantum beam splitter (BS) with transmission coefficient T and reflection coefficient R. Using the unitary operator representation and binomial expansion, we derive the exact probability distribution P_k for detecting k photons in one output port when the input ports are prepared in Fock states |N>_0 |M>_1. The balanced configuration R = T = 1/2 and N = M is examined in detail: we prove that destructive quantum interference suppresses all odd photon numbers at the outputs, leaving only even photon counts. The special case N = M = 1 reproduces the famous Hong-Ou-Mandel (HOM) dip with P_1 = 0. Beyond photon statistics, we demonstrate that the beam splitter functions as a deterministic, tunable source of photon-number entanglement. By tracing over one output mode, we construct the reduced density matrix rho_2 and compute the von Neumann entropy S = - sum_k P_k log2 P_k, a direct measure of entanglement between the two output ports. For the balanced case with N = M = 1, we obtain S = 1 ebit, i.e., maximal Bell-state entanglement. For larger equal photon numbers, the entropy increases monotonically. Our results transform the passive beam splitter into a deterministic source of photon-number entanglement, with direct applications in optical quantum computing, interferometry, and quantum information processing. Numerical three-dimensional plots of the probability distribution are provided, and all derivations are supported by rigorous algebra and a full appendix.
For optical homodyning, the matrix representation of a lossless beam splitter belongs to the SU(2) group of unimodular second-order unitary matrices. The connection between this group and the rotation group in three dimensions permits the field density operators at the input and output ports of the beam splitter to be related by means of well-known angular-momentum transformations. This, in turn, provides the joint output photon-number distribution, which may be written as a Fourier series in the relative phase shift imparted by the beam splitter, for a general joint state at its inputs. The series collapses to a single term if one of the input fields is diagonal in the number-state representation. If the inputs to both ports are further restricted to be pure number states, the joint, as well as the marginal photon-number distributions, turn out to be directly proportional to the square of Jacobi polynomials in the beam-splitter transmittance. These photon-number probabilities are invariant to a set of physical and time-reversal symmetries. When one of the input photon-number states is the vacuum, the beam splitter simply deletes photons from the other port in Bernoulli fashion, as if they were classical particles. The output photon number is then described by the binomial distribution. If the inputs at the two ports are different number states, neither of which is the vacuum, the photon-number distribution is expressible in terms of summed and weighted products of the results for photomixing with the vacuum. If the inputs at the two ports are identical number states, and a beam splitter of transmittance \ensuremath{\tau}=(1/2 is used, the photon-number distribution assumes a simple but interesting form. It vanishes for odd photon numbers, indicating that the photons assemble in pairs at each output port. Finally, it is shown that homodyning quantum fluctuations can be reduced by using a balanced photomixer for arbitrary input states.
In a previous paper Zhan et al. (2022) proposed the coherent state representation for linear operator, which shows a beautiful theoretical value. Inspired by the work, we further extend it to the nonlinear case, including single- and two-mode squeezing process. Based on squeezing operators’ transformation relation, we introduce their coherent state representations and derive their normal ordering forms by using the technique of integration within product of operator. As applications, we examine some operators’s normal ordering forms and show the entanglement property of prepared quantum states via the Schmidt decomposition. In particular, we further consider two cascaded two-mode squeezing operators, in which a generalized SU(1,1) interferometer is involved. In addition, the application of two-mode squeezing operator followed by photon-subtraction to generate non-Gaussian entangled states in quantum teleportation is discussed. Our results will be useful in preparation of quantum states and quantum metrology.