Two-mode Fresnel Operator Research Articles

The S-ordered Operator Expansions of One-mode and Two-mode Fresnel Operators and their Applications

By using the technique of integration within the s-ordered product of operators (IWSOP), we first deduce the s-ordered expansion of the one-mode and two-mode Fresnel operators. Employing the s-ordered operator expansion formula, the matrix elements of one-mode and two-mode Fresnel operator in the number state representation are also obtained, respectively.

New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property, we obtain new decomposition of the Fresnel operator as the product of the quadratic phase operator, the squeezing operator, and the fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems. The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.

Weyl Ordering of Two-Mode Fresnel Operator and Its Applications

By using the Weyl ordering operator formula and the Weyl transformation rule, we derive Weyl ordering of the two-mode Fresnel operator, and then obtain its matrix element in the coordinate representation, which is the integral kernel of the generalized two-mode Fresnel transformation in classical optics. By means of the optical operator method, we obtain a decomposition of the two-mode Fresnel operator’s Weyl ordering and an operator identity.

A DERIVATION OF THE WEYL ORDERED FORM OF TWO-MODE FRESNEL OPERATOR BY VIRTUE OF THE WEYL OPERATOR ORDERING METHOD

Based on the technique of integration with a Weyl ordered product (IWWOP) for the two-mode operator, we derive out the Weyl ordered form of two-mode Fresnel operator (TFO). The multiplication rule for TFO and the matrix element of Weyl ordered form of TFO in coordinate eigenstates are also discussed.

The two-mode quantum Fresnel operator and the multiplication rule of 2D Collins diffraction formula

By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.

Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.

New theorem relating two-mode entangled tomography to two-mode Fresnel operator

Based on the Fan—Hu's formalism, i.e., the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation, which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating the quantum tomogram of two-mode density operators, i.e., the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F2†ρF2, where F2 is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.

Relation between optical Fresnel transformation and quantum tomography in two-mode entangled case

Similar in spirit to the preceding work 9, where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e. where F 2 is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find where is the conjugated state to .

Application of the two-mode squeezed coherent state representation in deriving generalized optical Collins formula

We propose a new two-mode squeezed coherent state representation | z 1, z 2〉 g which is characteristic of the correlation between the squeezing and the displacement. Based on it and using the technique of integration within an ordered product of operators we obtain a generalized two-mode Fresnel operator (GTFO), which is an image of the mapping from ( z 1, z 2) to ( s z 1 + r z 2 * , r z 1 * + s z 2 ) in | z 1, z 2〉 g representation. The matrix element of GTFO in the coordinate representation leads to a generalized two-dimensional Collins formula (Huygens–Fresnel integration transformation describing optical diffraction) in entangled form.

Adaption of optical Fresnel transform to optical Wigner transform

Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.

Quantum Mechanical Version for Bessel Beam's Propagation in ABCD Optical System

By introducing the entangled state representation and the two-mode Fresnel operator we provide quantum mechanical version for Bessel beam's classical propagation in ABCD optical system. This provides the opportunity of studying various classical Fresnel transformations in the context of quantum optics.

Optical Four-Wave Mixing Operator, Fresnel Operators and Three-Mode Entangled State Representation

We analyse the optical four-wave mixing operator S and relate it to the two-mode Fresnel operator. It is shown that the direct product of the two-mode Fresnel operator and the single-mode Fresnel operator has a natural representation on the basis of a three-mode entangled state, which is constructed by S and a beam splitter transform.

Time Evolution Caused by Hamiltonian Composed of QuadraticCombination of Canonical Operators and Time-DependentTwo-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just thetime-evolutional unitary operator governed by the Hamiltonian composed ofquadratic combination of canonical operators in the way of exhibitingSU(1,1) algebra. This is an approach for obtaining the time-dependentHamiltonian from the preassigned time evolution in classical phase space,an approach which is in contrast to Lewis–Riesenfeld's invariant operatortheory of treating time-dependent harmonic oscillators.