Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Sinica

RELATIONSHIP BETWEEN FORMAL QUANTUM-OPERATOR METHODS AND MATRIX METHODS IN OPTICS

SUN FENG-JIU, 东北工学院物理系

Published Jan 1, 1985• 0 citations
Publisher

Abstract

This paper discusses the relationship between operator methods and matrix methods in optics. Based on the definition of optical variable operators and the hypothesis that optical operators of linear optical systems are unitary, the relation equation dealing with the relationship between operator methods and matrix methods in optics has been obtained from the general property of linear systems. For a given optical system, its optical operator can be derived from its optical matrix by solving the relation equation. And if the optical operator of a system is given, its optical matrix can also be derived from that equation. Thus the equivalency between operator methods and matrix methods in optics is demonstrated. Finally, the physical significance of the optical operator h has been pointed out.

DOI
10.7498/aps.34.368

Related papers

QUANTUM OPTICS NETWORKS, UNITARY OPERATORS AND COMPUTER ALGEBRA

Willi-Hans Steeb, Yorick Hardy

In linear quantum optics we consider phase shifters, beam splitters, displacement operations, squeezing operations etc. The evolution can be described by unitary operators using Bose creation and annihilation operators. This evolution can be reduced to matrix multiplication using unitary matrices. We derive these evolutions for the different unitary operators. Finally a computer algebra implementation is provided.

International Journal of Modern Physics C•Jul 1, 2008•1 citation

Method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

Juan Carlos García-Escartín, Vicent Gimeno, Julio José Moyano‐Fernández

The evolution of quantum light through linear optical devices can be described by the scattering matrix $S$ of the system. For linear optical systems with $m$ possible modes, the evolution of $n$ input photons is given by a unitary matrix $U={\ensuremath{\varphi}}_{m,M}(S)$, derived from a known homomorphism, ${\ensuremath{\varphi}}_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, $M$. We present a method to decide whether a given unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear optics or not and the inverse transformation ${\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

arXiv:1901.06178•Physical Review A•Aug 2, 2019•10 citations

Operator methods in physical optics

David Stoler

This paper is devoted to the exposition of operator algebraic methods that can be usefully applied to many areas of physical optics. These methods exploit the structural similarity between physical optics and quantum mechanics and make use of operator techniques widely used in quantum theory. We develop the basic approach here and illustrate its power by giving a compact derivation of the Fourier properties and the imaging properties of an ideal lens. We also use the operator approach to introduce and discuss operators and fields that are unaltered by passage through an optical system.

Journal of the Optical Society of America•Mar 1, 1981•133 citations

Theory of Many-Particle Systems. I

Paul C. Martin, Julian Schwinger

This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.

Physical Review•Sep 15, 1959•1,813 citations

Cases of structural equality between the scalar wave equations of optics and the quantum-mechanical Schrödinger equation

Hans Kragl

Given a set of special restrictions applying to the optical system and by restricting the solution space to one of its subspaces, one can transform the time-dependent and the stationary wave equations of scalar optics into an equation that is structurally equivalent to the quantum-mechanical Schrödinger equation. These conditions are specified, the subspace that makes that transformation possible is constructed, and the transformation is derived. The physical meaning of the new, transformed variables is also given. An application shows how the transformation can be applied in the fields of mode coupling of forward- and backward-traveling waves. Finally, the well-known beam propagation method is discussed from this point of view.

Journal of the Optical Society of America A•Jun 1, 1992•3 citations

Applications of quantum mechanical hypervirial theorems

John C. Nash, Nash, John C.

Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.

•Jan 1, 1972•0 citations
This paper discusses the relationship between operator methods and matrix methods in optics. Based on the definition of optical variable operators and the hypothesis that optical operators of linear optical systems are unitary, the relation equation dealing with the relationship between operator methods and matrix methods in optics has been obtained from the general property of linear systems. For a given optical system, its optical operator can be derived from its optical matrix by solving the relation equation. And if the optical operator of a system is given, its optical matrix can also be derived from that equation. Thus the equivalency between operator methods and matrix methods in optics is demonstrated. Finally, the physical significance of the optical operator h has been pointed out.
The evolution of quantum light through linear optical devices can be described by the scattering matrix SSS of the system. For linear optical systems with mmm possible modes, the evolution of nnn input photons is given by a unitary matrix $U={\ensuremath{\varphi}}_{m,M}(S)$, derived from a known homomorphism, ${\ensuremath{\varphi}}_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, MMM. We present a method to decide whether a given unitary evolution UUU for nnn photons in mmm modes can be achieved with linear optics or not and the inverse transformation ${\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.
This paper is devoted to the exposition of operator algebraic methods that can be usefully applied to many areas of physical optics. These methods exploit the structural similarity between physical optics and quantum mechanics and make use of operator techniques widely used in quantum theory. We develop the basic approach here and illustrate its power by giving a compact derivation of the Fourier properties and the imaging properties of an ideal lens. We also use the operator approach to introduce and discuss operators and fields that are unaltered by passage through an optical system.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
Given a set of special restrictions applying to the optical system and by restricting the solution space to one of its subspaces, one can transform the time-dependent and the stationary wave equations of scalar optics into an equation that is structurally equivalent to the quantum-mechanical Schrödinger equation. These conditions are specified, the subspace that makes that transformation possible is constructed, and the transformation is derived. The physical meaning of the new, transformed variables is also given. An application shows how the transformation can be applied in the fields of mode coupling of forward- and backward-traveling waves. Finally, the well-known beam propagation method is discussed from this point of view.
Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.