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

Chinese Physics Letters

A Class of New Nonlinear Bogoliubov Transformations

Xiaoxue Yang, Ying Wu

Published Nov 1, 2002• 5 citations
Publisher

Abstract

We find a class of new nonlinear Bogoliubov transformations and solve the eigenvalue problem of the model Hamiltonian describing the second harmonic generation with intensity-dependent coupling. These transformations differ from the previously known nonlinear quadrature-dependent Bogoliubov transformations.
DOI
10.1088/0256-307x/19/11/317

Related papers

A Class of New Nonlinear Bogoliubov Transformations

杨晓雪, 吴颖

We find a class of new nonlinear Bogolubov transformations and solve the eigenvalue problem of the model Hamiltonian describing the second harmonic generation with intensity-dependent coupling.These transformations differ from the previously known nonlinear quadrature-dependent bogoliubov transformations.

Chinese Physics Letters•Jan 1, 2002•0 citations

Quadrature-dependent Bogoliubov transformations and multiphoton squeezed states

Ying Wu, Robin Côté

We show that the nonlinear quadrature-dependent Bogoliubov transformations proposed by Siena et al. [Phys. Rev. A 64, 063803 (2001)] can in fact be constructed by the combination of two unitary transformations, a quadrature-dependent displacement followed by the standard squeezed transformation. Such a decomposition turns a nonlinear problem into an essentially linear one so that we are able to express explicitly the mean values and deviations of the quadrature operators and the photon variables under the multiphoton states in terms of those quantities averaged over the standard squeezed states involving only the quadrature-independent Bogoliubov transformation.

Physical Review A•Aug 5, 2002•27 citations

Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians

Clive Emary, R. F. Bishop

We present a new method for finding isolated exact solutions of a class of nonadiabatic Hamiltonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.

arXiv:quant-ph/0112061•Journal of Mathematical Physics•Aug 1, 2002•45 citations

Quadrature-dependent Bogoliubov transformations and multiphoton squeezed states

Silvio De Siena, Antonio Di Lisi, Fabrizio Illuminati

We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators ${X}_{1}$ and ${X}_{2}$ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function ${g}^{(2)}(0)$ and of the fourth-order correlation function ${g}^{(4)}(0)$ are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.

arXiv:quant-ph/0105070•Physical Review A•Nov 9, 2001•21 citations

Exact form of the Bogoliubov excitations in one-dimensional nonlinear Schrödinger equation

Dmitry L. Kovrizhin

arXiv:cond-mat/0109235•Physics Letters A•Sep 1, 2001•49 citations

The Analytic Aspects of The Bogoliubov Generating Functional Equations Within the Boboliubov Canonical Transformation Method

Nikolai N. Bogolubov, Anatoliy K. Prykarpatsky

The analytic aspects of the Bogoliubov generating functional equations and their transformation properties within the Bogoliubov canonical transformation method are studied. The classical Bogoliubov idea [2] to use the Wigner density operator transformation for studying the non equilibrium distribution functions within the Bogoliubov canonical transformation method is developed, a new analytic non-stationary solution to the classical N.N. Bogoliubov evolution functional equation is constructed.

Old and New Concepts of Physics•Jul 3, 2007•0 citations
We show that the nonlinear quadrature-dependent Bogoliubov transformations proposed by Siena et al. [Phys. Rev. A 64, 063803 (2001)] can in fact be constructed by the combination of two unitary transformations, a quadrature-dependent displacement followed by the standard squeezed transformation. Such a decomposition turns a nonlinear problem into an essentially linear one so that we are able to express explicitly the mean values and deviations of the quadrature operators and the photon variables under the multiphoton states in terms of those quantities averaged over the standard squeezed states involving only the quadrature-independent Bogoliubov transformation.
We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators X1{X}_{1}X1​ and X2{X}_{2}X2​ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function g(2)(0){g}^{(2)}(0)g(2)(0) and of the fourth-order correlation function g(4)(0){g}^{(4)}(0)g(4)(0) are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.