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Physical Review A

Quadrature-dependent Bogoliubov transformations and multiphoton squeezed states

Ying Wu, Robin Côté

Published Aug 5, 2002• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.66.025801

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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 present analytical results for the multiphoton squeezed states defined through nonlinear quadrature-dependent Bogoliubov transformations. These analytical results turn a nonlinear problem into an essentially linear one and they can be utilized 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 which only involves the quadrature-independent Bogoliubov transformation.
We present analytical results for the multiphoton squeezed states defined through nonlinear quadrature-dependent Bogoliubov transformations. These analytical results turn a nonlinear problem into an essentially linear one and they can be utilized to express explicitly the mean walues 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 which only involves the quadrature-independent Bogoliubov transformation.
We introduce the coordinate-dependent one- and two-mode squeezing transformations and discuss the properties of the corresponding one-and two-mode squeezed states. We show that the coordinate-dependent one-and two-mode squeezing transformations can be constructed by the combination of two transformations, a coordinate-dependent displacement followed by the standard squeezed transformation. Such a decomposition turns a nonlinear problem into a linear one because all the calculations involving the nonlinear one- and two-mode squeezed transformation have been shown to be able to reduce to those only concerning the standard one- and two-mode squeezed states.
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.
Extending the scheme developed for a single mode of the electromagnetic field in the preceding paper [F. Dell'Anno, S. De Siena, and F. Illuminati, Phys. Rev. A 69, 033812 (2004)], we introduce two-mode nonlinear canonical transformations depending on two heterodyne mixing angles. They are defined in terms of Hermitian nonlinear functions that realize heterodyne superpositions of conjugate quadratures of bipartite systems. The canonical transformations diagonalize a class of Hamiltonians describing nondegenerate and degenerate multiphoton processes. We determine the coherent states associated with the canonical transformations, which generalize the nondegenerate two-photon squeezed states. Such heterodyne multiphoton squeezed states are defined as the simultaneous eigenstates of the transformed, coupled annihilation operators. They are generated by nonlinear unitary evolutions acting on two-mode squeezed states. They are non-Gaussian, highly nonclassical, entangled states. For a quadratic nonlinearity the heterodyne multiphoton squeezed states define two-mode cubic phase states. The statistical properties of these states can be widely adjusted by tuning the heterodyne mixing angles, the phases of the nonlinear couplings, as well as the strength of the nonlinearity. For quadratic nonlinearity, we study the higher-order contributions to the susceptibility in nonlinear media and we suggest possible experimental realizations of multiphoton conversion processes generating the cubic-phase heterodyne squeezed states.