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Communications in Theoretical Physics

Multiphoton Squeezed States

Xiaoxue Yang, Wu Ying

Published Nov 15, 2003• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/40/5/585

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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 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.
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local-oscillator angle; for nnn-mode systems they depend on nnn heterodyne mixing angles. The canonical formalism realizes nontrivial mixing of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems, and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamiltonians of degenerate multiphoton processes and define a class of non-Gaussian, nonclassical multiphoton states that exhibit properties of coherence and squeezing. We show that such homodyne multiphoton squeezed states are generated by unitary operators with a nonlinear time evolution that realizes the homodyne mixing of a pair of conjugate quadratures. Tuning of the local-oscillator angle allows us to vary at will the statistical properties of such states. We discuss the relevance of the formalism for the study of degenerate (up-)down-conversion processes. In a companion paper [F. Dell'Anno, S. De Siena, and F. Illuminati, 69, 033813 (2004)], we provide the extension of the nonlinear canonical formalism to multimode systems, we introduce the associated heterodyne multiphoton squeezed states, and we discuss their possible experimental realization.
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.