Quadrature-dependent Bogoliubov transformations and multiphoton squeezed states - art. no. 063803

We introduce a linear, canonical transformation of the fundamental single-m ode field operators a and a(+) that generalizes the linear Bogoliubov trans formation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of a ny of the fundamental quadrature operators X-1 and X-2 to the linear transf ormation, thus making the original Bogoliubov transformation quadrature dep endent. Remarkably, the conditions of canonicity do not impose any constrai nt on the form of the nonlinear function. and lead to a set of nontrivial a lgebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum state s 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 ph ase space shows that besides coherence properties, these states exhibit a s queezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presenc e of the extra nonlinear term in the phase of the wave functions has also r elevant consequences on photon statistics and correlation properties. The n onquadratic structure of the associated Hamiltonians suggests that these st ates be generated in connection with multiphoton processes in media with hi gher nonlinearities. We dive a detailed description of the quadratic nonlin ear transformation, which defines four-photon squeezed states. In particula r, 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 e xhibits super-Poissonian statistics. while the latter indicates photon bunc hing in the four-photon emissions.