Realization of quantum nonlinearity in quadratic optomechanics, including optical and mechanical nonlinearity at the single-photon level, is crucial to explore extraordinary quantum features. Here we investigate an exponential enhancement of quadratic optomechanical interaction via parametric amplifications. The parametric process can form an in-phase amplified quadrature, and the rate of coupling to the in-phase quadrature will be exponentially increased even to the single-photon strong-coupling regime. Concretely, to achieve the optical nonlinearity mediated by the quadratic optomechanical interaction, we can employ a mechanical parametric process, and the in-phase mechanical quadrature can strongly couple to the cavity mode, leading to an enhanced optical nonlinearity such as a photon blockade. Similarly, via employing the optical parametric process, the in-phase optical quadrature can also effectively mediate the quadratic interaction, leading to an enhanced mechanical nonlinearity such as mechanical squeezing, bi-excitation exchange interaction and feasibility of quantum non-demolition detection of mechanical motion. However, the parametric process can also enlarge the noise, and a squeezing vacuum with the phase matching condition should be introduced to suppress the same.
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 and
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) 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.
Few-photon optomechanical effects not only are important physical evidence for understanding the radiation-pressure interaction between photons and mechanical oscillation, but also have wide potential applications in modern quantum technology. Here we study the few-photon optomechanical effects including photon blockade and generation of the Schr\"odinger cat states under the assistance of a cross-Kerr interaction, which is an inherent interaction accompanied by the optomechanical coupling in a generalized optomechanical system. By exactly diagonalizing the generalized optomechanical Hamiltonian and calculating its unitary evolution operator, we find the physical mechanism of the enhancement of photon blockade and single-photon mechanical displacement. The quantum properties in this generalized optomechanical system are studied by investigating the second-order correlation function of the cavity field and calculating the Wigner function and the probability distribution of the rotated quadrature operator for the mechanical mode. We also study the influence of the dissipations on the few-photon optomechanical effects.