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Optics Communications

Quasiclassical approach to the nonlinear Kerr dynamics

Mojdeh S. Najafabadi, A. B. Klimov, Luis Lorenzo Sánchez-Soto, Gerd Leuchs

Published Jun 28, 2023• 5 citations
Publisher

Abstract

We examine the quasiclassical approximation to the self-Kerr nonlinear effect. The corresponding dynamics appears as classical trajectories, with the quantumness of the state included via its Wigner function. We obtain analytical estimates of the optimal squeezing attainable that compare fairly well with the numerical quantum solution. We delimit the range of parameters for which the quasiclassical solution retains relevant quantum features.

DOI
10.1016/j.optcom.2023.129717

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We examine the quasiclassical approximation to the self-Kerr nonlinear effect. The corresponding dynamics appears as classical trajectories, with the quantumness of the state included via its Wigner function. We obtain analytical estimates of the optimal squeezing attainable that compare fairly well with the numerical quantum solution. We delimit the range of parameters for which the quasiclassical solution retains relevant quantum features.
A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (${\ensuremath{\chi}}^{(3)}$ nonlinearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian vs non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.
We study amplitude-squared squeezing in interaction of coherent light with a nonlinear Kerr medium modelled as an anharmonic oscillator with interaction Hamiltonian H = ½ λ a +2 a 2, where λ is proportional to χ(3) of the nonlinear medium and a is annihilation operator for the interacting field. We find the squeezing parameter S ( τ, r ) in terms of a dimensionless interaction time τ = λ t and Kerr parameter r , which is product of, τ and the average number of photons and obtain almost complete amplitude-squared squeezing (i.e., S ≈ 0) for very small interaction time and very large intensity of the interacting light. We optimize squeezing parameter S ( τ, r ) by an analytic estimation assuming high intensity of the interacting light and realistic values of Kerr nonlinearity following J.Bajer et al. [Czech. J. Phy. 52, 1313 (2002)] and obtain a scaling law for optimal amplitude-squared squeezing with minimum value S min , at r = r min for a given τ. The validity of the scaling law is checked numerically and analytically in the region of realistic values of Kerr nonlinearity and intensity of the interacting light.
Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.