Classical and quantum oscillators of quartic anharmonicities: second-order solution

Abstract

An analytical solution up to the second order in the coupling constant λ is obtained for a classical quartic anharmonic oscillator by using Taylor series method. Our solution yields, as a special instance, the corresponding results obtained by using Laplace transform. With the help of correspondence principle, the classical solution is used to obtain the solution corresponding to a quantum quartic anharmonic oscillator. In the weak coupling regime (i.e., anharmonic constant λ⪡1), the so-called secular terms in classical and quantum solutions are tucked in (summed up) to avoid the nonconvergence. Both the classical and quantum solutions are used to obtain the frequency shifts of the quartic oscillators. It is found that these frequency shifts coincide exactly with those of the earlier results obtained by other methods. From the quantum field theoretic point of view, our solution exhibits the so-called Lamb shift. As an application of the solution for the quantum oscillator, we examine the possibility of getting squeezed states out of the input coherent light interacting with a nonlinear medium of inversion symmetry.