The electromagnetic field coupled to a \((2l-1)\) order of nonlinear medium leads to the model of a 2lth anharmonic oscillator, \(l\ge 2\) being an integer. The oscillator with \(l=2\) corresponds to the model of a quartic anharmonic oscillator. In spite of the huge advancement of mathematical physics, we are still in search of exact analytical solution to the lowest order (i.e., \(l=2)\) of anharmonic oscillator. There are few interesting reports where the approximate solutions to the classical and quantum quartic oscillator problems are explored. In the present investigation, the exact analytical solutions of the quartic anharmonic oscillator under rotating wave approximation and hence the two-photon anharmonic oscillator with periodic forcing are exhibited. The solutions corresponding to the quantized two-photon anharmonic oscillator with periodic forcing is used to calculate the second-order variances of the canonically conjugate quadrature in terms of the initial coherent state. The effects of the forcing parameters on the second-order variances and hence on the squeezing effects are examined for various photon numbers and the dimensionless interaction constants. In these studies, we assume that the oscillator is in on-resonance with those of the external forcing term. The off-resonant contribution is found to be small and is neglected.

耦合到非线性介质\（（2l-1）\）阶的电磁场导致了第2 l非谐振子振荡器的模型，\（l \ ge 2 \）是整数。\（l = 2 \）的振荡器对应于四次非谐振荡器的模型。尽管数学物理学取得了巨大进步，但我们仍在寻找最低阶的精确解析解（即\（l = 2）\）非谐振荡器。很少有有趣的报告探讨经典和量子四次振荡器问题的近似解。在本研究中，展示了旋转波逼近下四阶非谐振子的精确解析解，并因此揭示了具有周期性强迫的双光子非谐振子。对应于具有周期性强迫的量化双光子非谐振荡器的解被用于计算就初始相干态而言的典范共轭正交的二阶方差。对于各种光子数和无量纲的相互作用常数，研究了强迫参数对二阶方差的影响，并因此对挤压效应的影响。在这些研究中，我们假设振荡器与外部强迫项的谐振。发现非谐振贡献很小并且被忽略了。