Using a perturbative approach, we investigate the parametric down-conversion process without the semiclassical approximation. A Wigner functional formalism, which incorporates both the spatiotemproal degrees of freedom and the particle-number degrees of freedom is used to perform the analysis. First, we derive an evolution equation for the down-conversion process in the nonlinear medium. Then we use the perturbative approach to solve the equation. The leading order contribution is equivalent to the semiclassical solution. The next-to-leading order contribution provides a solution that includes the evolution of the pump field as a quantum field.

中文翻译：

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超越半经典逼近的参数下转换

使用微扰方法，我们研究了不使用半经典逼近的参数下变频过程。结合时空自由度和粒子数自由度的Wigner功能形式主义用于进行分析。首先，我们推导了非线性介质中下变频过程的演化方程。然后，我们使用摄动法来求解方程。前导订单贡献等同于半经典解。次要的阶跃贡献提供了一种解决方案，其中包括将泵浦场演化为量子场。