We provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing and, also, include the effects both of self-phase modulation of the pump and of cross-phase modulation of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labeled by a frequency. We provide a proof that these fields satisfy bona fide commutation relations and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.

中文翻译：

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波导中高增益双光束生成的理论：从麦克斯韦方程到高效仿真

我们提供了一种有效的方法，用于计算麦克斯韦方程组的规范处理得出的波导中的高增益双光束生成。导出了运动方程，可以自然地适应通过自发参量下变频（SPDC）或自发四波混频产生的光子，并且还包括泵的自相位调制和孪生子的交叉相位调制的影响泵浦的光束。我们求解的方程涉及在空间中演化并由频率标记的场。我们提供的证明是，这些场满足了真正的换向关系，并且在遥远的过去和未来，它们都简化为标准的随时间变化的海森堡算子。解决了这些海森堡算子的输入输出关系后，我们还展示了如何构造描述双光束量子态的ket。最后，我们考虑具有平坦非线性分布的波导中高增益SPDC的示例，为此，我们的方法提供了一种仅需单个矩阵幂运算的显式解决方案。