A non-stationary one-dimensional cavity can be described by the time-dependent and multi-mode effective Hamiltonian of the so-called dynamical Casimir effect. Due to the non-adiabatic boundary conditions imposed in one of the cavity mirrors, this effect predicts the generation of real photons out of vacuum fluctuations of the electromagnetic field. Such photon generation strongly depends on the number of modes in the cavity and their intermode couplings. Here, by using an algebraic approach, we show that for any set of functions parameterizing the effective Hamiltonian, the corresponding time-dependent Schrödinger equation admits an exact solution when the cavity has one intermode interaction. With the exact time evolution operator, written as a product of eleven exponentials, we obtain the average photon number in each mode, a few relevant observables, and some statistical properties for the evolved vacuum state.

中文翻译：

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具有一个模间相互作用的非平稳腔的精确解

非平稳的一维腔可以通过所谓的动态卡西米尔效应的时间相关和多模有效哈密顿量来描述。由于在其中一个腔镜中施加了非绝热边界条件，这种效应预测了电磁场真空波动产生的真实光子。这种光子的产生很大程度上取决于腔中模式的数量及其模式间耦合。在这里，通过使用代数方法，我们表明对于任何参数化有效哈密顿量的函数，当腔具有一个模间相互作用时，相应的与时间相关的薛定谔方程允许精确解。使用精确的时间演化算子，写成 11 个指数的乘积，我们得到每个模式下的平均光子数，