We consider a harmonic oscillator (HO) with a time dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency \omega_{0}, then, at t = 0, its frequency suddenly increases to \omega_{1} and, after a finite time interval \tau, it comes back to its original value \omega_{0}. Contrary to what one could naively think, this problem is a quite non-trivial one. Using algebraic methods we obtain its exact analytical solution and show that at any time t > 0 the HO is in a squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from \omega_{0} to \omega_{1}), remaining constant after the second jump (from \omega_{1} back to \omega_{0}). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state.

中文翻译：

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具有两次跳频的瞬态谐波振荡器：精确的代数解

我们考虑一个具有时间相关频率的谐振子 (HO)，它经历了两次连续的突然变化。根据假设，HO 以频率 \omega_{0} 开始其基本状态，然后，在 t = 0，它的频率突然增加到 \omega_{1}，并且在有限的时间间隔 \tau 之后，它回到它的原始值 \omega_{0}。与人们天真地想的相反，这个问题是一个非常重要的问题。使用代数方法，我们获得了它的精确解析解，并表明在任何时间 t > 0 时，H2O 都处于压缩状态。我们在任意时刻明确计算相对于初始状态的相应挤压参数（SP），并表明，令人惊讶的是，它在第​​一次频率跳跃（从 \omega_{0} 到 \omega_{1}）后表现出振荡，第二次跳跃后保持不变（从 \omega_{1} 回到 \omega_{0}）。我们还计算了正交方差的时间演化。最后，但并非最不重要的是，我们计算了 HO 的真空（基态）持续概率幅度，以及其任何激发态的跃迁概率幅度。