论文标题
具有两个频率跳跃的时间依赖性的谐波振荡器：精确的代数解决方案
A time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution
论文作者
论文摘要
我们考虑了一个谐波振荡器（HO），其时间依赖于时间，该频率经历了两个连续的突然变化。通过假设HO以频率ω_{0}开始以其基本状态开始，然后，在t = 0时，其频率突然增加到ω__{1}，并且在有限的时间间隔τ之后，它返回其原始值ω__{0}。与人们可以天真地想到什么相反，这个问题是一个非常不平凡的问题。使用代数方法，我们获得了其确切的分析解决方案，并表明在任何时候t> 0 ho处于挤压状态。我们在任意瞬间与初始状态明确计算相应的挤压参数（SP），并表明，令人惊讶的是，它在第一个频率跳后表现出振荡（从ω__{0}到ω__{1}），到第二跳后保持不变（从ω__{1}回到ω_{1}回到ω______________________{0}）。我们还计算了正交方差的时间演变。最后但并非最不重要的一点是，我们计算了HO的真空（基本状态）持久性概率幅度，以及其在任何激发态的过渡概率幅度。
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 ω_{0}, then, at t = 0, its frequency suddenly increases to ω_{1} and, after a finite time interval τ, it comes back to its original value ω_{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 ω_{0} to ω_{1}), remaining constant after the second jump (from ω_{1} back to ω_{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.
