The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.

中文翻译：

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多面倒频谐振器：混沌与复杂性

谐波振荡器是物理模型的代名词。从概念上和计算上讲简单，但又足够丰富，可以教我们有关物理学的知识，涉及范围从经典力学到量子场论。这种多面性也延伸到其倒数对应，在该倒数对应中，振荡器频率在分析上一直延续到纯虚数。在本文中，我们使用最新开发的量子混沌诊断技术（例如失序相关器（OTOC）和电路复杂性）来探究反向谐波振荡器（IHO）。尤其是，我们研究了有和没有非高斯三次扰动的IHO位移算子的OTOC，以分别探索真实扰码和准扰码。此外，我们计算了倒立振荡器的完整量子李雅普诺夫频谱，在李雅普诺夫指数之间找到配对结构。我们还使用Heisenberg组来计算时间演化位移算子的复杂度，该算子显示出混沌行为。最后，我们将分析扩展到N反相谐波振荡器，以研究在耗散，加扰和渐近状态下编码的不同时间尺度上的复杂性行为。