Abstract
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.
Highlights
One would be hard-pressed to find a physical system that we have collectively learnt more from than the harmonic oscillator
Except in some special cases like bosonic quantum mechanics where it can be shown that the two-point spectral form factor (SFF) is obtained by averaging the four-point of-time-order correlator (OTOC) over the Heisenberg group [10], computing the SFF is a difficult task, compounded by various subtleties inherent to the spectral analysis of the chaotic Hamiltonian
We will consider the OTOC for the displacement operators in the inverted harmonic oscillator (IHO), which are defined in terms of the creation and annihilation operators as
Summary
One would be hard-pressed to find a physical system that we have collectively learnt more from than the harmonic oscillator. Except in some special cases like bosonic quantum mechanics where it can be shown that the two-point SFF is obtained by averaging the four-point OTOC over the Heisenberg group [10], computing the SFF is a difficult task, compounded by various subtleties inherent to the spectral analysis of the chaotic Hamiltonian This diagnostic toolbox has been further expanded with the introduction of a number of more information-theoretic resources with varying degrees of utility. It is fitting that we begin with a brief overview of the inverted harmonic oscillator
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