The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity

TL;DR

The paper investigates the inverted harmonic oscillator (IHO) as a tractable platform to study quantum chaos diagnostics and circuit complexity. It analyzes displacement-operator OTOCs, showing quasi-scrambling in the Gaussian IHO and genuine scrambling under a cubic perturbation, and derives a paired quantum Lyapunov spectrum with exponents $\pm\Omega$ that mirror the classical instability. Complexity is explored via two routes: Nielsen’s geometry on the Heisenberg group for single-operator evolution, and a multi-mode approach revealing three distinct time-regimes (dissipation, scrambling, and asymptotic growth) with a scrambling time $t_d \sim \delta^{2}\log(1/\sqrt{N})$. These results illustrate how OTOC and complexity complement each other in a simple yet rich model, offering guidance for analyzing chaos and scrambling in more complex quantum systems and quantum field theories.

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.

Figures (2)

• Figure 1: Time evolution of complexity of the displacement operator (computed from the operator method), for different choice of parameters. The red and the blue dotted curves correspond to $\{ \text{Im} (\alpha) = 0.1, \text{Re} (\alpha) = 0.1, \Omega = 0.1 \}$ and $\{ \text{Im} (\alpha) = 5, \text{Re} (\alpha) = 0.1, \Omega= 0.5 \}$ respectively
• Figure 2: Universal growth of the complexity in Eq. (\ref{['eq:complexityN']}) at different time scales. Top, middle, and bottom figures show, respectively, the power-law dissipation, the exponential scrambling in semi-log scale, and the intermediate linear growth. Blue dotted, black, and red dashed curves correspond to {$\delta= 0.4$, $N=100$}, {$\delta= 0.5$, $N=200$}, and {$\delta= 0.5$, $N=100$}, respectively. Other parameters are fixed as $m=1$, $\lambda=10$, $\delta \lambda=0.01$.