## The multi-faceted inverted harmonic oscillator: Chaos and complexity

Arpan Bhattacharyya, Wissam Chemissany, S. Shajidul Haque, Jeff Murugan, Bin Yan

SciPost Phys. Core 4, 002 (2021) · published 8 February 2021

- doi: 10.21468/SciPostPhysCore.4.1.002
- Submissions/Reports

### 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.

### Cited by 57

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Arpan Bhattacharyya, -
^{2}Wissam Chemissany, -
^{3}S. Shajidul Haque, -
^{3}Jeff Murugan, -
^{4}Bin Yan

^{1}இந்திய தொழில்நுட்பக் கழகம் காந்திநகர் / Indian Institute of Technology Gandhinagar [IITGN]^{2}California Institute of Technology [CalTech]^{3}University of Cape Town [UCT]^{4}Los Alamos National Laboratory [LANL]

- Department of Science and Technology, Ministry of Science and Technology (through Organization: विज्ञान एवं प्रौद्योगिकी विभाग / Department of Science and Technology [DST])
- Los Alamos National Laboratory [LANL]
- National Research Foundation
- Conseil de Recherches en Sciences Naturelles et en Génie / Natural Sciences and Engineering Research Council [NSERC / CRSNG]
- United States Department of Energy [DOE]