Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Instantons and Quantum Chaos

Hamza Jirari, H. Kröger, X.Q. Luo, Kevin J. M. Moriarty, S. G. Rubin

Published Jan 1, 2008• 16 citations
OpenAlex

Abstract

We suggest a closed form expression for the path integral of quantum transition amplitudes to construct a quantum action. Based on this we propose rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential.

Related papers

Closed Path Integrals and the Quantum Action

Hamza Jirari, H. Kröger, X.Q. Luo +2 more

We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with parameters different from the classical action. We present numerical results for the harmonic oscillator with weak perturbation, the quartic potential, and the double well potential. The quantum action is relevant for quantum chaos and quantum instantons.

arXiv:hep-th/9910141•Physical Review Letters•Jan 8, 2001•25 citations

Nonlinear Dynamics In Quantum Physics -- Quantum Chaos and Quantum Instantons

H. Kröger

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos.

arXiv:quant-ph/0302169•Feb 21, 2003•0 citations

Krylov complexity, path integrals, and instantons

Cameron Beetar, Eric Lobato Graef, Jeff Murugan +2 more

Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.

arXiv:2507.13226•Jul 17, 2025•0 citations

Alternative perspective on quantum tunneling and instantons

Frédérik Paradis, H. Kröger, G. Melkonyan +1 more

We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.

arXiv:quant-ph/0408051•Physical Review A•Feb 16, 2005•1 citation

From Quantum Action to Quantum Chaos

Hamza Jirari, H. Kröger, G. Melkonyan +2 more

We have recently suggested a quantum action, which has the form of a classical action and takes into account quantum effects via renormalized action parameters. Here we apply it to quantum chaos. We study a system in 2-D with weak anharmonic coupling ($V_{coupl} \propto x^{2}y^{2}$) being classically chaotic. We construct the quantum action at finite temperature. We compute Poincaré sections of the quantum action at that temperature and compare it with those of the classical action. We observe chaotic behavior in both cases.

arXiv:quant-ph/0108094•Aug 21, 2001•0 citations

Quantum Chaos via the Quantum Action

H. Kröger

We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling - which is classically a chaotic system. We compare Poincaré sections obtained from the quantum action with those from the classical action.

arXiv:quant-ph/0212093•Dec 16, 2002•1 citation
Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity K(t)K(t)K(t) for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for K(t)K(t)K(t), the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.
We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.
We have recently suggested a quantum action, which has the form of a classical action and takes into account quantum effects via renormalized action parameters. Here we apply it to quantum chaos. We study a system in 2-D with weak anharmonic coupling (Vcoupl∝x2y2V_{coupl} \propto x^{2}y^{2}Vcoupl​∝x2y2) being classically chaotic. We construct the quantum action at finite temperature. We compute Poincaré sections of the quantum action at that temperature and compare it with those of the classical action. We observe chaotic behavior in both cases.