Quantum Neimark-Sacker bifurcation

I I Yusipov,M V Ivanchenko

doi:10.1038/s41598-019-53526-2

Abstract

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

Highlights

It has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations
In this paper we find and study the quantum Neimark-Sacker bifurcation, which classical counterpart is the birth of a torus due to instability of a limit cycle[2]
Exemplifying in the experimentally relevant open quantum periodically modulated dimer model, we show that the stroboscopic Husimi distribution exhibits a transition from the unimodal to Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod, Russia. *email: mikhail. ivanchenko@itmm.unn.ru www.nature.com/scientificreports/

Summary

Introduction

It has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. In this paper we find and study the quantum Neimark-Sacker bifurcation, which classical counterpart is the birth of a torus (an invariant curve in the Poincaré/stroboscopic section) due to instability of a limit cycle (fixed point of the Poincaré map)[2]. Www.nature.com/scientificreports bagel-shaped form – a section of “quantum torus” – for the boson interaction strength close to the bifurcation value for the mean-field model.

Results

Conclusion