English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We deal with the dynamical system properties of aGorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation with mean-fieldHamiltonian that models a simple laser by applying a mean field approximationto a quantum system describing a single-mode optical cavity and a set of twolevel atoms, each coupled to a reservoir. We prove that the mean field quantummaster equation has a unique regular stationary solution. In case a relevantparameter $C_\mathfrak{b} $, i.e., the cavity cooperative parameter, is lessthan $1$, we prove that any regular solution converges exponentially fast tothe equilibrium, and so the regular stationary state is a globallyasymptotically stable equilibrium solution. We obtain that a locallyexponential stable limit cycle is born at the regular stationary state as$C_\mathfrak{b} $ passes through the critical value $1$. Then, the mean-fieldlaser equation has a Poincar\'e-Andronov-Hopf bifurcation at $C_\mathfrak{b} =1$ of supercritical-like type. Namely, we derive rigorously, at the level ofdensity matrices --for the first time--, the transition from a global attractorquantum state, where the light is not emitted, to a locally stable set ofcoherent quantum states producing coherent light. Moreover, we establish thelocal exponential stability of the limit cycle in case a relevant parameter isbetween the first and second laser thresholds appearing in the semiclassicallaser theory. Thus, we get that the coherent laser light persists over timeunder this condition. In order to prove the exponential convergence of thequantum state, we develop a new technique for proving the exponentialconvergence in open quantum systems that is based in a new variation ofconstant formula. Applying our main results we find the long-time behavior ofthe von Neumann entropy, the photon-number statistics, and the quantum varianceof the quadratures.

Supercritical Poincar\'e-Andronov-Hopf bifurcation in a mean field quantum laser equation