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arXiv:1403.5783 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Heavy quark master equations in the Lindblad form at high temperatures

Yukinao Akamatsu

Published Mar 3, 2015• 133 citations
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Abstract

We derive the quantum master equations for heavy quark systems in a high-temperature quark-gluon plasma in the Lindblad form. The master equations are derived in the influence functional formalism for open quantum systems in perturbation theory. These master equations have a wide range of applications, such as decoherence of a heavy quarkonium and Langevin dynamics of a heavy quark in the quark-gluon plasma. We also show the equivalence between the quarkonium master equations in the recoilless limit and the Schr\"odinger equations with stochastic potential.

DOI
10.1103/physrevd.91.056002

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We derive the quantum master equations for heavy quark systems in a high-temperature quark-gluon plasma in the Lindblad form. The master equations are derived in the influence functional formalism for open quantum systems in perturbation theory. These master equations have a wide range of applications, such as decoherence of a heavy quarkonium and Langevin dynamics of a heavy quark in the quark-gluon plasma. We also show the equivalence between the quarkonium master equations in the recoilless limit and the Schr"odinger equations with stochastic potential.
Accurate modelling and understanding of quarkonium production in ultrarelativistic heavy ion collisions requires a formalism that preserves the quantum properties of microscopic QˉQ\bar{Q}QQˉ​Q systems while treating the interaction of such pairs with the quark-gluon plasma (QGP). The open quantum system approach has recently emerged as one of the most fruitful schemes to meet such requirements. However, the quantum master equations obtained so far in this context are derived assuming a strict ordering between the QGP temperature TTT and the QˉQ\bar{Q}QQˉ​Q energy gaps (ΔEΔEΔE) of the quarkonia bound states. This limits their predictive power since, as the QGP expands and cools down, the system traverse all regimes between the quantum Brownian motion (QBM) regime for T≳ΔET\gtrsim ΔET≳ΔE to the quantum optical (QO) regime for ΔE≳TΔE\gtrsim TΔE≳T. In this paper, we derive and present a more general non-abelian quantum master equation of the Lindblad type, which does not suffer from these limitations and thus allows to faithfully describe the quantum evolution of the QQˉQ\bar{Q}QQˉ​ pairs during the whole QGP-evolution. We also provide some illustration of the key quantities governing this equation.
In this paper we study the real-time evolution of heavy quarkonium in the quark-gluon plasma (QGP) on the basis of the open quantum systems approach. In particular, we shed light on how quantum dissipation affects the dynamics of the relative motion of the quarkonium state over time. To this end we present a novel nonequilibrium master equation for the relative motion of quarkonium in a medium, starting from Lindblad operators derived systematically from quantum field theory. In order to implement the corresponding dynamics, we deploy the well established quantum state diffusion method. In turn we reveal how the full quantum evolution can be cast in the form of a stochastic nonlinear Schr\"odinger equation. This for the first time provides a direct link from quantum chromodynamics to phenomenological models based on nonlinear Schr\"odinger equations. Proof of principle simulations in one dimension show that dissipative effects indeed allow the relative motion of the constituent quarks in a quarkonium at rest to thermalize. Dissipation turns out to be relevant already at early times well within the QGP lifetime in relativistic heavy ion collisions.
Abstract We study the open system dynamics of a heavy quark in the quark-gluon plasma using a Lindblad master equation. Applying the quantum state diffusion approach by Gisin and Percival, we derive and numerically solve a nonlinear stochastic Schrödinger equation for wave functions, which is equivalent to the Lindblad master equation for the density matrix. From our numerical analysis in one spatial dimension, it is shown that the density matrix relaxes to the Boltzmann distribution in various setups (with and without external potentials), independently of the initial conditions. We also confirm that quantum dissipation plays an essential role not only in the long-time behavior of the heavy quark but also at early times if the heavy quark initial state is localized and quantum decoherence is ineffective.
\nIn recent years, a significant theoretical effort has been made towards a dynamical description of quarkonia inside the Quark-Gluon Plasma (QGP), using the open quantum systems formalism. In this framework, one can get a real-time description of a quantum system (here the quarkonium) in interaction with a thermal bath (the QGP) by integrating out the bath degrees of freedom and studying the system reduced density matrix. We investigate the real-time dynamics of a correlated heavy quark-antiquark pair inside the QGP using a quantum master equation previously derived from first QCD principles in [4]. The full equation is directly resolved in 1D to lessen computing costs and is used for the first time to gain insight on the dynamics in both a static and evolving medium following a Björken-like temperature evolution.\n
In recent years, a significant theoretical effort has been made towards a dynamical description of quarkonia inside the Quark-Gluon Plasma (QGP), using the open quantum systems formalism. In this framework, one can get a real-time description of a quantum system (here the quarkonium) in interaction with a thermal bath (the QGP) by integrating out the bath degrees of freedom and studying the system reduced density matrix. We investigate the real-time dynamics of a correlated heavy quark-antiquark pair inside the QGP using a quantum master equation previously derived from first QCD principles in [4]. The full equation is directly resolved in 1D to lessen computing costs and is used for the first time to gain insight on the dynamics in both a static and evolving medium following a Björken-like temperature evolution.
I review recent applications of the open quantum system framework in the understanding of quarkonium suppression in heavy-ion collisions, which has been used as a probe of the quark–gluon plasma for decades. The derivation of the Lindblad equations for quarkonium in both the quantum Brownian motion and the quantum optical limits and their semiclassical counterparts is explained. The hierarchy of time scales assumed in the derivation is justified from the separation of energy scales in nonrelativistic effective field theories of QCD. Physical implications of the open quantum system approach are also discussed. Finally, I list some open questions for future studies.