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arXiv:2205.13302 · Chinese Physics C

Open quantum system approach for heavy quark thermalization*

Zhuoxuan Xie, Baoyi Chen

Published Feb 15, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract We treat heavy quark as an open quantum system in a hot medium and rederive the stochastic Schrödinger equation (SSE) from the full Schrödinger equation for both heavy quarks and the medium. We apply the SSE to the dynamical evolutions of a heavy quark (as a system) in the static hot medium (as an environment). Heavy quarks interact with the medium via random scatterings, which exchange the momentum and phase factor randomly between two wave functions of the system and the environment. The exchange of momentum and phase factor results in the transition between different eigenstates of the system. These are included via an external stochastic potential in the Hamiltonian of SSE. Stochastic wave functions of a heavy quark are evolved with the stochastic external potential. The mean wave functions and corresponding momentum distributions of heavy quarks are obtained after the ensemble average over a large set of stochastic wave functions. We present the thermalization of heavy quarks in the static medium with different coupling strengths.

DOI
10.1088/1674-1137/acbc0f

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hep-phCross-listedarXiv:2610.22501•Oct 19, 2026
Abstract We treat heavy quark as an open quantum system in a hot medium and rederive the stochastic Schrödinger equation (SSE) from the full Schrödinger equation for both heavy quarks and the medium. We apply the SSE to the dynamical evolutions of a heavy quark (as a system) in the static hot medium (as an environment). Heavy quarks interact with the medium via random scatterings, which exchange the momentum and phase factor randomly between two wave functions of the system and the environment. The exchange of momentum and phase factor results in the transition between different eigenstates of the system. These are included via an external stochastic potential in the Hamiltonian of SSE. Stochastic wave functions of a heavy quark are evolved with the stochastic external potential. The mean wave functions and corresponding momentum distributions of heavy quarks are obtained after the ensemble average over a large set of stochastic wave functions. We present the thermalization of heavy quarks in the static medium with different coupling strengths.
The heavy quarks present in the quark-gluon plasma (QGP) can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history. In a previous study, a stochastic Schrödinger equation (SSE) has been applied to describe the transport process of heavy quarks, where an external field with random phases is used to simulate the thermal medium. In this work, we study the connection between the SSE and the Boltzmann equation (BE) approach in the Keldysh Green's function formalism. By comparing the Green's function of the heavy quark from the SSE and the Keldysh Green's functions leading to the Boltzmann equation, we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit. We subsequently confirm their consistency through numerical calculations.
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.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
Fakult¨at fu ¨r Physik, Universit t Bielefeld, D-33615 Bielefeld, Germany(Dated: November 15, 2011)We propose an open quantum systems approach to the physics of heavy quarkonia in a thermalmedium, based on stochastic quantum evolution. This description emphasizes the importance ofcollisions with the environment and focuses on the concept of spatial decoherence of the heavyquarkonium wave function. It is shown how to determine the parameters of the dynamical evolution,i.e. the real potential and the noise strength, from a comparison with quantities to be obtained fromlattice QCD. Furthermore the imaginary part of the lattice QCD heavy quark potential is found tobe naturally related to the strength of the noise correlations. We discuss the fate of QQ¯ analyticallyin a limiting scenario for the spatial decoherence and provide a qualitative 1-dimensional numericalsimulation of the real-time dynamics.
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.
Heavy quarks are often treated as Brownian probes of the quark-gluon plasma. We formulate the irreversible entropy balance of their momentum distribution and separate the production generated by collisions from the driving caused by an evolving medium. For linear dynamics the production splits into two positive contributions, from directed momentum and from the momentum width. A static bath obeys an exact H-theorem. A cooling bath adds an explicit driving term, and longitudinal Bjorken expansion enters as a separate kinematic drive rather than as collisional production. For a Bjorken temperature protocol we derive closed-form strong- and weak-coupling limits of the total production. At strong coupling it exceeds the initial relative entropy by a term linear in the relaxation time, whose quasistatic part has the thermodynamic-length form. At weak coupling it vanishes inversely with the relaxation time. A positive strong-coupling slope therefore guarantees maximal dissipation at intermediate coupling. The same structure holds for a relativistic Fokker-Planck equation with a Jüttner stationary state, solved numerically for charm and bottom quarks under the temperature protocol. Linear-response coefficients computed from the static generator reproduce both limits, and the numerical entropy balance is satisfied to high accuracy.