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arXiv:2508.11743

Thermalization of Bottomonium in the Quark-Gluon Plasma

Nora Brambilla A, Tom Magorsch, Antonio Vairo

Published Aug 15, 2025• 0 citations
arXivHTMLPDF

Abstract

We study the approach to equilibrium of bottomonium in the quark-gluon plasma within the open quantum system framework. We perform large-scale simulations of the long-time behavior in three dimensions using the quantum trajectory method to observe the emergence of steady states and determine the timescale of thermalization in position-, angular-momentum-, and color-space. We find that the thermalization timescale increases with decreasing temperature and decreasing coupling to the medium, which is given by transport coefficients of the medium. Additionally, we observe that the steady states exhibit small corrections to the Gibbs state due to medium interactions and show that these corrections diminish for weaker medium coupling and higher temperature. At a temperature of $450\,$MeV, quarkonium relaxes to a state that is approximately thermal, with the most significant correction being a smaller overlap of the $1S$ state with respect to the Gibbs state. We compare these findings with the master equation obtained at leading order in the expansion of the binding energy over the temperature, which we find to have a trivial steady state.

DOI
10.48550/arxiv.2508.11743

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We study the approach to equilibrium of bottomonium in the quark-gluon plasma within the open quantum system framework. We perform large-scale simulations of the long-time behavior in three dimensions using the quantum trajectory method to observe the emergence of steady states and determine the timescale of thermalization in position-, angular-momentum-, and color-space. We find that the thermalization timescale increases with decreasing temperature and decreasing coupling to the medium, which is given by transport coefficients of the medium. Additionally, we observe that the steady states exhibit small corrections to the Gibbs state due to medium interactions and show that these corrections diminish for weaker medium coupling and higher temperature. At a temperature of 450 450\,450MeV, quarkonium relaxes to a state that is approximately thermal, with the most significant correction being a smaller overlap of the 1S1S1S state with respect to the Gibbs state. We compare these findings with the master equation obtained at leading order in the expansion of the binding energy over the temperature, which we find to have a trivial steady state.
We revisit the pNRQCD open-quantum-system description of bottomonium in a quark-gluon plasma, writing the full master equation in a spherical basis and establishing a framework for its numerical treatment. The equation does not rely on an expansion in the ratio of the binding energy over the temperature, while retaining the Born, Markov, and dilute approximations. It depends on suitable convolutions of Coulombic Green's functions with real-time chromoelectric correlators, rather than a finite set of transport coefficients. It contains jump-operator contributions that enter with both positive and negative signs, so it is not of a Lindblad type. We utilize optimal operator rotations to minimize the negative sector and define a positive-sector approximation that retains only the optimized positive terms, obtaining in this way a Lindblad equation that can be simulated efficiently with the open-source code QTraj for arbitrary chromoelectric spectral functions. We treat the discarded sector as a systematic uncertainty associated with positivity-breaking effects of the pseudo-Lindblad generator. Positivity breaking effects can be attributed to the approximations made in the derivation of the quarkonium master equation, such as the Born--Markov approximations. Using a spectral model constrained by known infrared and ultraviolet limits, we compute the in-medium width and Bjorken-evolved survival probabilities of the Υ(1S,2S,3S)Υ(1\mathrm{S},2\mathrm{S},3\mathrm{S})Υ(1S,2S,3S). The computations show the convergence of the binding-energy-over-temperature expansion at high temperatures and provide the first large-scale quantum-trajectory simulation results for bottomonium in the low temperature regime. We can thus follow the bottomonium evolution over the full temperature range relevant in heavy-ion collisions.
\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
We study how much quantum structure bottomonium retains during its evolution in the quark-gluon plasma. Within the open quantum system framework, we simulate the Lindblad equation obtained from potential nonrelativistic QCD in the quantum Brownian regime. From the resulting real-time evolution of the bottomonium density matrix, we compute the Wigner transform and study its negativity as a measure of nonclassicality and a test of a key condition underlying classical Langevin-type approximations. We find that the medium suppresses the negativity, which nevertheless approaches a finite plateau close to that of the 1S1\mathrm{S}1S state. The overlaps with the excited 2S2\mathrm{S}2S and 3S3\mathrm{S}3S states arise from near-canceling positive and negative phase-space contributions, driven by negative regions in the Wigner transforms of the bound-state projectors, while the 1S1\mathrm{S}1S overlap is insensitive to such contributions. We further show that the position-space coherence is similarly suppressed and plateaus close to the 1S value. We trace the finite residual quantum structure to conditional purification: the medium preferentially dissolves weakly bound and unbound modes with large ⟨r2⟩\langle r^2\rangle⟨r2⟩, driving the surviving singlet S\mathrm{S}S-wave ensemble toward the 1S1\mathrm{S}1S state. More broadly, these results reveal that classicalization is partial and observable dependent, a feature that is relevant across applications of open quantum system methods in fundamental physics.
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.
Quantum computing provides a powerful framework for simulating real-time dynamics in open quantum systems, offering key advantages for modeling heavy-quarkonium transport in high-energy nuclear collisions. In this work, we perform quantum simulations of the isotropic next-to-leading-order Lindblad equation for bottomonium in the quark-gluon plasma using a reduced spherical coordinate representation. We discretize operators and wavefunctions, map the physical state onto qubits, and execute time evolution via parameterized quantum gate operations. By extracting the Υ(1S)Υ(1S)Υ(1S) survival probability, we quantitatively isolate the color-octet contribution, demonstrating that its overall impact is small in the final production of the bottomonium ground state Υ(1S)Υ(1S)Υ(1S) in the hot QCD medium at temperatures accessible at the Large Hadron Collider. Additionally, we have further optimized the quantum simulation algorithm for the Lindblad equation. The improved algorithm requires only a single ancillary qubit to realize the Lindblad evolution, thereby minimizing the circuit significantly.
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.