[Paper Review] Chaos simplifies quantum friction

Gabriel G. Carlo, Leonardo Ermann+1 more|arXiv (Cornell University)|Dec 6, 2018

Quantum chaos and dynamical systems3 citations

TL;DR

This paper demonstrates that chaos in quantum dissipative systems simplifies quantum friction by averaging out momentum-dependent quantum corrections, allowing a classical model with $ \hbar$-sized Gaussian noise to accurately reproduce quantum Wigner distribution dynamics. The key result is that chaotic dynamics suppresses phase-space dependence in quantum friction, making the system's behavior effectively classical in the presence of noise scaled to Planck's constant.

ABSTRACT

By means of studying the evolution equation for the Wigner distributions of quantum dissipative systems we derive the quantum corrections to the classical Liouville dynamics, taking into account the standard quantum friction model. The resulting evolution turns out to be the classical one plus fluctuations that depend not only on the $\hbar$ size but also on the momentum and the dissipation parameter (i.e. the coupling with the environment). On the other hand, we extend our studies of a paradigmatic system based on the kicked rotator, and we confirm that by adding fluctuations only depending on the size of the Planck constant we essentially recover the quantum behaviour. This is systematically measured in the parameter space with the overlaps and differences in the dispersion of the marginal distributions corresponding to the Wigner functions. Taking into account these results and analyzing the Wigner evolution equation we propose that the chaotic nature of our system is responsible for the independence on the momentum, while the dependence on the dissipation is provided implicitly by the dynamics.

Motivation & Objective

• To understand how quantum friction influences the evolution of Wigner distributions in classically chaotic and dissipative systems.
• To investigate the role of chaos in simplifying quantum corrections to classical Liouville dynamics.
• To determine whether a simple classical model with $ \hbar$-sized noise can reproduce the statistical features of quantum equilibrium states in dissipative systems.
• To identify the origin of discrepancies between classical and quantum marginal distributions in parameter space.

Proposed method

• Derive the Wigner function evolution equation for quantum dissipative systems using the Weyl-Wigner formalism and star product calculus.
• Specialize the evolution equation to the standard quantum friction model, identifying quantum corrections dependent on $\hbar$, momentum $p$, and dissipation parameter $\gamma$.
• Apply the formalism to the dissipative modified kicked rotator map (DMKRM), a paradigmatic model of quantum dissipative chaos.
• Systematically compare classical and quantum marginal distributions using overlap and dispersion metrics across the full parameter space of $k$ and $\gamma$.
• Analyze the Wigner evolution equation to isolate the role of chaotic dynamics in averaging out $p$-dependent quantum corrections.
• Use numerical simulations to measure agreement between classical noise models and quantum Wigner functions for finite $\hbar_{\rm eff}$.

Experimental results

Research questions

• RQ1 How do quantum corrections to classical Liouville dynamics depend on phase space variables like momentum $p$ and dissipation $\gamma$ in the presence of quantum friction?
• RQ2 To what extent can a classical model with $\hbar$-sized Gaussian noise reproduce the asymptotic Wigner distributions of quantum dissipative systems?
• RQ3 Why do discrepancies between classical and quantum distributions primarily occur in large regular regions of parameter space?
• RQ4 How does chaotic dynamics influence the effective dependence of quantum corrections on momentum and dissipation?

Key findings

• The quantum corrections to classical Liouville dynamics depend non-trivially on both momentum $p$ and the dissipation parameter $\gamma$, as derived from the Wigner evolution equation.
• A simple classical model with $\hbar$-sized Gaussian noise accurately reproduces the main features of quantum marginal Wigner distributions across most of the parameter space.
• Discrepancies between classical and quantum distributions are localized in the largest regular (isoperiodic stable structure) regions of parameter space, where quantum dynamics explore a larger effective basin of attraction.
• The overlap between classical and quantum marginal distributions remains high across most of the parameter space, indicating strong agreement.
• The difference in dispersion between classical and quantum distributions is most pronounced in the inner regions of large regular domains, where quantum distributions exhibit broader spreads.
• The dependence on $p$ in quantum corrections is effectively erased by chaotic dynamics, which averages these terms over time, while the $\gamma$-dependence is implicitly encoded in the map's iterative dynamics.