Motivating semiclassical gravity: a classical-quantum approximation for bipartite quantum systems (2306.01060v1)

Abstract: We derive a "classical-quantum" approximation scheme for a broad class of bipartite quantum systems from fully quantum dynamics. In this approximation, one subsystem evolves via classical equations of motion with quantum corrections, and the other subsystem evolves quantum mechanically with equations of motion informed by the evolving classical degrees of freedom. Using perturbation theory, we derive an estimate for the growth rate of entanglement of the subsystems and deduce a "scrambling time" - the time required for the subsystems to become significantly entangled from an initial product state. We argue that a necessary condition for the validity of the classical-quantum approximation is consistency of initial data with the generalized Bohr correspondence principle. We illustrate the general formalism by numerically studying the fully quantum, fully classical, and classical-quantum dynamics of a system of two oscillators with nonlinear coupling. This system exhibits parametric resonance, and we show that quantum effects quench parametric resonance at late times. Lastly, we present a curious late-time scaling relation between the average value of the von Neumann entanglement of the interacting oscillator system and its total energy: $S\sim 2/3 \ln E$.

• The paper derives a 'classical-quantum' approximation for bipartite systems, treating one part classically and the other quantum mechanically to model their interaction.
• This approximation is shown to be valid under specific conditions, including small coupling between the subsystems and the quantum state adhering to the generalized Bohr correspondence principle.
• Numerical examples demonstrate the approximation's efficacy and reveal how quantum effects can modulate classical dynamics, offering insights for semiclassical gravity and hybrid classical-quantum systems.

Motivating Semiclassical Gravity: A Classical-Quantum Approximation for Bipartite Quantum Systems

This paper explores the derivation of a "classical-quantum" approximation for bipartite quantum systems, a method where one part of the system is treated classically and the other quantum mechanically. The discourse emerges from the broader context of seeking approximations in the ongoing pursuit to reconcile quantum theory with gravitational interactions—a domain still replete with theoretical challenges.

Context and Motivation

In the field of theoretical physics, the quest to formulate a cohesive quantum theory of gravity remains unresolved. The complexity principally arises from integrating quantum mechanics, typically formulated in flat spacetime, with the dynamic spacetime of general relativity. Traditional approaches such as quantum field theory (QFT) in curved spacetime provide approximations by treating matter quantum mechanically while keeping spacetime a fixed classical background. However, these models do not incorporate backreaction effects—the influence of quantum matter on spacetime geometry—which limits their scope.

The semiclassical Einstein equations attempt to address this by introducing a coupling between the quantum expectation values of matter and classical spacetime geometry. Yet, these equations are fraught with challenges—prominent among them is the breakdown of energy conservation due to the lack of a self-consistent feedback loop between matter and geometry.

Classical-Quantum Approximation

The authors derive a classical-quantum approximation that aims to address some of the limitations mentioned above. Starting from fully quantum dynamics, they propose an approximation scheme for bipartite systems, wherein one part evolves via classical equations with quantum corrections, and the other part evolves quantum mechanically but under the influence of classical variables. This framework draws inspiration from traditional approaches like the Born-Oppenheimer approximation, often used in molecular dynamics, where heavy nuclei are treated classically due to their relatively slow dynamics compared to electron clouds.

In this classical-quantum context, the paper establishes necessary conditions for the validity of the approximation. These include small coupling between the quantum and classical systems and a quantum state in subsystem 1 that adheres to the generalized Bohr correspondence principle—implying that quantum expectations closely align with classical predictions.

Results and Implications

The paper provides a perturbative analysis to quantify the "scrambling time," which is the timescale over which initial product state assumptions hold before significant entanglement disrupts the approximation's validity. Numerical illustrations involving coupled harmonic oscillators—where one oscillator is treated classically—demonstrate the efficacy of the classical-quantum approximation under these constraints. Importantly, these studies reveal how quantum effects modulate classical phenomena such as parametric resonance, indicating potential applications in gravitational contexts like black hole evaporation or early universe cosmology, where quantum and classical interactions are tightly coupled.

Notably, a curious long-term scaling relation between the system's entanglement entropy and total energy is observed, suggesting further lines of inquiry into entropy production in hybrid systems.

Future Directions

The implications of this approximation for semiclassical gravity are significant, offering insights into scenarios where classical spacetime could be dynamically influenced by quantum fields. The work suggests potential pathways for probing semiclassical gravitational models using this approximation, particularly in symmetry-reduced contexts like cosmology. Moreover, extending the classical-quantum framework to more complex gravitational systems could yield novel insights into quantum gravity phenomena that evade traditional methodologies.

Overall, this paper adds a vital piece to the puzzle in understanding how classical and quantum worlds interact, with the potential to influence our approach to quantum gravity. It opens avenues for experimental and theoretical advancements, particularly in the emerging exploration of hybrid quantum-classical systems.