Quantization of Lagrangian Descriptors

Abstract

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.

Figures (2)

• Figure 1: Difference between the classical and quantum Lagrangian descriptor for the Hamiltonian saddle given in Eq. \ref{['eq:saddle']} calculated with A) $10$ modes and B) $800$ modes in the decomposition of the operator in Eq. \ref{['eq:operator_saddle']}. As predicted by Eq. \ref{['eq:width']}, the broadening of the manifolds grows as the number of modes considered in the calculation becomes larger. The numerical calculation was done with $\lambda = 3$, $T = 8/3$ and the initial conditions where sampled from a $400\times 400$ regular grid with $1200$ samples per initial condition. Note that while the quantum LD faithfully encodes the effective broadening through the path-integral average, the visual contrast is constrained by the finite grid resolution and colormap dynamic range; structures narrower than the grid spacing are captured analytically by Eq.\ref{['eq:width']} but cannot be visually resolved.
• Figure 2: Width of the invariant manifolds computed as a function of the number of modes $N$ for $1200$ samples and $T\lambda = 8$. In blue we depict the theoretical curve given in Eq. \ref{['eq:width']} and in orange the result from the simulation. The agreement between the theoretical prediction and the numerical result is within $1\%$.