Quantum Fluctuations Broaden Pathways for Faster Particle Transport

A new method to understand particle movement in complex systems has been formulated by extending Lagrangian descriptors into quantum mechanics. Javier Jiménez-López and V. J. García-Garrido, from the Universidad Complutense de Madrid, formulate these descriptors within a path integral framework, incorporating quantum effects into the analysis of particle transport. The formulation reveals that traditionally key boundaries defining movement become blurred due to quantum fluctuations, offering a geometric explanation for tunneling phenomena. Applying this approach with a Hamiltonian saddle establishes a new framework for studying phase space transport and potentially using Lagrangian descriptors in field theory. Quantifying broadened boundaries via path integral averaging of Lagrangian descriptors Invariant manifold widths, a key indicator of quantum tunneling probability, have increased by a factor of 200 when analysed using a new path integral formulation. Previously, accurately quantifying these widths was impossible due to the computational demands of resolving structures smaller than the sampling resolution; this method circumvents that limitation by characterising broadening through path integral averaging. This work introduces a quantum formulation of Lagrangian descriptors, extending their geometric description of classical transport into the quantum realm and establishing a direct link between quantum mechanics and dynamical systems theory. The broadening was demonstrated for a Hamiltonian saddle, a standard problem in physics used to model energy landscapes, where path integral sampling revealed the extent of manifold widening and subsequent barrier penetration. Quantitative analysis showed agreement within 1% between theoretical predictions and Monte Carlo simulations of the manifold width as a function of the number of modes used in the calculation, up to 800 modes. Furthermore, the ratio of broadenings between different systems remained consistent regardless of the resolution used, providing a reliable basis for comparison and enabling analysis of a range of systems based on a single reference point. Poincaré maps and the geometric characterisation of nonlinear phase space transport Dynamical systems theory is increasingly recognised as the natural arena in which dynamical phenomena unfold. Its systematic study dates back to H. Poincaré’s seminal work on the three-body problem, whose qualitative approach to differential equations laid the foundations of modern dynamical systems’ theory. By introducing geometric and topological methods, including invariant manifolds, periodic orbits, and recurrence, Poincaré revealed that the long-term evolution of trajectories is governed by intricate structures embedded in phase space. In nonlinear systems, transport is thus mediated by these invariant phase space structures. Powerful analytical and computational tools are required to understand and characterise these structures, including Poincaré maps, which reduce continuous flows to discrete-time dynamical systems and expose the geometry of phase space, and Lyapunov exponents, which quantify the rate of separation of nearby trajectories. Lagrangian descriptors (LDs) have emerged as a simple yet powerful framework that addresses this challenge by revealing phase space structures directly from trajectory information. LDs are defined as a scalar functional evaluated along trajectories of a dynamical system and are computationally efficient, having been successfully applied across disciplines. In quantum mechanics, transport is typically analysed through quasi-probability distributions such as the Wigner and Husimi functions, through semiclassical propagators, or Entangled Trajectory Molecular Dynamics (ETMD). While these approaches successfully capture interference and tunneling phenomena, they do not directly identify the underlying geometric phase space structures. Consequently, a geometric formulation of quantum transport barriers, analogous to the invariant manifolds of classical mechanics, remains largely unexplored. This work presents a quantum formulation of LDs based on Feynman’s path integral, averaging the classical LD over quantum fluctuations to incorporate quantum effects that endow classical invariant manifolds with a finite width, transforming sharp transport barriers into geometric objects with intrinsic thickness. This delocalization effect yields a direct phase-space interpretation of tunneling: classically disjoint regions become connected through the overlap of broadened manifolds. The width of the quantum-delocalized invariant manifolds admits an analytical estimate controlled by the fluctuation spectrum. These results establish a geometric framework for quantum transport that bridges dynamical systems theory and path-integral quantum mechanics, and naturally suggest extensions of LDs to classical and quantum field theories. The trajectory-based technique of Lagrangian descriptors (LDs) originated in Geophysics to analyse Lagrangian transport and mixing processes in the ocean and the atmosphere. In recent years, it has been shown to reveal phase space structures, such as equilibria, stable and unstable manifolds, invariant tori, and periodic orbits, that are closely related to the dynamical behaviour of trajectories in phase space. For a Hamiltonian system H(x) with n degrees of freedom (DoF), the evolution of the state vector x = (q, p) ∈S is governed by Hamilton’s equations of motion: x = f(x) = J ∇H(x), where J is the Poisson matrix. A Lagrangian descriptor L is a non-negative scalar functional that depends on a non-negative function F(x(t; x0), t) and the initial condition x0 at time t0 and which is defined as: L(x0, t0, T) = ∫t0+T t0−T dt F(x(t; x0), t). In this work, the following integrand is adopted to define the LD functional: F(x, t) = 2n ∑i=1 |fi(x, t)|1/2, as it has been shown to be very effective for unveiling phase space structures. Considering a Hamiltonian function of the form: H(q, p) = 1/2pTM−1p + V(q), where M is a symmetric positive-definite mass matrix and V(q) is the potential, the corresponding configuration space action is: S[q] = ∫T −T dt 1/2qTMq −V(q). Fluctuations around the classical trajectory qcl(t) are introduced as: q(t) = qcl(t) + η(t), with η(±T) = 0. The action difference functional, which encodes all the physical processes of the system, is then given by: ∆S[η; qcl] := S[qcl + η] −S[qcl], and the path integral associated with the fluctuations is: Z Y Dη exp i ħ∆S[η; qcl], where Y denotes the space of real paths satisfying Dirichlet boundary conditions. Since the integrand is oscillatory, a well-defined mathematical formulation is achieved by deforming the integration contour into a complexified space. The integral is expressed as a sum over Lefschetz thimbles: Z Y Dη e i ħ∆S[η;qcl] = ∑σ nσ Z Jσ Dη e i ħ∆S[η;qcl], where Jσ are the steepest descent cycles associated with the relevant complex saddle points and nσ are intersection numbers. Then, for any observable L[q, p] defined as a scalar functional over the trajectory, the corresponding expectation value conditioned on the sector associated with qcl is defined as: ⟨L⟩= P σ nσ R Jσ Dη L[q, p] e i ħ∆S[η;qcl] P σ nσ R Jσ Dη e i ħ∆S[η;qcl]. In order to analyse how the invariant manifolds of the system are broadened due to the fluctuations, a transverse coordinate is used. Assuming that the structure is locally characterised by the vanishing of a smooth scalar function: g(q, p, t) = 0, one can define a transverse coordinate u(q, p, t) as a function which is zero over the manifold and such that |u(q, p, t)| is a perpendicular distance to the manifold. A natural way of doing this is to define the variable: u(q, p, t) = g(q, p, t) ∥∇g(q, p, t)∥, and calculate its variance within the path integral formalism as: σ2 u(t) = P σ nσ R Jσ Dη |u(q, p, t)|2 e i ħ∆S[η;qcl] P σ nσ R Jσ Dη e i ħ∆S[η;qcl], which yields the broadening of the invariant manifolds: σrms = s 1/2T ∫T −T dt σ2u(t). Consider an arbitrary Hamiltonian vector field with DoF, and suppose it has a saddle equilibrium point at the origin at which the matrix associated with the linearization of Hamilton’s equations about the equilibrium point has eigenvalues ±λ. Using Hamiltonian normal form theory, one can construct the Hamiltonian function that describes the linearized dynamics of the system in phase space about the saddle equilibrium: H(q, p) = λ/2 p2 −q2, λ > 0. The invariant stable (Ws) and unstable (Wu) manifolds emanating from the saddle equilibrium act as transport barriers in phase space, and are given by the sets: Wu/s = { (q, p) ∈R2 | p = ±q }, where the plus and minus signs correspond, respectively, to the unstable and stable manifolds. The real-time phase-space action, in natural units (ħ= 1), is: S[q, p] = ∫T −T dt (p q − H(p, q)), from which the configuration-space action is recovered after integrating the momentum dependency: S[q] = ∫T −T dt 1/2λ q2 + λ/2 q2. Decomposing the path as a sum of the classical trajectory and fluctuations that satisfy Dirichlet boundary conditions, the action can be separated as: S[q, q] = S[qcl, qcl] + S[η, η], where S[qcl, qcl] corresponds to the classical action and S[η, η] is the action of the fluctuations, given by: S[η, η] = 1/2λ ∫T −T dt η2 + λ/2 η2, which is directly linked to the following Sturm-Liuville operator: O = −d2 dt2 + λ2. This allows us to express the fluctuations in terms of the eigenfunctions of O as: η(t) = ∞ ∑n=1 cnφn(t), where cn are the Fourier coefficients and the eigenfunctions of O are: φn(t) = 1/√T sin(kn(t + T)), kn = nπ/2T. Since the eigenfunctions are orthogonal, the action of the fluctuations can be rewritten as: S[η, η] = 1/2λ ∞ ∑n=1 (k2 n + λ2)c2 n. To make the integral converge, each mode is deformed into its steepest-descent contour by letting: cn = eiπ/4yn with yn ∈R, so that the oscillatory weight becomes a real Gaussian: exp i 2λ k2 n + λ2 c2 n →exp −1/2λ k2 n + λ2 y2 n, and the fluctuation field becomes complex: η(t) = eiπ/4 ∞ ∑n=1 ynφn(τ). This method allows for the dynamical evolution to take place in real time while the integration contour lives in a complexified space. Then, 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. A coordinate is defined to analyse how these manifolds broaden due to fluctuations: u(t) = g(q, p, t) ∥∇g(q, p, t)∥, where g(q, p, t) is zero on the manifold and represents a perpendicular distance to it. The variance of this coordinate is calculated within the path integral framework to quantify the broadening of the invariant manifolds. For a Hamiltonian system with one degree of freedom and a saddle equilibrium point at the origin, the Hamiltonian function describing the linearized dynamics is: H(q, p) = λ/2 p2 − q2, where λ is a positive constant. The real-time phase-space action is: S[q, p] = ∫T −T dt (p q − H(p, q)). Decomposing the path into a classical trajectory and fluctuations satisfying Dirichlet boundary conditions separates the action into a classical component and the action of the fluctuations: S[η, η] = 1/2λ ∫T −T dt η2 + λ/2 η2. This is linked to a Sturm-Liouville operator. The broadening of the invariant manifolds is then determined by calculating the root-mean-square width: σrms = √1/2T ∫T −T dt σ2u(t). Visualising quantum particle trajectories using path integrals and dynamical systems theory Establishing a geometric understanding of quantum transport offers a powerful new way to visualise how particles navigate complex energy landscapes. This work successfully marries dynamical systems theory with path integral quantum mechanics, revealing how quantum fluctuations blur the boundaries that define classical movement. However, the current demonstration relies heavily on the Hamiltonian saddle, a simplified model system; extending this quantum Lagrangian descriptor approach to more intricate Hamiltonians remains a significant challenge. This geometric approach offers a valuable new perspective through which to view quantum behaviour, particularly ‘tunneling’, where particles pass through barriers they classically shouldn’t. Path integrals, a method for calculating quantum probabilities, are central to this visualisation; they allow researchers to map particle movement as a spread of possibilities rather than fixed paths. Researchers have developed a geometric method to visualise how quantum particles traverse energy landscapes. This approach utilises ‘path integrals’ to map movement as a range of possibilities, not fixed routes, offering insights into quantum ‘tunneling’. The research successfully combined dynamical systems theory with path integral quantum mechanics to visualise quantum particle behaviour. This provides a geometric framework for understanding how quantum fluctuations broaden invariant manifolds and facilitate tunneling, where particles penetrate energy barriers. By averaging classical Lagrangian descriptors over fluctuations, researchers created a quantum descriptor that incorporates these effects, demonstrated using a Hamiltonian saddle. The authors note that extending this approach to more complex Hamiltonian systems presents a future challenge. 👉 More information 🗞 Quantization of Lagrangian Descriptors 🧠 ArXiv: https://arxiv.org/abs/2604.04128 Tags: