The stochastic wave function method is proposed to study the diffusion regimes of alkali atoms on metallic surfaces. The Lindblad approach, based on the microscopic Hamiltonian information in the Caldeira-Leggett model, is presented and numerical calculations of the dynamics are carried out to characterize surface diffusion for two different systems: Na-Cu(111) and Li-Cu(111). Calculations of the intermediate scattering function for an isolated adsorbate are compared, in the Brownian limit, with results deduced from helium spin-echo (HeSE) experiments after reducing them to single adsorbate dynamics. To illustrate the method we present the dependence on momentum transfer and the temperature dependency. Results show that the experiment can be described at a quantitative level by the 1-D quantum model (reduced dimensionality).
The stochastic wave function method is proposed to study the diffusion regimes of alkali atoms on metallic surfaces. The Lindblad approach, based on the microscopic Hamiltonian information in the Caldeira-Leggett model, is presented and numerical calculations of the dynamics are carried out to characterize surface diffusion for two different systems: Na-Cu(111) and Li-Cu(111). Calculations of the intermediate scattering function for an isolated adsorbate are compared, in the Brownian limit, with results deduced from helium spin-echo (HeSE) experiments after reducing them to single adsorbate dynamics. To illustrate the method we present the dependence on momentum transfer and the temperature dependency. Results show that the experiment can be described at a quantitative level by the 1-D quantum model (reduced dimensionality).
Stochastic wave function formalism is briefly introduced and applied to study the dynamics of open quantum systems; in particular, the diffusion of Xe atoms adsorbed on a Pt(111) surface. By starting from a Lindblad functional and within the microscopic Caldeira–Leggett model for linear dissipation, a stochastic differential equation (Ito^-type differential equation) is straightforwardly obtained. The so-called intermediate scattering function within the ballistic regime is obtained, which is observable in Helium spin echo experiments. An ideal two-dimensional gas has been observed in this regime, leading to this function behaving as a Gaussian function. The influence of surface–adsorbate interaction is also analyzed by using the potential of two interactions describing flat and corrugated surfaces. Very low surface coverages are considered and, therefore, the adsorbate–adsorbate interaction is safely neglected. Good agreement is observed when our numerical results are compared with the corresponding experimental results and previous standard Langevin simulations.
This thesis investigates quantum surface diffusion of adsorbates on metallic substrates with particular emphasis on the interplay between coherence, dissipation, and environmental interactions. To this end, the stochastic wave function (SWF) method is employed to study the diffusion regimes. Its formulation relies on the stochastic Schrödinger equation in Itô form, developed within the Caldeira-Leggett model and its Lindblad extension. A key feature is the derivation of the intermediate scattering function from the Fourier transform of the diagonal elements of the density matrix. This establishes a direct connection between simulated dynamics and experimentally accessible observables in incoherent neutron scattering. For the first time, the SWF method and the density matrix formalism are applied to surface diffusion, moving beyond the Langevin and generalized Langevin formulations commonly employed in earlier studies. Simulations are carried out to characterize surface diffusion of Xe on Pt(111), Na on Cu(111), Li on Cu(111), and H and D on Pt(111) across different physical conditions. Two extreme time regimes are analyzed, ballistic (very short times) and Brownian or diffusive (very long times). Calculations of the intermediate scattering function for an isolated adsorbate are compared, in both limits, with results from helium spin-echo (HeSE) experiments. In the short-time regime, Xe/Pt(111) shows the expected transition from free-particle to dissipative motion as the friction coefficient and energy barrier increase. Na/Cu(111) exhibits the dynamics of a classical atom undergoing single hops between nearest-neighbor sites. In contrast, Li/Cu(111) shows quantum deviations from the simple Chudley-Elliott model, indicating that its diffusion cannot be explained by classical particle behavior. This agreement with experiment demonstrates the accuracy of the method in regimes where quantum effects are non-negligible. For the lightest adsorbates, H and D on Pt(111), no theoretical framework has so far provided a unified description of the classical and quantum regimes. Previous attempts reproduced experimental hopping rates, yet the estimated values of the friction coefficients were either unreasonably large or far too small. In contrast, the SWF method provides reliable tunneling rates as a function of inverse surface temperature, together with consistent and physically meaningful estimates of friction. Overall, this thesis demonstrates that the SWF method offers a robust and physically consistent formulation for modeling quantum surface diffusion. This approach successfully reproduces analytical theory and experimental observations, capturing classical diffusion at high temperatures and quantum tunneling at low temperatures and providing the first unified description across classical and quantum regimes. This work lays a solid foundation for future research involving multiple interacting adsorbates, two-dimensional surface potentials, and non-Markovian regimes. RESUMEN Esta tesis investiga la difusión superficial cuántica de adsorbatos en sustratos metálicos, con especial énfasis en la coherencia, disipación e interacciones con el entorno. Para estudiar los diferentes regímenes de difusión se emplea el método de la función de onda estocástica (SWF). Su formulación se basa en la ecuación estocástica de Schrödinger en forma de Itô, desarrollada dentro del modelo de Caldeira-Leggett y su extensión de Lindblad. Una característica clave es la derivación de la función de dispersión intermedia a partir de la transformada de Fourier de los elementos diagonales de la matriz densidad. Esto establece una conexión directa entre la dinámica simulada y los observables accesibles experimentalmente en dispersión incoherente de neutrones. Por primera vez, el método SWF y el formalismo de la matriz densidad se aplican a la difusión superficial, superando las formulaciones de Langevin y de Langevin generalizado comúnmente empleadas en estudios anteriores. Se realizaron simulaciones para caracterizar la difusión superficial de Xe en Pt(111), Na en Cu(111), Li en Cu(111), H y D en Pt(111) en distintas condiciones físicas. Se analizan dos regímenes temporales extremos: el balístico (tiempos muy cortos) y el browniano o difusivo (tiempos muy largos). Los cálculos de la función de dispersión intermedia para un adsorbato aislado se comparan, en ambos límites, con los resultados experimentales de la técnica de dispersión inelástica mediante eco de espín de helio (HeSE). En el régimen de tiempos cortos, Xe/Pt(111) muestra la transición esperada de movimiento libre a movimiento disipativo a medida que aumentan el coeficiente de fricción y la barrera de energía. Na/Cu(111) presenta la dinámica de un átomo clásico que realiza saltos individuales entre sitios vecinos. En contraste, Li/Cu(111) muestra desviaciones cuánticas respecto al modelo simple de Chudley-Elliott, lo que indica que su difusión no puede explicarse mediante un comportamiento clásico de partículas. Esta concordancia con los experimentos demuestra la precisión del método en regímenes donde los efectos cuánticos no son despreciables. Para los adsorbatos más ligeros, H y D en Pt(111), ningún marco teórico ha logrado hasta ahora una descripción unificada de los regímenes clásico y cuántico. Intentos previos reprodujeron tasas experimentales de salto, pero los valores estimados de los coeficientes de fricción resultaron ser o bien excesivamente grandes o demasiado pequeños. En contraste, el método SWF proporciona tasas de tunelamiento fiables en función del inverso de la temperatura superficial, junto con estimaciones consistentes y físicamente significativas de la fricción. En conjunto, esta tesis demuestra que el método SWF ofrece una formulación sólida y físicamente coherente para modelar la difusión superficial cuántica. Este enfoque reproduce con éxito la teoría analítica y las observaciones experimentales, capturando la difusión clásica a altas temperaturas y el tunelamiento cuántico a bajas temperaturas, y proporcionando la primera descripción unificada en los regímenes clásico y cuántico. Este trabajo sienta una base sólida para futuras investigaciones que involucren múltiples adsorbatos interactuantes, potenciales superficiales bidimensionales y regímenes no markovianos.
Surface diffusion is described in terms of the intermediate scattering function in the time domain and reciprocal space. Two extreme time regimes are analyzed, ballistic (very short times) and Brownian or diffusive (very long times). This open dynamics is studied from the master equation for the reduced density matrix within the Caldeira-Leggett formalism. Several characteristic magnitudes in this decoherence process such as the coherence length, ensemble width and purity of the density matrix are analyzed. Furthermore, for flat surfaces, the surface diffusion is considered for the Schrödinger cat states and identical adsorbates or adparticles, bosons and fermions. The analytical results are compared with those issued from solving the Lindblad master equation through the stochastic wave function method. This numerical analysis is extended to be applied to corrugated surfaces.
Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.
Abstract The so-called quantum-to-classical transition is studied in the surface diffusion dynamics. This continuous and smooth transition is carried out from the Liouville-von Neumann equation by scaling Planck’s constant. As an application, the Brownian motion of an adsorbate on a flat surface is analyzed within the Caldeira–Leggett master equation formalism for the reduced density matrix and the two extreme time behaviors, the ballistic and diffusive motions. In the ballistic time regime, the intermediate scattering function displays a Gaussian function which is governed by the thermal velocity in the classical regime, whereas in the quantum regime, by the initial spreading velocity of the wave packet. Adsorbates behave as a classical or quantum ideal gas. Finally, in the diffusive regime, and starting from the Chudley–Elliott model, the well-known H-function is calculated for three different surface temperatures in the diffusion of H and D on a Pt(111) surface. The main goal of this analysis is whether one can discriminate the irreversibility coming from tunneling or thermal activation diffusion.
A Lindblad approach is presented to describe quantum dynamics of open systems. It is based on a construction of a Lindblad functional from the microscopic Caldeira-Leggett model for linear dissipation. It leads to a master equation for the reduced density matrix, which preserves positive evolution on short times and asymptotically approaches equilibrium at high and low temperatures. This master equation is applied to study the femtosecond dynamics of vibrational heating, relaxation, and bond breaking at a metal surface. Both a direct solution of the density matrix and an indirect solution with stochastic wave functions are presented. The latter, besides its statistical equivalence to the density matrix, provides a more physical picture about quantum jumps of individual open systems.