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Physical review. B, Condensed matter

Dissipative quantum dynamics at surfaces: A nonlinear-coupling model

Shiwu Gao

Published Dec 15, 1999• 10 citations
Publisher

Abstract

The recently developed Lindblad approach for quantum dynamics of an open system with linear dissipation has been extended to a model of nonlinear coupling. The coupling term ${H}_{i}={\ensuremath{\sum}}_{k}{C}_{k}{f(x)q}_{k}$ is an arbitrary function of the system coordinate, but remains to be linear in bath coordinates. A Lindblad master equation and its associated stochastic differential equation have been obtained. To study dissipative surface dynamics, a general form of $f(x)$ has been proposed. It is shown that such a nonlinear-coupling model is crucial to qualitatively describe gas-surface dynamics, when a gas molecule impinging or leaving a surface goes beyond the linear-coupling regime.

DOI
10.1103/physrevb.60.15609

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The recently developed Lindblad approach for quantum dynamics of an open system with linear dissipation has been extended to a model of nonlinear coupling. The coupling term Hi=\ensuremath∑kCkf(x)qk{H}_{i}={\ensuremath{\sum}}_{k}{C}_{k}{f(x)q}_{k}Hi​=\ensuremath∑k​Ck​f(x)qk​ is an arbitrary function of the system coordinate, but remains to be linear in bath coordinates. A Lindblad master equation and its associated stochastic differential equation have been obtained. To study dissipative surface dynamics, a general form of f(x)f(x)f(x) has been proposed. It is shown that such a nonlinear-coupling model is crucial to qualitatively describe gas-surface dynamics, when a gas molecule impinging or leaving a surface goes beyond the linear-coupling regime.
When a gas phase atom or molecule hits a nonrigid surface, both elastic and inelastic scattering and sticking can occur. We suggest treating the dynamics of these processes using open-system density matrix theory. For the “free→free” and “free→bound” events at hand, both fundamental and numerical problems arise. The fundamental problem is that the adsorbate “system” is anharmonic and the coupling between the system and the substrate “bath” has to be nonlinear at least in the system coordinates. Here we propose a new Lindblad-type open-system density matrix approach which accounts for system anharmonicity and nonlinearity of the system–bath coupling. The numerical problem is that for a dissipative scattering process large basis sets or grids are required, making the storage and direct propagation of a density matrix difficult. To overcome this problem we use a mapped Fourier method which reduces the grid size and hence the storage requirements significantly. We apply the new methods and techniques to a simple model resembling the simultaneous scattering and sticking of an O2 molecule at a metal surface.
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.
We propose a method which combines the quantum-classical mapping approach to surface hopping (MASH) with the dissipative quantum dynamics of the Lindblad master equation. Like conventional surface-hopping methods, our approach is based on classical trajectories coupled to the dynamics of a quantum subsystem. However, instead of evolving the subsystem wavefunction according to the time-dependent Schrödinger equation, we use stochastic quantum trajectories derived from secular Redfield theory. This enables the simulation of open quantum systems coupled simultaneously to Markovian quantum baths and anharmonic non-Markovian classical degrees of freedom. Applications to the spin--boson model and to the cavity-enhanced fluorescence of an electronically nonadiabatic molecule show excellent agreement with fully quantum-mechanical benchmarks.
We develop a perturbation scheme to treat the dynamics of a quantum Brownian particle (coordinate xxx) coupled nonlinearly to a bath of oscillators (coordinates qn{q}_{n}qn​) with an interaction action in the form $\ensuremath{\lambda}f(x){q}_{n}^{k}$. We derive the influence functionals for the k=2,3,4k=2, 3, 4k=2,3,4 cases to order ${\ensuremath{\lambda}}^{2}$ and derive the master equations for the special cases of local dissipation and white noise, as well as the general cases of nonlocal dissipation and colored noise for f(x)=xandx2f(x)=x \mathrm{and} {x}^{2}f(x)=xandx2. We show that a generalized fluctuation-dissipation relation exists between the lthl\mathrm{th}lth-order noise kernels of the kthk\mathrm{th}kth-order coupling and their corresponding dissipation kernels.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.