Intermediate scattering function and quantum recoil in non-Markovian quantum diffusion

Abstract

Exact expressions are derived for the intermediate scattering function (ISF) of a quantum particle diffusing in a harmonic potential and linearly coupled to a harmonic bath. The results are valid for arbitrary strength and spectral density of the coupling. The general, exact non-Markovian result is expressed in terms of the classical velocity autocorrelation function, which represents an accumulated phase during a scattering event. The imaginary part of the exponent of the ISF is proportional to the accumulated phase, which is an antisymmetric function of the correlation time $t$. The expressions extend previous results given in the quantum Langevin framework where the classical response of the bath was taken as Markovian. For a special case of non-Markovian friction, where the friction kernel decays exponentially in time rather than instantaneously, we provide exact results relating to unconfined quantum diffusion, and identify general features that allow insight to be exported to more complex examples. The accumulated phase as a function of the t has a universal gradient at the origin, depending only on the mass of the diffusing system particle. At large t the accumulated phase reaches a constant limit that depends only on the classical diffusion coefficient and is therefore independent of the detailed memory properties of the friction kernel. Non-Markovian properties of the friction kernel are encoded in the details of how the accumulated phase switches from its $t\to -\infty$ to its $t\to -\infty$ limit, subject to the constraint of the universal gradient. When memory effects are significant, the transition from one limit to the other becomes non-monotonic, owing to oscillations in the classical velocity autocorrelation. The result is interpreted in terms of a solvent caging effect, in which slowly fluctuating bath modes create transient wells for the system particle.

References in corpus (13)

• Quantum Computing
• The density-matrix renormalization group
• Review article: Linear optical quantum computing
• Non-Markovian dynamics in open quantum systems
• Quantum Non-Markovianity: Characterization, Quantification and Detection
• An open--quantum--system formulation of particle decay
• Electron-Phonon Coupling and Surface Debye Temperature of Bi${}_2$Te${}_3$(111) from Helium Atom Scattering
• Energy Dissipation during Diffusion at Metal Surfaces: Disentangling the Role of Phonons vs. Electron-Hole Pairs
• Diffusion Rates for Hydrogen on Pd(111) from Molecular Quantum Dynamics Calculations
• Linear response theory of activated surface diffusion with interacting adsorbates
• Quantum Markovian activated surface diffusion of interacting adsorbates
• Polarisation in Spin-Echo Experiments: Multi-point and Lock-in Measurements
• Phonon lineshapes in atom-surface scattering