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 $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ to its $t\ensuremath{\rightarrow}+\ensuremath{\infty}$ limit, subject to the constraint of the universal gradient. When memory effects are significant, the transition from one limit to the other becomes nonmonotonic, 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.
Assuming that “quantum spacetime” is fundamentally discrete, how might this discreteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particular to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rectilinear motion are still possible. Such “swerves” have been described in and. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the corresponding analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves . Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further generalization to a “second-quantized field” might have more dramatic, if less obvious, consequences.
We address memory effects and diffusive properties of a continuous-time quantum walk on a one-dimensional percolation lattice affected by spatially correlated random telegraph noise. In particular, by introducing spatially correlated time-dependent fluctuations in nearest-neighbor hopping amplitudes, we describe random domains characterized by global noise. The resulting open dynamics of the walker is then unraveled by an ensemble average over all the noise realizations. Our results show that time-dependent noise assisted by spatial correlations leads to strong memory effects in the walker dynamics and to robust diffusive behavior against the detrimental action of uncorrelated noise. We also show that spatially correlated classical noise enhances localization breaking, thus making a quantum particle spread on longer distances across the lattice.