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 its t → +∞ 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.

Disentangling theorem and scattering functions for long-range coherent tunneling

“…Equation 31, together with the relations (29) and (30) and the definition of the function g(i Ka) and its real and imaginary parts, constitutes the central analytical result of the current study.…”

“…hδ. An alternative perspective on the imaginary part, which draws a close connection to quantum recoil in continuous systems [30,51,52], can be seen by casting the integral representation (28) in terms of shifted energy and crystal momentum. Namely, by using standard double-angle trigonometric identities and the explicit forms of g and g , one finds an integral of the form (32) wherehω(k) is the energy dispersion (9).…”

“…A common and relevant nonseparable example is provided by the hexagonal close-packed lattice. If the Cartesian unit vectors arex andŷ, the primitive unit cell of the real-space lattice can be specified by the lattice vectors a 1 = ax ; a 2 = a [x cos(60) +ŷ cos (30)].…”

“…The honeycomb system is described in real space by a unit cell as described in the previous section for a hexagonal surface, but with tight-binding sites at positions (a/2, c/3) and (a, 2c/3) relative to the lower left corner of each unit cell, where c = a cos (30). The tight-binding sites therefore form two interleaved sublattices.…”

“…High-resolution measurement of reciprocal-space surface dynamics is possible using helium-3 surface spin-echo spectroscopy (HeSE) [20], and the interpretation of HeSE data generally relies on models for the ISF. Scientific progress associated with the technique is not driven simply by an accumulation of systems measured, but by the development of analytical models for potentially observable surface dynamical processes [25][26][27][28][29][30] in parallel with ongoing improvements in instrumentation [31][32][33][34][35], data acquisition routines [36,37], and analysis methods [38,39]. The main analytical result and method presented here provides a new class of contribution in that context.…”