Description:

What is the fate of microscopic quantum (probability) density and phase fluctuations (i.e., the spatio-temporal dynamics and survival statistics of local density and phase fluctuations) on regular lattices, disordered systems, and random/complex interconnects? Classical persistence has been widely studied in a variety of diffusive and interacting systems [1]. In diffusive systems, the persistence probability is defined as the probability that the relevant field (e.g., the diffusive field or the density fluctuations) has stayed above the mean or has not reached a threshold at a location up to time t. More generally, including interacting systems or fluctuating interfaces, the persistence probability can be defined as the probability that a location or region remained “active”, or a spin has not flipped, or the interface has not crossed zero up to time t [1,2].

Here I’ll discuss quantum persistence on regular lattices by analyzing amplitude and phase fluctuations of the wave function governed by the time-dependent free-particle Schrödinger equation [3]. The quantum system is initialized with local random uncorrelated Gaussian amplitude and phase fluctuations. In analogy with classical diffusion, the persistence probability is defined as the probability that the local (amplitude or phase) fluctuations have not changed sign up to time t. Our results show that the persistence probability in quantum diffusion exhibits exponential-like tails. We also provide some insights by analyzing the two-point spatial and temporal correlation functions in the limit of small fluctuations. In particular, in the long-time asymptotic limit, the temporal correlation functions for both local amplitude and phase fluctuations become time-homogeneous. Hence, the zero-crossing events correspond to those governed by a stationary Gaussian process, with an autocorrelation-function power-law tail decaying sufficiently fast to imply an exponential-like tail of the persistence probabilities [2].

[1] A.J. Bray, S.N. Majumdar, and G. Schehr, “Persistence and first-passage properties in nonequilibrium systems” Advances in Physics, 62, 225, (2013).

[2] G.F. Newell and M. Rosenblatt, “Zero crossing probabilities for Gaussian stationary processes”, Ann. Math. Stat. 33, 1306 (1962).

[3] C. Ma, O. Malik, and G. Korniss, “Fluctuations and persistence in quantum diffusion on regular lattices”, Phys. Rev. E 111, 024126 (2025).

Gyorgy Korniss, Professor, Rensselaer Polytechnic Institute

Host: Istvan Kovacs