Quantum systems generally lack precise trajectories. Hence, the notion of a quantum random walk is somewhat problematic. One way to make the idea precise is to consider quantum evolution which is repeatedly interrupted by position measurements, each of which localises the system to a particular state. The sequence of measurements then yields a path defined as a sequence of states. How do such walks differ from classical random walks?
In a recent paper, Avihai Didi of Bar-Ilan University in Israel and LML Fellow Eli Barkai consider this question, showing that key elements of the quantum dynamics show up in statistical quantities such as the mean return time or mean passage time between two states. Moreover, they find, for some special time intervals between the measurements, the resulting random walk reveals surprising behaviour similar to that observed in classical non-ergodic systems.
Didi and Barkai analyse a model in which the time evolution involves two alternating elements – quantum dynamics described by the Schrödinger equation and a Hamiltonian H, and position measurements. The measurements break the continuous quantum evolution in a periodic way, at moments given as multiples of a basic time interval, τ. The Hamiltonian, H, is represented here by a graph where nodes describe states and edges describe the hopping amplitudes between states. This can be represented as an adjacency matrix, chosen to reflect walks in different geometries such as an infinite line or a finite ring of states.
In the so-called Zeno limit of nearly continuous measurements, it is well known that any trajectory remains localised. More generally, as Didi and Barkai show, the system dynamics for finite measurement intervals superficially appear to show Gaussian statistics similar to a classical random walk. However, their analysis using large deviation theory and an Edgeworth expansion yields subtle quantum corrections to this behaviour. In particular, they find that particular sampling rates – certain frequencies of repeated measurements – cause remarkable changes.
In finite systems, the consequences include the divergence of the mean detection passage time for transitions between particular states. Namely, there can be a decomposition of the phase space into mutually exclusive regions, the system unable to flow from one to the other. As they note, the latter effect closely mimics ergodicity breaking in classical systems. As the authors show here, however, the quantum effect originates as a result of destructive interference linked to resonant interactions generated as measurements occur at frequencies which are multiples of differences among the energy eigenvalues of different states of the system.
The paper is available at https://journals.aps.org/pre/abstract/10.1103/PhysRevE.105.054108