Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

Abstract

In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. The formalism is very general in that it applies to any quantum system, to any form of non-unitary interaction, and to any probability distribution for the random times. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, e.g. in modelling nano wires. Considering the case of one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, owing to spontaneous quantum fluctuations tunnels between the nearest-neighbour sites. We consider two representative forms of interactions, one that implements a stochastic reset of quantum dynamics in which the density operator is at random times reset to its initial form, and one in which projective measurements are performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement (MSD) of the particle about its initial location. This stands in stark contrast to the behavior in the absence of interactions, when the particle has an unbounded growth of the MSD in time, with no signatures of localization. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which it does not require us to perform measurements at exactly regular intervals that are hallmarks of such scenarios.

Stochastic resets in the context of a tight-binding chain driven by an oscillating field

“…For the particular case of no field, implying w(t) = t, we recover using the above equation the known result [5,7]: S(t) = ∆ 2 t 2 /2. From Eq.…”

“…It is easily checked that in the absence of forcing, when one has w(t) = t, the above equation reduces to the following result known in the literature (see, e.g., [7]): P m (t) = J 2 m−n 0 (∆t). Note that since one has J −m (x) = (−1) m J m (x), the quantity P m (t) in Eq.…”

“…In this section, we briefly discuss following Ref. [7] our formalism of Stochastic Liouville equation to study analytically unitary evolution interspersed at random times with instantaneous non-unitary interactions in the context of a general quantum system, and then apply the formalism to obtain explicit results for the TBC subject to a periodic forcing field in the case in which the interactions implement stochastic resets. The quantum system while starting with a density operator ρ(0) is assumed to be undergoing unitary evolution in time for a fixed time t. The evolution is interspersed with instantaneous interaction with the environment at random times, which induces non-unitarity in the evolution of the system, and which is implemented by a given interaction operator T .…”

“…The evolution ends with unitary evolution for time duration t − t µ . We will later in the paper consider the form of T that implements stochastic resets of quantum dynamics [3,7], whereby the density operator is at random times reset to a given form (see later). We feel it pertinent at this stage to cite a number of recent work on themes related to resets of quantum dynamics.…”

“…In recent times, we ourselves have considered the paradigmatic case of free tunnelling motion in the TBC in the context of projective quantum measurements as well as stochastic resets in Ref. [7], hereafter designated as I. In I, we encountered an interesting effect of localization of the TBC particle on the sites of the underlying lattice, whereby the free particlelike unbounded motion gets localized because of random interruptions occasioned by the resetting process.…”