Abstract In this work, we study in the framework of the so-called driven tight-binding chain (TBC) the issue of quantum unitary dynamics interspersed at random times with stochastic resets mimicking non-unitary evolution due to interactions with the external environment, the driven TBC involves a quantum particle hopping between the nearest-neighbour sites of a one-dimensional lattice and subject to an external forcing field that is periodic in time. We consider the resets to be taking place at exponentially-distributed random times. Using the method of stochastic Liouville equation, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We establish the remarkable effect of localization of the TBC particle on the sites of the underlying lattice at long times. The system in the absence of stochastic resets exhibits delocalization of the particle, whereby the particle does not have a time-independent probability distribution of being found on different sites even at long times, and, consequently, the mean-squared displacement of the particle about its initial location has an unbounded growth in time. One may induce localization in the bare model only through tuning the ratio of the strength to the frequency of the field to have a special value, namely, equal to one of the zeros of the zeroth order Bessel function of the first kind. We show here that localization may be induced by a far simpler procedure of subjecting the system to stochastic resets.
Abstract In this work, we study in the framework of the so-called driven tight-binding chain (TBC) the issue of quantum unitary dynamics interspersed at random times with stochastic resets mimicking non-unitary evolution due to interactions with the external environment, the driven TBC involves a quantum particle hopping between the nearest-neighbour sites of a one-dimensional lattice and subject to an external forcing field that is periodic in time. We consider the resets to be taking place at exponentially-distributed random times. Using the method of stochastic Liouville equation, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We establish the remarkable effect of localization of the TBC particle on the sites of the underlying lattice at long times. The system in the absence of stochastic resets exhibits delocalization of the particle, whereby the particle does not have a time-independent probability distribution of being found on different sites even at long times, and, consequently, the mean-squared displacement of the particle about its initial location has an unbounded growth in time. One may induce localization in the bare model only through tuning the ratio of the strength to the frequency of the field to have a special value, namely, equal to one of the zeros of the zeroth order Bessel function of the first kind. We show here that localization may be induced by a far simpler procedure of subjecting the system to stochastic resets.
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.
We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbor sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a setup that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both the presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.
We address here the issue of quantum measurement in which the interaction between the system at hand and the measuring apparatus causes a collapse of the wave function or the density operator. The analysis is carried out for the prototypical tight-binding chain involving a quantum particle hopping between nearest-neighbour sites on a lattice, with the system subject to repeated projective measurements to a detector site. However, unlike previous work, the measurements are performed at random times. In the absence of measurement, the particle while starting from a site is known to spread out to farther sites as time progresses, leading to complete delocalization of the particle in time. In presence of measurements, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We consider the representative case in which the time gap between successive measurements are sampled independently from an exponential distribution. The novel result of our findings is the revelation of the striking effect of Stochastic Localization, whereby the particle has at long times time-independent probabilities to be on different sites. This implies localization of the particle at long times, achieved through classical stochasticity involved with the random times for measurements. An extreme case of such localization is attained if the measurements are done very frequently, when the particle is completely localized at the detector site! We also point out the stochastic version of the Zeno effect, whereby dynamical evolution of a quantum system is arrested through repeated measurements at frequent-enough times. In the wake of recent experiments on projective measurements at random times achieved through randomly-pulsed sequences, our proposition is amenable to experimental realization.
Reduced transport and localization in isolated quantum systems are typically attributed to spatially-extended disorder, but may also emerge from the influence of a few controllable defects. We show here how a single defect profoundly reshapes wave-function spreading on a finite and periodic tight-binding lattice. Adapting the defect technique from classical random-walk studies, we obtain exact time-resolved site-occupation probabilities and several observables of interest. Even a single defect induces remarkable nonlinear effects, including non-monotonic suppression of transport, enhanced localization at distant sites, and strong sensitivity to the initial particle position at long times. These results demonstrate that minimal perturbations can generate nontrivial long-time transport signatures, giving rise to a microscopic defect-driven mechanism of quantum localization. Although the main results presented pertain to a single isolated defect, we show that the developed formalism may naturally extend to multiple as well as to a wider class of defects.
We study nonequilibrium dynamics of integrable and nonintegrable closed quantum systems whose unitary evolution is interrupted with stochastic resets, characterized by a reset rate r, that project the system to its initial state. We show that the steady-state density matrix of a nonintegrable system, averaged over the reset distribution, retains its off-diagonal elements for any finite r. Consequently a generic observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}$, whose expectation value receives a contribution from these off-diagonal elements, never thermalizes under such dynamics for any finite r. We demonstrate this phenomenon by exact numerical studies of experimentally realizable models of ultracold bosonic atoms in a tilted optical lattice. For integrable Dirac-like fermionic models driven periodically between such resets, the reset-averaged steady state is found to be described by a family of generalized Gibbs ensembles characterized by r. We also study the spread of particle density of a noninteracting one-dimensional fermionic chain, starting from an initial state where all fermions occupy the left half of the sample, while the right half is empty. When driven by resetting dynamics, the density profile approaches at long times to a nonequilibrium stationary profile that we compute exactly. We suggest concrete experiments that can possibly test our theory.
We investigate the effect of a two-level jump process or random telegraph noise on a square wave driven tight-binding lattice. In the absence of the noise, the system is known to exhibit dynamical localization for specific ratios of the amplitude and the frequency of the drive. We obtain an exact expression for the probability propagator to study the stability of dynamical localization against telegraph noise. Our analysis shows that in the presence of noise, a proper tuning of the noise parameters destroys dynamical localization of the clean limit in one case, while it induces dynamical localization in an otherwise delocalized phase of the clean model. Numerical results help verify the analytical findings. A study of the dynamics of entanglement entropy from an initially half-filled state offers complementary perspective.