We study the role played by noise on the bounded state of a two-particle QW with interaction, as introduced in [1]. The bounded (molecular) state can be effectively described as a one-particle QW in 1D, with a coin operator which depends on the extra phase that is acquired by the interaction at each time step. The noise is introduced by a random change in the value of the phase during the evolution, from a constant probability distribution within a given interval. The consequences of introducing such kind of noise depend on both the center value and the width of that interval: a wider interval manifests as a higher level of noise. The center value is also important for the fate of the molecular state (as noise increases). Depending on this value, the molecule can be destroyed. More interestingly, one can also find some range of parameters for which the bound state survives under the form of a localized state, a situation that can be related to the violation of the stability criterion of the molecule. The existence of this localized state for such kind of time-dependent noise is, to the best of our knowledge, totally new, since localization (i.e., Anderson localization) is linked in the literature to a spatial random noise.
We study the role played by noise on the bounded state of a two-particle QW with interaction, as introduced in [1]. The bounded (molecular) state can be effectively described as a one-particle QW in 1D, with a coin operator which depends on the extra phase that is acquired by the interaction at each time step. The noise is introduced by a random change in the value of the phase during the evolution, from a constant probability distribution within a given interval. The consequences of introducing such kind of noise depend on both the center value and the width of that interval: a wider interval manifests as a higher level of noise. The center value is also important for the fate of the molecular state (as noise increases). Depending on this value, the molecule can be destroyed. More interestingly, one can also find some range of parameters for which the bound state survives under the form of a localized state, a situation that can be related to the violation of the stability criterion of the molecule. The existence of this localized state for such kind of time-dependent noise is, to the best of our knowledge, totally new, since localization (i.e., Anderson localization) is linked in the literature to a spatial random noise.
Quantum walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phenomena not described by the single-particle model, due to entanglement and interaction among the walkers. In this context, we present here a thorough numerical study of an one dimensional system of two quantum walkers exhibiting rich collective dynamics controlled by simple time-dependent unitary coins proposed in [Phys. Rev. A \textbf{80}, 042332(2009)] and [Phys. Rev. A \textbf{73},062304(2006)]. We study how the interplay of coin time-dependence, simple interaction schemes, entanglement and the relative phase between the coin states of the particles influences the evolution of the quantum walk. The results show that the system offers a rich variety of collective dynamical behavior while being controlled by time dependent coins. In particular, we find and characterize fascinating two-body localization phenomena with tunable quasiperiodic dynamics of correlations and entanglements which are quantities of quantum origin.
The time evolution of one- and two-dimensional discrete-time quantum walk with increase in disorder is studied. We use spatial, temporal and spatio-temporal broken periodicity of the unitary evolution as disorder to mimic the effect of disordered/random medium in our study. Disorder induces a dramatic change in the interference pattern leading to localization of the quantum walks in one- and two-dimensions. Spatial disorder results in the decreases of the particle and position entanglement in one-dimension and counter intuitively, an enhancement in entanglement with temporal and spatio-temporal disorder is seen. The study signifies that the Anderson localization of quantum state without compromising on the degree of entanglement could be implement in a large variety of physical settings where quantum walks has been realized. The study presented here could make it feasible to explore, theoretically and experimentally the interplay between disorder and entanglement. This also brings up a variety of intriguing questions relating to the negative and positive implications on algorithmic and other applications.
We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator U(k). An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position x. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of U(k). Finally, for fixed time t sufficiently large, we examine the dependence on t of the probability of finding a particle at a given location x.
We consider two particles hopping on a chain with a contact interaction between them. At strong interaction, there is a molecular bound state separated by a direct gap from a continuous band of atomic states. Introducing weak disorder in the interaction, the molecular state becomes Anderson-localized. At stronger disorder, part of the molecular band delocalizes and dissociates due to its hybridization to the atomic band. We characterize these different regimes by computing the density of states, the inverse participation ratio, the level-spacing statistics, and the survival probability of an initially localized state. The atomic band is best described as that of a rough billiard for a single particle on a square lattice that shows signatures of quantum chaos. In addition to typical ``chaotic states,'' we find states that are localized along only one direction. These ``separatrix states'' are more localized than chaotic states, and similar in this respect to scarred states, but their existence is due to the separatrix isoenergy line in the interaction-free dispersion relation, rather than to unstable periodic orbits.
The interplay between disorder and topology triggers interesting effects for the propagation of quantum particles in low-dimensional network structures. For instance, the occurrence of Anderson localization, a commonly observed effect in a quantum network with static disorder, can be suppressed by introducing topology-a phenomenon dubbed as topological Anderson localization transition [1] . In one-dimensional systems, for example, disorder induces localization on a short length scale while topology counters that through long-range correlations, yielding divergent correlation length and extremely slow spreading at quantum criticality. However, its experimental demonstration still remains an outstanding challenge as that would require, in contrast to Anderson localization transition, addi¬tional control over internal degrees of freedom, which is hard to implement, for instance, in cold-atom systems [2] . Quantum walks (QWs), a quantum analog of random walks, comprising quantum coin toss and conditioned step operations are already established for studying quantum transport phenomena, such as, disorder and topology [5] . Based on our theoretical proposal [3] , here, we provide an experimental route using time-multiplexed quantum walk [4] for evidencing the targeted phenomenon.
Abstract We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin. The rotation angle, which depends upon the position of a quantum particle, parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classical-like, semi-classical-like, and quantum-like. Further, we study the Shannon entropy associated with position and the coin space of a quantum particle, and compare them with the case of the position-independent coin. Our results show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy by considering a quantum walk with two identical position-dependent entangled coins. We observe that in general, a wave function becomes more localized as compared to the case of the position-independent coin and hence the corresponding Shannon entropy is lower. Our results show that a position-dependent coin can be used as a controlling tool of quantum walks.