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arXiv:2011.07198

Quantum walkers in a disordered lattice with power-law hopping

G. A. Domínguez‐Castro, R. Paredes

Published Nov 14, 2020• 10 citations
arXivHTMLPDF

Abstract

We study the effects of interparticle interactions and power-law tunneling couplings on quantum walks executed by both a single one and a pair of hard-core bosons moving in clean and disordered one-dimensional lattices. For this purpose, we perform exact diagonalization to explicitly evaluate the short and long time probabilities of finding the walkers within a surveillance area. Our main conclusions, summarized in phase diagrams in the disorder-power-law and interaction-disorder spaces, allowed us to discern two different scenarios for the single and two quantum walkers dynamics. While in the single particle case the transition to localized and extended regimes is identified for well defined values of the disorder amplitude and power law hopping, those frontiers are replaced by diffuse contours in the interacting two particle case. In fact, counterintuitive transport regimes as diffusion enhanced by disorder, and space constrained dynamics assisted by both interactions and short tunneling range are found. Our results are of direct relevance for quantum systems with long-range interactions that are currently realized in the laboratory.

DOI
10.48550/arxiv.2011.07198

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We study the effects of interparticle interactions and power-law tunneling couplings on quantum walks executed by both a single one and a pair of hard-core bosons moving in clean and disordered one-dimensional lattices. For this purpose, we perform exact diagonalization to explicitly evaluate the short and long time probabilities of finding the walkers within a surveillance area. Our main conclusions, summarized in phase diagrams in the disorder-power-law and interaction-disorder spaces, allowed us to discern two different scenarios for the single and two quantum walkers dynamics. While in the single particle case the transition to localized and extended regimes is identified for well defined values of the disorder amplitude and power law hopping, those frontiers are replaced by diffuse contours in the interacting two particle case. In fact, counterintuitive transport regimes as diffusion enhanced by disorder, and space constrained dynamics assisted by both interactions and short tunneling range are found. Our results are of direct relevance for quantum systems with long-range interactions that are currently realized in the laboratory.
We study the effects of long-range hopping and long-range interparticle interactions on the quantum walk of hard-core bosons in ideal and disordered one-dimensional lattices. We find that the range of hopping has a much more significant effect on the particle correlation dynamics than the range of interactions. We illustrate that long-range hopping makes the correlation diagrams asymmetric with respect to the sign of the interaction. We examine the relative role of repulsive and attractive interactions on the dynamics of scattering by isolated impurities and Anderson localization in disordered lattices. We show that weakly repulsive interactions increase the probability of tunneling through isolated impurities and decrease the localization.
The quantum walk of two hard-core bosons in one-dimensional lattice under the effect of long-range inter-particle interaction is studied in detail. We also simulate the influence of an isolated defect that may exist in the lattice on the quantum walk of two particles by adding an additional potential energy to a certain lattice site. Using exact diagonalization method, the continuous-time quantum walk is directly simulated. The numerical simulations show that the range of interaction (long-range or short-range), the strength of the inter-particle interaction, the initial state of the two particles and the presence of the isolated defect have great influences on the quantum walk. Under the effect of strong long-range interaction, the particles initially located on the non-adjacent lattice sites have a co-walking behavior, while under the short-range interactions (nearest-neighbor interactions) only two particles initially located on the neighboring lattice sites can exhibit co-walking. After introducing the isolated defect into the system with strong interaction, two particles residing on the same side of the isolated defect keep co-walking, while two particles located on either sides of the isolated defect or one particle located on the isolated defect and the other particle staying on the side of the isolated defect, the two particles keep stationary or co-walking near the defect, displaying the characteristics of localization. By using the second-order perturbation theory of degenerate quantum system, a comprehensive theoretical analysis of the above numerical results is given. The theoretical analysis reveals the underlying physical law of quantum walks of two particles in one-dimensional lattice under the effects of strong long-range interaction and isolated defect in the lattice.
Two interacting particles in lattices, in the absence of dissipation, can not distin- guish between attractive or repulsive interaction when the range of their tunnelling is limited to nearest neighbor sites. However, we find that, in the case of long-range tunnelling, the particles exhibit different dynamics for different types of interactions of the same strength. The nature of dynamical correlations between particles also becomes significantly different. For weak interactions, particles develop a character in correlation which is in between that of antiwalking and cowalking when the tunnelling is long-range. For strong interactions, particles cowalk independently of their statistics. A few recent experiments have demonstrated such effects of interactions on quantum walk of photons, atoms and spin excitations on various lattice platforms. In disordered lattices the effect of coherent backscattering makes particles localize to their initial position. We find that a weak repulsive interaction reduces localization and a strong interaction enhances localization. We also calculate the correlations between the particles in the disordered 1D and 2D systems. The effect of long-range tunnelling on localization of particles in disordered 1D systems has been explored. For large ordered or disordered lattices, computation of localization parameters becomes difficult. In these cases, an efficient recursive algorithm is used to calculate Green’s functions exactly. We extend such algorithm to disordered systems in both one and two dimensions. We also illustrate that this recursive algorithm maps directly to some graph structures like binary trees. We perform calculations for quantum walk of interacting particles on such graphs. The method is also used to calculate the properties of interacting particles on lattices with gauge fields. For disordered 2D lattices, we introduce and test approximations which produce accurate results and make the calculations more efficient. We examine the localization parameters for a broad range of interaction and disorder strengths and try to find differences among parameters within the range.
The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.
We consider a one-dimensional fermionic lattice system with long-ranged power-law decaying hopping with exponent $\ensuremath{\alpha}$. The system is further subjected to dephasing noise in the bulk. We investigate two variants of the problem: (i) an open quantum system where the setup is further subjected to boundary reservoirs enabling the scenario of a nonequilibrium steady-state charge transport, and (ii) time dynamics of an initially localized single-particle excitation in the absence of boundary reservoirs. In both variants, anomalous superdiffusive behavior is observed for $1<\ensuremath{\alpha}<1.5$, and for $\ensuremath{\alpha}>1.5$ the setup is effectively short ranged and exhibits conventional diffusive transport. Our findings are supported by analytical calculations based on the multiple scale analysis technique that leads to the emergence of a fractional diffusion equation for the density profile. Our study unravels an interesting interplay between long-range interaction and dephasing mechanism that could result in the emergence of unconventional behavior in open quantum systems.
Quantum walk (QW) on a disordered lattice leads to a multitude of interesting phenomena, such as Anderson localization. While QW has been realized in various optical and atomic systems, its implementation with superconducting qubits still remains pending. The major challenge in simulating QW with superconducting qubits emerges from the fact that on-chip superconducting qubits cannot hop between two adjacent lattice sites. Here we overcome this barrier and develop a gate-based scheme to realize the discrete time QW by placing a pair of qubits on each site of a one-dimensional (1D) lattice and treating an excitation as a walker. It is also shown that various lattice disorders can be introduced and fully controlled by tuning the qubit parameters in our quantum walk circuit. We observe a distinct signature of transition from the ballistic regime to a localized QW with an increasing strength of disorder. Finally, an eight-qubit experiment is proposed where the signatures of such localized and delocalized regimes can be detected with existing superconducting technology. Our proposal opens up the possibility of exploring various quantum transport processes with promising superconducting qubits.