Light-cone and local front dynamics of a single-particle extended quantum walk

Abstract

We study the light-cone and front dynamics of a single particle continuous time extended quantum walk on a one dimensional lattice with finite range hopping. We show that, in general, for an initially localized state, propagating wave fronts can be characterized as ordinary or extremal fronts with the latter exhibiting an anomalous sub-diffusive scaling behaviour in the front region. We investigate the dynamical global and local scaling properties of the cumulative probability distribution function for the extended walk with nearest and next-nearest neighbour hopping using analytical and numerical methods. The global scaling shows the existence of a 'causal light-cone' corresponding to excitations travelling with a velocity smaller than a maximal 'light velocity'. Maximal fronts moving with fixed 'light velocity' bound the causal cone. The front regions spread with time sub-diffusively exhibiting a local Airy scaling which leads to an internal staircase structure. At a certain critical next-nearest neighbour hopping strength, there is a transition from a phase with one 'causal cone' to a phase with two nested 'causal cones' and the existence of an internal staircase structure in the corresponding cumulative distribution profiles. We also connect the study to that in spin chain systems and indicate that a single particle quantum walk on the one dimensional lattice already captures the many body physics of a spin chain system. In particular, we suggest that the time evolution of a single particle quantum walk on the one dimensional lattice with an initially localized state is equivalent to the time evolution of a domain wall initial state in a corresponding spin chain system. probability density p(n, t) = |ψ(n, t)| 2 scales as

Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps

“…We have comprehensively explored an extension of the EQW, a non-Markovian quantum walk process where the jump sites are selected from a uniform distribution giving rise to hyperballistic diffusion. Even though jumps in QWs have been introduced either in discrete-time 34–40 and continuous-time 41–45 models, we highlight that none of these previous works have found the richness of dynamical transitions we have presented herein with our new time-dependent protocol of jumps and two types of coin operators. Our flexible distribution, the q -exponential, allows us to recover both the standard QW in the limit q → 1/2 and the previous proposal q → ∞ 28 , providing a more general framework.…”

Long time dynamics of a single particle extended quantum walk on a one dimensional lattice with complex hoppings: a generalized hydrodynamic description

“…In a walk with nearest and next nearest hopping, we showed that there is a transition from a regime with one causal cone to a regime with two nested causal cones. Further, we showed that the local probability densities exhibit anomalous subdiffusive scaling near extremal fronts and the nature of the scaling depends on the order of the front [20]. We also connected the study to that of spin-chains where the existence of such upper bounds on the spread or Lieb-Robinson (LR) bounds for the speed with which information propagates has been earlier studied [24][25][26].…”

“…They have been shown to exhibit ballistic propagation instead of the diffusive behaviour expected of a classical random walk [18][19][20][21][22][23]. In recent work [20], we showed that interesting 'causal light cone structures' appear in a single particle continuous time quantum walk with a finite range of hopping. In the bulk, the walk exhibits ballistic propagation of wave fronts.…”

“…There has been particular interest in the study of continuous-time quantum walks [2,5,[17][18][19][20] because they provide alternate ways to model and study many body lattice Hamiltonians. They have been shown to exhibit ballistic propagation instead of the diffusive behaviour expected of a classical random walk [18][19][20][21][22][23].…”

“…There has been particular interest in the study of continuous-time quantum walks [2,5,[17][18][19][20] because they provide alternate ways to model and study many body lattice Hamiltonians. They have been shown to exhibit ballistic propagation instead of the diffusive behaviour expected of a classical random walk [18][19][20][21][22][23]. In recent work [20], we showed that interesting 'causal light cone structures' appear in a single particle continuous time quantum walk with a finite range of hopping.…”