Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exponential tail estimates for quantum lattice dynamics

Christopher Cedzich, Alain Joye, Albert H. Werner, Reinhard F. Werner

TL;DR

This work establishes exponential tail bounds for the leakage of quantum lattice dynamics outside the propagation cone defined by the group-velocity spectrum. By extending the symbols to complex shifts and analyzing the strong limits of conjugated evolution, the authors prove a state-independent large-deviation rate function $I$ via a Legendre transform of $R$, computable from $H(p)$ or $W(p)$. The framework unifies discrete- and continuous-time dynamics and yields explicit rates and boundary behavior, with illustrative examples across 1D, 2D, and 3D lattice models. The results provide practical, uniform estimates for the fuzziness of propagation cones and connect propagation geometry to quantitative exponential suppression of super-velocity signals, bearing on bounds analogous to Lieb–Robinson bounds in nonrelativistic lattice systems.

Abstract

We consider the quantum dynamics of a particle on a lattice for large times. Assuming translation invariance, and either discrete or continuous time parameter, the distribution of the ballistically scaled position $Q(t)/t$ converges weakly to a distribution that is compactly supported in velocity space, essentially the distribution of group velocity in the initial state. We show that the total probability of velocities strictly outside the support of the asymptotic measure goes to zero exponentially with $t$, and we provide a simple method to estimate the exponential rate uniformly in the initial state. Near the boundary of the allowed region the rate function goes to zero like the power 3/2 of the distance to the boundary. The method is illustrated in several examples.

Exponential tail estimates for quantum lattice dynamics

TL;DR

This work establishes exponential tail bounds for the leakage of quantum lattice dynamics outside the propagation cone defined by the group-velocity spectrum. By extending the symbols to complex shifts and analyzing the strong limits of conjugated evolution, the authors prove a state-independent large-deviation rate function via a Legendre transform of , computable from or . The framework unifies discrete- and continuous-time dynamics and yields explicit rates and boundary behavior, with illustrative examples across 1D, 2D, and 3D lattice models. The results provide practical, uniform estimates for the fuzziness of propagation cones and connect propagation geometry to quantitative exponential suppression of super-velocity signals, bearing on bounds analogous to Lieb–Robinson bounds in nonrelativistic lattice systems.

Abstract

We consider the quantum dynamics of a particle on a lattice for large times. Assuming translation invariance, and either discrete or continuous time parameter, the distribution of the ballistically scaled position converges weakly to a distribution that is compactly supported in velocity space, essentially the distribution of group velocity in the initial state. We show that the total probability of velocities strictly outside the support of the asymptotic measure goes to zero exponentially with , and we provide a simple method to estimate the exponential rate uniformly in the initial state. Near the boundary of the allowed region the rate function goes to zero like the power 3/2 of the distance to the boundary. The method is illustrated in several examples.
Paper Structure (15 sections, 8 theorems, 80 equations, 6 figures)

This paper contains 15 sections, 8 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Setting and statement of the main estimate
  4. Group velocity and propagation region
  5. Proof of Theorem \ref{['mainprop']}
  6. Discussion of the bound and general properties
  7. Elementary properties
  8. Large $x$ and relation to trivial propagation bounds
  9. Small $\lambda$ and behavior near $\partial\Gamma$
  10. Examples
  11. 1D Qubit walk
  12. A 2D walk with circular light cone
  13. A 2D walk with non-convex propagation region
  14. 3D cones and rate functions
  15. Continuity of the supremum

Key Result

Theorem 2.1

In the setting described above, a rate function $I$ in LDbound can be taken as the Legendre transform of a function $R: {\mathbb R}^s \mapsto {\mathbb R}$ that can be computed directly from $H(p)$, respectively $W(p)$, as where ${\mathbb B}=[-\pi,\pi]^s$ denotes the Brillouin zone.

Figures (6)

  • Figure 1: Subsets needed in the proof of the basic estimate. The red hatched patch is $\widetilde{M}$ from \ref{['Mtilde']}. The straight lines indicate the half spaces characterized by the $\lambda_i$.
  • Figure 2: The functions $R$ and $I$ after \ref{['1Dprerate']} and \ref{['1Drate']} for $a=.5$.
  • Figure 3: Left: Dispersion relation $\omega_\pm(p)$ according to \ref{['2DWeylOmega']}. Right: Probability density of the group velocity for particles starting at the origin. The mesh on the graph is in polar coordinates for velocity, shown up to $\vert v\vert=.8$. Bottom right: Propagation region, i.e., the unit disc, shaded according to probability density.
  • Figure 4: Propagation region for the Hamiltonian \ref{['nonconvexH']}. The plot is generated by computing the group velocity on a regularly spaced grid in momentum space. The density of points thus corresponds to the asymptotic probability density for any reasonably well localized initial state.
  • Figure 5: Propagation region for the walk \ref{['bb94W']}. This convex set is the intersection of three orthogonal cylinders.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 5 more