toplogo
Sign In

Upper Bounds on Outside Probabilities for Non-Autonomous Quantum Dynamics


Core Concepts
We prove sub-ballistic upper bounds on the outside probabilities of wavepackets evolving according to generic non-autonomous Schrödinger equations, assuming the momentum vanishes sufficiently fast in the propagation front.
Abstract

The paper studies the dynamics of non-autonomous Schrödinger equations on lattices, where the Hamiltonian is given by H(t) = H0 + V(t) with H0 a long-range self-adjoint operator satisfying a power-law decay condition.

The key results are:

  1. Theorem 1.1 provides sub-ballistic upper bounds on the outside probabilities P(N,t), i.e., the probability that the wavepacket is found outside a region |x| ≥ Ntα, for 1/2 < α ≤ 1. The main assumption is that the momentum vanishes sufficiently fast in the propagation front.

  2. Theorem 1.2 establishes a general ballistic upper bound on P(N,t), independent of the solution. This extends the classical Radin-Simon bound to the non-autonomous setting.

  3. Theorem 1.3 shows that higher moments of the position operator are also confined to the linear light cone, provided the initial state has bounded higher moments.

  4. Theorem 1.4 extends these results to generic non-linear, non-autonomous Schrödinger equations on lattices via a frozen-coefficient argument.

The proofs rely on commutator estimates and the construction of suitable propagation observables that track the spacetime localization of the evolving wavepackets. The results apply to a wide class of long-range Hamiltonians and can be useful for understanding the dynamics of quantum many-body systems at the mean-field level.

edit_icon

Customize Summary

edit_icon

Rewrite with AI

edit_icon

Generate Citations

translate_icon

Translate Source

visual_icon

Generate MindMap

visit_icon

Visit Source

Stats
The Schrödinger equation studied is i∂tu = (H0 + V(t))u, where H0 is a bounded self-adjoint operator on ℓ2(Zd) with power-law decaying off-diagonal terms. The outside probability is defined as P(N,t) = Σ|x|>N |u(x,t)|^2. The main assumption is that the momentum vanishes sufficiently fast in the propagation front, i.e., ∥i[H, distBR]Wt(distBR)ut∥ ≤ (vα - δ)tα-1 ∥Wt(distBR)ut∥ for some v, δ > 0.
Quotes
"Sub-ballistic upper bounds are obtained, assuming that momentum vanishes sufficiently fast in the front of the wavepackets." "A special case gives a refinement of the general ballistic upper bound of Radin-Simon's, showing that the evolution of wavepackets are effectively confined to a strictly linear light cone with explicitly bounded slope."

Key Insights Distilled From

by Jingxuan Zha... at arxiv.org 10-01-2024

https://arxiv.org/pdf/2409.13762.pdf
Upper bounds in non-autonomous quantum dynamics

Deeper Inquiries

Potential Applications of the Dynamical Upper Bounds Established in This Work to the Study of Quantum Many-Body Systems

The dynamical upper bounds established in this work have significant implications for the study of quantum many-body systems, particularly in understanding particle transport and localization phenomena. One of the primary applications is in the analysis of the spreading of wavepackets in many-body systems, where the results can provide insights into how quantum information propagates through a system. The upper bounds suggest that under certain conditions, the dynamics of wavepackets can be effectively confined to linear light cones, which is crucial for understanding the speed of information transfer in quantum systems. Moreover, these bounds can be applied to investigate the effects of long-range interactions in many-body systems, where the decay conditions on the Hamiltonian can lead to different transport behaviors. For instance, in systems like the Bose-Hubbard model, the results can help elucidate the transition between localized and delocalized phases, particularly in the presence of disorder or nonlinearity. The findings may also inform the design of quantum algorithms and protocols that rely on coherent transport of quantum states, enhancing the efficiency of quantum computing and communication systems.

How to Determine Sufficient Conditions on the Initial State and Potential V(t) to Ensure the Key Assumption (1.12) Holds for Concrete Hamiltonians

To ensure that the key assumption (1.12) holds for concrete Hamiltonians, one must analyze the interplay between the initial state and the time-dependent potential ( V(t) ). Sufficient conditions can be derived by examining the decay properties of the initial state and the behavior of the potential over time. Specifically, one can impose conditions on the initial state ( u_0 ) to ensure that it is well-localized and has a sufficiently rapid decay in momentum space. For instance, if the initial state is chosen to be compactly supported or has a rapid decay, it can help satisfy the requirement that the dynamics push the solution to lower frequencies as time progresses. Additionally, the potential ( V(t) ) should be uniformly bounded and exhibit certain regularity properties, such as smoothness or decay, to avoid introducing instabilities in the evolution. By analyzing the spectral properties of the Hamiltonian and ensuring that the momentum operator's bounds are satisfied, one can establish the necessary conditions for (1.12) to hold.

Can the Techniques Developed Here Be Extended to Study the Dynamics of Other Types of Non-Autonomous Partial Differential Equations Beyond the Schrödinger Setting?

Yes, the techniques developed in this work can be extended to study the dynamics of other types of non-autonomous partial differential equations (PDEs) beyond the Schrödinger setting. The core methodologies, such as the commutator estimates and the use of adiabatic spacetime localization observables (ASTLOs), are versatile and can be adapted to various contexts. For example, similar approaches can be applied to non-autonomous wave equations or nonlinear dispersive equations, where the evolution of wave packets and their localization properties are of interest. The key is to identify appropriate operators and observables that capture the dynamics of the system under consideration. By leveraging the established upper bounds and the techniques for analyzing the propagation of observables, researchers can gain insights into the behavior of solutions to a broader class of non-autonomous PDEs, potentially leading to new results in mathematical physics and applied mathematics.
0
star