[Paper Review] Telegraph signals as a solution to the time dependent Schrödinger equation

D. Drakova, G. Doyen|arXiv (Cornell University)|Apr 25, 2012

Optical and Acousto-Optic Technologies1 references3 citations

TL;DR

This paper demonstrates that telegraph signals—random, sudden transitions between two states—can emerge from the time-dependent Schrödinger equation without collapse postulates, via weak, local coupling of a particle to a high-density continuum of environmental states. The resulting coherent dynamics produce telegraph-like behavior due to sharp, Lorentzian-shaped resonances in the spectral function, with time scales ranging from picoseconds to seconds.

ABSTRACT

A particle switching between two sides of a symmetric system in interaction with a continuum exhibits a telegraph-like time development without the need of the Born-Bohr principle of reduction on eigenstates of the measuring equipment. The origin of the telegraph signal is a very weak local coupling of the particle to the continuum which is connected with an enormous slow down of the particle motion. The proposed mechanism might serve as a useful simple model for studying decoherence effects due to coupling to the environment.

Motivation & Objective

• To explain the origin of random telegraph signals in quantum systems without invoking the Born-Bohr collapse postulate.
• To investigate whether coherent time evolution governed by the time-dependent Schrödinger equation can produce telegraph-like state transitions.
• To model how weak, local coupling to a continuum of environmental states leads to apparent statistical switching behavior.
• To provide a minimal, analytically tractable framework for studying decoherence effects in open quantum systems.

Proposed method

• The system is modeled as a particle tunneling between two symmetric sites (α and β), each connected to a local environment via gateway and remote states.
• The Hamiltonian includes on-site energies, tunneling between remote and gateway states, and weak, local coupling between gateway states and a quasi-continuum of environmental states.
• The Green's function formalism is used to compute the self-energy and spectral functions, revealing Lorentzian-shaped resonances due to weak coupling.
• The time evolution of the wave packet amplitude is derived from the imaginary part of the Green's function, linked to resonance width and decay rate.
• The spectral weight on on-shell eigenstates derived from environmental components (|κ−⟩) is shown to alternate, leading to telegraph-like behavior.
• Numerical solutions of the time-dependent Schrödinger equation are used to visualize the state-switching dynamics, confirming the emergence of telegraph signals.

Experimental results

Research questions

• RQ1 Can a purely coherent solution of the time-dependent Schrödinger equation produce telegraph-signal-like dynamics without wavefunction collapse?
• RQ2 What physical conditions—specifically coupling strength and environmental structure—are necessary for such coherent telegraph signals to emerge?
• RQ3 How does the time scale of the switching behavior relate to the energy spacing of environmental states?
• RQ4 Why does the spectral weight of the initial state alternate between on-shell and off-shell components in the presence of a continuum?
• RQ5 To what extent can this model serve as a minimal description of decoherence in open quantum systems?

Key findings

• Telegraph signals emerge from coherent time evolution due to weak, local coupling to a high-density continuum of environmental states, without requiring measurement-induced collapse.
• The spectral function exhibits sharp, Lorentzian-shaped resonances centered on the energy shell, which arise only when coupling is both weak and spatially localized.
• The time development of the wave packet amplitude is directly linked to the imaginary part of the Green's function, showing alternating large and small values consistent with sudden state transitions.
• The spectral weight of the initial state |gα⟩ is non-zero and larger on eigenstates derived from |κ−⟩ components, explaining the observed alternating behavior.
• The switching time scale is determined by the energy separation between adjacent environmental states, spanning from picoseconds to seconds depending on the system.
• The model reproduces experimentally observed telegraph signals in low-temperature STM experiments, such as hydrogen tautomerization and adsorbate switching on surfaces, via a unitary, deterministic evolution.