Spiking and collapsing in large noise limits of SDEs

Abstract

We analyze the strong noise limit of one-dimensional stochastic differential equations (SDEs). Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.

v1: Preliminary version. v2: Expanded version treating the general two-boundary case and the case of Rabi oscillations. 29 pages, 3 figures

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• Computing the Rates of Measurement-Induced Quantum Jumps
• Zooming in on Quantum Trajectories
• Emergence of jumps in quantum trajectories via homogeneization
• Stochastic spikes and strong noise limits of stochastic differential equations
• Perturbation Theory for Weak Measurements in Quantum Mechanics, I -- Systems with Finite-Dimensional State Space
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Cited by in corpus (5)

• Quantum resetting in continuous measurement induced dynamics of a qubit
• The Open Quantum Brownian Motion and continual measurements
• Spikes in Poissonian quantum trajectories
• To spike or not to spike: the whims of the Wonham filter in the strong noise regime
• Continuous collapse models on finite dimensional Hilbert spaces