Repeatedly readable state, spontaneous collapse, and quantum/classical boundary

TL;DR

This paper introduces a state-dependent spontaneous collapse model to delineate the quantum/classical boundary, predicting when superpositions are forbidden or allowed based on measurement readability and system parameters.

Contribution

The model uniquely ties collapse to the potential for repeated measurement, without actual monitoring, differing from existing collapse theories.

Findings

Superposition forbidden if ED >> 4πħc in trapped systems

Superposition allowed if pθD ≥ 8ħ in double-slit scenarios

No limit on coherence length for massless photons

Abstract

We propose a model to identify the quantum/classical boundary. The model introduces a spontaneous collapse of state superposition: $\frac{d}{dt} \rho_{ij} =-\frac{i}{\hbar}[H,\rho]_{ij}-\rho_{ij}/\tau_{ij}$. Different from other collapse models, the collapsing scale $\tau_{ij}$ here does not contain a universal parameter, but is specified by the two states $| i\rangle $ and $ | j\rangle$: If each state is {\em in principle} repeatedly readable (typically by a QND measurement), then $\tau_{ij}$ is the {\em potentially} needed measuring time to discriminate the two states, and the collapse occurs spontaneously {\em without} any actual monitoring. Otherwise, $\tau_{ij}=\infty$, which means no collapse and everlasting superposition. This happens if one state is not repeatedly readable, or if the two states cannot possibly be discriminated in a particular circumstance (for example in the Rabi oscillation). Detailed analysis shows that for a "trapped Schr{\"o}dinger's cat", the superposition of $|{\rm here} \rangle$ and $| {\rm there} \rangle $ is forbidden if $E D \gg 4\pi \hbar c$, and allowed if $E D \le 4\pi \hbar c$, where $D$ is the trap separation and $ E$ is the energy gap, which can be estimated with $ M v^2$. The model also constrains a "free Schr{\"o}dinger's cat" to display double-slit interference if $p\theta D\ge 8\hbar$, where $p= Mv$, $\theta $ is the angle spanned by the two trajectories, and $D$ is the slit separation. In contrast, this model sets no limit on the coherent length of massless photon, thus the arm of a Michelson interferometer can be arbitrarily long. The spontaneous collapse which we propose can occur for an isolated system, and parallels the decoherence induced by interaction with environment.