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

Abstract

We propose a model to identify the quantum/classical boundary. The model introduces a spontaneous collapse of state superposition: $\frac{d}{dt}{\rho }_{ij}=-\frac{i}{\hslash }[H,\rho {]}_{ij}-{\rho }_{ij}/{\tau }_{ij}$. Different from other collapse models, the collapsing scale ${\tau }_{ij}$ here does not contain a universal parameter, but is specified by the two states $|i⟩$ and $|j⟩$: 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{ö}dinger's cat", the superposition of $|\mathrm{here}⟩$ and $|\mathrm{there}⟩$ is forbidden if $ED\gg 4\pi \hslash c$, and allowed if $ED\le 4\pi \hslash c$, 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{ö}dinger's cat" to display double-slit interference if $p\theta D\ge 8\hslash$, 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.

18 pages, 0 figures