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: $\frac{d}{dt} ρ_{ij} =-\frac{i}{\hbar}[H,ρ]_{ij}-ρ_{ij}/τ_{ij}$. Different from other collapse models, the collapsing scale $τ_{ij}$ 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 $τ_{ij}$ is the {\em potentially} needed measuring time to discriminate the two states, and the collapse occurs spontaneously {\em without} any actual monitoring. Otherwise, $τ_{ij}=\infty$, 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 $|{\rm here} \rangle$ and $| {\rm there} \rangle $ is forbidden if $E D \gg 4π\hbar c$, and allowed if $E D \le 4π\hbar c$, 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{ö}dinger's cat" to display double-slit interference if $pθD\ge 8\hbar$, where $p= Mv$, $θ$ 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.