Complementarity in the Bohr-Einstein Photon Box

TL;DR

This paper analyzes the Bohr-Einstein photon box thought experiment, demonstrating how quantum dynamics ensure the complementarity of measurements and revealing a fine structure in uncertainty distribution that relates to the EPR experiment.

Contribution

It provides a detailed calculation showing how seemingly simultaneous measurements become non-commuting when traced back to the photon escape time, confirming Bohr's qualitative arguments.

Findings

Explicit calculation of non-vanishing commutators supports complementarity.

Uncertainty distribution depends on the timing of the box measurement.

Results connect the photon box experiment to the foundations of the EPR paradox.

Abstract

The photon box thought experiment can be considered a forerunner of the EPR-experiment: by performing suitable measurements on the box it is possible to ``prepare'' the photon, long after it has escaped, in either of two complementary states. Consistency requires that the corresponding box measurements be complementary as well. At first sight it seems, however, that these measurements can be jointly performed with arbitrary precision: they pertain to different systems (the center of mass of the box and an internal clock, respectively). But this is deceptive. As we show by explicit calculation, although the relevant quantities are simultaneously measurable, they develop non-vanishing commutators when calculated back to the time of escape of the photon. This justifies Bohr's qualitative arguments in a precise way; and it illustrates how the details of the dynamics conspire to guarantee the requirements of complementarity. In addition, our calculations exhibit a ``fine structure'' in the distribution of the uncertainties over the complementary quantities: depending on \textit{when} the box measurement is performed, the resulting quantum description of the photon differs. This brings us close to the argumentation of the later EPR thought experiment.