The Heisenberg limit for laser coherence

Abstract

To quantify quantum optical coherence requires both the particle- and wave-natures of light. For an ideal laser beam [1,2,3], it can be thought of roughly as the number of photons emitted consecutively into the beam with the same phase. This number, $\mathfrak{C}$, can be much larger than $\mu$, the number of photons in the laser itself. The limit on $\mathfrak{C}$ for an ideal laser was thought to be of order ${\mu }^2$ [4,5]. Here, assuming nothing about the laser operation, only that it produces a beam with certain properties close to those of an ideal laser beam, and that it does not have external sources of coherence, we derive an upper bound: $\mathfrak{C}=O({\mu }^4)$. Moreover, using the matrix product states (MPSs) method [6,7,8,9], we find a model that achieves this scaling, and show that it could in principle be realised using circuit quantum electrodynamics (QED) [10]. Thus $\mathfrak{C}=O({\mu }^2)$ is only a standard quantum limit (SQL); the ultimate quantum limit, or Heisenberg limit, is quadratically better.

6 pages, 4 figures, and 31 pages of supplemental information. v2: This paper is now published [Nature Physics DOI:10.1038/s41567-020-01049-3 (26 October 2020)]. For copyright reasons, this arxiv paper is based on a version of the paper prior to the accepted (21 August 2020) version