The Heisenberg limit for laser coherence

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.