The Heisenberg limit for laser coherence

Quantum optical coherence can be quantified only by accounting for both the particle- and wave-nature of light. For an ideal laser beam ^1–3, the coherence can be thought of roughly as the number of photons emitted consecutively into the beam with the same phase. This number, ℭ, can be much larger than the number of photons in the laser itself, μ. The limit for an ideal laser was thought to be of order μ² (refs. ^4,5). Here, assuming only that a laser produces a beam with properties close to those of an ideal laser beam and that it has no external sources of coherence, we derive an upper bound on ℭ, which is of order μ⁴. Moreover, using the matrix product states method ⁶, we find a model that achieves this scaling and show that it could, in principle, be realized using circuit quantum electrodynamics ⁷. Thus, ℭ of order μ² is only a standard quantum limit; the ultimate quantum limit—or Heisenberg limit—is quadratically better.