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 μ 2 (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 μ 4 . 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 μ 2 is only a standard quantum limit; the ultimate quantum limit—or Heisenberg limit—is quadratically better.