Abstract: 

To quantify quantum optical coherence requires both the particle- and wave-natures of light. For an ideal laser beam, it 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 μ, the number of photons in the laser itself. The limit on C for an ideal laser was thought to be of order μ². Here, assuming nothing about the laser operation, only that it produces a beam with properties close to those of an ideal laser beam, and that it does not have external sources or stores of coherence, we derive an upper bound: C = O(μ⁴). Moreover, using the matrix product states method, we find a model that achieves this scaling. Thus C = O(μ²) is only a standard quantum limit; the ultimate quantum limit, or Heisenberg limit, is quadratically better.

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