# How do we write the state of a continuous-wave laser beam?

Many of us are probably familiar with the notion that the output of a laser is a coherent state $$|\alpha\rangle$$. But this simplistic description falls short in many respects. For instance, how can I describe the output of a CW laser? Even omitting the transverse spatial variables and polarization, there are several properties that the state should have:

1. For each 'point' in the beam, there is a coherence region around it
2. Points far enough have random phases with respect to each other
3. The linewidth is a Lorentzian

I thought that one possible description could be along the lines of (but I'm still very unhappy with it):

$$\rho = \int dt \left(\int d\tau |\tau\rangle\otimes|\alpha e^{i\phi(\tau-t)}\rangle\right)\left(\int d\tau' \langle\tau|\otimes\langle\alpha e^{i\phi(\tau'-t)}|\right)$$ where all the integrals are from $$T_{on}$$ to $$T_{off}$$ and the normalization constant is $$(T_{off}-T_{on})^{-2}$$ and $$\phi(\tau)$$ is a random phase which is roughly constant for intervals of the order of the coherence time. The kets $$|\tau\rangle$$ are intended to be "time eigenstates", whose function is purely operational, not physical$${}^1$$. Here $$\alpha$$ depends on the coherence time, but it's proportional to the square root of the intensity of the laser.

This seems to satisfy the first two properties, but I'm not sure how to enforce the third. I suspect that it has something to do with the autocorrelation function of the random function $$\phi(\tau)$$.

$${}^1$$ Akin for instance to the "position eigenstates" that one could use to write out a wave function $$|\psi(x)\rangle := \int dx\, \psi(x)|x\rangle$$.

• Please see this: arxiv.org/abs/quant-ph/0104036 and there should be some useful references within this paper
– Ilya
Commented Mar 16, 2017 at 9:38