# Laser power and coherent state amplitude

How it is possible to relate the power $$P$$ of a laser light with frequency $$\omega$$, to the amplitude $$\alpha$$ of its description as a coherent state $$\vert \alpha \rangle$$ ? For a massive particle in an harmonic potential one has $$A = \sqrt{\dfrac{2 \hbar}{m \omega}} \vert\alpha\vert$$ where $$A=\sqrt{P}$$ is the oscillation amplitude. But what about a masses particle?

Is it possible to use the mean photon number during time $$T$$, $$\dfrac{P T}{\hbar \omega}=\bar{n}(T)$$ ?

• This is not an answer, just thinking: Depending on the Q of the laser cavity, the ratio of the emitted beam to the power of the light inside the cavity can be anything between 1 and 0. So, the frequency and amplitude inside the cavity is not sufficient to calculate the emitted power of the laser. – S. McGrew Dec 16 '18 at 15:16
• Is your $\alpha$ the coherent state inside the laser's resonant cavity? – DanielSank Dec 16 '18 at 18:14
• If possible, I would prefer not talking about the laser cavity. Imagine there is just a propagating laser light, impinging on some optical elements. @S.McGrew, in this sense I'm directly referring to the power of the light emitted from the laser cavity. – m137 Dec 17 '18 at 7:08
• @DanielSank, $\alpha$ is then the coherent state associated to the propagating field. – m137 Dec 17 '18 at 7:09
• The power of the light in the emitted beam is directly proportional to the number of photons emitted per unit time, and to the energy per photon. But it seems you might be asking something different-- – S. McGrew Dec 17 '18 at 14:32

First, you seem to take the analogy with the massive harmonic oscillator too literally. The analogy is pretty strong, but purely formal, so as you guessed, the mass as no meaning here. Furthermore, if you speak of amplitude, you may think of the electromagnetic field, the exact prefactor will depend whether you look at the electric field $$\left(\frac{\hbar}{2\omega V \epsilon_0}\right)$$ or the magnetic field $$\left(\frac{\hbar\omega}{2 V \epsilon_0}\right)$$ (if I made no mistake).
Since you are interested in power, the question is simpler: the average energy of a coherent state is $$\lvert\alpha\rvert^2\hbar\omega$$ in Joules (or $$\lvert\alpha\rvert^2$$ in photons). If this is a coherent state trapped in a cavity, it is the total energy of the coherent state. For a propagating laser, this information is also sufficient if $$\left|\alpha\right>$$ describes the coherent state of a pulse, since it gives the total (average) energy of the pulse. For a square pulse of duration $$\Delta t$$, just divide by $$\Delta t$$ to have the power. If the pulse is not square, you should use the pulse profile to have the correct evolution of the power over time.
For a continuous wave (CW) laser, it is slightly more subtle. $$\left|\alpha\right>$$ describes the state in a single temporal mode and is associated with an energy, not a power. You can chose to see a CW laser of power $$P$$ as a succession of square pulses of duration $$\Delta t$$. In which case, each of these pulse has an energy $$P\Delta t$$ and would be described by a coherent state $$\left|\sqrt{\frac{P\Delta t}{\hbar \omega}} e^{i \phi(t)}\right>$$. The choice of $$\Delta t$$ is arbitrary, but the different description of the state it leads are in fact equivalent.