How does a laser emit light in a coherent state?

Question

Lasers work by stimulated emission of atomic transitions. Stimulated emission produces two photons which, because the particle number is well-defined, projects the field into a Fock state. However, it is a known fact that lasers emit light in a coherent state. How does the field evolve from a particle-state to a superposition of particle-states? Omitting normalization:

$$ | n \rangle \rightarrow \sum_{n=0}^{\infty}\frac{\alpha^n }{\sqrt{n!}}| n \rangle $$

I guess one way of looking at it is that the field shifts according to $\Delta n \Delta \phi \geq 1$ from certain particle number to certain phase but it feels like a superficial answer to me. What I want to understand is the mechanism that allows this to happen. Is it the reflection with the mirror? Is it the imposed boundaries of the resonating cavity? Pumping method?

Answer 1

You are making an incorrect assumption in your question: There is no physical evolution from a number state (aka Fock state). This evolution happened purely inside physicists' heads as it was realized that laser light is not properly described by number states. The problem is your assumption that the particle number ever is well-defined.

Lasing action is an inherently quantum-mechanical process: A photon interacts with a two-level system in its upper state. Unlike the simplified description you seem to be using, this does not always result in two photons and the two-level system in its lower state. What really happens is that a superposition between that result and the boring one, with no interaction at all, is created. Hence you have a superposition between a light field with one and with two photons. Continue this to the (theoretical, but sensible) limit of infinitely many such interactions (with interaction strength tuned to give your desired mean photon number), and you get coherent states.

Answer 2

I'm going to stir things a bit and say that laser light is actually not a coherent state.

Because the emission events are random and independent to good approximation, this leads to a Poisson process. Consequently, the laser light will be in a classical mixture of Fock states with Poissonian number statistics (as is the number statistics for coherent states, but without a well-defined phase) . I don't think that this part is really controversial, I believe standard quantum optics books (e.g. Walls-Millburn) mention it. The common explanation to describe them with coherent states later on is spontaneous symmetry breaking: the mixed states interact weakly with an environment, and since the coherent states are pointer states, $U(1)$ phase symmetry is broken and the photon field assumes a pure coherent state. This is not so different from the onset of Bose-Einstein condensation, I believe.

There has also been an alternative claim, in the paper

"Optical coherence: A convenient fiction", Klaus Mølmer, Phys. Rev. A 55, 3195 (1997)

which to my understanding says that the symmetry breaking never really occurs, and everything we think to know about laser light having a well-defined phase is merely an illusion because of a circularity in reasoning about interference experiments.

I'm not enough of an expert to really say that I can completely agree with the latter claim, but based on the number of citations and not being aware of anyone really debunking it, it's tempting to believe that it might hold some truth.

Answer 3

It is true that the light coming from a laser is not precisely a coherent state. One can measure the photon statistics to see that it only approximates the Poisson statistics. However, the OP is not concerned with the precise modeling of the light coming form a laser. Instead, the question is how a coherent state can come about as a result of the stimulated emission that occurs in a laser. To address this aspect, I present here a simplistic view of the process. It neglects the possibility that the excited atom remains excited and not radiate.

When the laser is switch on, it starts with the creation of a population inversion. Then one of the excited atoms decays spontaneously. The photon that is produced by the spontaneous decay stimulates other atoms to decay producing more photons. However, one can also see the subsequent stimulated decays together with the initial spontaneous decay as multiple spontaneous decays together with a normalization process.

Each spontaneous decay will effectively produce a superposition $$ |\psi_{spon}\rangle = |\text{vac}\rangle\beta + |1\rangle\zeta , $$ where $|\text{vac}\rangle$ is the vacuum state, $|1\rangle$ is a single photon state with the correct cavity mode, and $\beta$ and $\zeta$ are coefficients. The reason for the vacuum state is not because the atom did not radiated, which I exclude here, but because some of the photons are produced in modes that would not survive in the long run. These I remove and replace by the vacuum state.

For $n$ such spontaneous decays, one gets $$ |\psi_{n-spon}\rangle = \mathcal{N}\left(|\text{vac}\rangle\beta + |1\rangle\zeta\right)^n = \mathcal{N}\sum_{p=0}^n \frac{n!}{p! (n-p)!} \left(|1\rangle\right)^p \frac{\zeta^p}{\beta^p}, $$ where $\mathcal{N}$ is a normalization constant. In the last expression I absorbed $\beta^n$ in $\mathcal{N}$ and removed the tensor products with the vacuum states. Note that a tensor product of $p$ single photon states becomes a $p$-photon Fock state $$\left(|1\rangle\right)^p= \sqrt{p!} |p\rangle . $$

In the limit $n\rightarrow\infty$, we have $$ \frac{n!}{(n-p)!}\rightarrow 1 . $$ So, the state becomes $$ |\psi\rangle = \mathcal{N}\sum_{p=0}^{\infty} \frac{\alpha^p}{\sqrt{p!}} |p\rangle , $$ where we defined $$ \frac{\zeta}{\beta} \equiv \alpha . $$ What remains is to compute the normalization constant of the state, which would then lead to the well-known expression for the coherent state.