Why does stimulated emission not contribute to linewidth?

Question

The rough quantum mechanical explanation for linewidth is that the lifetime $\tau$ of an excited level is associated with an uncertainty $\Delta E$ in its energy satisfying $$\Delta E\tau=\hbar$$ and so there is a linewidth $$\Delta \omega=\frac{\Delta E}{\hbar}=\frac{1}{\tau}.$$ However, under stimulated emission, the rate of depopulation of the upper level increases and so its lifetime effectively decreases. Why does the linewidth not increase correspondingly?

Classically, it is obvious that stimulated emission does not contribute because it is coherent whereas spontaneous emission is incoherent. However, quantum mechanically, I can't see why the reduced lifetime would not be included in the uncertainty principle.

Answer 1

The short answer: the relevant lifetime is not the one of an individual excitation, but the first-order coherence time, the timescale at which the phase diffuses.

The longer answer is very interesting. On the quantum level, both the photons coming in and out of the cavity are described by a Lindblad process. For the gain, the jump operators are $\sqrt{R}a^\dagger$ (leading to rate $Raa^\dagger=R(n+1) $) and for the losses they are $\sqrt{\gamma}a$ (leading to rate $\gamma a^\dagger a=\gamma n $). The +1 for the gain is what is typically attributed to the spontaneous emissions. Note that all the absorptions and emissions would invoke some kind of fluctuations, but they are much smaller.

The semiclassical picture, such as introduced in Theory of the linewidth of semiconductor lasers (Henry,'82) attributes all the fluctuations to the spontaneous emissions, but this makes them unphysically large. In this picture, each spontaneous emission doesn't increase the particle number deterministically with +1, but can change it either way up to the order of $\sqrt{n}$!

I puplished a paper related to this quantum-classical correspondence in a slightly different system, but should be related to this, especially sections III and IV.

Answer 2

Emission means different things when talking about the lifetime of an excited state and about stimulated emission.

Finite lifetime of an excited state means that, if an atom in an excited state is left alone, it will eventually relax to the ground state, spontaneously emitting a photon. This results from the fact that we have an infinite number of states with equal energy:

• one excited atomic state and photon vacuum: $|e, 0\rangle$
• infinite number of states with a photon of wave vector $\mathbf{k}$ and polarization $\lambda$ and the atom in the ground state: $|g, 1_{\mathbf{k},\lambda}\rangle$.

The probability of finding the atom in the excited state is thus descreasing till it becomes negligeably small, approximately as $$ P_e(t)=e^{-t/\tau}, $$ where we call $\tau$ the lifetime. It is typically governed by the strength of the coupling between the atom and the photon field (See this answer for more detaild discussion.)

In some more complex situations the decay may be not exponential and we may want to include into the lifetime the effects related to the uncertainty of the atomic frequency (inhomogeneous broadening).

Stimulated emission is essentially formation of a coherent superposition between the excited states of atoms and a single photon mode. The effetctive strength of the coupling is proportional to the square root of the number of photons, as per well-known boson operators relations $$ a|n\rangle =\sqrt{n}|n-1\rangle, a^\dagger|n\rangle =\sqrt{n+1}|n+1\rangle. $$ The atoms and photons are never in a state with a definite number of photons or excited atoms, but rather in a state permanent Rabi oscillations (that is, till the light escapes the laser and we observe it). The most suitable model for this case is Dicke model, described as a critical phenomenon. There is no meaningful way to speak about an atom lifetime here, although one could meaningfully speak of the lifetime of the whole photon condensate as it leaks from the cavity and discussed the uncertainty of atomic frequencies due to various inhomogeneous effects (like random movement of atoms in a gas laser, random Stark shifts in a solid state, or random initial and final kinetic energies of the recombining electron and hole in a semiconductor laser.)