The natural linewidth is, remarkably, related to the lifetime of the excited state. To understand the lifetime of an atomic state, first imagine that all of the atoms from the above thought experiment are in the excited state. Next, we turn off the laser beam. As time progresses, the atoms will each emit a photon to go back to the ground state at a random time. Spectroscopists often use the word “decay” when talking about an atom emitting a photon to go back to the ground state. This is a random process, so some atoms decay back to the ground state quickly while others take their time. Statistically, each atom has a probability of decaying out of the excited state that will look like Fig. 1. At t = 0, all the atoms are in the excited state. As time marches on, atoms start to decay and the excited state fraction gets smaller. For the graph in Fig.1, about 63% of the atoms have decayed back to the ground state in 6.25 ns, and about 80% of the atoms have decayed back to the ground state in 10 ns. The mathematical form of the decay curve is:
(1)
where τ (this is the Greek letter lowercase tau) is called the lifetime of the excited state. This function is called an exponential decay. Like the natural linewidth of a transition, τ has a unique value for every transition in an atom. The lifetime of the excited state used for the decay in the graph is τ = 6.25 ns, which means that after 6.25 ns about 63% of the atoms have decayed. There is nothing special about 63%; it is just what physicists decided to define as the lifetime. Mathematically it is a nice definition because when t = τ , the fraction of atoms that have not decayed is e−1 = 0.368, or 36.8%. After two lifetimes (2×6.25 ns = 12.5 ns), about 86.5% of the atoms have decayed leaving e−2 = 0.135, or 13.5%, of the atoms in the excited state.
Amazingly, the natural linewidth and the lifetime are related! The formula relating the two quantities is:
(3.8)
Fig. 1The excited state fraction as a function of time for an excited state that has a lifetime of 6.25 ns
As you can imagine, atomic physicists and spectroscopists often use the angular frequency form of natural linewidth when thinking about lifetime.2 These two quantities are inversely related to each other. This means:
• If the lifetime is small, the natural linewidth is large. This means that excited states with a small lifetime have absorption and transmission plot spectral features that are relatively wide.
• If the lifetime is large, the natural linewidth is small. This means that excited states with a large lifetime have absorption and transmission plot spectral features that are relatively narrow.
