Excitation of an atom, by either absorption or collision, leaves it in an excited state. From there it jumps back to a lower energy level by emitting a photon. How long does this process take? There are actually two questions here. First, how long does an atom remain in an excited state before undergoing a quantum jump to a lower state? Second, how long does the transition last as the quantum jump is occurring?

Our best understanding of the quantum physics of atoms is that quantum jumps are instantaneous. The absorption or emission of a photon is an all-or-nothing event, so there is not a time when a photon is “half emitted.” The prediction that quantum jumps are instantaneous has troubled many physicists, but careful experimental tests have never revealed any evidence that the jump itself takes a measurable amount of time.

The time spent in the excited state, waiting to make a quantum jump, is another story. FIGURE 41.28 shows experimental data for the length of time that doubly charged xenon ions ${\text{Xe}}^{++}$ spend in a certain excited state. In this experiment, a pulse of electrons was used to excite the atoms to the excited state. The number of excited-state atoms was then monitored by detecting the photons emitted—one by one!—as the excited atoms jumped back to the ground state. The number of photons emitted at time t is directly proportional to the number of excited-state atoms present at time $t$. As the figure shows, the number of atoms in the excited state decreases exponentially with time, and virtually all have decayed within 25 ms of their creation.

FIGURE 41.28 Experimental data for the photon emission rate from an excited state in $\text{Xe}{}^{++}$.

Figure 41.28 has two important implications. First, atoms spend time in the excited state before undergoing a quantum jump back to a lower state. Second, the length of time spent in the excited state is not a constant value but varies from atom to atom. If every excited xenon ion lived for 5 ms in the excited state, then we would detect no photons for 5 ms, a big burst right at 5 ms as they all decay, then no photons after that. Instead, the data tell us that there is a range of times spent in the excited state. Some undergo a quantum jump and emit a photon after 1 ms, others after 5 ms or 10 ms, and a few wait as long as 20 or 25 ms.

Consider an experiment in which $N_0$ excited atoms are created at time $t=0$. As the curve in Figure 41.28 shows, the number of excited atoms remaining at time t is well described by the exponential function

where $\tau$ is the point in time at which $e^{-1}=0.368=36.8\text{\%}$ of the original atoms remain in the excited state. Thus 63.2% of the atoms, nearly two-thirds, have emitted a photon and jumped to the lower state by time $t=\tau$. The interval of time $\tau$ is called the lifetime of the excited state. From Figure 41.28 we can deduce that the lifetime of this state in ${\text{Xe}}^{++}$ is $\approx 4\text{ms}$ because that is the point in time at which the curve has decayed to 36.8% of its initial value.

This lifetime in ${\text{Xe}}^{++}$ is abnormally long, which is why the state was studied. More typical excited-state lifetimes are a few nanoseconds. TABLE 41.3 gives some measured values of excited-state lifetimes. Whatever the value of $\tau$, the number of excited-state atoms decreases exponentially. Why is this?