Spontaneous Emission - Theory

Theory

Spontaneous transitions was not explainable within the framework of the old quantum theory, that is a theory in which the atomic levels are quantized, but the electromagnetic field is not. In fact, using the machinery of the usually called "first-quantized" quantum mechanics and computing the probability of spontaneous transitions from one stationary state to another, one finds that it is zero. In order to explain spontaneous transitions, quantum mechanics must be extended to a "second-quantized" theory, wherein the electromagnetic field is quantized at every point in space. Such a theory is known as a quantum field theory; the quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.

In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom (for more information, see Ref. ). As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.

Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the different directions in which the photon can be emitted. Equivalently, one might say that the phase space offered by the electromagnetic field is infinitely larger than that offered by the atom. Since one must consider probabilities that occupy all of phase space equally, the combined system of atom plus electromagnetic field must undergo a transition from electronic excitation to a photonic excitation; the atom must decay by spontaneous emission. The time the light source remains in the excited state thus depends on the light source itself as well as its environment. Imagine trying to hold a pencil upright on the end of your finger. It will stay there if your hand is perfectly stable and nothing perturbs the equilibrium. But the slightest perturbation will make the pencil fall into a more stable equilibrium position. Similarly, vacuum fluctuations cause an excited atom to fall into its ground state.

In spectroscopy one can frequently find that atoms or molecules in the excited states dissipate their energy in the absence of any external source of photons. This is not spontaneous emission, but is actually nonradiative relaxation of the atoms or molecules caused by the fluctuation of the surrounding molecules present inside the bulk.