Two-State Systems: The Atomic Operators

In the digital world, the concepts of on and off or high and low or 0 and 1 are common classical two-state systems. Quantum systems can be similarly configured, as we saw in Chapter 9 with the demonstration of Rabi oscillations. Two-state or few-state systems are so important that a powerful algebra has been developed to study and explore these systems. A similar algebra emerged from the algebra developed for spin ½ particles. While Chapter 10 discussed the spinors and spin matrices and the corresponding Pauli matrices, in this chapter the corresponding commutators are determined for the various atomic operators first introduced in Chapter 15. We then move to the Heisenberg picture including the operators for the vacuum field. The Heisenberg equations of motion are derived following the rules in Chapter 8 when a classical electromagnetic field is present and then in the presence of the quantum vacuum to include the effects of decay. This provides the first means of handling the return of an excited population back to the ground state which is very challenging to deal with in the amplitude picture. This chapter is enormously important because it sets the stage for much more advanced studies in advanced texts that determine the impact of fluctuations of the field and correlations measured from single photon emitters.