Thus far we have derived the semiclassical equations for atom--radiation interaction in two different pictures. There is a third equivalent picture called the Bloch vector picture whose interpretation is most useful for this system. The Bloch vector was created for the problem of nuclear magnetic resonance in which a two--state spin system (up and down) interacts with a magnetic field. These problems are equivalent in many respects. The only difference in the fields is the magnitude of the frequencies. Nuclear magnetic resonance has a radio frequency field while the semiclassical model has an optical frequency field. Nonetheless, the fact that an atom is not a true two--state system produces many distinctions between these two. As more of the physics of the semiclassical model is discussed, it will become obvious that much of the terminology has been taken directly from nuclear magnetic resonance.
Richard Feynman was one of the first scientists to utilize the Bloch vector for a problem other than nuclear magnetic resonance. He generated the Bloch vector from the elements of the density matrix. These definitions are:
Each component is a real number. It is easy to see this by using our knowledge of the nature of these elements
w is merely the difference in the probability of being in the upper and lower states. This difference is often called the inversion. u and v concern the phase of the system and are directly related to the dipole moment of the atom. They are the components of the dipole moment which are, respectively, in--phase and in--quadrature with the electric field. u is the dispersive component of the dipole moment. v is the magnitude of the absorptive component of the dipole moment; however, the sign of the absorptive element is negative. Therefore, -v is the true absorptive component.
The Bloch vector has a number of interesting properties. If the density matrix is in a pure state, , then . In this situation, the Bloch vector is confined to a sphere of radius one called the Bloch sphere. The equations of motion can also be written in this frame in a compact vector equation form. Let , then the equations of motion are , where is a constant vector around which the Bloch vector is precessing.
Figure: The Bloch Vector
The exact definition of depends on the frame of reference. When we changed into , we moved into the rotating frame. The rotating frame can be clearly understood in the Bloch vector picture as transforming into a frame in which the Bloch vector processes about , which is static.