This problem will be analyzed with the density matrix. The equations of motion for the density matrix can be constructed with eq:T10 and eq:j3. The derivation is quite algebraic but is more or less straightforward. The key to finding the equations of motion lies in understanding the coupled system's Hilbert space. The wavefunction will be assumed to be
where |a> is the atom's eigenket, and |n> is the field's eigenket. The identity operator discussed in chapter three is now a double sum over the atomic and field wavefunctions. Given this definition the derivation of the density matrix can be performed:
where the function returns one if x is the upper atomic state and negative one if x is the lower atomic state.
eq:j4 completely models a single, undamped mode of radiation interacting with an atom quantum mechanically. A simulation of this model is currently beyond the computational resources of a personal computer for a significant number of field frequency states, say twenty, because the entire density matrix must be solved. The special case of a two--state atom interacting with a two--state field will be discussed later in this chapter. Therefore, an approximation must be made.
eq:j5 contains four operators. Substitute and and their conjugates into eq:j5
Two of the terms in eq:j6 oscillate with frequency , while the other two have a frequency of . If the field mode's frequency is very close to the atom's transition frequency, then the slowly oscillating terms will be far more important than the fast terms. If these fast terms are dropped, one obtains
as the new interaction potential. The complete hamiltonian is now:
eq:j7 is called the Jaynes--Cummings hamiltonian.
The approximation made in the Jaynes--Cummings model should be familiar. It is the Rotating Wave Approximation in another guise. From previous work, we know that this approximation is valid only when the field is close to resonance and when the field's intensity is low. The first assumption is easily satisfied, as is the second, because the types of fields for which a fully quantum mechanical treatment is required are very weak, i.e., fields who would miss a single or a few quanta of energy. Therefore, this approximation is quite good.
The equations of motion for this hamiltonian can be constructed from eq:j4 by simply dropping all the terms which both raise and lower the atomic and field variables:
eq:j8 is much friendlier--looking than eq:j4, and for good reason. Let us find what the diagonal terms of the density matrix depend upon
The a index can take on only two values: u and d, where d+1=u and u+1 and d-1 are undefined. Therefore
An interesting simplification occurs if n in eq:j9 is replaced by n+1
Rewriting eq:j10 and using the fact that the derivative is a linear operator allows one to state that
The sum of these two diagonal terms must be a constant which can be shown to equal one. This sum is normalized, and both terms in the sum depend on the same off--diagonal term, and its conjugate. Using eq:j8, one can show that this term depends only on itself and these two terms
The equations of motion for the Jaynes--Cummings model can thus be summarized as:
where is the detuning discussed in chapter two. There is a set of these equations for every value of n. Each set of equations is referred to as a manifold in the literature. In terms of real and imaginary parts, these three equations translate into four equations of motion for every manifold. Let , and remember that, since the density matrix is hermitian, . Therefore
Each manifold is independent. The equations which describe a manifold's time evolution are like those for the semiclassical model. The frequency of the classical wave for each one is where n is the field's eigenstate number, i.e., n starts at zero.
It is important to remember that this density matrix describes how the atom and field interact. The matrix is initialized by taking the outer product of the initial field and atom state. In order to later measure the state of either the atom or field, the interaction matrix must be collapsed either with respect to its atomic or field variables according to
where P(u) and P(d) are the probability of being in the upper or lower state, and P(n) is the probability of a given field state.