Optical Phase Space - Background Information

Background Information

When developing a quantum theory of light it is very common to use an electromagnetic oscillator as a model. An electromagnetic oscillator is an oscillation of the electric field due to an excitation of the electric field. Since the magnetic field is proportional to the rate of change of the electric field, this too oscillates. This gives rise to such phenomena as light and these are the systems being studied when looking at an optical phase space. These systems obey and evolve according to Maxwell's Equations.

Let u(x,t) be a vector function describing a single mode of an electromagnetic oscillator. For simplicitity, it is assumed that this electromagnetic oscillator is contained in a vacuum. One such example is given by

where u₀ is the polarization vector, k is the wave vector, w the frequency, and AB denotes the dot product between the vectors A and B. This is the equation for a plane wave and is a simple example of such an electromagnetic oscillator. The oscillators being examined could either be free waves in space or some normal mode contained in some cavity.

A single mode of the electromagnetic oscillator is isolated from the rest of the system and examined. Such an oscillator can be described by the annihilation operator as the Hamiltonian is strictly a function of the 'annihilation operator' which is in turn responsible for the time evolution of the system. This can be interpreted as the quantized amplitude with which u can be excited. It can then be shown that the electric field strength (i.e. the electromagnetic oscillator used to model the system) is given by:

(where x_i is a single component of x, position). The Hamiltonian for an electromagnetic oscillator is found by quantizing the electromagnetic field for this oscillator and the formula is given by:

(where w is the frequency of the spatial-temporal mode). The annihilation operator is the bosonic annihilation operator and so it obeys the commutation relation given by:

(which is shown in the article Creation and annihilation operators). The eigenstates of the annihilation operator are called coherent states:

It is important to note that the annihilation operator is not Hermitian, therefore its eigenvalues can be complex, an important consequence.

Finally, the photon number is given by the operator which gives the number of photons in the given spatial-temporal mode, u.