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### The Coherent State

A coherent state is the most important field state because it mimics a classical field. As the average number of photons in the field increases, the fields become more and more classical, because the uncertainties become negligible with respect to the amplitude of the electric and magnetic fields. Because of this result, photon statistics are discussed in terms of how they vary from that of a coherent state.

A coherent state was first analyzed by Schrödinger. It was a theoretical construct discovered inside the analysis of the quantum harmonic oscillator. A coherent state is a superposition of states which is an eigenfunction of the destruction operator

where is the eigenvalue associated with this eigenfunction.[15]

The density matrix which will model a coherent state must be set according to the equation

The first point to notice about this equation is that it sets every entry in the density matrix, not just the diagonal. Let us examine the diagonal of the density matrix

The coherent state obeys a common form of statistics called Poisson or small number statistics.[16] This identification allows one to state that is the average number of photons in the system where is the coherent state's eigenvalue for the destruction operator. In general, is a complex number [17]

However, can be defined as a real number. This definition sets the initial momentum of the system equal to zero and maximizes the position, i.e., an initial phase of zero has been set in phase space.

Another hallmark of Poisson statistics is that the variance of the photon number is equal to the average photon number. The ratio is often used in analyzing a field's photon statistics. If the ratio is one, then the statistics are Poissonian. If the ratio is greater than one, then the statistics are called super--Poissonian. The statistics are called sub--Poissonian if the ratio is less than one. This classification is common. The central properties of a coherent state are listed in the following table[18]

Next: The Squeezed Coherent Up: Radiation Sources Previous: Minimum Uncertainty States

Andy Antonelli
Wed May 17 14:34:24 EDT 1995