Q-Representation of Number States

Quasi-probability densities represent the density operator of optical fields. Here the special quasi-probability densities are plotted as functions of and for the first eleven photon numbers . The complex variable is an eigenvalue of the non-Hermitian annihilation operator . The number state of the field is an energy eigenstate of the same field also. This means that in these plots the optical fields can be characterized by exactly photons.

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As usual in quantum optics, the density operators of light fields can be represented by normalized real-valued functions. There are three types of functions: , and are the P-representation (Glauber-Sudarshan representation), the Q-representation and the Wigner-Weyl distribution, respectively. For example, the diagonal elements of the density operator define the Q-representation: is a real, well-behaved, non-negative definite and bounded function of the complex variable ; . The set of states forms the basis of coherent states, the eigenstates of the annihilation operator . Note that the quasi-probability densities are not genuine probability densities, but they are suitable to calculate expectation values (mean values) of ordered operator products. In our case, expectation values of antinormal ordered operator products can be calculated with the help of the Q-representation.
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