P-Representation of Laser Light
Quasi-probability densities are representations of the density operator ρ of optical fields. Here the special quasi-probability density of laser light near the laser threshold is plotted as a function of and . The complex variable is an eigenvalue of the non-Hermitian annihilation operator . The constants and represent optical damping and amplification (gain), respectively. The expression for the function can be derived from the equation of motion for the laser field density matrix. This so-called "master equation" considers both the interaction of active atoms, which are resonant with the single-mode laser field, and the decay of the atomic levels. This Demonstration shows the influence of the parameters and on the shape of the real-valued function . The threshold condition of the laser is .
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: the -representation (Glauber–Sudarshan representation), ; the -representation, ; and the Wigner–Weyl distribution, . For example, the -representation is a diagonal representation of the density operator in terms of coherent states . is a real-valued function of the complex variable . The coherent states are eigenstates of the annihilation operator . Note that all the quasi-probability densities are not genuine probability densities, but are suitable to calculate expectation values (mean values) of ordered operator products. In this case, expectation values of all normal ordered operator products can be calculated with the help of the -representation.
Snapshot 1: laser operation below threshold:
Snapshot 2: laser operation at threshold:
Snapshot 3: laser operation above threshold: