Analytical treatment of quantized electromagnetic field (Mathematical and physical aspects)

Abstract

In this thesis we have introduced and studied the problem of three mutually time- dependent coupled oscillators. The system consists of three different electromagnetic fields injected within a cavity with unequal frequencies. As a result of the interaction between the fields, the cavity material changes and the system acquires the time through the coupling parameters. The problem has been handled from different points of view, for example, in chapter two we have considered a particular form for the third coupling parameter to connect up the other two couplings and field frequencies. The wave function in the SchrÄodinger picture and in the coherent states are calculated besides the Green's function and the Wigner function. Also, the phenomenon of squeezing is discussed in detail, where we have used different kinds of squeezing. For example, the normal squeez- ing for different initial states, as well as the principal squeezing are considered. It has been shown that when the system is initially in even coherent states collapse and revival phenomenon can be seen.

In chapter three we have considered the same problem, however, with a restriction on the third field frequency. Contrary to the previous chapter we have assumed that the third field frequency contains beside the other two field frequencies all the coupling parameters of the system. The main concentration was on the nonclassical properties as well as on the transition amplitude of the system. Therefore we have calculated the dynamical quantities using the solution of the equations of motion in the Heisenberg picture. Also the Wigner function is calculated via the wave function in the coherent states. The phenomenon of squeezing is discussed where the normal squeezing and the amplitude squared squeezing are involved. The transition amplitude between different states are also considered and a complete section is devoted to this purpose. We have shown that the rate of exchange of the energy between the states is always sensitive to the variations in the initial mean photon numbers in addition to the field frequencies and the coupling parameters.

In the fourth chapter the problem has been reformulated and the rotating-wave approximation is applied by discarding the nonconservative terms from the Hamiltonian of the model. The problem is converted into parametric amplifier and frequency converter models coupled to each other through a time-dependent coupling parameter. In this case the time-dependence is apparent and our consideration is extended to discuss the effect of the pump phase on the whole system. Due to the difficulty on obtaining the wave function in both the number states and the coherent states, our starting point was the solution of the equations of motion in the Heisenberg picture. The nonclassical properties have been examined by discussing the squeezing phenomenon and the photon bunching and antibunching through the correlation function. In addition we have also considered the quasiprobability distribution functions Wigner and Q-function as well as the phase distribution. For a large value of the coupling parameter which is responsible for the parametric amplifier, the system gets very sensitive to any variation in the pump phase. In the chapter five we have turned our attention to consider the problem of the time- dependent frequency converter model. The Hamiltonian model we have considered in this case consists of an arbitrary time-dependent coupling and pump phase in addition to an external random force. The wave function is calculated in both coherent and number state representations. Discussions related to the nonclassical properties are given. Also we have managed to get the connection between the integrability condition and the random force to reduce the noise in the system.