The time-dependent harmonic oscillator with variable mass under the action of external driving force

Abstract

In this thesis we study the time-dependent harmonic oscillator which has attracted much attention since the mid of the last century. After all this time, it is still an open problem. This is due to the fact that the problem is strongly connected with the second harmonic generation as well as with SU(1; 1) Lie algebra. There is no doubt that the second harmonic generation can be regarded as one of the fundamental problem in the field of quantum optics. Also the generators of SU(1; 1) play a role of connecting up the quantum mechanics with Lie algebra. The main concentration of the previous study restricted on a pure quantum mechanical treatment without considering the effect of the driving force in presence of some time-dependent masses. Therefore, we devote the present thesis to fill this gap and study the effect of external driving force on a certain time-dependent masses. More precisely, we introduce two different time-dependent masses law. The first is the modified mass law for the damped harmonic oscillator where the damping factor is taken to be equal to the oscillator frequency, however, it is shifted by an arbitrary factor to avoid the critical case. The second case is the pulsating law which has been introduced during the study of the interaction between atom-field within Fabry-P´ erot Cavity. We extend our study to include the constants of the motion beside the statistical properties of such system. Furthermore we have dealt with the problem from quantum optics point of view and used different initial states in our consideration. This has been built up on the derivation of the wave function and the solution of the Heisenberg equations of motion which can be regarded as the key of dealing with any quantum mechanical problem. Further, we calculated the Green’s function for both models from which the density matrix and, consequently, the energy eigenvalue as well as the entropy can be obtain. We concentrated on the non-classical properties and discuss the phenomenon of squeezing as well as the second order correlation function introduced by Roy Glauber. It has been shown that the squeezing phenomenon is pronounced in all cases. However, it is also sensitive to the variation in the involved parameters. Similar conclusion can be reported for the correlation function, where we can see super-Poissonian and sub-Poissonian beside coherence behavior. In our consideration for the constants of motion, we introduced two classes of invariants and obtained the eigenvalue and the corresponding eigenfunction by employing a new definition of the Dirac operators. In the meantime we direct our work to involve the generators of SU(1; 1) Lie algebra to discuss the nonclassical properties too.