Application Of Algebraic Dynamical Method In Quatum Mechanics And Quantum Optics

Man-made quantum systems are in fact microscopic particles controlled by the man-made potentials or boundary conditions. With the development of the science1 and technology, we have entered into a new era to control a single atom, electron, and other particles. The man-made quantum system is one of the most attractive systems, which can be divided into two kinds: autonomous and nonautonomous ones according to the time-independence or time-dependence of the potentials and boundary conditions. Most of the man-made quantum systems are nonautonomous, so that our task is how to systemically solve such systems in order to realize the control of the particles and to make them serve us. Algebraic dynamical method is an effective method to deal with the dynamical evolution of such systems by making use of its algebraic structure. The method has several key steps as follows: firstly, we rewrite the Hamiltonian into a linear combination of the generators of some algebra; secondly, we use a gauge transformation to change the Hamiltonian into a function of the Cartan operators of the dynamical Lie algebra; finally, the exact solution to the system is obtained by using the inverse transformation.Most of the quantum systems in practice are affected bytheir environments, which make the systems dissipative. In past, there have been two approaches to deal with such systems. One is based on solving the Schrodinger equation by adding the Hamiltonian a phenomenological imaginary potential to simulate the dissipative effects. This method is not from the first principle. A better approach is based on the reduced density matrix, which eliminate the infinite number of the freedoms of the dissipative environment. However, this second approach is limited to some specific representations as it solves the equation of the reduced density matrix. Furthermore, it usually deals with the problem in the autonomous case, i.e. the Hamiltonian or Liouville superoperator are time-independent. The key problem is how to solve the dissipative nonautonomous systems in an arbitrary representation. To solve the problem, one can resort to the algebraic dynamical method which has been generalized from quantum mechanical systems to quantum statistical systems. The generalized method contains the following steps: The first step based on the sandwich structure where the destruction (and creation) operators of the system stand to the left and right sides of the density operator , is to construct the right and left algebras together with their composite algebras, then the Liouville super-operator (dissipation operator) in the master equation can be written as a linear combination of the compositealgebras. Therefore, the master equation becomes a Schrodinger-like equation. Subsequently, we can solve the system according to the algebraic dynamical method in the quantum mechanics.The two-mode optical system and the dissipative harmonic oscillator driven by an external field studied in this article are very important in the quantum mechanics and quantum optics. The two-mode optical system can be used to discuss the interaction-free measurement, reversible decoherence of the Schrodinger cat state, and quantum information transfer between quantum cavities and so on. The dissipative harmonic oscillator driven by an external field describes energy loss in a cavity pumped by an external field or dissipation by phonons in a solid. In the present article we apply the algebraic dynamical method in quantum mechanics and quantum statistical mechanics to obtain the exact solutions of such systems, and discuss the related physical phenomena such as quantum geometrical phase, decoherence, photon transfer between cavities, and so on. Meanwhile, we compare this method with those used by other authors and show the advantages of the algebraic dynamical method. For example, it can deal with the systems with time-dependent boundaries and environments, and it can yield analytical solutions which directly display relevant physical meaning such as conservative quantum number, and d...