Applications Of Continuous Entangled State Representation In Theoretical Quantum Optics

Dirac symbolic method is a "language", which must be used by everyone for studying quantum physics. In some sense, its profoundly reflect to physical nature has beyond the era, and its connotation and beauty still needs further awareness. The technique of integration within an ordered product of operators (IWOP for short) and continuous entangled state representation (originated from the idea of entanglement by EPR) are the innovative developments to Dirac symbolic theory, enrich the basic mathematical and physical theory of quantum mechanics, and not only further reveal the scientific beauty of Dirac symbolic method, but opened up some new applications of continuous variable entangled state representation in a number of physics research areas. In this work, we will introduce its applications in theoretical quantum optics.First, taking creation and annihilation operator mapping into differential operatorsin entangled state representation into account, we will be able to discuss the Fokker-Planck equation of a random process in the sense of entangled state represention, and show its Li algera structure and physical meaning clearly. On one hand, by virtue of "pair coherent state", We find that the wavefunction of a pair coherent state (or SU(1,1) coherent state) in the entangled state representation is just the eigenfunction of a type of Fokker-Planck differential operator. On the other hand, we point out that some complex variables Fokker-Planck differential equations, which includes a number of differential operators of SU(1,1) algebra, can be solved by using the operator decomposition formula.Second, the reduced denity master equation (ME) is an important method in the study of quantum optics. We can employ various methods (Superoperator method, Characteristic function, etc.) to deal with ME. Here we have proposed a new method -entangled state representation method to solve ME. Using it we can conveniently and simply convert a ME into its corresponding differential equation and from solution of which we can extract the reduced density operator. In this work, we have made a damping harmonic oscillator in a squeezed vacuum bath for an example to interpret this new method and then calculated the Wigner function of the reduced density operator to specifically analyse the system's decoherence process. Comparing to the traditional method of solving ME, our method is simple and effective.Third, there exists many transformations in quantum mechanics. By virtue of the technique of IWOP, we can construct a" bridge " between classical canonical tranformation and quantum unitary transformation, and promote a further development of Dirac's representation transformation theory. Dirac himself admired the theory of canonical transformations in quantum mechanics very much," I think that is the piece of work which has most pleased me of all the works that I've done in my lif.... The transformation theory (became) my darling." and that shows the importance of the Dirac's representation transformation theory. Coherent-entangled state is a relatively unique one which is of coherence and entanglement characteristics. Making the representation transformation to the coherent-entangled state, we have derived a new squeezing operator in a nature way and found this unitary operator can satisfy the group multiplication rule. And we have proved that this operator and the classical Lenz-Fresnel transformation are related in such a manner that the matrix element of it in the entangled state representation is just the kernel of the Lenz-Fresnel transformation and established a link between the classical Lenz-Fresnel optical transformation and its counterparts in quantum mechanics.Fourth, we can apply the entangled state representation for not only studying theoretical quantum optics, but for the analysis of various theoretical and experimental in quantum information, as well as for deeply studying the interference between two Bose-Einstein condensates. In this work, by using the two-mode entangled Wigner operator, we have made a deep analysis to the optical beam splitter entanglement rules. In order to demonstrate the interference patterns of BECs, we have proposed that the atomic coherent state (ACS) in Schwinger bosonic realization is a faithful representation for describing the steady relative phase of interference BECs. The so-called "phase state ", which has been introduced in before literature, can be replaced by ACS. That can enrich the way of dealing with problem and clarify the physics meaning.