| Controlling quantum phenomena is a great challenge to mankind. In recent years quantum control has become a major topic in the fields of quantum optics and quantum information. Both photon subtraction and photon addition of opti-cal field states have been suggested for quantum control. Nevertheless, quantum control is always disturbed by decoherence. Because quantum systems are usually surrounded by thermal reservoir, dissipation is inevitable and decoherence natu-rally happens. The theory of quantum open systems is a fundamental approach to the understanding of quantum decoherence. This theory describes dynamics of a system interacting with its environment. One can construct a differential equation for time evolution of reduced density operator of the system, the master equation, by performing trace over the degree of freedom of reservoir. Recently Professor Fan Hong-yi et al. proposed a new method, which utilizes the thermal entangled state representations (TESR) to map density operators onto a state-vector in two-mode Fock space whose first mode is the system mode while the second mode is a fictitious one, and consequently the master equation appears as a Schrodinger-like time evolution equation. By virtue of TESR theory and the technique of integration within an ordered product (IWOP) of operators, one can obtain so-called infinite operator sum representation of exact solution of some master equation.Very recently, by analyzing theoretical scheme of quantum control through photon addition, Professor Fan Hong-yi et al. propose a new kind of optical field (a mixed state) whose density operator is p=λ(1-λ)l:Ll(-λ2α+α/1-λ)e-λα+α:(here::denotes normal ordering symbol, and Li is the l-th Laguerre polynomial), which is named Laguerre-polynomial-weighted photon fields (LPWCPF). Remark-ably, when l=0, L0(x)=1,ρ'λ:e-λα+α:=λeα+αln(1-λ). In this par- ticular case, p reduces to the chaotic photon field, so the chaotic photon field is a special case of LPWCPF, which is the solution to the master equation d/dtρ=-ε(a+aρ+paa+-aρa+-a+ρa), describing a diffusion channel in the case that the initial state is a number state|l><l|,and λ=1/1+εt Experimentally, LPWCPF may be implemented by letting the photon number state|l><l|pass through a diffusion channel.In addition, the evolution law of the mean photon number of LPWCPF is l+εt, so we can control photon number by adjusting the diffusion parameter ε. Due to this merits, LPWCPF plays a very key role in quantum control. How-ever, quantum control is always disturbed by decoherence due to dissipation, so exploring how the LPWCPF evolves in the amplitude damping channel is very mandatory. Experimentally, in quantum cascade control, a beam of LPWCPF light, as an output from a diffusion channel, enters into a cavity surrounded by a thermal reservoir, and then we are concerned how the LPWCPF and its pho-ton number evolves with time in the amplitude damping channel. Theoretically, by using a newly derived generating function of two-variable Hermite polynomi-als, IWOP technique and the entangled state representation theory, we obtain its evolution law, which turns out to be a new LPWCPF with a new parameter, depending on T=1-e-2Kt, where κ is decay rate. What’s more, in our discus-sions the physical difference between the diffusion and the amplitude damping is pointed out.Further, in this article we study the decoherence of the other special opti-cal field named negative binomial states (NBS) in a laser channel,whose density operator is∑n=0∞(n+s)!/N!S!γs+1(1-γ)n|n><n|. As a special state between thermal chaotic states and pure coherent states, NBS has many interesting nonclassical properties. The NBS can be produced when some photons of a beam of chaotic light are detected by a photocounter. Thus it will be interesting to know how NBS evolves in a laser channel involving both dissipation and pumping. Supposing that a beam of NBS light is prepared and stored in a cavity surrounded by a thermal reservoir and there is also a pumping for emitting photons into the cavity, then we explore how the NBS field evolves with time. By using IWOP technique and a newly derived negative-binomial theorem involving Laguerre polynomials, we ob-tain the evolution law of NBS, which is an infinite operator-sum of photon-added negative binomial state with a new negative-binomial parameter, and the mean photon number in the final state evolving with e-2(κ-g)t, where g and κ represent the cavity gain and loss respectively.Considering damping channel is a special case of laser channel, we also study how the NBS dephase in a damping channel and found that a negative binomial state evolves into a new negative binomial state only dynamically adjusting its parameter after passing through the amplitude damping channel. In addition, we also obtain the mean number of photons, relative fluctuation of number of photons, quantum mechanical second-order degrees of coherence.This thesis is organized as follows:In Chap.1, we briefly introduce some basic theories of quantum optics and the technique of integration within ordered product (IWOP) of operators, which is proposed by Prof. Fan first. We also introduce the quantum mechanical mixed state representation as well as pure state representation and prove that the pure state representation can be derived from the mixed state representation. Espe-cially we introduced the special pure state representation-entangled state repre-sentation. Based on IWOP technique we try to reacquaint fundamental quantum states in optical field from new point of view, such as Fock state, coherent state and chaotic state.In Chap.2, by virtue of IWOP technique and the theory of thermo field dynamics, we introduce a new thermo entangled state representation|η> with continuous variables to tackle the decoherence of open quantum systems. By em-ploying177) one can conveniently convert the master equation of density operator p into a c-number equation about function <η|ρ>, and then we can obtain Kraus infinite operator-sum form of solution of master equation of amplitude damping channel, diffusion channel and laser channel.In Chap.3, briefly reviewing the properties of Hermite polynomials, Laguerre polynomials and two-variable Hermite polynomials, such as definition, generating function formula, related properties and transforming relationship, we derive some new generating function formula on Hermite polynomials and binomial theorem involving Hermite polynomials as well as negative-binomial theorem involving Laguerre polynomials through operator Hermite polynomials method. Further-more, for tackling some questions in entangled state representation we also derive binomial theorem involving two-variable Hermite polynomials.In Chap.4, focusing on the decoherence of a new optical field called LPWCPF in a damping channel, which play an important role in quantum control, we discuss the properties as well as preparation of LPWCPF. By virtue of the new generating function of Hermite polynomials we derive the evolution law and photon number formula in the damping channel.In Chap.5, we also discuss the decoherence of the other optical field named NBS in a laser channel.Using the negative-binomial theorem regarding to Laguerre polynomials, we derive the evolution law as well as photon number formula of NBS passing through a laser channel. Furthermore, we also discuss how a NBS dephases in the damping channel when the laser channel degrades into an am-plitude damping channel. In this case, by virtue of IWOP technique we obtain the mean number of photons, relative fluctuation of number of photons, quantum mechanical second-order degrees of coherence.In Chap.6, we introduce some our other work on quantum theory, such as detaching two single-mode squeezing operators from the two-mode squeezing op-erator, which points out that two-mode squeezing mechanism also involves some single-mode squeezing and new3-mode bosonic operator realization of SU(2) Lie algebra studied from the point of view of squeezing in which we find the natural representation (the eigenvectors of J+or J_) of the3-mode squeezing operator e2λJz. Another work is quantum mechanical state equation for describing evolution of projects of financial investment in which exploiting the methods of studying the decoherence in quantum control, we investigate how financial assets evolve in the financial market and obtain some results in line with the real market firstly. In the last chapter, we summary the research work above and give some possible research directions of these fields in the future. |
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