Time Evolution Of Distribution Function Of Quantum Phase Space In Some Quantum Channels

Phase space is a word first appeared in Hamiltonian dynamics of classical physics. Coordinate frame constituted by coordinate q and momentum p spans a phase space, which vividly illustrates the Hamiltonian canonical equations. The concept of phase volume in phase space is used to describe the number of state, ensemble and thermodynamic probability in statistical mechanics, phase volume invariance in evolution is called Liouville’s theorem. However, in quantum theory, people can’t accurately measure the position and momentum of microscopic particles simultaneously according to the Heisenberg uncertainty principle, that is to say, one can’t determine a phase point. So it is naturally to think that one can define a quasi distribution function in phase space to study the quantum states of microscopic particles and their motion. In view of this, Wigner introduced the density operator p corresponding to the classical distribution function W(q,p), its marginal distributions are corresponding to the observed probabilities of the particle in coordinate space and momentum space respectively, which gives new meaning to the phase space. But the Wigner function itself is not always positive definite, so it can’t be as a probability distribution function (can only be called quasi probability distribution function). On the basis of the definition of the Wigner function Husimi introduced a new distribution function——Husimi function. Because it is always positive definite, thus it can be used as a new probability distribution function. This article will study how the Wigner function and Husimi function in various kinds of quantum channel evolute over time by using the thermo entangled state representation method and based on IWOP technique. The main contents are as follows:1.1)The new technique of quantum representation integration is briefly introduced. We discussed the technique of integration within an ordered product (IWOP) of operators and the technique of integration within the Weyl ordered product (IWWOP) of operators.2) The basic ideas and research methods of quantum open system are expounded and the Kraus operator sum reprensentation of density matrix is briefly illustrated.3) We review how to solve the master equations of density operators under a variety of quantum decoherence models by using the method of thermo entangled state representation.2. As the application of IWWOP technique in the quantum phase space theory, we discussed what kind of quantization scheme can make a classical phase space ray (expressed with δ(x-λq-σp)) keep the form of δ-function, namely δ(x-λQ-vP). We found that it is Weyl quantization scheme.3. The quantization of a mesoscopic circuit is an important subject of studying quantum computer and superconducting quantum circuit. Using a new viewpoint and a new method, we replaced the problem of the quantum dissipation of mesoscopic RLC circuit by studying the evolution of the corresponding density operator in the amplitude dissipative channel and obtained the analytic form of the density matrix of the circuit was.4. We found that no matter what kind of decoherence modes time evolution of their Wigner functions can be always ascribed to time evolution of their Wigner operators. Based on which, we used the Kraus operator sum representation of density operator to derive the time evolution law of the Wigner operator in several quantum channels such as amplitude dissipative channel, laser process, phase diffusion (or damping) model and noise channel in diffusion limit.5. Time evolution of Husimi functions can be always ascribed to time evolution of their coarse-Guassin-smoothing-Wigner-operator. Based on which, we derive the time evolution law of Husimi function in amplitude dissipative channel.6. For an harmonic oscillator with a field intensity related external source we establish the nonlinear number-phase squeezed state, in this state we find that while the number fluctuation increases, the phase fluctuation decreases correspondingly. The number-phase uncertainty relationship is exactly derived.