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理论物理通讯：英文版

Topological Effect of Reduced Quantum Trajectory in Coherence and Decoherence

史旭光, 段一士, 秦志杰

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The topological expression of the reduced quantum trajectory is given by using φ-mapping topologicalcurrent theory.The topological expression is used in study of the coherence and the decoherence.We find the expressionof vorticity for decoherence has a similar form as that for coherence.A topological reason leading to the decoherenceis given and a new parameter is defined to indicate the coherence degree.The parameter is different from the dampingfactor because it relates to the topological structure of the reduced quantum trajectory.

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The topological expression of the reduced quantum trajectory is given by using φ-mapping topologicalcurrent theory.The topological expression is used in study of the coherence and the decoherence.We find the expressionof vorticity for decoherence has a similar form as that for coherence.A topological reason leading to the decoherenceis given and a new parameter is defined to indicate the coherence degree.The parameter is different from the dampingfactor because it relates to the topological structure of the reduced quantum trajectory.
The topological expression of the reduced quantum trajectory is given by using ϕ-mapping topological current theory. The topological expression is used in study of the coherence and the decoherence. We find the expression of vorticity for decoherence has a similar form as that for coherence. A topological reason leading to the decoherence is given and a new parameter is defined to indicate the coherence degree. The parameter is different from the damping factor because it relates to the topological structure of the reduced quantum trajectory.
We have studied the topological expression of the reduced quantum trajectory using ϕ-mapping topological current theory and reduced density matrix theory. Moreover, we have proposed a new topological expression called the reduced quantum topological trajectory. Ignoring the coefficients of the winding number, we have found that the expression for the reduced quantum topological trajectory is the same for coherence as it is for decoherence. Under certain conditions, the vorticity of the reduced quantum topological trajectory is a topological invariant.
The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.
Through the application of ϕ-mapping topological theory, the properties of vortices in quantum Röntgen effect are thoroughly studied. The explicit expression of the vorticity is obtained, in which the δ function indicates that the vortices can only stem from the zero points of ϕ, and the magnetic flux of the consequent monopoles is quantized in terms of the Hopf indices and Brouwer degrees. The evolution of vortex lines is discussed. The reduced dynamic equation and a conserved dynamic quantity on stable vortex lines are obtained.
Using the phi-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the delta function indicates that the vortices can only be generated from the zero points of Phi and are quantized in terms of the Hopf indices and Brouwer degrees. The phi-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.
The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.