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Topological structures in the Husimi flow

Matheus Veronez, Marcus A. M. de Aguiar

Published Jan 1, 2016• 4 citations
OpenAlex

Abstract

Abstract: We study the topological properties of the quantum phase space current in the Husimi representation, focusing on the dynamical differences, induced by these properties, between the quantum and the classical flows. We show that the zeros of the Husimi function are stagnation points of the current and have a nonzero topological charge. Due to overall charge conservation, new stagnation points with opposite charge appear in pairs in the Husimi current and they have important roles in dynamical processes. As an example we show the topological effect of the zeros in the transmission rate of a particle through a potential barrier

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From Quantum Chaos to Irreversibility

Mikito Toda

New ideas are presented which relate dynamical features of quantum systems corresponding to chaos (so called “quantum chaos”) with the origin of irreversibility. They are based on studying topological defects of phase in the Husimi representation. Dynamical process leading to random phase is found in “quantum chaos”. The role of this for the origin of irreversibility is discussed from the following two points: (i) phase retrieval problem; (ii) entanglement of phase defects.

Progress of Theoretical Physics Supplement•Jan 1, 1994•1 citation
Abstract: We study the topological properties of the quantum phase space current in the Husimi representation, focusing on the dynamical differences, induced by these properties, between the quantum and the classical flows. We show that the zeros of the Husimi function are stagnation points of the current and have a nonzero topological charge. Due to overall charge conservation, new stagnation points with opposite charge appear in pairs in the Husimi current and they have important roles in dynamical processes. As an example we show the topological effect of the zeros in the transmission rate of a particle through a potential barrier
We derive a continuity equation for the Husimi function evolving under a general non-Hermitian Hamiltonian and identify the phase space flow associated with it. For the case of unitary evolution we obtain explicit formulas for the quantum flow, which can be written as a classical term plus quantum corrections. The quantum terms can be expanded in powers of Planck's constant providing a series of semiclassical corrections to the classical flow. We test the exact and semiclassical formulas for a particle in a double well potential and find numerical evidence that the zeros of the Husimi function are always saddle points of the flow. Merging or splitting of stagnation points, reported for the Wigner flow, was not observed.
We compare quantum phase space distributions of individual quantum states of the diamagnetic hydrogen atom obtained by means of Wigner functions with those given by Husimi functions. The comparison is carried out at effective h(cross) approximately 0.035 and at a fixed scaled energy ( epsilon =-0.316) which corresponds to an especially interesting mixed classical phase-space structure. The object of the comparison is to establish which of the two distributions best correlates with the classical Poincare surface of section for a representative set of single states. As expected, the states investigated display the strongest positive intensity on a single local invariant phase-space structure (such as a scar or torus) and at this level there is little difference between the Husimi and the Wigner function. However the weak structures (fringes of the Wigner function or zeros of the Husimi function) are shown here to be radically different for the Wigner relative to the Husimi representation. In both cases the weak structures permeate all of phase space. In addition they show different character for integrable and chaotic dynamics and so reflect the global structure of phase-space to a much greater extent than the localized strong structure. The Wigner functions of selected states are shown to have additional dynamically significant structures which are not apparent in the Husimi functions. These include negative intensity features seen in the Wigner-but not the Husimi-functions which are very well correlated with structures of the classical Poincare surfaces of section. Despite its complex oscillatory nature the Wigner function delineates better classical features, for example showing the outline of islands of stability avoided by states supported by chaotic regions with greater sharpness than the Husimi function.
Phase-space representations reveal the geometry of quantum mechanics and distinguish quantum features from classical ones. However, the positivity of a density operator is among the properties of quantum states that phase-space representations capture least directly: deciding whether a phase-space function describes a physical quantum state has so far required global conditions coupling distant points in phase space. Here we show that these conditions can in fact be made local, by considering the Husimi representation in phase space. We establish a complete set of conditions which captures when a phase-space function fff is the Husimi function of a quantum state, with each condition involving only the derivatives of the function fff evaluated at a single phase-space point. We further show that each condition has an equivalent algebraic characterization, which takes the form of the nonnegativity of a determinant of the underlying operator in a displaced Fock basis. As a result, these conditions obey a recurrence relation organizing them into a differential hierarchy, which terminates based on the rank of the operator. Our results thus show that operator rank and positivity are local properties in phase space, reinforcing the role of phase-space representations as a geometric description of quantum mechanics.
Abstract We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman–von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing (Skodje et al 1989 Phys. Rev. A 40 2894). We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical–quantum theories are discussed.