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arXiv:1306.1502 · Journal of Physics A Mathematical and Theoretical

Phase space flow in the Husimi representation

M Veronez, Marcus A. M. de Aguiar

Published Nov 12, 2013• 14 citations
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Abstract

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.

DOI
10.1088/1751-8113/46/48/485304

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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.
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
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.
The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i) the initial Husimi distribution evaluated along phase-space trajectories and (ii) the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.
The Hamiltonian flow of a classical, time-independent, conservative system is incompressible, it is Liouvillian. The analog of Hamilton's equations of motion for a quantum-mechanical system is the quantum-Liouville equation. It is shown that its associated quantum flow in phase space, Wigner flow, is not incompressible. It gives rise to a quantum analog of classical Hamiltonian vector fields: the Wigner phase space velocity field~w\bm ww, the divergence of which can be unbounded. The loci of such unbounded divergence form lines in phase space which coincide with the lines of zero of the Wigner function. Along these lines exist characteristic pinch points which coincide with stagnation points of the Wigner flow.
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.
The localization transition in the Hermitian Aubry-André model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-André case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.