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arXiv:1411.3511 · The European Physical Journal Plus

Wigner’s quantum phase-space current in weakly-anharmonic weakly-excited two-state systems

Dimitris Kakofengitis, Ole Steuernagel

Published Sep 1, 2017• 36 citations
arXivHTMLPDF

Abstract

There are no phase-space trajectories for anharmonic quantum systems, but Wigner’s phase-space representation of quantum mechanics features Wigner current J . This current reveals fine details of quantum dynamics —finer than is ordinarily thought accessible according to quantum folklore invoking Heisenberg’s uncertainty principle. Here, we focus on the simplest, most intuitive, and analytically accessible aspects of J. We investigate features of J for bound states of time-reversible, weakly-anharmonic one-dimensional quantum-mechanical systems which are weakly-excited. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd potentials. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline patterns can be characterised by J’s discrete stagnation points, how these arise and how a quantum system’s dynamics is constrained by the stagnation points’ topological charge conservation. We additionally show that quantum dynamics in phase space, in the case of vanishing Planck constant ℏ or vanishing anharmonicity, does not pointwise converge to classical dynamics.

DOI
10.1140/epjp/i2017-11634-2

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There are no phase-space trajectories for anharmonic quantum systems, but Wigner’s phase-space representation of quantum mechanics features Wigner current J . This current reveals fine details of quantum dynamics —finer than is ordinarily thought accessible according to quantum folklore invoking Heisenberg’s uncertainty principle. Here, we focus on the simplest, most intuitive, and analytically accessible aspects of J. We investigate features of J for bound states of time-reversible, weakly-anharmonic one-dimensional quantum-mechanical systems which are weakly-excited. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd potentials. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline patterns can be characterised by J’s discrete stagnation points, how these arise and how a quantum system’s dynamics is constrained by the stagnation points’ topological charge conservation. We additionally show that quantum dynamics in phase space, in the case of vanishing Planck constant ℏ or vanishing anharmonicity, does not pointwise converge to classical dynamics.
The behaviour of classical systems is characterised by their phase portraits; the collections of their trajectories. In quantum mechanics phase portraits are still considered impossible to compute due to the complexity of quantum trajectories arising from the introduction of the quantum correction terms. Instead, in this thesis, we identify the Wigner current (the rate of flow per unit area of the Wigner distribution), as the quantum analogue of the classical phase-space current, and through Wigner current's fieldline portraits we show that it reveals hidden features of quantum dynamics and extra complexity. In our analysis, we focus on the simplest, most intuitive, and analytically accessible aspects of the Wigner current. We investigate its features for weakly-anharmonic weakly-excited bound-states of time-reversible one-dimensional quantum-mechanical systems. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline portraits can be characterised by the Wigner current's discrete stagnation points, how these arise and how a quantum system's dynamics is constrained by the stagnation points' topological charge conservation. We additionally demonstrate the conceptual power of the Wigner current by addressing some confusion found in the literature. We also stress the usefulness of the integral form of Wigner's representation as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold starproducts. The associated integral form of the Wigner current is used here in an elementary proof which shows that only systems up to quadratic in their potential fulfil Liouville's theorem of volume preservation in quantum mechanics.
Phase-space features of the Wigner flow for an anharmonic quantum system driven by the harmonic oscillator potential modified by the addition of an inverse square (one-dimensional Coulomb-like) contribution are analytically described in terms of Wigner functions and Wigner currents. Reporting about three correlated continuity equations which quantify the flux of quantum information in the phase space, the nonclassicality profile of such an anharmonic system can be consistently obtained in terms of the fluxes of probability, purity, and von Neumann--like entropy. Considering that quantum fluctuations can be identified from distortions over the classical regime, they can be quantified through the above-mentioned information fluxes whenever some classically bounded volume of the phase space is selected. Our results suggest that the Wigner flow approach works as a probe of quantumness and classicality for a large set of anharmonic quantum systems driven by quantum wells.
Phase space is the unity of position and momentum configuration space. It allows for an effective description of dynamical systems and is particularly useful when it comes to studying chaos theory and statistical mechanics. After the advent of quantum physics early in the 20th century, E. Wigner [91], J. E. Moyal [62] and H. J. Groenewold [31] introduce a quantum theory in phase space. Despite the apparent added complexity of the mathematics involved in this new framework, the underlying classical and quantum equations show many similarities. The probability distribution in classical physics becomes the Wigner distribution, a probability distribution usually featuring negative values. In 2013, O. Steuernagel and D. Kakofengitis, inspired by the work of H. Bauke [7] and E. Wigner [91], identified the quantum analogue of the classical phase space flow: the Wigner current J [83]. This Wigner current allows the visualisation of quantum dynamics through a quantum fluid dynamics perspective in phase space. This thesis is written by collection of five articles. They are prefaced by an introduction into the basics of quantum phase space theory and its link with both classical phase space dynamics and the standard Schrödinger approach, followed by the articles published during this PhD. Article 1 shows the importance of the integral form of the Wigner current. We use it to derive the Ehrenfest’s theorem, as well as to refute some propositions made within the community. Article 2 shows that, using the Wigner current, an Eulerian and Lagrangian point of view do not always give the same results for the quantum case. We demonstrate that the negativities of the Wigner distribution, sign of quantumness of the system, are created by the Wigner velocity field singularities. The Wigner velocity field is the quantum analogue of the classical phase space velocity field. In Article 3, we see that even though Wigner distributions of quantum systems feature spotty structures which saturate on scales ɑZ [97], the construction of a superoscillating Wigner distribution allows one to generate much smaller structures, of the order of ɑZ /α with α a positive constant potentially very large. In Article 4, we introduce the concept of quantum shear suppression in phase space. The Wigner current features an effective quantum “viscosity”, suppressing classical dynamics fine details. This viscosity is the mechanism by which the Zurek scale is enforced dynamically onto the state in phase space. In Article 5, we apply the previous ideas to Kerr-type oscillators. Its Wigner current is derived, and using it we show that its values are conserved on a ring during the time evolution of the Kerr oscillator. The shear suppression is also studied.
We experimentally reconstruct Wigner's current of quantum phase-space dynamics. We reveal the ``push-and-pull'' associated with damping and diffusion due to the coupling of a squeezed vacuum state to its environment. In contrast to classical dynamics, where (at zero temperature) dissipation only ``pulls'' the system toward the origin of phase space, we also observe an outward ``push'' because our system has to obey Heisenberg's uncertainty relations. With squeezed vacuum states generated by an optical parametric oscillator at variable pumping levels, we identify the pure squeezing dynamics and its central stagnation point with a topological charge of ``$\ensuremath{-}1$''. This work demonstrates high resolving power and establishes an experimental paradigm for measuring the quantumness and nonclassicality of the dynamics of open quantum systems.
The behaviour of classical mechanical systems is characterised by their phase portraits, the collections of their trajectories. Heisenberg's uncertainty principle precludes the existence of sharply defined trajectories, which is why traditionally only the time evolution of wave functions is studied in quantum dynamics. These studies are quite insensitive to the underlying structure of quantum phase space dynamics. We identify the flow that is the quantum analog of classical particle flow along phase portrait lines. It reveals hidden features of quantum dynamics and extra complexity. Being constrained by conserved flow winding numbers, it also reveals fundamental topological order in quantum dynamics that has so far gone unnoticed.
The creation of quantum coherences requires a system to be anharmonic. The simplest such continuous one-dimensional quantum system is the Kerr oscillator. It has a number of interesting symmetries we derive. Its quantum dynamics is best studied in phase space, using Wigner's distribution WWW and the associated Wigner phase space current J\mathbf{J}J. Expressions for the continuity equation governing its time evolution are derived in terms of J\mathbf{J}J and it is shown that J\mathbf{J}J for Kerr oscillators follows circles in phase space. Using J\mathbf{J}J we also show that the evolution's classical shear in phase space is quantum suppressed by an effective ``viscosity.'' Quantifying this shear suppression provides measures to contrast classical with quantum evolution and allows us to identify special quantum states.