https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

The ability to generate particles from the quantum vacuum is one of the most profound consequences of Heisenberg's uncertainty principle. Although the significance of vacuum fluctuations can be seen throughout physics, the experimental realization of vacuum amplification effects has until now been limited to a few cases. Superconducting circuit devices, driven by the goal to achieve a viable quantum computer, have been used in the experimental demonstration of the dynamical Casimir effect, and may soon be able to realize the elusive verification of analog Hawking radiation. This Colloquium article describes several mechanisms for generating photons from the quantum vacuum and emphasizes their connection to the well-known parametric amplifier from quantum optics. Discussed in detail is the possible realization of each mechanism, or its analog, in superconducting circuit systems. The ability to selectively engineer these circuit devices highlights the relationship between the various amplification mechanisms.

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Colloquium: Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits

Nonadiabatic mixed quantum–classical (NA-MQC) dynamics methods form a class of computational theoretical approaches in quantum chemistry tailored to investigate the time evolution of nonadiabatic phenomena in molecules and supramolecular assemblies. NA-MQC is characterized by a partition of the molecular system into two subsystems: one to be treated quantum mechanically (usually but not restricted to electrons) and another to be dealt with classically (nuclei). The two subsystems are connected through nonadiabatic couplings terms to enforce self-consistency. A local approximation underlies the classical subsystem, implying that direct dynamics can be simulated, without needing precomputed potential energy surfaces. The NA-MQC split allows reducing computational costs, enabling the treatment of realistic molecular systems in diverse fields. Starting from the three most well-established methods—mean-field Ehrenfest, trajectory surface hopping, and multiple spawning—this review focuses on the NA-MQC dynamics...

https://hal-amu.archives-ouvertes.fr/hal-01965458/document

Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics.

We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set---i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone---the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states---e.g., photon-added, photon-subtracted, cubic-phase, and cat states---and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to subuniversal and universal quantum information processing over continuous variables.

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Resource theory of quantum non-Gaussianity and Wigner negativity

The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

Recent advances in Wigner function approaches

Wigner functions and Weyl transforms of operators offer a formulation of quantum mechanics that is equivalent to the standard approach given by the Schrodinger equation. We give a short introduction and emphasize features that give insight into the nature of quantum mechanics and its relation to classical physics. A careful discussion of the classical limit and its difficulties is also given. The discussion is self-contained and includes complete derivations of the results presented.

Wigner functions and Weyl transforms for pedestrians

The pumping of a swing from a standing position is modeled as a rigid object forced to rotate back and forth at the lower ends of supporting ropes. This model after some approximations leads to a harmonic oscillator with driving and parametric terms. It is then argued that in the regime of the common playground swing the driving terms dominate and the pumping of a swing in the standing position is best characterized as a driven oscillator. Examination of the relative phase of the swinger and the swing also supports this conclusion. This model is compared with earlier work which claimed that the swing pumped by a standing swinger is characterized as a parametric oscillator. Simple demonstrations of both mechanisms are described. A comparison of pumping while standing and seated is also made.

The pumping of a swing from the standing position

The pumping of a playground swing from the seated position is modeled using a dumbbell consisting of three masses. The equation of motion of the system when the dumbbell is forced to rock back and forth is obtained. When reasonable approximations for amplitudes are made, the equation of a linear harmonic oscillator with a driving and three parametric terms is obtained. Based on an analysis of this equation using numerical values of the parameters, it is concluded that the pumped swing is a driven harmonic oscillator for small‐to‐moderate amplitudes. Growth rates and relative phase of pumping and swing motion are determined and discussed. The source of the driving term is traced to the conservation of total angular momentum (motion of the center of mass and motion about the center of mass) about the point of support of the swing.

The pumping of a swing from the seated position

A relativistic electron beam propagating along a constant magnetic field has four normal modes: the slow and fast space charge and the slow and fast cyclotron modes. When propagating through a dielectric where the beam velocity is greater than the velocity of light in the dielectric, the slow modes may become unstable. We call these the space‐charge‐Cerenkov and cyclotron‐Cerenkov instabilities. In this paper we use the cold fluid model in the small signal limit to obtain a general dispersion relation for propagation at an arbitrary direction with respect to the magnetic guide field. This describes both of the above instabilities as special cases. This result is then generalized to include the effects of a thermal spread. Implications for possible application for generation or amplification of microwaves, where the beam passes near the dielectric, are also considered. We conclude that the cyclotron‐Cerenkov instability is a strong candidate for the short wavelength microwave regime.

Space-charge-Cerenkov and cyclotron-Cerenkov instabilities in an electron-beam dielectric system

We describe a nonlinear mechanical system that is easy to construct and demonstrates most of the nonlinear effects associated with mechanical systems. The equation of motion for the system is easily derived through a geometrical argument and is found to be Duffing’s equation. The relative strengths of the linear and nonlinear terms can be easily varied and it is possible, in principle, to make the linear term vanish completely. The system is also considered in a driven form. Periodic motions of the system are analyzed theoretically and the results are compared with experiment. Nonperiodic motions are also considered.

William B. Case

Papers

Wigner functions and Weyl transforms for pedestrians

The pumping of a swing from the standing position

The pumping of a swing from the seated position

Space-charge-Cerenkov and cyclotron-Cerenkov instabilities in an electron-beam dielectric system

Nonlinear effects in a simple mechanical system