arXiv subjects

Jonathan Berkheim

Publications and source records attributed to Jonathan Berkheim.

7 recordsLinked to original sources

arXiv2026

Quantum Pointwise Dimension (QUPID): An algorithm for identifying classical chaos in quantum dynamics

The Pointwise Dimension is a diagnostic that reveals the effective dimensionality of classical trajectories in a $2N$-dimensional phase space. The essence of this diagnostic is to find a scaling according to which density changes with radius $r$ around a point of interest in phase space. Particularly, regular trajectories are confined to an $N$-dimensional invariant torus, while chaotic trajectories explore a $(2N-1)$-dimensional energy surface. In classical mechanics, this trajectory density can be found by counting recurrences to within the sphere of radius $r$. In quantum mechanics, probability density is intrinsic, making the Pointwise Dimension compelling, but surprisingly this has not been studied in the quantum regime. In this work, we implement the Pointwise Dimension in the quantum regime using the Husimi representation of the quantum wavepacket. This implementation entails a long-time average of the time-dependent Husimi representation to obtain a stationary phase space density, and a radial spatial integration of this density over a sphere of radius $r$. The classical-quantum correspondence is excellent for both autonomous and non-autonomous systems, adding a new diagnostic to the study of quantum chaos. Finally, we explore the possibility, both classically and quantum mechanically, of obtaining the Pointwise Dimension from a time correlation function.

quant-ph↗

arXiv2026

The robustness of composite pulses elucidated by classical mechanics. II. The role of initial state imperfection

In nuclear magnetic resonance (NMR), Composite Pulses (CPs) are widely used to correct for pulse imperfections, e.g., RF field inhomogeneity and resonance offset. Although robust pulse sequences have been developed throughout the years, the imperfection of the initial state has not been widely discussed in the literature as an additional systematic error. In previous work, we developed a classical canonical framework to perform stability analysis and used this as a measure of CP robustness. In that work, a single initial condition was allowed to evolve under various pulse imperfections. The current work extends this approach to $2D$ distributions of initial conditions on the Bloch Sphere; the objective is to minimize the area in order to preserve coherence, while maximizing population inversion of the entire distribution. As a case study, we investigate Levitt's $90(x)180(y)90(x)$ pulse sequence, when there is a spread in initial conditions. The canonical framework enables us to assess the robustness of Levitt's pulse sequence, and we find that it is maintained to a great extent even when considering a spread of initial conditions. Nevertheless, by conducting a numerical optimization, we have identified several variants of Levitt's pulse sequence that produce a larger coherent population inversion when there is a spread in initial conditions.

physics.atom-ph↗

arXiv2025

The robustness of composite pulses elucidated by classical mechanics: Stability around the globe

Composite Pulses (CPs) are widely used in Nuclear Magnetic Resonance (NMR), optical spectroscopy, optimal control experiments and quantum computing to manipulate systems that are well-described by a two-level Hamiltonian. A careful design of these pulses can allow the refocusing of an ensemble at a desired state, even if the ensemble experiences imperfections in the magnitude of the external field or resonance offsets. Since the introduction of CPs, several theoretical justifications for their robustness have been suggested. In this work, we suggest another justification based on the classical mechanical concept of a stability matrix. The motion on the Bloch Sphere is mapped to a canonical system of coordinates and the focusing of an ensemble corresponds to caustics, or the vanishing of an appropriate stability matrix element in the canonical coordinates. Our approach highlights the directionality of the refocusing of the ensemble on the Bloch Sphere, revealing how different ensembles refocus along different directions. The approach also clarifies when CPs can induce a change in the width of the ensemble as opposed to simply a rotation of the axes. As a case study, we investigate the $90(x)180(y)90(x)$ CP introduced by Levitt, where the approach provides a new perspective into why this CP is effective.

quant-ph↗

arXiv2024

The Gaussian Kicked Rotor: Periodic forcing with finite-width pulses and the role of shifting the kick

The Kicked Rotor is perhaps the simplest physical model to illuminate the transition from regular to chaotic motion in classical mechanics. It is also widely applied as a model of light-matter interactions. In the conventional treatment, the infinitesimal width of each kick allows an immediate integration of the equations of motion. This in turn allows a full description of the dynamics via a discrete mapping, the Standard Map, if one looks at the dynamics only stroboscopically. It turns out that this model is only part of a much richer story if one accounts for finite temporal width of the kick. In this letter, we formulate a general model of finite-width periodic forcing and derive a continuous set of maps that depend on a parameter shift $Δ$ that allows one to capture the motion in both the driven and kicked regimes. The fixed points and symmetry of the mapping are shown analytically and numerically to depend on the value of the shift parameter.

nlin.CD↗

arXiv2024

Attochaos I: The classically chaotic postcursor of high harmonic generation

Attosecond physics provides unique insights into light-matter interaction on ultrafast time scales. Its core phenomenon, High Harmonic Generation (HHG), is often described by a classical recollision model, the simple-man or three-step model, where the atomic potential is disregarded. Many features are already well explained using this model; however, the simplicity of the model does not allow the possibility of classical chaotic motion. We show that beyond this model, classical chaotic motion does exist albeit on timescales that are generally longer than the first recollision time. Chaos is analyzed using tools from the theory of dynamical systems, such as Lyapunov exponents and stroboscopic maps. The calculations are done for a one-dimensional Coulomb potential subjected to a linearly polarized electric field.

physics.class-ph↗

arXiv2022

High Harmonic Generation without Tunnel-Ionization

A new High Harmonic Generation (HHG) scheme, which doesn't rely on Tunnel-Ionization as the ionization mechanism but rather on Single-Photon Ionization, is theoretically proposed and numerically demonstrated. The scheme uses two driver fields: an extreme-ultraviolet driver which induces the ionization, and a circularly-polarized, co-rotating, two-color infrared driver carried at a fundamental frequency and its second harmonic which induces the recollision. Using Classical and time-dependent Schrödinger equation simulations of a model Argon atom, we show that in this scheme ionization is essentially decoupled from recollision. Releasing the process from being Tunneling-dependent reduces its degree of nonlinearity, which offers new capabilities in attosecond science, such as generation of High Harmonics from highly-charged ions, or from specific deep core electronic levels. It is shown that the emitted high harmonics involve the absorption of photons of one color of the infrared driver, and the emission of photons of the second color. This calls for future examination of the possible correlations between the emitted high harmonics.

physics.atom-ph↗

arXiv2021

Variational solutions for Resonances by a Finite-Difference Grid Method

We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as resonances and provide the energies and inverse lifetimes of the system under study (e.g., molecules) in metastable states. This approach allows incorporating finite grid methods in the study of resonance phenomena in chemistry. Possible applications include the calculation of electronic autoionization resonances which occur when ionization takes place as the bond lengths of the molecule are varied. Alternatively, the method can be applied to calculate nuclear predissociation resonances which are associated with activated complexes with finite lifetimes.

physics.comp-ph↗