arXiv subjects

David J. Tannor

Publications and source records attributed to David J. Tannor.

At least 19 recordsLinked to original sources

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↗

arXiv2025

Audio Compression using Periodic Gabor with Biorthogonal Exchange: Implementation Using the Zak Transform

An efficient new approach to signal compression is presented based of a novel variation on the Gabor basis set. Following earlier work by Shimshovitz and Tannor, we convolve the conventional Gabor functions with Dirichlet functions to obtain a Periodic Gabor basis set (PG). The PG basis is exact for continuous functions that are periodic band-limited. Using the orthonormality of the Dirichlet functions, the calculation of the PG coefficients becomes trivial and numerically stable, but its representation does not allow compression. Large compression factors are achieved by exchanging the PG basis with its biorthogonal basis, thereby using the localized PG basis to calculate the coefficients (PGB). Here we implement the PGB formalism using the Fast Zak Transform and obtain very high efficiency with respect to both CPU and memory. We compare the method with the state of the art Short-Time Fourier Transform (STFT) and Discrete Wavelet Transform (DWT) methods on a variety of audio files, including music and speech samples. In all cases tested our scheme surpasses the STFT by far and in most cases outperforms DWT.

eess.AS↗

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↗

arXiv2023

Two Hundred Years After Hamilton: The Simple Axiom That Underlies Classical Mechanics

In 1834-1835, Hamilton published two papers that revolutionized classical mechanics. In these papers, he introduced the Hamilton-Jacobi equation, Hamilton's equations of motion and the principle of least action. These three formulations of classical mechanics became the forerunners of quantum mechanics, but none of these is what Hamilton was looking for: he was looking for what he called the principal function, $S(q',q'',T)$, from which the entire trajectory history can be obtained just by differentiation. Here we show that all of Hamilton's formulations can be derived just by assuming that the principal function is additive, $S(q',q'',T)=S(q',Q,t_1)+S(Q,q'',t_2)$ with $t_1+t_2=T$. This simple additivity axiom can be considered the fundamental principle of classical mechanics and shows that analytical mechanics is essentially just a footnote to the problem of finding the shortest path between two points. The simplicity of the formulation could provide new perspectives on some of the major themes in classical mechanics including symplectic geometry, periodic orbit theory and Morse theory, as well as giving new perspectives on quantum mechanics. Moreover, it could potentially provide a unified description of different areas of physics, leading to insight for example, into the transition from deterministic dynamics to statistical mechanics.

physics.class-ph↗

arXiv2022

Optimal control for maximally creating and maintaining a superposition state of a two-level system under the influence of Markovian decoherence

Reducing decoherence is an essential step toward realizing general-purpose quantum computers beyond the present noisy intermediate-scale quantum (NISQ) computers. To this end, dynamical decoupling (DD) approaches in which external fields are applied to qubits are often adopted. We numerically study DD using a two-level model system (qubit) under the influence of Markovian decoherence by using quantum optimal control theory with slightly modified settings, in which the physical objective is to maximally create and maintain a specified superposition state in a specified control period. An optimal pulse is numerically designed while systematically varying the values of dephasing, population decay, pulse fluence, and control period as well as using two kinds of objective functionals. Although the decrease in purity due to the decoherence gives rise to the upper limit of the target expectation value, i.e., the saturated value, the optimally shaped pulse effectively deals with the decoherence by gradually creating the target superposition state to realize the saturated value as much as possible.

quant-ph↗

arXiv2021

Control of concerted back-to-back double ionization dynamics in helium

Double ionization (DI) is a fundamental process that despite its apparent simplicity provides rich opportunities for probing and controlling the electronic motion. Even for the simplest multielectron atom, helium, new DI mechanisms are still being found. To first order in the field strength, a strong external field doubly ionizes the electrons in helium such that they are ejected into the same direction (front-to-back motion). The ejection into opposite directions (back-to-back motion) cannot be described to first order, making it a challenging target for control. Here, we address this challenge and optimize the field with the objective of back-to-back double ionization using a (1 + 1)-dimensional model. The optimization is performed using four different control procedures: (1) short-time control, (2) derivative-free optimization of basis expansions of the field, (3) the Krotov method, and (4) control of the classical equations of motion. All four procedures lead to fields with dominant back-to-back motion. All the fields obtained exploit essentially the same two-step mechanism leading to back-to-back motion: first, the electrons are displaced by the field into the same direction. Second, after the field turns off, the nuclear attraction and the electron-electron repulsion combine to generate the final motion into opposite directions for each electron. By performing quasi-classical calculations, we confirm that this mechanism is essentially classical.

physics.atom-ph↗

arXiv2021

Duality of the Principle of Least Action: A New Formulation of Classical Mechanics

A dual formalism for Lagrange multipliers is developed. The formalism is used to minimize an action function $S(q_2,q_1,T)$ without any dynamical input other than that $S$ is convex. All the key equations of analytical mechanics -- the Hamilton-Jacobi equation, the generating functions for canonical transformations, Hamilton's equations of motion and $S$ as the time integral of the Lagrangian -- emerge as simple consequences. It appears that to a large extent, analytical mechanics is simply a footnote to the most basic problem in the calculus of variations: that the shortest distance between two points is a straight line.

physics.class-ph↗

arXiv2019

A Three-step Model of High Harmonic Generation using Complex Classical Trajectories

We present a new trajectory formulation of high harmonic generation that treats classically allowed and classically forbidden processes within a single dynamical framework. Complex trajectories orbit the nucleus, producing the stationary Coulomb ground state. When the field is turned on, these complex trajectories continue their motion in the field-dressed Coulomb potential and therefore tunnel ionization, unbound evolution and recollision are described within a single, seamless framework. The new formulation can bring mechanistic understanding to a broad range of strong field physics effects.

quant-ph↗

arXiv2018

Gradient optimization of analytic controls: the route to high accuracy quantum optimal control

Quantum computation places very stringent demands on gate fidelities, and experimental implementations require both the controls and the resultant dynamics to conform to hardware-specific constraints. Superconducting qubits present the additional requirement that pulses must have simple parameterizations, so they can be further calibrated in the experiment, to compensate for uncertainties in system parameters. Other quantum technologies, such as sensing, require extremely high fidelities. We present a novel, conceptually simple and easy-to-implement gradient-based optimal control technique named Gradient Optimization of Analytic conTrols (GOAT), which satisfies all the above requirements, unlike previous approaches. To demonstrate GOAT's capabilities, with emphasis on flexibility and ease of subsequent calibration, we optimize fast coherence-limited pulses for two leading superconducting qubits architectures - flux-tunable transmons and fixed-frequency transmons with tunable couplers.

quant-ph↗

arXiv2018

Systematic elimination of Stokes divergences emanating from complex phase space caustics

Stokes phenomenon refers to the fact that the asymptotic expansion of complex functions can differ in different regions of the complex plane, and that beyond the so-called Stokes lines has an unphysical divergence. An important special case is when the Stokes lines emanate from phase space caustics of a complex trajectory manifold. In this case, symmetry determines that to second order there is a double coverage of the space, one portion of which is unphysical. Building on the seminal but laconic findings of Adachi, we show that the deviation from second order can be used to rigorously determine the Stokes lines and therefore the region of the space that should be removed. The method has applications to wavepacket reconstruction from complex valued classical trajectories. With a rigorous method in hand for removing unphysical divergences, we demonstrate excellent wavepacket reconstruction for the Morse, Quartic, Coulomb and Eckart systems.

quant-ph↗

arXiv2017

Dynamical pruning of the multiconfiguration time-dependent Hartree method (DP-MCTDH): An efficient approach for multidimensional quantum dynamics

We present two strategies for combining dynamical pruning with the multiconfiguration time-dependent Hartree method (DP-MCTDH), where dynamical pruning means on-the-fly selection of relevant basis functions. The first strategy prunes the primitive basis that represents the single-particle functions (SPFs). This is useful for smaller systems that require many primitive basis functions per degree of freedom, as we will illustrate for NO$_2$. Furthermore, this allows for higher-dimensional mode combination and partially lifts the sum-of-product-form requirement onto the structure of the Hamiltonian, as we illustrate for nonadiabatic 24-dimensional pyrazine. The second strategy prunes the set of configurations of SPF at each time step. We show that this strategy yields significant speed-ups with factors between 5 and 50 in computing time, making it competitive with the multilayer MCTDH method.

physics.chem-ph↗

arXiv2017

Wavepacket revivals via complex trajectory propagation

Complex-valued semiclassical methods hold out the promise of treating classically allowed and classically forbidden processes on the same footing. In addition, they provide a natural way to describe optical excitation with complex fields within the trajectory framework. Despite their promise, these methods have until now been limited to short time propagation, due to the numerical difficulties introduced by the complexification. Using a new Final Value Representation of the Coherent State Propagator (FINCO), combined with an analysis of the complex classical phase space, we achieve accurate wavepacket propagation all the way to the revival time of a strongly anharmonic system.

quant-ph↗

arXiv2016

Efficient molecular quantum dynamics in coordinate and phase space using pruned bases

We present an efficient implementation of dynamically pruned quantum dynamics, both in coordinate space and in phase space. We combine the ideas behind the biorthogonal von Neumann basis (PvB) with the orthogonalized momentum-symmetrized Gaussians (Weylets) to create a new basis, projected Weylets, that takes the best from both methods. We benchmark pruned dynamics using phase-space-localized PvB, projected Weylets, and coordinate-space-localized DVR bases, with real-world examples in up to six dimensions. We show that coordinate-space localization is most important for efficient pruning and that pruned dynamics is much faster compared to unpruned, exact dynamics. Phase-space localization is useful for more demanding dynamics where many basis functions are required. There, projected Weylets offer a more compact representation than pruned DVR bases.

physics.chem-ph↗

arXiv2015

Reply to Comment by Brown and Carrington on "Phase-Space Approach to Solving the Time-Independent Schrödinger Equation"

The Comment of Brown and Carrington Jr. (BC) has two main points: 1) that the contraction idea of Shimshovitz and Tannor (ST) can be used for any DVR basis, not necessarily periodic, and 2) that the biorthogonal basis introduced by ST is unnecessary. We fully agree with the first point and in fact have several works that were in press at the time of the Comment that illustrate this. On the second point we show that BC implicitly use the biorthogonal basis of ST.

quant-ph↗

arXiv2015

Are there traps in quantum control landscapes?

There has been great interest in recent years in quantum control landscapes. Given an objective $J$ that depends on a control field $\varepsilon$ the dynamical landscape is defined by the properties of the Hessian $δ^2 J/δ\varepsilon^2$ at the critical points $δJ/δ\varepsilon=0$. We show that contrary to recent claims in the literature the dynamical control landscape can exhibit trapping behavior due to the existence of special critical points and illustrate this finding with an example of a 3-level $Λ$-system. This observation can have profound implications for both theoretical and experimental quantum control studies.

quant-ph↗

arXiv2015

Quantum control landscape for a $Λ$-atom in the vicinity of second order traps

We show that the second order traps in the control landscape for a three-level $Λ$-system found in our previous work {\it Phys. Rev. Lett.} {\bf 106}, 120402 (2011) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow --- at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of the $\varepsilon(t)=0$ second order trap, performed using the GRAPE and BFGS algorithms.

quant-ph↗