Localized Wave Packets Research Articles

On the quantum coherent states of microparticle in a viscous medium

Using a canonical quantization, the motion of a micro-particle in a medium with the viscous resistance is analyzed. Localized wave packets of the type of coherent states are considered in detail. It has been established that viscous resistance suppresses the quantum properties of a micro-particle, which makes it possible to consider a viscous medium as a classical device.

Stripe Soliton in All‐Fiber Lasers

AbstractStripe solitons (SSs), the special localized wavepackets exhibiting periodic modulations, are identified in condensed matter physics and nonlinear science. Here, giant‐chirp and chirp‐free SSs in all‐fiber lasers are demonstrated composed of single‐mode fiber and polarization‐maintaining fiber. The chirp property of SSs depends on the cavity dispersion while the stripe period relies on the fiber birefringence. Theoretical and numerical results coincide with experimental findings, revealing that the filtering and frequency shift of two vector modes is governed by the mode coupling related to birefringence and the nonlinear coupling induced by the phase modulation, respectively, which disrupts their complementarity and ultimately leads to modulation of the overall spectrum. With the aid of the saturable absorption and cross‐phase modulation effects, the multi‐color pulses synergistically overlap, giving rise to the unique SSs. These results present a simple approach for generating chirp‐controllable SSs in all‐fiber lasers, which is crucial for applications in difference frequency generation and soliton physics.

Generation of Soliton Frequency Combs in NEMS.

In this study, we present an innovative approach leveraging combination internal resonances within a NEMS platform to generate mechanical soliton frequency combs (FCs) spanning a broad spectrum. In the time domain, the FCs take the form of a periodic train of narrow pulses, a highly coveted phenomenon within the realm of nonlinear wave-matter interactions. Our method relies on an intricate interaction among multiple vibration modes of a bracket-nanocantilever enabled by the strong nonlinearity of the electrostatic field. Through numerical simulation and experimental validation, we demonstrate that by amplifying the motions of the NEMS with the external electrostatic forcing tuned to excite the superharmonic resonance of order-n of the fundamental mode and exploiting combination internal resonances, we can generate multiple stable localized mechanical wave packets with different lobe sizes embodying soliton states I and II. This represents a significant breakthrough with profound implications for quantum computing and metrology.

Entanglement in Bound States of Two Particles with a Localized Center-of-Mass Wave-Function

Abstract We explore some entanglement-related features of systems consisting of two particles coupled through a central potential. We consider two types of systems, constituted by particles in one spatial dimension interacting through an attractive Dirac Delta potential, or through a harmonic quadratic potential. The degree of freedom corresponding to the system’s center of mass is either regarded as confined within a one dimensional box, or described by a localized Gaussian wave-packet. As a quantitative indicator of the degree of entanglement between the particles, we use the linear entropy of the one-particle marginal density matrix. In general, this quantity cannot be calculated analytically (except in some special cases related to the Harmonic interaction). Consequently, the linear entropy has to be evaluated numerically. Since the concomitant numerical problem consist in evaluating a multi-dimensional integral in four dimensions, the Monte Carlo integration method is an appropriate one for this task. We explore numerically how the system’s entanglement depends on the different interaction potentials, on the different types of confinement for the system’s center of mass, and on the associated parameters describing the size and geometry of the system.

Complexity Measures in the Tight-Binding Model

Abstract The deformation of a wave packet is a significant topic in classical and quantum mechanics. Understanding this phenomenon is relevant in the study of various physical systems. In this work, we characterize the evolution of a highly localized wave packet in a tight-binding lattice. We investigate the behavior of the probability distribution associated with the wave packet and the accompanying complexity measures. We take information entropy, disequilibrium, disorder, and complexity measures to describe the localization-delocalization process from a highly localized initial pulse, showing the particles moving in a lattice. The main result is obtained from the entropy definition (Logarithmic and Linear) and the inverse of the participant ratio to describe the expected localization-delocalization process, evoking two definitions of Complexity: CLMC and CSDL .

Anomalous diffusion, prethermalization, and particle binding in an interacting flat band system

We study the broadening of initially localized wave packets in a quasi one-dimensional diamond ladder with interacting, spinless fermions. The lattice possesses a flat band causing localization. We place special focus on the transition away from the flat band many-body localized case by adding very weak dispersion. By doing so, we allow propagation of the wave packet on significantly different timescales which causes anomalous diffusion. Due to the temporal separation of dynamic processes, an interaction-induced, prethermal equilibrium becomes apparent. A physical picture of light and heavy modes for this prethermal behavior can be obtained within Born–Oppenheimer approximation via basis transformation of the original Hamiltonian. This reveals a detachment between light, symmetric and heavy, anti-symmetric particle species. We show that the prethermal state is characterized by heavy particles binding together mediated by the light particles.

Quantum isotropic Universe in RQM analogy: The cosmological horizon

We investigate the quantum dynamics of the isotropic Universe in the presence of a free massless scalar field, playing the role of a physical clock. The Hilbert space is constructed via a direct analogy between the Wheeler-DeWitt equation in the minisuperspace and a relativistic scalar one in physical space. In particular, we show how the introduction of a “turning point” in the Universe evolution allows to overcome an intrinsic ambiguity in representing the expanding and collapsing Universe. In this way, the positive and negative frequencies are simply identified with time reversed states. The main subject of the present analysis is the construction of a horizon operator, whose quantum behavior is investigated when Polymer Quantum Mechanics is implemented to describe the asymptotic evolution near the initial singularity. The reason of this choice is motivated by the intrinsic spreading of localized wavepackets when the polymer dispersion relation governs the quantum dynamics. The evidence that the mean value of the quantum horizon operator follows its semiclassical behavior (corrected for polymerization) is a clear indication that a concept of causality can be restored also in the quantum cosmological picture.

Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Plasma-infused solitary waves: Unraveling novel dynamics with the Camassa–Holm equation

This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm ([Formula: see text]) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the [Formula: see text] equation has exhibited versatility across various disciplines, including nonlinear optics and elasticity theory. It intricately delineates the interplay between nonlinear and dispersive effects in wave systems, with nonlinearity arising from component interactions and dispersion rooted in the temporal spreading of waves. Furthermore, the [Formula: see text] equation governs the spatiotemporal evolution of wave profiles, encompassing both nonlinear and dispersive influences. Notably, the equation allows for soliton solutions — localized wave packets sustaining their form over extended distances. The identification of precise solitary wave solutions holds paramount significance for comprehending the [Formula: see text] equation’s behavior in diverse physical contexts, such as fluid dynamics and nonlinear optics. Moreover, this study establishes a correlation between the investigated model and plasma physics, demonstrating the efficacy and efficiency of the employed computational techniques through benchmarking against alternative computational methods. This augmentation underscores the broader relevance of the [Formula: see text] equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics.

Quasi-classical dynamics of a charged particle under conditions of ionization losses in a weak magnetic field

The quasi-classical dynamics of a charged particle in a weak magnetic field in the presence of dissipative losses caused by ionization of the medium is studied. The approximate approach proposed here is a generalization of the Caldirola—Kanai method for quantizing the translational motion of particles in dissipative media. It is shown that a weak curvature of a classical trajectory by the magnetic field is accompanied by an isotropic increase of uncertainty of the particle coordinates to the some maximum value at the moment the localized probability density wave packet stops. The limitation of the increase of coordinate uncertainty is due to irreversible ionization losses.

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics.

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

A Brief Historical Overview and Mathematical Equations of Soliton

bstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that maintains its shape even after colliding with other similar localised wave packets and propagates freely at a constant speed. The medium's balanced cancellation of nonlinear and dispersive effects is responsible for its extraordinary stability. Dispersive effects are a characteristics of several systems where a wave frequency determines how fast it travels. Subsequently, it was shown that soliton solutions offer stable solutions for a broad range of weakly nonlinear dispersive partial differential equations that characterise physical systems. Solitons have a self-restoring quality that makes them desirable for long-distance, high-speed transmissions. The strain on networks is quickly reaching a critical threshold as transmission speeds over longer distances are now approaching 40 Gbit/sec. In the future, solitons might offer intriguing alternatives.

The Study on the Theoretical Perspectives of Soliton and Its Applications

Abstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that propagates freely at a constant speed and keeps its shape even after colliding with other similar localised wave packets. Its exceptional stability arises from the balanced cancellation of nonlinear and dispersive effects in the medium. In engineering, a soliton wave is a self-reinforcing solitary wave packet that keeps its shape as it travels at a steady speed. It results from the medium's non-linear and dispersive effects cancelling each other out. Many systems exhibit dispersive effects, in which the wave frequency controls the wave's speed of propagation. It was then demonstrated that soliton solutions provide stable solutions for a wide class of dispersive partial differential equations that are weakly nonlinear and describe physical systems. Solitons are preferred for high-speed, longdistance transmissions because of their self-restoring characteristics. Given that transmission speeds over longer distances are now close to 40 Gbit per second, the strain on networks is rapidly approaching a critical threshold. Solitons may provide interesting alternatives in the future. Non-linear wave equations that describe how waves propagate in specific physical systems are solved to get soliton solutions. These waves appear as solutions in mathematical models of various systems, such as optical wave-guides, crystal lattice vibrations, and water waves.

The Stability and Behaviour of the Superposition of Non-Linear Waves in Space

The superposition of non-linear waves in space refers to the phenomenon where two or more waves overlap and combine to form a new wave pattern. Non-linear waves are characterized by their ability to interact with each other, leading to complex behaviors that are not observed in linear wave systems. Understanding the stability and behavior of the superposition of non-linear waves in space is crucial in various fields such as physics, engineering, and oceanography. When non-linear waves superpose, their interactions can lead to a range of behaviors, including wave breaking, formation of solitons (localized wave packets), and the generation of harmonics. The stability of the superposition is determined by the balance between the non-linear effects and dispersive effects, which can either stabilize or destabilize the wave pattern. In addition, the behavior of non-linear waves in space is influenced by external factors such as boundaries, dissipation, and external forcing. In this paper, we study the behavior and characteristics of waves when they interact with each other. Superposition refers to the phenomenon where multiple waves combine to form a resultant wave. In the case of linear waves, this superposition occurs according to the principles of linear superposition, which states that the displacement or amplitude at any point is the algebraic sum of the displacements or amplitudes of the individual waves. Understanding the superposition of linear waves in space has various applications in fields such as physics, engineering, acoustics, optics, and signal processing. By studying how waves interact and combine, researchers can gain insights into wave propagation, interference patterns, wave reflections, diffraction, and other phenomena that occur when waves meet.

Quasi-classical motion of a particle in a bulk dissipative medium

A version of the quasi-classical approach is proposed, which makes it possible to describe the straight-line motion of a micro-particle in a bulk dissipative medium. From the side of the medium, the particle is acted upon by the force of viscous friction and the drag force, which are proportional to the velocity and the square of the velocity, respectively. In addition, an external conservative force is applied to the particle. The Green’s function of the particle and the quasi-classical coherent state in the form of a three-dimensional localized wave packet are found. It is shown that the translational motion of the wave packet is accompanied by a monotonic increase in the uncertainties of the particle coordinates up to certain maximum asymptotic values. Due to the drag force, these asymptotic uncertainties contain information about the corresponding initial uncertainties in the coordinates of the micro-particle, about its initial velocity, as well as about the external conservative force.

The exotic behavior of the wave evolution in Lévy crystals within a fractional medium

We investigate a traveling Gaussian wave packet transport through a rectangular quantum barrier of lévy crystals in fractional quantum mechanics formalism. We study both standard and fractional Schrödinger equations in linear and nonlinear regimes by using a split-step finite difference (SSFD) method. We evaluate the reflection, trapping, and transmission coefficients of the wave packet and the wave packet spreading by using time-dependent inverse participation ratio (IPR) and second moment. By simultaneously adjusting the fractional and nonlinear terms, we create sharp pulses, which is an essential issue in optoelectronic devices. We illustrate that the effects of barrier height and width on the transmission coefficient are strangely different for the standard and fractional Schrödinger equations. We observe fortunately soliton-like localized wave packets in the fractional regime. Thus, we can effectively control the behavior of the wave evolution by adjusting the available parameters, which can excite new ideas in optics.

Effect of relativistic acceleration on tripartite entanglement of Gaussian states

The performance of quantum information protocols can be impacted by the noninertial motion of parties such as in a gravitational field. Previous studies have primarily focused on the degradation of entanglement for global modes, revealing that bipartite entanglement vanishes in the limit of large acceleration, whereas tripartite entanglement does not. In this study, we examine the impact of acceleration on tripartite Gaussian resource states encoded in spatially localized wave packets. Contrarily to prior findings, we observe that genuine tripartite entanglement (GTE) vanishes in the limit of large acceleration for various initial states and observer trajectories.

Transformation of a fully nonlinear breather-like package of internal waves over a bottom step in a layered fluid

In this paper, we study the process of transformation of a localized wave packet over a bottom step in a three-layer fluid, in which the height of the step is equal to or exceeds the thickness of the lower layer; therefore, density stratification becomes two-layer in the shallow water zone. In numerical experiments, both the height of the step and the width of the step were varied. The problem is solved in the framework of a fully nonlinear model of hydrodynamics of an inviscid incompressible stratified fluid. The primary analysis consisted in estimating the values of dimensionless parameters used, as a rule, in runup problems: the Froude and Iribarren numbers, the ratio of the characteristic wavelength to the characteristic slope width, the ratio of the topographic slope to the characteristic wave beam angle. Since the “cutoff” line for the lower pycnocline is partially or completely located on a step, one could expect the effects of run-up, breaking or reflection of waves propagating along the lower pycnocline, but this doesn’t happen. It is shown that the reflection of the wave packet from the step is minimal in all cases considered, a strong steepening of the wave is observed, but no breaking occurs in this case — the wave then just quickly decays on the lower pycnocline. An analysis of the spectral amplitudes and energy fields allows us to conclude that there is a transfer of energy from the lower pycnocline to the upper one. The breather in a two-layer fluid cannot exist, but the wave packet formed in the upper pycnocline after its destruction has much higher energy than it has before the step.

Coupling dynamics of dissipative localized structures: From polarized vector solitons to soliton molecules

Dissipative solitons are well known as representative nonlinear localized wave packets resulting from the composite balance of dispersion, nonlinearity, gain and loss in optical cavities. Since the last three decades, dissipative solitons keep as one of the most active research topics in the framework of nonlinear physics, not only for fundamental researches but also for various industrial applications. From one hand, due to the rich nonlinear dynamics, dissipative solitons have been considered as an ideal testbed to study the pumping-dissipation-bifurcation complexities. On the other hand, the emergences of dissipative solitons are tightly linked with the emission of ultrashort laser pulses, as well as the formation of broadband optical frequency combs. Compared with single-soliton counterpart, the multi-component dissipative solitons have exhibited even more fascinating properties and much wider range of applications, therefore the coupling dynamics of dissipative solitons is the blooming and emergent research area in the last decade. From this point of view, this review article covers the latest developments of compound dissipative solitons, not only in passive Kerr resonators, but also in mode-locked laser cavities. Ranging from polarized vector solitons to soliton molecules, the rising degree of freedom of these multi-component localized structures is boosting the discovery of novel nonlinear dynamics and underpin the various photonic applications in light science.

Solitary wave characteristics on the fine structure of the mesospheric sporadic sodium layer

The so-called sporadic sodium layers (SSLs or NaS) are proposed to be strongly related to wave fluctuations. The solitary wave is a particular solution of the partial differential equation whose energy travels as a localized wave packet. A soliton, on the other hand, is a special type of solitary wave that exhibits a particle-like behavior with a strong stable form. For the first time, the solitary wave theory has been used in this research to study the fine structure of SSL/NaS. We performed soliton fitting processes on the observed data from the Andes Lidar Observatory and found out that 24/27 NaS events had exhibited similar features/characteristics to a soliton. Time series of the net anomaly of the NaS revealed the same variation process to the solution of a generalized five-order KdV equation. Our results, therefore, suggested that the NaS phenomenon would be a pertinent tracer for non-linear wave studies in the atmosphere.