Harmonic Potential Research Articles

Exploring outputs from concatenated stochastic heat engines

Abstract Recent work on the concatenation of two simple heat engines has shown that it may lead to non-monotonic variations in the efficiency and power with parameters like driving amplitudes and asymmetries in cycle periods. Motivated by this study, we investigate the effect of the concatenation between two stochastic heat engines where colloidal particles have been trapped in harmonic potentials. The stiffness parameters of each engine are varied cyclically, but with different cycle periods, with a common thermal bath that acts as a sink for the first engine but as a source for the second. We consider two types of protocol, the first where the trap strength undergoes sudden jumps, and the second where it varies linearly with time. In both we find several non-trivial effects, like the non-monotonic functional dependence of the engine outputs on several parameters used in the setup. For the protocol that varies linearly with time, the concatenation leads to enhanced output power as compared to a single effective engine, in a suitable range of parameters. It has been shown that the output from the combined system shows a peak with respect to the asymmetry in the cycle times of the engines that have been concatenated. A general relation for the efficiency of an arbitrary number of concatenated engines driven quasistatically has been provided.

Dynamic and static properties of quantum Hall and harmonic oscillator systems on the non-commutative plane

We study two quantum mechanical systems on the noncommutative plane using a representation independent approach. First, in the context of the Landau problem, we obtain an explicit expression for the gauge transformation that connects the Landau and the symmetric gauge in noncommutative space. This lead us to conclude that the usual form of the symmetric gauge A⃗=−β2Ŷ,β2X̂, in which the constant β is interpreted as the magnetic field, is not true in noncommutative space. We also be able to establish a precise definition of β as a function of the magnetic field, for which the equivalence between the symmetric and Landau gauges is held in noncommutative plane. Using the symmetric gauge, we obtain results for the spectrum of the quantum Hall system, its transverse conductivity in the presence of an electric field, and other static observables. These results amend the literature on quantum Hall effect in the noncommutative plane in which the incorrect form of the symmetric gauge, in noncommutative space, is assumed. We also study the non-equilibrium dynamics of simple observables for this system. On the other hand, we study the dynamics of the harmonic oscillator in non-commutative space and show that, in general, it exhibits quasi-periodic behavior, in striking contrast with its commutative version. The study of dynamics reveals itself as a most powerful tool to characterize and understand the effects of non-commutativity.

One-loop radiative corrections in bumblebee-Stueckelberg model

This work aims to study the radiative corrections in a vector model with spontaneous Lorentz symmetry violation, known in the literature as the bumblebee model. We consider such a model with self-interaction quadratic smooth potential responsible for spontaneous Lorentz symmetry breaking. The spectrum of this model displays a transversal nonmassive mode, identified as Nambu-Goldstone, and a massive longitudinal mode. Besides the Lorentz symmetry, this model also exhibits gauge symmetry violation. To restore the gauge symmetry, we introduce the Stueckelberg field and calculate the two-point function by employing the principal-value (PV) prescription. The result is nontransversal, leading to a massive excited mode.

Decoherence quantification through commutation relations decay for open quantum harmonic oscillators

Decoherence quantification through commutation relations decay for open quantum harmonic oscillators

Ground state of Tonks–Girardeau gas under density-dependent gauge potential in a one-dimensional harmonic potential

Ground state of Tonks–Girardeau gas under density-dependent gauge potential in a one-dimensional harmonic potential

Evolution of the wave function's shape in a time-dependent harmonic potential

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

Correlation between parameters in the microstructural vector theory and Hill's plastic potential

The dissipation inequality provides a useful method for specifying constitutive equations of plastic deformation. There are two main ways to use the dissipation inequality to determine the plastic deformation path: by utilizing a plastic potential with the normality rule, or by not utilizing the plastic potential to determine the plastic flow according to the elastic distortion of the material coordinate axis (microstructural vectors). While both methods satisfy the inequality condition and predict experimental results well, they have been developed separately without discussion on relationship. This study presents a physics-based mathematical discussion on the relative dependence between the two methods of determining plastic flow and their point of connection, using Hill's quadratic plastic potential. For this, the conversion of the distortional length and angle changes in microstructural vectors to deviatoric strain tensor under small elastic deformation conditions is mathematically demonstrated. Using these equations, the rate of inelastic deformation derived from microstructural vectors exhibited a similar form to that derived from Hill's quadratic potential. Moreover, the model parameters of the microstructural vectors can be calibrated using the Hill's function. The plastic flow derived from the microstructural vectors and Hill's function exhibited a remarkably similar trend under the assumption of small elastic deformation, although not identical. The newly defined relationships were implemented numerically and validated through simulations. The parameters of the microstructural vector theory derived from Hill's plastic potential can be utilized in materials science and engineering.

English

English

Dynamical evolutions of optical smooth positons in variable coefficient nonlinear Schrödinger equation with external potentials

We construct the optical smooth positon solutions for the generalized variable coefficient nonlinear Schrödinger equation with longitudinally varying dispersion, nonlinearity, linear and harmonic potentials. By employing the similarity transformation technique, we derive this solution along with the homogeneous smooth positon solutions of the basic NLS equation and the integrable requirement. The dynamical properties of the constructed second- and third-order positon solutions are investigated considering the two physical scenarios. In the first situation, we consider a longitudinally variable linear potential while keeping the dispersion parameter constant. In the second scenario, we take into account three different types of modulated dispersion parameters, namely quadratic, exponential, and linear, to analyze the various features in the intensity profile of inhomogeneous optical smooth positons. In both situations, we observe a wide range of significant nonlinear phenomena in the intensity profiles of positons. These phenomena include amplification, deformation, compression, prolongation, suppression, and distortion. Our findings could make a significant contribution to intriguing nonlinear optics experiments focused on investigating optical positons.

Optimal performance of the stochastic thermodynamic engine with a periodic heat bath

Stochastic thermodynamics provides a conceptual framework for describing the fluctuating behavior of small systems like colloids or biomolecules far from thermodynamic equilibriums but still contacted with a heat bath. In contrast to most literature focusing on the classical paradigm of Carnot engines, we herein study the optimal performance of the thermodynamic heat engine with a heat bath that periodically changes temperature, which is outside controllable by a time-dependent harmonic potential. Under reasonable assumptions on the control actuation, we derive the achievable upper bound for the maximal power and also the optimal control protocol. In addition, we also obtain the corresponding efficiency at maximal power, which only depends on the ratio of the minimal and maximal value of the temperature profile.

Theoretic impetuses and lengths of Feinberg-Horodecki equation

The exact solution of the Feinberg-Horodecki equation for time-dependent harmonic vector potential has been investigated under a one-dimensional system. The quantized momentum and its corresponding un-normalized wave functions were explicitly obtained. The Fisher information (for time and momentum) and variance (for time and momentum) were calculated using expectation values of time and momentum via Hellman-Feynman Theory (HFT). The time and momentum Shannon entropy were obtained using an existing formula. Numerical results were for time and momentum Fisher information to confirm the Cramer-Rao inequality. Another numerical results were obtained for time and momentum Shannon entropy to confirm Bialynick-Birula, Mycielski (BBM) inequality. The effects of the potential parameters such as mass of the spring and the frequency on the theoretic quantities were fully examined. The new variance inequality was established using the inequalities of Fisher information. The established inequality was confirmed by numerical results which also satisfied the popular Cramer-Rao inequality. The theoretic impetuses for Fisher information, variance and Shannon entropy respectively, were calculated and their variation with some potentials were studied.

Carnot, Stirling, and Ericsson stochastic heat engines: Efficiency at maximum power.

This work uses the low-dissipation strategy to obtain efficiency at maximum power from a stochastic heat engine performing Carnot-, Stirling- and Ericsson-like cycles at finite time. The heat engine consists of a colloidal particle trapped by optical tweezers, in contact with two thermal baths at different temperatures, namely hot (T_{h}) and cold (T_{c}). The particle dynamics is characterized by a Langevin equationwith time-dependent control parameters bounded to a harmonic potential trap. In a low-dissipation approach, the equilibrium properties of the system are required, which in our case, can be calculated through a statelike equationfor the mean value 〈x^{2}〉_{eq} coming from a macroscopic expression associated with the Langevin equation.

Imaging of the charge-transfer reaction of spin-orbit state-selected Ar+(2P3/2) with N2 reveals vibrational-state-specific mechanisms.

Charge-transfer reactions are ubiquitous and play important roles in various gaseous environments, but, despite a long history of extensive research, our understanding of their dynamics at the quantum state-to-state level is still lacking. Here we report quantum-state-resolved experiments for the paradigmatic charge-transfer reaction Ar+ + N2 → Ar + N2+ using a three-dimensional velocity-map imaging crossed-beam apparatus with the Ar+ beam prepared exclusively in the spin-orbit state 2P3/2. High-resolution scattering images show strong dependence of rotational and angular distributions on the vibrational quantum number of the N2+ product. Trajectory surface-hopping calculations, which semi-quantitatively reproduce the experimental observations, support the existence of two distinct charge-transfer mechanisms. One of these, in the dominant N2+(v' = 1) channel, is the well-known long-distance harpooning mechanism. However, the highly rotationally excited products in the forward direction are attributed to a hard-collision glory scattering mechanism, which occurs on account of the strong attraction between the collisional partners counterbalanced by the short-range repulsive interaction.

Generalized Langevin equation for solute dynamics in fluids with time-dependent friction

The generalized Langevin equation (GLE) for a Brownian particle (BP) in a bath under the influence of a moving external harmonic potential is derived. It is assumed that the friction coefficients of the bath particles depend on time. The found GLE has the usual form but its memory kernel is a generalization of the expression known as a Prony series used to approximate a number of memory functions from the literature. Analytical solutions are obtained for the mean and mean squared displacements assuming the overdamped character of motion of the BP.

Bonding and Acidity of the Formal Hydride in the Prototypical Au9 (PPh3 )8 H2+ Nanocluster.

The role of hydrogen atoms as surface ligands on metal nanoclusters is of profound importance but remains difficult to directly study. While hydrogen atoms often appear to be incorporated formally as hydrides, evidence suggests that they donate electrons to the cluster's delocalized superatomic orbitals and may consequently behave as acidic protons that play key roles in synthetic or catalytic mechanisms. Here we directly test this assertion for the prototypical Au9 (PPh3 )8 H2+ nanocluster, formed by addition of a hydride to the well-characterized Au9 (PPh3 )8 3+ . Using gas-phase infrared spectroscopy, we were able to unambiguously isolate Au9 (PPh3 )8 H2+ and Au9 (PPh3 )8 D2+ , revealing an Au-H stretching mode at 1528 cm-1 that shifts to 1038 cm-1 upon deuteration. This shift is greater than the maximum expected for a typical harmonic potential, suggesting a potential governing cluster-H bonding that has some square-well character consistent with the hydrogen nucleus behaving as a metal atom in the cluster core. Complexing this cluster with very weak bases reveals a redshift of 37 cm-1 in the Au-H vibration, consistent with those typically seen for moderately acidic groups in gas phase molecules and providing an estimate of the acidity of Au9 (PPh3 )8 H2+ , at least with regard to its surface reactivity.

Bonding and Acidity of the Formal Hydride in the Prototypical Au9(PPh3)8H2+ Nanocluster

AbstractThe role of hydrogen atoms as surface ligands on metal nanoclusters is of profound importance but remains difficult to directly study. While hydrogen atoms often appear to be incorporated formally as hydrides, evidence suggests that they donate electrons to the cluster's delocalized superatomic orbitals and may consequently behave as acidic protons that play key roles in synthetic or catalytic mechanisms. Here we directly test this assertion for the prototypical Au9(PPh3)8H2+ nanocluster, formed by addition of a hydride to the well‐characterized Au9(PPh3)83+. Using gas‐phase infrared spectroscopy, we were able to unambiguously isolate Au9(PPh3)8H2+ and Au9(PPh3)8D2+, revealing an Au−H stretching mode at 1528 cm−1 that shifts to 1038 cm−1 upon deuteration. This shift is greater than the maximum expected for a typical harmonic potential, suggesting a potential governing cluster‐H bonding that has some square‐well character consistent with the hydrogen nucleus behaving as a metal atom in the cluster core. Complexing this cluster with very weak bases reveals a redshift of 37 cm−1 in the Au−H vibration, consistent with those typically seen for moderately acidic groups in gas phase molecules and providing an estimate of the acidity of Au9(PPh3)8H2+, at least with regard to its surface reactivity.

High-performance Kerr quantum battery

We propose and investigate the performance of a hybrid quantum battery, the so-called Kerr quantum battery, which consists of two interacting quantum oscillators, i.e., the charger is a harmonic oscillator and the battery is an anharmonic oscillator involving the Kerr nonlinearity. Such a setup creates nonuniform spacing between energy levels of the quantum oscillator that increases with the energy level. We find that the Kerr quantum battery can store more energy than the qubit battery and reaches maximum stored energy faster than the harmonic oscillator battery. In particular, the average charging power of the Kerr quantum battery is larger than the qubit battery. Furthermore, most of the stored energy in the Kerr quantum battery can be extracted for work. All of the properties of the Kerr quantum battery are controlled by the strength of nonlinearity, in which the enhancement of the nonlinearity transforms the battery from a harmonic oscillator to a qubit.

Supersymmetric Reshaping and Higher‐Dimensional Rearrangement of Photonic Lattices

AbstractIntegrated Jx photonic lattices, inspired by the quantum harmonic oscillator and due to their equidistant eigenvalue spectrum, have proven extremely useful for various applications, such as perfect imaging and coherent transfer of quantum states. However, to date their large‐scale implementation remains challenging. This study applies concepts from supersymmetry (SUSY) to construct two‐dimensional (2D) systems with spectra identical to that of one‐dimensional (1D) Jx lattices. While exhibiting different dynamics, these 2D equivalent systems retain the key imaging and state transfer properties of their 1D Jx counterpart. The method extends to all systems with separable spectra and facilitates experimental fabrication of large‐scale photonic circuits.

Feshbach–Villars oscillator in Kaluza-Klein theory

This research focuses on exploring the relativistic quantum dynamics of spin-0 scalar massive charged particles within the framework of the relativistic Feshbach-Villars oscillator (FVO) in the context of the Kaluza-Klein Theory (KKT). Specifically, we investigate the behavior of these particles in the presence of a cosmic string space-time within the framework of the Kaluza-Klein theory. To begin, we solve the Feshbach-Villars equation in the background of a cosmic string space-time by considering the principles of the Kaluza-Klein theory. We obtain the eigenvalue solution, which provides valuable insights into the behavior of the system. Next, we re-examine this system by considering the Feshbach-Villars quantum oscillator, a specific mathematical representation of the Feshbach-Villars equation. By employing this oscillator, we are able to analytically determine the eigenvalues of the system. This analytical solution allows for a deeper understanding of the system's characteristics. In the subsequent stage, we investigate the interaction between the Feshbach-Villars equation and oscillator in the presence of a cosmic dislocation within the framework of the Kaluza-Klein Theory. We analytically solve the wave equation associated with this system, shedding light on the behavior of the wave function. Furthermore, we analyze the influence of the topological defect (cosmic dislocation) on the quantification of energy and the wave function of the Feshbach-Villars oscillator. We also explore the impact of external fields on the system. These analyses provide valuable insights into how these factors affect the behavior and properties of the Feshbach-Villars oscillator.

Generalized dynamics and fluctuation-dissipation theorem for a parabolic potential.

This article revisits the fluctuation-dissipation relationship of a generalized Brownian particle constrained in a harmonic potential and immersed in a thermal bath whose degrees of freedom also interact with the external field. The generalized Langevin equationput forth herein keeps its original form. The different terms now depend on a compound strength related to those of the field and bath oscillators. The resulting theorem keeps Kubo's original form with the memory kernel depending on the frequency of the field. It has the consequence that friction is not a constant but is a function of the field frequency as predicted by molecular dynamics simulations. The position particle distribution and the Fokker-Planck equationassociated with the generalized Langevin equationare also derived.