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We study the quantum interference between the excitation pathways in a three-level $\Lambda$-type atom interacting with two short laser pulse trains under the conditions of electromagnetically induced transparency. The probability amplitude equations which describe the interaction of a three-level $\Lambda$-type atom with two laser pulse trains are numerically solved. We derive analytical expressions for the population of the upper excited state for resonant laser pulse trains with a rectangular temporal profile. By varying the parameters of the laser pulse trains such as area of a single pulse, detuning, repetition period, and number of individual pulses, we analyze the quantum interference between the excitation pathways in terms of the upper excited state population.

В даній роботі розв’язана задача дифракції природного, неполяризованого світла на ґратці, що утворена нескінченною періодичною послідовністю нескінченно тонких металевих стрічок. Застосовано квантовий підхід до її розв’язання. Розв’язок задачі дифракції природного, неполяризованого світла базується на результатах отриманих окремо для випадків дифракції Н- та Е-поляризованих фотонів, розв’язаних строгим методом крайової задачі Рімана-Гільберта. Робота присвячена розрахунку густини ймовірності знаходження Н- чи Е- поляризованого фотона в даній точці простору для того випадку коли потік фотонів, що падає на ґратку являє собою суміш рівної густини фотонів Н- та Е-поляризації. Оскільки дифракційна картина повторюється з періодом ґратки l, в роботі представлені результати розрахунків густини ймовірності |Ψ|2 в залежності від координати y для одного періоду в межах зміни y/l від нуля до одиниці. Із порівняння графіків |Ψ|2 для Н- та Е-поляризованих фотонів і їх суперпозиції встановлена залежність між числом максимумів в дифракційній картині природного світла та відношенням періоду ґратки до довжини хвилі фотона. Як виявилось число максимумів зростає пропорційно вказаному відношенню. Основні максимуми при цьому розташовуються проти щілин. Для неполяризованого, природного світла дифракційна картина, якісно більше схожа на відповідну картину для Н-поляризованих фотонів, але відрізняється висотою максимумів. При зростанні відношення періоду ґратки до довжини хвилі фотона спостерігається тенденція до вирівнювання їх висоти. In this paper, the problem of diffraction of natural light on a grating of infinite periodic sequence of infinitely thin metal strips is solved. A quantum approach was applied to its solution. The solution of the diffraction problem of a natural, unpolarized wave is based on the results obtained separately for the cases of diffraction of H- and E-polarized photons, solved by the strict method of the Riemann-Hilbert boundary value problem. The work is devoted to the calculation of the density of the probability of finding an H- or E-polarized photon at a given point in space for the case when the photon flow falling on the grating is a mixture of equal densities of H- and E-polarized photons. Since the diffraction pattern repeats with the grating period l, the paper presents the results of calculations of the probability density |Ψ|2 depending on the y coordinate for one period within the limits of the change of y/l from zero to one. From the comparison of graphs |Ψ|2 for H- and E- of polarized photons and their superposition, the relationship between the number of maxima in the diffraction pattern of natural light and the ratio of the grating period to the photon wavelength is established. As it turned out, the number of maxima increases proportionally to the specified ratio. At the same time, the main masses are located against the gaps. For unpolarized, natural light, the diffraction pattern is qualitatively similar to the corresponding pattern for H-polarized photons, but differs in the height of the maxima.

An approach for evaluating spectra from ground state probability amplitudes (GSPA) obtained from diffusion Monte Carlo (DMC) simulations is extended to improve the description of excited state energies and allow for coupling among vibrational excited states. This approach is applied to studies of the protonated water trimer and tetramer, and their deuterated analogs. These ions provide models for solvated hydronium, and analysis of these spectra provides insights into spectral signatures of proton transfer in aqueous environments. In this approach, we obtain a separable set of internal coordinates from the DMC ground state probability amplitude. A basis is then developed from products of the DMC ground state wave function and low-order polynomials in these internal coordinates. This approach provides a compact basis in which the Hamiltonian and dipole moment matrix are evaluated and used to obtain the spectrum. The resulting spectra are in good agreement with experiment and in many cases provide comparable agreement to the results obtained using much larger basis sets. In addition, the compact basis allows for interpretation of the spectral features and how they evolve with cluster size and deuteration.

In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.

In the present paper, we have proposed the experimentally achievable method for the characterization of the collective states of qubits in a linear chain. We study a temporal dynamics of absorption of a single-photon pulse by three interacting qubits embedded in a one-dimensional open waveguide. Numerical simulations were performed for a Gaussian-shaped pulse with different frequency detunings and interaction parameters between qubits. The dynamic behavior of the excitation probability for each qubit is investigated. It was shown that the maximum probability amplitudes of excitation of qubits are reached when the frequency of external excitation coincides with the frequency of excitation of the corresponding eigenstate of the system. In this case, the magnitude of the probability amplitude of each qubit in the chain unambiguously correlates with the contribution of this qubit to the corresponding collective state of the system, and the decay of these amplitudes is determined by the resonance width arising from the interaction of the qubit with the photon field of the waveguide. Therefore, we show that the pulsed harmonic probe can be used for the characterization of the energies, widths, and the wave functions of the collective states in a one dimensional qubit chain.

We investigate a periodicity of Grover walk on complete graphs with a self-loop at each vertex. We study an evolution matrix which steps forward a state of probability amplitude vector by using algebraic method. Then we find a periodic behaviour that the probability amplitude vector at each vertex gets back to initial state after some steps. It is shown that Grover walk on complete graphs on n vertices with a self-loop at each vertex is periodic with period 2n.

This paper argues that the path integral formulation of quantum mechanics suggests a form of holism for which the whole (total ensemble of paths) has properties that are not strongly reducible to the properties of the parts (the single trajectories). Feynman’s sum over histories calculates the probability amplitude of a particle moving within a boundary by summing over all the possible trajectories that the particle can undertake. These trajectories and their individual probability amplitudes are thus necessary in calculating the total amplitude. However, not all possible trajectories are differentiable, thus suggesting that they are not physical possibilities, but only mathematical entities. It follows that if the possible differentiable trajectories are taken to be part of the physical system, they are not sufficient to calculate the total probability amplitude. The conclusion is that the total ensemble is weakly non-supervenient upon the physically possible trajectories.

In Quantum ElectroDynamics (QED) the propagator is a function that describes the probability amplitude of a particle going from point A to B. It summarizes the many paths of Feynman’s path integral approach. We propose a reverse propagator (R-propagator) that, prior to the particle’s emission, summarizes every possible path from B to A. Wave function collapse occurs at point A when the particle randomly chooses one and only one of many incident paths to follow backwards with a probability of one, so it inevitably strikes detector B. The propagator and R-propagator both calculate the same probability amplitude. The R-propagator has an advantage over the propagator because it solves a contradiction inside QED, namely QED says a particle must take EVERY path from A to B. With our model the particle only takes one path. The R-propagator had already taken every path into account. We propose that this tiny, infinitesimal change from propagator to R-propagator would vastly simplify the mathematics of Nature. Many experiments that currently describe the quantum world as weird, change their meaning and no longer say that. The quantum world looks and acts like the classical world of everyday experience.

Розв’язана задача про дифракцію Е-поляризованих фотонів при нормальному падінні на ґратку, утворену необмеженою послідовністю нескінченно тонких металевих стрічок. Світло представляється як потік частинок – фотонів. Задача знаходження псі-функції фотона, розсіяного ґраткою, зводиться до граничної задачі Рімана-Гільберта. Розв’язок задачі отримано у вигляді нескінченної системи лінійних алгебраїчних рівнянь, що сходиться. Система придатна для будь-яких співвідношень між довжиною хвилі та періодом структури та будь-яких співвідношень між шириною щілини та періодом ґратки. Як витікає із порівняння де Бройлівського представлення псі-функції з її розкладом у ряд Фур’є, можливі значення складової імпульсу фотона, перпендикулярної до первинного напрямку руху, визначаються парними значеннями "кванта" імпульсу, величина яких залежить від періода ґратки. Фотони, пропущені або відбиті ґраткою, отримують дискретні значення імпульсу внаслідок взаємодії з ґраткою і відхиляються на дискретні кути від первинного напрямку. Як випливає з чисельних розрахунків, дифракційні максимуми розташовуються перед щілиною і мають деяку внутрішню структуру залежно від співвідношення між довжиною хвилі та періодом ґратки. При збільшенні відношення періоду ґратки до довжини хвилі дифракційний пік розділяється. Дифракційна картина спостерігається, коли це відношення більше одиниці. Коли воно стає менше одиниці, дифракційна картина зникає, маємо однорідну освітленість. Отже, значення вказаного відношення рівне одиниці є пороговим. The problem of diffraction of E-polarized light at normal falling on a grating of infinitely thin metallic strips is solved. Light is represented as a flux of particles – photons. The problem of determining the psifunction of a photon scattered by the grating is led down to the Riemann-Hilbert boundary problem. A strict solution is obtained in the form of a convergent infinite system of linear algebraic equations. The system equations are valid for any relation between wavelength and period of the structure and any relation between slit width and strip width. As follows from a comparison of the de Broglie representation of the psi-function and its decomposition into Fourier series, the possible values of the photon momentum component perpendicular to its initial direction of motion are determined by even values of the "quantum" of momentum, whose magnitude is determined by the grating period. Photons passed through or reflected by the grating get discrete values of momentum when interacting with the grating and deviate at discrete angles. Numerical calculations show that the diffraction maxima are located in front of the slit and have some internal structure that depends on the ratio between the grating period and the photon wavelength. As the ratio of the grating period to the photon wavelength increases, the diffraction peak splits. When the ratio becomes less than unity, the diffraction pattern disappears, we have a uniform illumination. Therefore, the value of the specified ratio, equal to one, is the threshold.

We compute the four-momentum radiated during the scattering of two spinless bodies, at leading order in the Newton's contant $G$ and at all orders in the velocities, using the Effective Field Theory worldline approach. Following \cite{Mougiakakos:2021ckm}, we derive the conserved stress-energy tensor linearly coupled to gravity generated by localized sources, at leading and next-to-leading order in $G$, and from that the classical probability amplitude of graviton emission. The total emitted momentum is obtained by phase-space integration of the graviton momentum weighted by the modulo squared of the radiation amplitude. We recast this as a two-loop integral that we solve using techniques borrowed from particle physics, such as reverse unitarity, reduction to master integrals by integration-by-parts identities and canonical differential equations. The emitted momentum agrees with recent results obtained by other methods. Our approach provides an alternative way of directly computing radiated observables in the post-Minkowskian expansion without going through the classical limit of scattering amplitudes., Comment: 41 pages, corrected typos

We discuss the problem of filtering out abnormal states from a larger number of quantum states. For this type of problem with [Formula: see text] items to be searched, both the traditional search by enumeration and classical Grover search algorithm have the complexity about [Formula: see text]. In this letter a novel quantum search scheme with exponential speed up is proposed for abnormal states. First, a new comprehensive quantum operator is well-designed to extract the superposition state containing all abnormal states with unknown number [Formula: see text] with complexity [Formula: see text] in probability 1 via well-designed parallel phase comparison. Then, every abnormal state is achieved respectively from [Formula: see text] abnormal states via [Formula: see text] times’ measurement. Finally, a numerical example is given to show the efficiency of the proposed scheme.

The formation of quantum vortices in photodetachment by a sequence of left- and right-handed circularly polarized laser pulses in various configurations is analyzed using either the strong-field approximation or numerical solution of the time-dependent Schr\"odinger equation. Two types of pulse sequences are considered: reducible and irreducible. While the former can be decomposed into series of consecutive and identical (irreducible) subtrains of pulses, the latter cannot be decomposed in that way. As we show, the vortex pattern in the probability amplitude of photodetachment is fully determined by the irreducible pulse configuration. Additional repetitions of an irreducible train create, in the three-dimensional momentum space, nonvortex nodal surfaces, the position of which is estimated. The conditions for the experimental observation of quantum vortices are also determined.

An efficient and accurate approach for using diffusion Monte Carlo (DMC) to study molecular clusters is described and applied to an investigation of (H2O)n with n = 1, 2, 3. In this approach, importance sampled DMC is used to describe the intramolecular degrees of freedom of the water monomers. The intermolecular degrees of freedom are treated using standard unguided DMC approaches. This hybrid approach allows us to use significantly smaller simulations to obtain converged zero-point energies and ground state wave functions than would be required for a fully unguided simulation of the same system. We demonstrate the efficiency, accuracy, and utility of this approach through a study of nine isotopologues of water trimer, focusing on how the ground state probability amplitude for these clusters samples the various low-energy minima on the potential surface.

Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (Yang in Sci Rep 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born’s Rule, Schrödinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to calculate measurement probability. Second, we provide a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also allows us to elegantly explain the double slit experiment, to describe the interaction history between the measured system and a series of measuring systems, and to calculate entanglement entropy based on path integral and influence functional. In return, the implementation brings back new insight to path integral itself by completing the explanation on why measurement probability can be calculated as modulus square of probability amplitude. Lastly, we clarify the connection between our reformulation and the quantum reference frame theory. A complete relational formulation of quantum mechanics needs to combine the present works with the quantum reference frame theory.

This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected.

We calculate the probability amplitude for tree-level elastic electron-muon scattering in Minkowski spacetime with carefully prepared initial and final wave packets. The obtained nonzero amplitude implies a nonvanishing probability for detecting a recoil electron outside the light-cone of the initial muon. Transposing this Minkowski-spacetime scattering result to a near-horizon spacetime region of a massive Schwarzschild black hole and referring to a previously proposed Gedankenexperiment, we conclude that, in principle, it is possible to have information transfer from inside the black-hole horizon to outside., Comment: 18 pages, 3 figures, v5: published version

We study the gravitational radiation emitted during the scattering of two spinless bodies in the post-Minkowskian Effective Field Theory approach. We derive the conserved stress-energy tensor linearly coupled to gravity and the classical probability amplitude of graviton emission at leading and next-to-leading order in the Newton's constant $G$. The amplitude can be expressed in compact form as one-dimensional integrals over a Feynman parameter involving Bessel functions. We use it to recover the leading-order radiated angular momentum expression. Upon expanding it in the relative velocity between the two bodies $v$, we compute the total four-momentum radiated into gravitational waves at leading-order in $G$ and up to an order $v^8$, finding agreement with what was recently computed using scattering amplitude methods. Our results also allow us to investigate the zero frequency limit of the emitted energy spectrum., 11 pages, 1 figure

Розв’язана задача про дифракцію Н-поляризованого світла при нормальному падінні на гратку, утворену необмеженою послідовністю нескінченно тонких металевих смуг. Застосовано квантово-механічний підхід до задачі дифракції. Світлова хвиля представлена у вигляді потоку частинок – фотонів. Ймовірність фотона в області перед та за граткою описується двовимірною псі-функцією – однозначною, неперервною та обмеженою, яка задовольняє двовимірному рівнянню Шредінгера. Строгий розв’язок задачі про визначення псі-функції фотона, розсіяного граткою, зводиться до граничної задачі Рімана-Гільберта. Розв’язок задачі отримано у вигляді нескінченної системи лінійних алгебраїчних рівнянь. Система придатна для будь-яких співвідношень між довжиною хвилі та періодом структури та будь-яких співвідношень між шириною щілини та шириною смуги, зручна для проведення чисельних розрахунків за допомогою ПК. Аналіз виразу для псі-функції дає можливість зробити наступний висновок. Фотони, пропущені або відбиті граткою, отримують дискретні значення імпульсу внаслідок взаємодії з граткою і відхиляються на дискретні кути від первинного напрямку. Максимуми інтенсивності на дифракційній картині з’являються в точках, куди надходять фотони, а мінімуми – в точках, куди фотони не надходять. Як випливає з аналізу значень імпульсу фотона, можливі значення складової імпульсу фотона, перпендикулярної до його початкового напрямку руху, визначаються парними значеннями "кванта" імпульсу, величина якого визначається періодом гратки. Цей результат може розглядатися як деяке правило відбору можливих значень складової імпульсу фотона. Як випливає з чисельних розрахунків, дифракційні максимуми розташовуються перед щілиною і мають деяку внутрішню структуру, залежно від співвідношення довжини хвилі та періоду гратки. При зменшенні відношення довжини хвилі до періоду гратки дифракційний пік виявляється злегка промодульованим. Коли відношення довжини хвилі до періоду гратки стає більше одиниці, дифракційна картина зникає, маємо однорідну освітленість. The problem of diffraction of H-polarized light at normal incidence on an unlimited sequence of infinitely thin metallic strips is solved. A quantum-mechanical approach to the problem of diffraction is applied. Light wave is represented as a flow of particles – photons. Probability of a photon in front of and behind the grating is described by two-dimensional psi-function – single-valued, continuous and restricted, which satisfies the two-dimensional Schrödinger equation. A strict solution of the problem about determination of psi-function of a photon dissipated by the grating is led down to the boundary Riemann-Hilbert problem. The solution is obtained in the view of the convergent infinite system of linear algebraic equations. The system is suitable for any relation between the wavelength and the period of the structure and any relation between the width of a slit and the width of a strip and is convenient for numerical calculations with the help of PC. Analysis of the expressions obtained for psi-function gives the possibility to conclude the following. Photons passed through or reflected by the grating get the discrete values of momentum during the interaction with the grating and deviate at discrete angles which are determined by the obtained expressions. There are intensity maxima in the points where photons come and minima – in the points where photons do not come. As follows from the analysis of the values of the photon momentum, the possible values of the constituent of the photon momentum perpendicular to its initial direction of motion are determined by even values of the "quantum" of momentum which magnitude is determined by the grating period. This result may be examined as a rule of selection of possible values of the perpendicular constituent of the photon momentum. As follows from numerical calculations, the diffraction maxima are located in front of a slit and have some internal structure, depending on the relation between the wavelength and the grating period. When the ratio of the wavelength to the grating period decreases, the diffraction peak turns out to be slightly modulated. When the ratio of the wavelength to the grating period is more than one, the diffraction pattern vanishes and we have homogeneous illuminance.

We discuss the decay of a two-level system into an engineered reservoir of coupled harmonic oscillators in the single-excitation manifold and propose its optical simulation with an homogeneous chain of coupled waveguides where individual elements couple to an external waveguide. We use two approaches to study the decay of the optical analogue for the probability amplitude of the two-level system being in the excited state. A Born approximation allows us to provide analytic closed-form amplitudes valid for small propagation distances. A Fourier-Laplace approach allows us to estimate an effective decay rate valid for long propagation distances. In general, our two analytic approximations match our numerical simulations using coupled mode theory and show non-Markovian decay into the engineered reservoir. In particular, we focus on two examples that provide enhancement or suppression of the decay decay rate using flat-top or Gaussian coupling distributions., 17 pages, 8 figures

Generation of electron vortices in photodetachment of ${\mathrm{H}}^{\ensuremath{-}}$ by circularly polarized laser pulses is analyzed by means of strong-field approximation and by numerically solving the time-dependent Schr\"odinger equation. A very good agreement is shown for the magnitude and the phase of the probability amplitude of photodetachment from both approaches. We demonstrate that spiral-like patterns in the probability amplitude of detachment, observed for a pair of counter-rotating circularly polarized laser pulses, cannot be associated with nonvanishing topological charge vortices. The latter can be generated, on the other hand, by a circularly polarized laser pulse or a sequence of such pulses with corotating polarizations. Such interpretation of our results follows from the hydrodynamical formulation of quantum mechanics and its generalization to arbitrary parametric spaces.

Generation of electron vortices in photodetachment driven by linearly polarized laser pulses is demonstrated, both within the strong-field approximation and by numerically solving the time-dependent Schr\"odinger equation. The sensitivity of the resulting vortex patterns in the momentum distributions of photoelectrons to the laser pulse parameters (including the pulse duration, its intensity, and the carrier-envelope phase) is analyzed. It is shown that, for a nonzero carrier-envelope phase of the driving pulse, vortex structures in the probability amplitude of detachment occur, similar to von K\'arm\'an vortex streets in fluid mechanics. In addition, hexagonal regions of high probability (honeycomb patterns) are observed, with vortex-antivortex pairs located at their vertices.

We introduce the Quantum Mechanics formalism given by the different operations at the Hilbert space yielding the probability amplitude from a primary source to an target receptor that is seen as the fusion center of all signals. More than an abstract application, we develop a closed-form theory that is applied to certain cases where standards theoretical approaches might fail to describe nonlinear phenomenology of wireless-based events. In concrete we calculate the probability amplitude and therefore it is interpreted as the chance for getting a pure signal from a primary source. Taking into account the completeness property of the orthogonal states, we proceed to implement the noises that would appear as a spontaneous mechanism more than an “adhoc” procedure. Finally, we pass to determinate numerically the signals along the bands that commonly take place the wireless connectivity.

The accurate identification and characterization of microbiological species is a must that allows us to design engineered pharmacology to tackle down the diversity of diseases derived of them. This paper presents a study about the usage of both classical dynamical and electrodynamics in conjunction to the Feynman’s path integral to describe as well as identify the displacement of bacteria in closed spaces. Our methodology consists in the in the usage of the usage of probability amplitude to investigate the theoretical motility of bacteria can be disturbed through electrical interactions from the fact that them contain ions in their biochemical composition.Our study yields that bacteria might exhibit a pattern of probabilities as function of space and time. It would be advantageous for an engineered nanodevice that would sense them uniquely through electrical interactions. © 2020 ASTES Publishers. All rights reserved.

In a prospective scenario of nanomedicine, an engineered cargo of nanoparticles would be crucial to alleviate diseases that might to require of a precise doses of biochemical compounds in situ. In this paper, the technique of Feynman's propagator to estimate the probability amplitude in the different paths of a charged nanoparticle that would opt to reach a beta cell, is employed. The aim is to generate the artificial releasing of granules of insulin in beta cells to avoid hyperglycemia in those diabetes patients. For this end the quantum mechanics evolution operator is employed. Once the square of the amplitudes are calculated, it leads to analyzed the cases in where repulsive electric forces might to cause to decrease the chance of nanoparticle to go through the cell. Computational simulations are presented.

This thesis concerns quantum states that are superpositions of discrete or of continuum energy levels. It develops mathematical methods to characterize the physical properties of these states and finds important consequences. Regarding rational extensions of the harmonic oscillator, no other authors had constructed the coherent states associated with the ladder operators of Marquette and Quesne. It was our aim to proceed with this construction, then to calculate physical properties of these coherent states, looking for classical or quantum behaviour. The method of partial wave analysis was known to be not applicable to the Coulomb potential for the scattering of plane waves. It was my aim to repeat the calculation for wavepacket scattering, to investigate whether the method became applicable. Other authors have constructed relativistic probability amplitudes and compared them to the scalar Klein-Gordon amplitude. In this context, it was my aim to question the conclusion of a minimum width for a massive particle. In the same context, I investigated the possibility of a four-vector probability current with the Newton-Wigner position probability density as the zero component. In Chapter 12, my aim was to construct a measure of localization for the photon, knowing from the work of other authors that a point-localized photon state is not possible.In Chapters 3, 4 and 5, our measures of classicality for the Barut-Girardello coherent states included plotting energy expectations as functions of the magnitude of the coherent state para- meter, z, position probability densities for the coherent states and the even and odd cat states, the Wigner functions for particular choices of z, the two-photon number probability distribu- tion and the linear entropy for a coherent state on a beamsplitter, position spreads to look for squeezing and the Mandel Q parameter. In Chapter 6, it was necessary to approximate an integral containing a particular Wigner rotation matrix. An approximation of general Wigner rotation matrices, at low angle, was derived and its precision evaluated numerically in Chapter 7. The exact values for the Coulomb phase shifts were used to construct the incoming wavefunction. Time reversal and a rotation were used to construct the outgoing wavefunction. The sums over l were evaluated numerically. In Chapter 10, four arguments were used against the minimum width. Comparison of the scalar amplitude and the relativistic probability amplitude for a Newton-Wigner localized state showed the root of the problem. A Lorentz transformation was applied to the scalar function to see if the “minimum” width became even smaller. A simple construction produced scalar functions with arbitrarily small widths. A relevant argument from Almeida and Jabs was included here. In Chapter 11, the algebra involving the four components of a four-vector current (at the origin) and the boost generators was derived. All of these commutators must be satisfied by a four-vector, providing a test for the claimed four-vector probability currents of other authors. In Chapter 12, field expectations were taken in a coherent state with a mean photon number of unity, built from identical single-photon states.In Chapters 3, 4 and 5, we found a mix of classical and quantum behaviour for the coherent states. In none of the systems studied did we see the high degree of classicality of the unmodified harmonic oscillator. The investigation of cat states was relevant to the fundamental problem of measurement in quantum mechanics. In Chapter 6, the use of wavepackets introduced a convergence factor into the sum over l, which diverges in the plane wave treatment. The elimination of a divergence is the most significant result of this thesis. Scattering probabilities remained finite at all angles, in disagreement with the diverging Rutherford formula, but only on small angular regions around forward scattering. Future work will investigate the measurability of the phenomena predicted here. In Chapter 10, all four arguments pointed to no minimum width for a massive particle. In Chapter 11, it was found that there can be no four-vector current density with the Newton-Wigner position probability density as the zero component, for any spin. Other authors were found to be in error when they claimed to have produced such a four-vector current density. This result strengthens the position of the Dirac current as the appropriate current for an electron. In Chapter 12, the coherent state field expectations were found to provide a meaningful measure of localization. In summary, wavepackets give a realism to the description of a scattering experiment that is absent with the use of plane waves. Phenomena not captured by the plane wave scattering amplitude, such as time delays, are calculable in the wavepacket context. Relativistic probability amplitudes provide a complete description of free particles, consistent with quantum mechanics and with special relativity. They predict agreement with the Heisenberg uncertainty principle and relativistic effects such as Lorentz contraction. The absence of relativistic probability amplitudes from standard textbooks on relativistic quantum mechanics is a serious omission.

The wavefunction is an important element of the multiscale quantum harmonic oscillator algorithm (MQHOA). To verify the physical model and the multimode optimization performance of the MQHOA for multimode optimization in this paper, we define a multidimensional harmonic-Gaussian potential function. When the wavefunction of the MQHOA for multimode optimization and the probability amplitude of an ammonia molecule are compared, the ground-state wavefunction can reflect the probability distribution of the two-state ammonia molecule. We obtain the extrema of the proposed function using the Hessian matrix and optimize the proposed function with the multimode algorithm. The experiments show that the multimode algorithm can determine the extrema of the proposed function with appropriate parameters. Changes in the proposed function barriers have little effect on the optimization ability of the multimode algorithm. The optimization ability of the multimode algorithm is determined by its own wavefunction.

We investigate the response of a quantum particle in the Tietz-Hua quantum potential driven by biharmonic fields: a low-frequency force and a very high frequency force. The response is characterized by the occurrence of a maximum in the first-order transition probability amplitude ${|s|}^{2}$ under the influence of the applied fields. It is shown that in the absence of the high-frequency component of the applied fields, ${|s|}^{2}$ shows a distinct sequence of resonances, whereas an increase in the amplitude of the high-frequency field induces minima in ${|s|}^{2}$. However, the ${|s|}^{2}$ maximum occurs in the low-frequency regime where it may be considered otherwise weak in the presence of a single harmonic force.

Rabi oscillation, originally proposed in nuclear magnetic resonance, is a well-known phenomenon associated with a driven two-level system. Although magnetic fields typically can bring about chirality into unusual phenomena such as chiral edge states in the quantum Hall effect, it is not clear if chirality exists in Rabi oscillations. Here we unveil the intrinsic chirality carried by the phase in a Rabi problem. For opposite detuning of the driving field, the phase evolution of the probability amplitude exhibits a mirror symmetry. Consequently, constructive or destructive interference of two off-resonant Rabi processes under different initial conditions is level dependent and symmetry protected. Experimentally, we demonstrate such features in a photonic setting with adjustable detuning, yet our results may prove pertinent to the study of similar phenomena in other driven two-level systems beyond photonics.

The probability amplitudes for quantum entanglement, also known as Bell sates, are utilized to arrive explicitly at the identity matrix I and the $$\sigma_{x}$$σx, $$\sigma_{y}$$σy, and $$\sigma_{z}$$σz Pauli matrices, via a straight-forward $$2 \times 2$$2×2 matrix representation that utilizes the vector direct product. It is also indicated that this approach is completely equivalent to the utilization of the Kronecker product $$\otimes$$⊗ to multiply the relevant ket vectors. Furthermore, it is shown that the polarization rotation matrix R, operating on the various versions of the probability amplitude for quantum entanglement, yields four identities elegantly interconnecting these probability amplitudes.

Quantum measurement and quantum operation theory is developed here by taking the relational properties among quantum systems, instead of the independent properties of a quantum system, as the most fundamental elements. By studying how the relational probability amplitude matrix is transformed and how mutual information is exchanged during measurement, we derive the formulation that is mathematically equivalent to the traditional quantum measurement theory. More importantly, the formulation results in significant conceptual consequences. We show that for a given quantum system, it is possible to describe its time evolution without explicitly calling out a reference system. However, description of a quantum measurement must be explicitly relative. Traditional quantum mechanics assumes a super observer who can instantaneously know the measurement results from any location. For a composite system consists space-like separated subsystems, the assumption of super observer must be abandoned and the relational formulation of quantum measurement becomes necessary. This is confirmed in the resolution of EPR paradox. Information exchange is relative to a local observer in quantum mechanics. Different local observers can achieve consistent descriptions of a quantum system if they are synchronized on the information regarding outcomes from any measurement performed on the system. It is suggested that the synchronization of measurement results from different observers is a necessary step when combining quantum mechanics with the relativity theory.

We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which could be made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are actually made, the paths, which may now be called “real”, acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensembles, whose conflicting attributes may, although by no means should, appear “paradoxical”. We describe in detail the ensembles, resulting from intermediate measurements of mutually commuting, or non-commuting, operators, in terms of the real paths produced. In the same manner, we analyse the Hardy’s and the “three box” paradoxes, the photon’s past in an interferometer, the “quantum Cheshire cat” experiment, as well as the closely related subject of “interaction-free measurements”. It is shown that, in all these cases, inaccurate “weak measurements” produce no real paths, and yield only limited information about the virtual paths’ probability amplitudes.

The main idea of this paper is motivated by a paradigm from quantum physics where the probability amplitude is built as a superposition of the wave functions of states, and the squared modulus of amplitude defines the probability of state membership. Similar linear aggregates are used in classical physics for description of wave interference effects. In contrast to regular techniques of probability estimation in social-economic research (such as logistic regression, multinomial-logit (MNL), discrete choice modeling (DCM), conjoint, best-worst scaling (BWS), and other methods), the proposed approach of probability amplitude modeling permits finding choice probabilities themselves and demonstrates possible interference phenomena of entanglement of different choices. Particularly, a BWS example evaluated by complex utility MNL demonstrates how a choice of each item is composed from its pure-state and mix-state probabilities. The obtained numerical results are supportive of theoretical considerations and practical applications of the probability amplitude modeling, and can serve for better understanding and evaluation of choice decisions.

For many physical systems, the fractional-order system can describe the dynamics more accurately comparing with the integer-order system. However, its parameters, which include not only the identification of model coefficients but also the estimation of fractional orders, are more difficult to be identified. This paper proposes a modified quantum bacterial foraging algorithm (MQBFA) for parameters identification of the fractional-order system, and the MQBFA algorithm is a relatively new issue in recent years with strong engineering backgrounds. Not only does the proposed MQBFA algorithm apply the real number encoding of the single gene to effectively perform encoding, but it also utilizes a continuously varying rotation angle to update the rotation gate. First, a real number encoding of the single gene is designed to improve the searching efficiency of the algorithm. Second, an improved quantum rotation angle is applied to update the rotation angle continuously and adaptively. Third, the modificatory factor of probability amplitude is introduced to enrich the population diversity. In addition, to prevent the generation of invariant solutions in the early stage of quantum evolution, probability of optimal rotation angle is presented. Furthermore, the convergence of the proposed algorithm is analyzed. Based on several benchmark functions and parameters identification of the fractional-order system, simulation results and comparisons demonstrate the effectiveness of the proposed MQBFA algorithm.

CS (compressive sensing) is adept in the signals of X-ray pulsars. The observation matrix in CS is a core technology. In the CS for X-ray pulsars, the observation matrix has three significant indexes: simple computation, arbitrary size and high precision. To optimize the observation matrix, we propose a quick quantum-based CS, and apply it to the pulsar positioning and velocimetry approach. In the quantum-based CS, the profile dictionary with multi-skewness-degrees and multi-phases are comprised of the X-ray pulsar profiles with multiple phases and skewness degrees. This pulsar profile dictionary is also adopted as the probability amplitude of quantum. The multiple quantum observations of the dictionary form the quantum observation matrix. The quantum observation matrix explores the uncertainty of quantum, and improves its diversity. As the super-resolution sparse recovery, the quantum matched estimation approach provides the pulsar positioning and the pulsar velocimetry simultaneously. Simulation results indicate that compared with the χ2 velocimetry, the quick quantum CS-based pulsar positioning and velocimetry has highly accurate and less calculation.

The Boyd Conjecture is that the amplitude of an elementary ray from the Theory of Elementary Waves (TEW) is thephysical analog of a probability amplitude from Quantum Mechanics (QM). Boyd learned TEW from his cousin Lewis E.Little, who had said that quantum math is the roadmap to the world of elementary waves, but we didn’t know how to readthe map. He suggested that Boyd, with a degree in mathematics, should work on deciphering the map. In 2014 Boydbegan thinking that amplitudes are the core of QM, and elementary rays the core of TEW: perhaps they were equivalent.Both decisively influence particle behavior without conveying any energy. This essay discusses the history, strengths,weakness and implications of this conjecture.

A specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations is presented. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closedform solutions each standing for a particular two-state model, which generally is only conditionally integrable in the sense that for these field configurations the amplitude and phase modulation functions are not varied independently. However, there exists at least one exception when the model is unconditionally integrable, that is the Rabi frequency and the detuning of the driving optical field are controlled independently. This is a constant-amplitude periodic level-crossing model, for which the detuning in a limit becomes a Dirac delta-comb configuration with variable frequency of the levelcrossings. The exact solution for this model is derived, the Floquet exponents are determined and study of the population dynamics in the system for various regions of the input parameters is done.

It has been argued the total vibrational probability amplitude for a molecular nonadiabatic eigenfunction should not have nodes unless required by symmetry. For a model with two nonadiabatically coupled electronic states, exact factorization of nonadiabatic eigenfunctions into a total vibrational probability amplitude and a normalized electronic factor reveals zero-dimensional nodes in nonadiabatic eigenfunctions over a two-dimensional vibrational space. These conical nodes have the shape of a right elliptical cone with the node at the vertex. Low dimensionality nodes are likely when the number of nonadiabatically coupled electronic states is less than or equal to the dimensionality of the vibrational space.

Analytic expressions for the probability density of states of a molecule with internal rotation and the probability of finding the state in the potential well are derived for the first time. Two methods are proposed for assigning conformers to potential wells. A quantitative measure of localization and delocalization of a state in the potential well is introduced. The rotational symmetry number is generalized to the case of asymmetric rotation. On the basis of the localization criterion, a model is developed for calculating the internal rotation contribution to thermodynamic properties of individual conformers with low rotational barriers and/or at a high temperature.

The well-known probability amplitude equation for two entangled quanta (n = 2) traveling in two different directions (N = 2) is explicitly extended to higher multiples of quanta (n = 4, 6, 8...)and higher even multiples of propagation channels (N = 4, 6, 8...). The case of N = 3, involves Hamilton’s quaternions, and is also outlined.

A new expression for the scattering matrix in local quantum field theory is obtained on the basis of a generalized formulation of the microcausality condition. This expression is relativistically covariant, unitary, and causal and coincides with the generally accepted expression (the T-exponential) in the case when ultraviolet (UV) divergences are absent in the latter. Conditions are formulated, under the fulfillment of which the elements of the constructed matrix do not contain UV divergences. By way of an example, the probability amplitude of the particle–particle transition for a model with ℒ(x) = λ : φ 3(x): in the second-order perturbation theory is found.

The problem of subsalt imaging in the presence of salt-boundary uncertainty is considered. In this case, the salt-body model should be considered by a probability density function rather than by a unique deterministic function. We discuss a way to look at seismic imaging using the path-integral concept. The method computes the image by summing the contributions of individual images computed for different statistical realizations of the salt-body boundaries. It samples different realizations instead of relying on only one model derived from manual or automatic picking. The focusing mechanism is achieved by a weighting function (probability amplitude), which is designed to emphasize contributions from models close to the stationary one and to suppress contributions from unlikely models. The presented examples demonstrate principles and feasibility of the new concept.

We have investigated the operation of a composite detector comprising of elements arranged in the shape of a well. A probabilistic formalism is introduced in terms of only one probability amplitude. Considering up to four detector hit events, gamma-ray absorptions and gamma-ray scatterings both inside and outside the composite detector are studied. Our results show that there are three groups of modules based on the values of the peak-to-background ratio of a module.

We consider a harmonic oscillator (HO) with a time dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency \omega_{0}, then, at t = 0, its frequency suddenly increases to \omega_{1} and, after a finite time interval \tau, it comes back to its original value \omega_{0}. Contrary to what one could naively think, this problem is a quite non-trivial one. Using algebraic methods we obtain its exact analytical solution and show that at any time t > 0 the HO is in a squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from \omega_{0} to \omega_{1}), remaining constant after the second jump (from \omega_{1} back to \omega_{0}). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state., Comment: 26 pages, 9 figures

Aiming at the problem that quantum genetic algorithm is easy to fall into local optimization in the process of function optimization under the original framework, an improved quantum genetic algorithm using population partition method and dynamic inverse probability amplitude strategy is proposed. In the process of optimization, the improved algorithm utilizes the superior individuals in each generation of populations to construct a population genotype suitable for evolution through the individual binary form. On this basis, replace the inferior individuals in the population with a certain proportion, and then construct a number of similar individuals that tend to the genotype of the superior individual. Therefore, the algorithm can continuously improve the adaptability of the problem and gradually converge to the optimal solution. At the same time, for the premature or local optimization problem that the algorithm may be trapped, in the period of algorithm stagnation, the population diversity is enriched by resetting the genotype of some individuals and narrowing the search space by using the current optimal solution.

We propose and demonstrate phonon-number-resolving detection of the multiple local phonon modes in a trapped-ion chain. To mitigate the effect of phonon hopping during the detection process, the probability amplitude of each local phonon mode is mapped to the auxiliary long-lived motional ground states. Sequential state-dependent fluorescence detection is then performed. In the experiment, we have successfully observed the time evolution of two local phonon modes in two ions, including the phonon-number correlation between the two modes., 5 pages, 3 figures

A bound state between a quantum emitter (QE) and surface plasmon polaritons (SPPs) can be formed, where the excited QE will not relax completely to its ground state and is partially stabilized in its excited state after a long time. We develop some theoretical methods for investigating this problem and show how to form such a bound state and its effect on the non-Markovian decay dynamics. We put forward an efficient numerical approach for calculating the analytical part of the self-energy for frequency below the lower energy threshold. We also propose an efficient formalism for obtaining the long-time value of the excited-state population without calculating the eigenfrequency of the bound state or performing a time evolution of the system, in which the probability amplitude for the excited state in the steady limit is equal to one minus the integral of the evolution spectrum over the positive frequency range. With the above two quantities obtained, we show that the non-Markovian decay dynamics of an initially excited QE can be efficiently obtained by the method based on the Green’s function expression for the evolution operator when a bound state exists. A general criterion for identifying the existence of a bound state is presented. The performances of the above methods are numerically demonstrated for a QE located around a metal nanosphere and in a gap plasmonic nanocavity. Numerical results show that these methods work well and the QE becomes partially stabilized in its excited state at a long time for the transition dipole moment beyond its critical value. In addition, it is also found that this critical value is heavily dependent on the distance between the QE and the metal surface, but nearly independent on the size of the nanosphere or the rod. Our methods can be utilized to understand the suppressed decay dynamics for a QE in an open quantum system and provide a general picture on how to form such a bound state.