Feynman Path Integral Approach Research Articles

On the fractional perturbation theory and optical transitions in bulk semiconductors: Emergence of negative damping and variable charged mass

This study is devoted to the construction of a fractional perturbation theory which represents the basic technique of approximation in quantum mechanics. Our analysis is based on the fractional actionlike variational approach (FALVA) implementation of fractionality. After discussing the basic properties of fractional harmonic oscillator through the fractional Feynman's path integral approach, we have discussed the Rabi oscillations within the context of two-level quantum systems for simulated optical transition. In particular the case α>1 is correlated to the emergence of negative damping and negative probabilities which have been obtained in several quantum out-of-equilibrium systems. The fractional Fermi Golden rule and the fractional Heisenberg's uncertainty principle have been also obtained and their main properties and features have been discussed accordingly. We have discussed optical transitions in bulk semiconductors. It was observed that the fundamental properties of transitions rates are recovered if the charged particle moving inside the semiconductor is variable with time.

Multiple phonon modes in Feynman path-integral variational polaron mobility

The Feynman path-integral variational approach to the polaron problem, along with the associated Feynman-Hellwarth-Iddings-Platzman (FHIP) linear-response mobility theory, provides a computationally amenable method to predict the frequency-resolved temperature-dependent charge-carrier mobility, and other experimental observables in polar semiconductors. We show that the FHIP mobility theory predicts non-Drude transport behavior, and shows remarkably good agreement with the recent diagrammatic Monte Carlo mobility simulations of Mishchenko et al. [Phys. Rev. Lett. 123, 076601 (2019)] for the abstract Fr\"ohlich Hamiltonian. We extend this method to multiple phonon modes in the Fr\"ohlich model action. This enables a slightly better variational solution, as inferred from the resulting energy. We carry forward this extra complexity into the mobility theory, which shows a richer structure in the frequency and temperature-dependent mobility, due to the different phonon modes activating at different energies. The method provides a computationally efficient and fully quantitative method of predicting polaron mobility and response in real materials.

Balance and Imbalance p-wave superfluid in BCS-BEC crossover regime at finite temperature: Feynman Path Integral approach

The effect of population imbalance in p-wave superfluids in the BCS-BEC crossover regime at finite temperatures is studied through the path integral formalism. Using the Hubbard–Stratonovich transformation, the partition function of the system is cast from quaternary to quadratic fields and the thermodynamic potential in the Saddle-point approximation in presence of fluctuation is derived. Similarly, the probability of condensation, the normal and superfluid densities are calculated and we find that in the weak coupling regime, the population imbalance is not favorable to superfluid density. Also, it is shown that in the presence of the population imbalance, the condensate fraction exhibits a phase transition and that from a certain value of the temperature the system deviates robustly against the population imbalance for p-wave pairing and that when one approaches the strong coupling, the atoms are strongly correlated and the configuration of the system is forbidden for a certain value of the coupling and tends to merge into evidence that the system is robust against population imbalance as one moves away from the strong coupling regime. Finally, we observed that the chemical potential increases with the population imbalance as we get closer to the strong coupling regime.

Ray-wave duality of electromagnetic fields: a Feynman path integral approach to classical vectorial imaging.

We consider how vectorial aspects (polarization) of light propagation can be implemented and their origin within a Feynman path integral approach. A key part of this scheme is in generalizing the standard optical path length integral from a scalar to a matrix quantity. Reparametrization invariance along the rays allows a covariant formulation where propagation can take place along a general curve. A general gradient index background is used to demonstrate the scheme. This affords a description of classical imaging optics when the polarization aspects may be varying rapidly and cannot be neglected.

Deep Learning for Feynman's Path Integral in Strong-Field Time-Dependent Dynamics.

Feynman's path integral approach is to sum over all possible spatiotemporal paths to reproduce the quantum wave function and the corresponding time evolution, which has enormous potential to reveal quantum processes in the classical view. However, the complete characterization of the quantum wave function with infinite paths is a formidable challenge, which greatly limits the application potential, especially in the strong-field physics and attosecond science. Instead of brute-force tracking every path one by one, here we propose a deep-learning-performed strong-field Feynman's formulation with a preclassification scheme that can predict directly the final results only with data of initial conditions, so as to attack unsurmountable tasks by existing strong-field methods and explore new physics. Our results build a bridge between deep learning and strong-field physics through Feynman's path integral, which would boost applications of deep learning to study the ultrafast time-dependent dynamics in strong-field physics and attosecond science and shed new light on the quantum-classical correspondence.

Path-integral approach to the calculation of the characteristic function of work.

Work statistics characterizes important features of a nonequilibrium thermodynamic process, but the calculation of the work statistics in an arbitrary nonequilibrium process is usually a cumbersome task. In this work, we study the work statistics in quantum systems by employing Feynman's path-integral approach. We derive the analytical work distributions of two prototype quantum systems. The results are proved to be equivalent to the results obtained based on Schrödinger's formalism. We also calculate the work distributions in their classical counterparts by employing the path-integral approach. Our study demonstrates the effectiveness of the path-integral approach for the calculation of work statistics in both quantum and classical thermodynamics, and brings important insights to the understanding of the trajectory work in quantum systems.

The sojourn time of the inverted isotropic harmonic oscillators in a tilted magnetic field

By employing the Feynman path integral approach, we investigate the sojourn time of a two-dimensional (2D) and a three-dimensional (3D) inverted isotropic harmonic oscillator under a tilted magnetic field. It is shown that the sojourn time in the 2D case behaves quite differently from the 3D case. In the 3D case, the sojourn time is a monotonically increasing function of the strength of the magnetic field, and its value approaches a finite value as the strength of the magnetic field approaches infinity. In the 2D case, the behavior of the sojourn time as a function of the strength and the angle θ of inclination of the magnetic field is much complicated. When the angle of inclination , the system is unstable and the sojourn time is always finite. However, when , the system possesses a stable region with infinite large sojourn time as the strength of the magnetic field varies. The reason behind is explained via stability region analysis.

Coarse-graining of many-body path integrals: Theory and numerical approximations.

Feynman's imaginary time path integral approach to quantum statistical mechanics provides a theoretical formalism for including nuclear quantum effects (NQEs) in simulation of condensed matter systems. Sinitskiy and Voth [J. Chem. Phys. 143, 094104 (2015)] have presented the coarse-grained path integral (CG-PI) theory, which provides a reductionist coarse-grained representation of the imaginary time path integral based on the quantum-classical isomorphism. In this paper, the many-body generalization of the CG-PI theory is presented. It is shown that the N interacting particles obeying quantum Boltzmann statistics can be represented as a system of N pairs of classical-like pseudoparticles coupled to each other analogous to the pseudoparticle pair of the one-body theory. Moreover, we present a numerical CG-PI (n-CG-PI) method applying a simple approximation to the coupling scheme between the pseudoparticles due to numerical challenges of directly implementing the full many-body CG-PI theory. Structural correlations of two liquid systems are investigated to demonstrate the performance of the n-CG-PI method. Both the many-body CG-PI theory and the n-CG-PI method not only present reductionist views of the many-body quantum Boltzmann statistics but also provide theoretical and numerical insight into how to explicitly incorporate NQEs in the representation of condensed matter systems with minimal additional degrees of freedom.

The Arbitrary l-state Solutions of the Hellmann Potential by Feynman Path Integral Approach

In this study, we used the Path integral method to obtain the bound state solutions of the Hellmann potential. Firstly we analytically derived the radial kernel expression of the Hellmann potential using the approximation of the centrifugal term and space-time transformations. Then we calculated the exact energy spectrum and the normalized eigenfunction from the poles of the Green function and their residues. We expressed normalized wave functions in terms of Jacobi polynoms and Hypergeometric functions.

Quantum option pricing and data analysis

The paper proposes to treat financial models using techniques of quantum mechanics. The methodology relies on the Dirac matrix formalism and the Feynman path integral approach. This leads us to reexamine in this framework the classical option pricing models of Cox-Ross-Rubinstein and Black-Scholes. Moreover, financial data are classified with respect to the spectrum of a certain observable and then analyzed to identify price jumps using supervised machine learning tools.

Channel Capacity Calculation at Large SNR and Small Dispersion within Path-Integral Approach

We consider the optical fiber channel modelled by the nonlinear Shrödinger equation with additive white Gaussian noise. Using Feynman path-integral approach for the model with small dispersion we find the first nonzero corrections to the conditional probability density function and the channel capacity estimations at large signal-to-noise ratio. We demonstrate that the correction to the channel capacity in small dimensionless dispersion parameter is quadratic and positive therefore increasing the earlier calculated capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region. Also for small dispersion case we find the analytical expressions for simple correlators of the output signals in our noisy channel.

Next-to-leading-order corrections to capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region.

We consider the optical fiber channel modeled by the nonlinear Schrödinger equation with zero dispersion and additive Gaussian noise. Using the Feynman path-integral approach for the model, we find corrections to conditional probability density function, output signal distribution, conditional and output signal entropies, and the channel capacity at large signal-to-noise ratio. We demonstrate that the correction to the channel capacity is positive for large signal power. Therefore, this correction increases the earlier calculated capacity for a nondispersive nonlinear optical fiber channel in the intermediate power region.

Generalized Hartree-Fock-Bogoliubov description of the Fröhlich polaron

We adapt the generalized Hartree-Fock-Bogoliubov (HFB) method to an interacting many-phonon system free of impurities. The many-phonon system is obtained from applying the Lee-Low-Pine (LLP) transformation to the Frohlich model which describes a mobile impurity coupled to noninteracting phonons. We specialize our general HFB description of the Frohlich polaron to Bose polarons in quasi-1D cold atom mixtures. The LLP transformed many-phonon system distinguishes itself with an artificial phonon-phonon interaction which is very different from the usual two-body interaction. We use the quasi-one-dimensional model, which is free of an ultraviolet divergence that exists in higher dimensions, to better understand how this unique interaction affects polaron states and how the density and pair correlations inherent to the HFB method conspire to create a polaron ground state with an energy in good agreement with and far closer to the prediction from Feynman's variational path integral approach than mean-field theory where HFB correlations are absent.

Double path integral method for obtaining the mobility of the one-dimensional charge transport in molecular chain.

We report on a theoretical investigation concerning the polaronic effect on the transport properties of a charge carrier in a one-dimensional molecular chain. Our technique is based on the Feynman's path integral approach. Analytical expressions for the frequency-dependent mobility and effective mass of the carrier are obtained as functions of electron-phonon coupling. The result exhibits the crossover from a nearly free particle to a heavily trapped particle. We find that the mobility depends on temperature and decreases exponentially with increasing temperature at low temperature. It exhibits large polaronic-like behaviour in the case of weak electron-phonon coupling. These results agree with the phase transition (A.S. Mishchenko et al., Phys. Rev. Lett. 114, 146401 (2015)) of transport phenomena related to polaron motion in the molecular chain.

A study on the optimization of finite volume effects of B K in lattice QCD by using the CUDA

Lattice quantum chromodynamics (QCD) is the non-perturbative implementation of field theory to solve the QCD theory of quarks and gluons by using the Feynman path integral approach. We calculate the kaon CP (charge-parity) violation parameter B K generally arising in theories of physics beyond the Standard Model. Because lattice simulations are performed on finite volume lattices, the finite volume effects must be considered to exactly estimate the systematic error. The computational cost of numerical simulations may increase dramatically as the lattice spacing is decreased. Therefore, lattice QCD calculations must be optimized to account for the finite volume effects. The methodology used in this study was to develop an algorithm to parallelize the code by using a graphic processing unit (GPU) and to optimize the code to achieve as close to the theoretical peak performance as possible. The results revealed that the calculation speed of the newly-developed algorithm is significantly improved compared with that of the current algorithm for the finite volume effects.

Numerical Path Integral Approach to Quantum Dynamics and Stationary Quantum States

AbstractApplicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined. Coherent quantum dynamics is demonstrated with one dimensional test cases (quantum dot models) and performance of the Trotter kernel as compared with the exact kernels is tested. Also, a novel approach for finding the ground state and other stationary sates is presented. This is based on the incoherent propagation in real time. For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling. We asses the numerical prerequisites for all of the above.

Unified treatment of the bound states of the Schiöberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schiöberg and Eckart potentials using an approximation of the centrifugal term. In order to determine the ℓ-states solutions, we use the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space–time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions. We show that the Eckart potential can be derived from the Schiöberg potential. The obtained results are compared to those produced by other methods and are found to be consistent.

The quenching field effect on the motion of an electron in an electromagnetic field potential

In this paper, the Feynman path integral approach based on the Gaussian wave packet approximation is employed to evaluate the coherence factor and the probability distribution for a single electron subjected to a magnetic field tailored by a variable on/off force. The quenching field effect is observed to create decoherence in the motion of the electron. The coherence factor is observed to decrease while the distribution probability function shows dephasing characteristics for weak and strong field strengths.

Classical-Quantum Correspondence for Above-Threshold Ionization

We measure high resolution photoelectron angular distributions (PADs) for above-threshold ionization of xenon atoms in infrared laser fields. Based on the Ammosov-Delone-Krainov theory, we develop an intuitive quantum-trajectory Monte Carlo model encoded with Feynman's path-integral approach, in which the Coulomb effect on electron trajectories and interference patterns are fully considered. We achieve a good agreement with the measured PADs of atoms for above-threshold ionization. The quantum-trajectory Monte Carlo theory sheds light on the role of ionic potential on PADs along the longitudinal and transverse direction with respect to the laser polarization, allowing us to unravel the classical coordinates (i.e., tunneling phase and initial momentum) at the tunnel exit for all of the photoelectrons of the PADs. We study the classical-quantum correspondence and build a bridge between the above-threshold ionization and the tunneling theory.

Feynman integral treatement of the Hyperbolical potential with a centrifugal term approximation

We obtain an analytical expression for the energy eigenvalues of the Hyperbolical potential using an approximation of the centrifugal term. In order to obtain the l-states solutions, we use the the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space-time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the eigenfunctions are given in terms of hypergeometric functions. The present results are consistent with those obtained by others methods.