Time-dependent Schrodinger Equation Research Articles

Eliminating the Second-Order Time Dependence from the Time Dependent Schrödinger Equation Using Recursive Fourier Transforms

A strategy is developed for writing the time-dependent Schrödinger Equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the first without using the time ordering operator. The energy distribution is calculated for a number of standard perturbation theory examples at first- and second-order. Possible applications include characterization of photonic spectra for bosonic sampling and four-wave mixing in quantum computation and Bardeen tunneling amplitude in quantum mechanics.

Accelerating wavepacket propagation with machine learning.

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.

A Modified Crank-Nicolson Method for Numerical Solution of the Linear Time-Dependent Schrodinger Equation

This paper presents the effectiveness of the solution of the time-dependent Schrodinger equation using the modified Crank-Nicolson Method (MCNM). The method was developed through finite difference approximation. To demonstrate the effectiveness of this approach, three different problems were tested with these approaches by using MATLAB soft code. Also, this research verifies that, in the presence or absence of potential energy, the modified Crank-Nicolson method for solving the time-dependent Schrödinger equation is accurate, convergent, and computationally efficient.

Entanglement in photo-ionization process.

We report a study of the entanglement between the quantized photon field and an atom arising in the photo-ionization process. Our approach is based on an ab initio solution of the time-dependent Schrödinger equation (TDSE) describing the quantum evolution of a bipartite system consisting of the atom and the quantized electromagnetic field. Using the solution of the TDSE, we calculate the reduced photon density matrix, which we subsequently use to compute entanglement entropy. We explain some properties of the entanglement entropy and propose an approximate formula for the entanglement entropy based on the analysis of the density matrix and its eigenvalues. We present the results of a comparative study of the entanglement in the photo-ionization process for various ionization regimes, including the tunneling and the multiphoton ionization regimes.

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

A realist interpretation of unitarity in quantum gravity

Unitarity is a difficult concept to implement in canonical quantum gravity because of state non-normalisability and the problem of time. We take a realist approach based on pilot-wave theory to address this issue in the Ashtekar formulation of the Wheeler–DeWitt equation. We use the postulate of a definite configuration in the theory to define a global time for the gravitational-fermionic system recently discussed in Alexander et al (2022 Phys. Rev. D 106 106012), by parameterising a variation of a Weyl-spinor that depends on the Kodama state. The total Hamiltonian constraint yields a time-dependent Schrodinger equation, without semi-classical approximations, which we use to derive a local continuity equation over the configuration space. We implement the reality conditions at the level of the guidance equation, and obtain a real spin-connection, extrinsic curvature and triad along the system trajectory. We obtain quantum corrections to deSitter spacetime from the guidance equation. The non-normalisable Kodama state is naturally factored out of the full quantum state in the conserved current density, opening the possibility for quantum-mechanical unitarity. We also give a pilot-wave generalisation of the notion of unitarity applicable to non-normalisable states, and show the existence of equilibrium density for our system. Lastly, we find unitary states in mini-superspace by finding an approximate solution to the Hamiltonian constraint.

Expression Time Dependent Schrodinger Equation from De Broglie Hypothesis and Hamiltonian Function

The Schrodinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis. Thus, depending on the quantum equations the quantum of total energy relation (E = T + V ) and de Broglie relation (? = h=p). Accordingly, new time dependent Schrodinger equation is obtained. As a result, by using the wave function and nonrelativistic wave function beside plank hypothesis. Moreover, TDSE was obtained, thus by used additional kinetic energy and potential energy (classical equation). Finally, a result corresponding to TDSE in 3D was confirmed by using De Broglie hypothesis.

Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential

We explore the possibility of adding complex absorbing potential at the boundaries when solving the one-dimensional real-time Schrödinger evolution on a grid using a quantum computer with a fully quantum algorithm described on a n qubit register. Due to the complex potential, the evolution mixes real- and imaginary-time propagation and the wave function can potentially be continuously absorbed during the time propagation. We use the dilation quantum algorithm to treat the imaginary-time evolution in parallel to the real-time propagation. This method has the advantage of using only one reservoir qubit at a time, that is measured with a certain success probability to implement the desired imaginary-time evolution. We propose a specific prescription for the dilation method where the success probability is directly linked to the physical norm of the continuously absorbed state evolving on the mesh. We expect that the proposed prescription will have the advantage of keeping a high probability of success in most physical situations. Applications of the method are made on one-dimensional wave functions evolving on a mesh. Results obtained on a quantum computer identify with those obtained on a classical computer. We finally give a detailed discussion on the complexity of implementing the dilation matrix. Due to the local nature of the potential, for n qubits, the dilation matrix only requires 2n CNOT and 2n unitary rotation for each time step, whereas it would require of the order of 4n+1 C-NOT gates to implement it using the best-known algorithm for general unitary matrices.

Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential

In this article, we investigate the global well-posedness of initial value problems of the time-dependent discrete nonlinear Schrodinger equation with a complex potential and sufficiently general nonlinearity on a multidimensional lattice in weighted \( l^p\) spaces for \( 1< p <\infty\). Thanks to our improved estimates we are able to prove the existence of global attractor for \( l^p\) solutions to the initial value problem.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/12/abstr.html

Excitation dynamics in molecule resolved by internuclear distance driven by the strong laser field.

Rydberg-state excitation of stretched model molecules subjected to near-infrared intense laser fields has been investigated based on a fully quantum model (QM) proposed recently and the numerical solutions of time-dependent Schrödinger equation (TDSE). Given the good agreement between QM and TDSE, it is found that, as the molecules are stretched, the electron tends to be trapped into low-lying Rydberg-states after its ionization from the core, which can be attributed to the shift of the ionization moments corresponding to maximum excitation populations. Moreover, the n-distribution is broadened for molecules with increasing internuclear distance, which results from the change of momentum distribution of emitted electrons. Analysis indicates that both of the above phenomena are closely related to the interference effect of electronic wave packets emitted from different nuclei. Our study provides a more comprehensive understanding of the molecular excitation in intense laser fields, as well as a means of possible applications to related experimental observations.

Dichroism Effects in the Ionization of Polarized Atoms by Short Circularly Polarized Laser Pulses

The dichroism effects, i.e., asymmetries of the ionization probability with respect to the inversion of either the atomic orientation (magnetic dichroism, MD) or the circular polarization degree of the photon beam (circular dichroism, CD), are investigated using the time-dependent perturbation theory (PT). It is shown that the magnitude of these effects depends not only on the polarization states of the atom and laser pulse but also on the intensity and duration of the latter. We find that the CD can also be observed in the ionization of oriented initial bound states, which is impossible in long-pulse ionization. Predictions of our PT analysis are supported by the results obtained by numerical solutions of the time-dependent Schrödinger equation (TDSE) describing the ionization of the excited 2P-states of the hydrogen atom.

Sparse adaptive basis set methods for solution of the time dependent Schrodinger equation

Scalable numerical solutions to the time dependent Schrodinger equation remain an outstanding goal in theoretical chemistry. Here we present a method which utilises recent breakthroughs in signal processing to consistently adapt a dictionary of basis functions to the dynamics of the system. We show that for two low-dimensional model problems the size of the basis set does not grow quickly with time and appears only weakly dependent on dimensionality. The generality of this finding remains to be seen. The method primarily uses energies and gradients of the potential, opening the possibility for its use in on-the-fly ab initio quantum wavepacket dynamics.

High-order harmonic generation, attosecond pulse, and non-sequential double ionization in the helium atom under high-intensity femtosecond laser pulses

High-order harmonic generation (HHG), attosecond pulse train (APT), isolated attosecond pulse (IAP), and non-sequential double ionization (NSDI) in the Helium atom under intense femtosecond laser pulses are computed using the time-dependent Schrodinger equation (TDSE) in one dimension (1D). By taking into account the electron–electron and electron-nucleus interactions, as well as determining the Helium atom’s ground state wave function through the imaginary time propagation (ITP) method, we are able to observe the widely recognized ‘knee structure’ in the ionization probability of the Helium atom as a function of intensity within an ionization boundary condition model. The results are in good agreement with the experimental data reported by (Walker et al (1994) Phys. Rev. Lett. 73, 1227).

Calibration of quantitative rescattering model for simulating vortex high-order harmonic generation driven by Laguerre–Gaussian beam with nonzero orbital angular momentum

We calibrate the macroscopic vortex high-order harmonic generation (HHG) obtained by the quantitative rescattering (QRS) model to compute single-atom induced dipoles against that by solving the time-dependent Schrödinger equation (TDSE). We show that the QRS perfectly agrees with the TDSE under the favorable phase-matching condition, and the QRS can accurately predict the main features in the spatial profiles of vortex HHG if the phase-matching condition is not good. We uncover that harmonic emissions from short and long trajectories are adjusted by the phase-matching condition through the time-frequency analysis and the QRS can simulate the vortex HHG accurately only when the interference between two trajectories is absent. This work confirms that it is an efficient way to employ the QRS model in the single-atom response for precisely simulating the macroscopic vortex HHG.

Magnetic exchange field signature in ultrafast electron dynamics in monolayer WSe2 by ultrashort optical pulse

We theoretically investigate the ultrafast electron dynamics in monolayer tungsten diselenide interacting with an irradiated ultrashort strong optical pulse in the presence of a magnetic exchange field. The dynamics of Dirac fermions in monolayer WSe2 is affected by spin and valley splitting, of course, enhanced by applied magnetic exchange field. There exists remarkable interaction between the exchange field and valley index because of huge spin–orbit coupling at the valence band. Regarding the optical pulse duration is smaller than the electron scattering time, the electron dynamics remains coherent and is described by the time-dependent Schrodinger equation. The strong electric field of the pulse, interacted with monolayer WSe2 causes appearance of a dipole moment connecting two valence and conduction bands. Because of the K and K′ valleys behave differently by the magnetic exchange field, the perfect valley-spin polarization effect is achieved. We explore the irreversible conduction band electron transition affected by a magnetic exchange field. The signature of the exchange field on such transitions is quite obvious. The conduction band electron redistribution around the Dirac points is represented, when the pulse ends. Asymmetric hot spots pattern in two different K and K′ valleys depends more or less on the magnitude of the magnetic exchange field. By reversing the direction of the exchange field, the process of spin polarization is completely reversed and it results in the electron distribution pattern. The high spin-valley resolved splitting in monolayer WSe2 provides a unique platform to investigate the magnetic exchange field driven phenomena, leading to valley polarization effect.

Time-dependent variational approach to solve multi-dimensional time-dependent Schrödinger equation

We present an efficient approach to solve multi-dimensional time-dependent Schrödinger equation (TDSE) in an intense laser field. In this approach, each spatial degree of freedom is treated as a distinguishable quasi-particle. The non-separable Coulomb potential is regarded as a two-body operator between different quasi-particles. The time-dependent variational principle is used to derive the equations of motion. Then the high-order multi-dimensional problem is broken down into several lower-order coupled equations, which can be efficiently solved. As a demonstration, we apply this method to solve the two-dimensional TDSE. The accuracy is tested by comparing the direct solutions of TDSE using several examples such as the strong-field ionization and the high harmonic generation. The results show that the present method is much more computationally efficient than the conventional one without sacrificing accuracy. The present method can be straightforwardly extended to three-dimensional problems. Our study provides a flexible method to investigate the laser-atom interaction in the nonperturbative regime.

Stability of Finite Difference Solution of Time-Dependent Schrodinger Equations

In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. Using the matrix representation for the numerical algorithm, it is shown that for the explicit finite difference method, the solution is conditionally stable while it becomes unconditionally stable for implicit finite difference methods. A 1D Harmonic Oscillator problem shall be used to illustrate this behaviour.

ITVOLT: An iterative solver for the time-dependent Schrödinger equation

We present a novel approach for solving the time-dependent Schrödinger equation (TDSE). The method we propose converts the TDSE to an equivalent Volterra integral equation; introducing a global Lagrange interpolation of the integrand transforms the equation to a linear system, which is then solved iteratively. In this paper, we derive the method, explore its performance on several examples, and discuss the corresponding numerical details. Program summaryProgram Title: Iterative Volterra Propagator (ITVOLT)CPC Library link to program files:https://doi.org/10.17632/jzd5xpv6t7.1Developer's repository link:https://github.com/ry-schneider/Iterative_Volterra_Propagator.gitLicensing provisions: MITProgramming language: Modern FortranNature of problem: ITVOLT is a solver for the time-dependent Schrödinger equation (TDSE). More broadly, it can be applied to any problem that can be written as a Volterra integral equation.Solution method: ITVOLT solves a Volterra integral equation representation of the TDSE by reducing it to a linear system via Lagrange interpolation of the integrand. It then solves the system iteratively with one of several iteration schemes.Additional comments including restrictions and unusual features: General subroutines assume that the Hamiltonian for the TDSE being solved can be represented by a symmetric banded matrix whose time-dependent component is some scalar function of time (for example a pulse) multiplied by a fixed matrix.

QPC-TDSE: A parallel TDSE solver for atoms and small molecules in strong lasers

The QPC-TDSE program serves as a general tool to study laser-driven dynamics of electrons in ideal isolated atoms and molecules by solving the full-dimensional non-relativistic time-dependent Schrödinger equation (TDSE) within single-active-electron approximation. It expands the full-dimensional electronic wavefunction in spherical coordinates by spherical harmonics and B-spline functions and employs a set of parallel Crank-Nicolson propagators combined with split-operator techniques to evolve the wavefunction in time, which support centrifugal and multi-polar static potentials to treat atomic and molecular scenarios and accepts arbitrary combinations of linearly or elliptically polarized lasers within the dipole approximation. The program is capable of extracting the photo-electron momentum distribution via t-SURFF approach or projection onto either the exact scattering states or the planewaves. Its applications in different scenarios are given as examples, e.g., above threshold ionization, attosecond clock, higher-order harmonic generation. Program summaryProgram Title: QPC-TDSECPC Library link to program files:https://doi.org/10.17632/xjm3kfgv75.1Licensing provisions: GPLv3Programming language: C++External libraries: HDF5, GSL, MKLNature of problem: Numerical solution of TDSE and extraction of various types of electron spectrum.Solution method: The electronic wavefunction is expanded by B-spline functions and spherical harmonics whose range is chosen elaborately to reduce the total number of partial waves for non-linearly polarized lasers. The Crank-Nicolson approach combined with an operator-splitting scheme is used to propagate the wavefunction in time, either in velocity gauge or length gauge. Matrix inversions are solved via either dense or sparse linear algebra solvers according to their structures. The t-SURFF method and projections onto either the scattering states or planewaves are provided for the accurate extraction of the momentum distributions.Additional comments including restrictions and unusual features: Only lasers within dipole approximation are supported. For the multi-polar potentials, only pure Coulombic ones are supported. Routines for solving exact scattering states have only been implemented for centrifugal potentials. The codes are written in C++17 and can only be compiled on the platforms that support the avx instruction sets. An extension for the propagation algorithm using the avx-512 intrinsics is provided as optional.

Entanglement Dynamics of Two V‐type Atoms with Dipole–Dipole Interaction in Dissipative Cavity

AbstractIn this work, a coupled system of two V‐type atoms with dipole–dipole interaction in a dissipative single‐mode cavity, which couples with an external environment, is studied. The analytical solution of this model is obtained by solving the time dependent Schrodinger equation after Hamiltonian of dissipative cavity is diagonalized by introducing a set of new creation and annihilation operators according to Fano theorem. It is also discussed in detail how the entanglement dynamics of different initial states are influenced by the cavity‐environment coupling, the spontaneously generated interference (SGI) parameter, and the dipole–dipole interaction between two atoms . The results show that the SGI parameter has different effects on entanglement dynamics under different initial states. Namely, the SGI parameter will increase the decay rate of the initially maximal entangled state and reduce that of the initially partial entangled state. For the initially product state, the larger SGI parameter corresponds to the more entanglement generated. The entanglement monotonically decreases under the weak cavity‐environment coupling, while the oscillation of entanglement will occur under the strong cavity‐environment coupling. The larger the dipole–dipole interaction is, the slower the entanglement decays and the more the entanglement will be generated. So the dipole–dipole interaction can not only protect and generate entanglement very effectively, but also enhance the regulation effect of the SGI parameter on entanglement.