Imaginary Time Propagation Method Research Articles

Asynchronous quantum Kármán vortex street in two-component Bose-Einstein condensate with PT symmetric potential

The dynamics of a miscible two-component Bose-Einstein condensate (BEC) with PT (parity-time) symmetric potential are investigated numerically. The dynamical behaviors of the system is described by Gross-Pitaevskii (GP) equations under the mean-field theory. Firstly, the ground state of the system is obtained by the imaginary-time propagation method. Then dynamical behaviors are numerically simulated by the time-splitting Fourier pseudo-spectral approach under periodic boundary conditions. By adjusting the width and velocity of the obstacle potential, various patterns such as no vortex, oblique drifting vortex dipole, V-shaped vortex pairs, irregular quantum turbulence and combined modes are studied. It is noted that the shedding vortex pairs in components 1 and 2 are staggered, which is called “the asynchronous quantum Kármán vortex street”. Here, the ratio of the distance between two vortex pairs in one row to the distance between vortex rows is approximately 0.18, which is less than the stability criterion 0.28 of classical fluid. We calculated the drag force acting on the obstacle potential during generation of the asynchronous quantum Kármán vortex street. It is found that periodical oscillation of the drag force is generated via drifting up or down of the vortex pairs. Meanwhile, we analyzed the influence of the imaginary part of the PT symmetric potential with gain-loss for wake. The trajectory and frequency of the vortex are changed, due to the imaginary part breaks the local symmetry of the system. In addition, the imaginary part affects the stability of the asynchronous quantum Kármán vortex street. Lots of numerical simulations are carried out to determine the parameter regions of various vortex shedding modes. We also proposed an experimental protocol to realize the asynchronous quantum Kármán vortex street in the miscible two-component BEC with PT symmetric potential.

A novel tetrahedral spectral element method for Kohn-Sham model

High order computational approach for the electronic structure calculation is a long-standing research area. Among all the discretization methods, spectral element method stands out for its flexibility to capture the singularity and high accuracy from the spectral perspective. In this work, we propose a tetrahedral spectral element method for full-potential electronic structure calculation. In the method, a C0 space is constructed for the numerical solution, based on a modification of basis functions proposed in [L. Jia et al. (2022) [48]] with the aid of generalized Koornwider polynomials, and a special designed tetrahedral mesh. Moreover, the interpolation and visualization are developed to provide numerical insights of our method. To demonstrate its potential towards practical simulations, we apply our method for calculating ground states of given electronic structures using the Kohn-Sham model. The method consists of a combination of an imaginary time propagation method and a self-consistent field iteration. For accelerating the simulation, a p-multigrid solver is developed which takes the advantage of the high order property of our method. A number of numerical experiments validate the effectiveness of our method, i.e., the expected spectral accuracy can be observed from results, and convincing ground states of various atoms and molecules are obtained.

Vortex formation and quench dynamics of rotating quantum droplets

We investigate vortex formation and quench dynamics in rotating quantum droplets. We use the variational method to analytically show that vortex formation requires the breakdown of the rotational symmetry of the system via shape deformations and calculate the bifurcation point and lowest energy surface mode of the system. In our numerical simulations, we first demonstrate the systematic stationary vortex states and their critical points with the imaginary time propagation method, in which the average angular momentum acts as the convergence condition of the final state, we also show that the critical points for vortex formation via simulation are consistent with variational results. Then, we present two methods for generating vortex states via the real-time quench dynamics with symmetric and asymmetric rotating quantum droplet traps, by choosing different quench times and rotation strengths, one obtains the final states with arbitrary vortices. Our study provides a dynamic method for experimentally investigating vortex formation in rotating quantum droplets.

Quantum phases and dynamics of dipolar spin-1 ferromagnetic Bose–Einstein condensates with spin–orbit coupling in a double-well potential

We consider a spin-1 ferromagnetic Bose–Einstein condensate with Rashba spin–orbit coupling (SOC) and dipole–dipole interaction (DDI) in a double-well potential. For the nonrotating case, the ground state of the system is obtained by using the imaginary time propagation method, and a ground-state phase diagram is given with respect to the SOC strength and the DDI strength. It is shown that the nonrotating system sustains rich exotic ground-state phases including the ox-horn phase with hidden vortices and antivortices, density droplet phase with vortex–antivortex pairs, modulated stripe phase with vortex–antivortex chains, checkerboard phase with visible vortex–antivortex chains and hidden vortex–antivortex pairs, and the triangular vortex–antivortex lattice phase. For the rotating case, the dynamics of the system is investigated by using a phenomenological dissipation model. The effects of SOC, DDI and rotation on the unique dynamic behaviors of the rotating system and the final steady-state structures are revealed and analyzed. In particular, the rotating system supports fascinating novel spin textures and skyrmion excitations, and experiences a series of topological structure transitions such as the transitions from an interlaced skyrmion–antiskyrmion lattice pair to a skyrmion–meron lattice pair and then to a meron lattice pair.

The Wigner function of ground state and one-dimensional numerics

In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schrödinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum hydrodynamic model Cai et al. (2013) [7] to reduce the dimension of the problem, ii) an imaginary time propagation method is designed for solving the model, and numerical techniques such as solution reconstruction are proposed for the feasibility of the method. Results of several numerical experiments verify our method, in which the potential application of the method for large scale system is demonstrated by examples with density functional theory.

Pressureless static solutions in a Newton-Yukawa gravity model

Non-minimally coupled curvature-matter gravity models are an interesting alternative to the Theory of General Relativity and to address the dark energy and dark matter cosmological problems. These models have complex field equations that prevent a full analytical study. Nonetheless, in a particular limit, the behavior of a matter distribution can, in these models, be described by a Schr\"odinger-Newton system. In nonlinear optics, the Schr\"odinger-Newton system can be used to tackle a wide variety of relevant situations and several numerical tools have been developed for this purpose. Interestingly, these methods can be adapted to study General Relativity problems as well as its extensions. In this work, we report the use of these numerical tools to study a particular non-minimal coupling model that introduces two new potentials, an attractive Yukawa potential and a repulsive potential proportional to the energy density. Using the imaginary-time propagation method we have shown that stationary solutions arise even at low energy density regimes.

Iterative Power Algorithm for Global Optimization with Quantics Tensor Trains.

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution ρ0(x) and iteratively applies the recurrence relation ρk+1(x) = U(x) ρk(x)/∥Uρk∥L1, where U(x) = e-V(x) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function δ(x - x*), so the global minimum can be obtained as the position expectation value x* = Tr[x δ(x - x*)]. QTT representations of V(x) and ρ(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine-thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., pmax} folded as a d-dimensional 21 × 22 × ··· × 2d tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

Vortex-lattice in a uniform Bose–Einstein condensate in a box trap

We study numerically the vortex-lattice formation in a rapidly rotating uniform quasi-two-dimensional Bose–Einstein condensate (BEC) in a box trap. We consider two types of boxes: square and circle. In a square-shaped 2D box trap, when the number of generated vortices is the square of an integer, the vortices are found to be arranged in a perfect square lattice, although deviations near the center are found when the number of generated vortices is arbitrary. In case of a circular box trap, the generated vortices in the rapidly rotating BEC lie on concentric closed orbits. Near the center, these orbits have the shape of polygons, whereas near the periphery the orbits are circles. The circular box trap is equivalent to the rotating cylindrical bucket used in early experiment(s) with liquid He II. The number of generated vortices in both cases is in qualitative agreement with Feynman’s universal estimate. The numerical simulation for this study is performed by a solution of the underlying mean-field Gross–Pitaevskii (GP) equation in the rotating frame, where the wave function for the generated vortex lattice is a stationary state. Consequently, the imaginary-time propagation method can be used for a solution of the GP equation, known to lead to an accurate numerical solution. We also demonstrated the dynamical stability of the vortex lattices in real-time propagation upon a small change of the angular frequency of rotation, using the converged imaginary-time wave function as the initial state.

Energetics of pure and doped helium droplets - application to interpreting pick-up experiments

We use helium density functional theory to calculate the energies of spherically symmetric 4He helium droplets both with and without heteroatom dopants. Self-consistent calculations using an imaginary time propagation method are used to compute structural and energetic properties of these droplets ranging in size from 50 to 9500 atoms. Particular attention is given to the solvation energies of the resident dopant atoms, as these values play an important role in experimental superfluid helium calorimetry techniques. We also suggest a method of predicting new droplet size distributions following dopant pickup using the chemical potential values obtained from our calculations.

Ground state of a rotating Bose-Einstein condensate with in-plane quadrupole field

Compared with the scalar Bose-Einstein condensate, the spinor Bose-Einstein condensate, in which internal degrees of freedom are essentially free, has aroused the great interest in the study of topological excitations. In particular, the spinor Bose-Einstein condensate with rotation provides a new opportunity for studying novel quantum states including a coreless vortex and vortex lattice. To date, in the presence of rotation, a great many of studies on the topological excitations have focused on the Bose-Einstein condensate system with the uniform Zeeman field or without external magnetic field. However, the ground state structure of a rotating Bose-Einstein condensate in the presence of in-plane gradient-magnetic-field remains an open question. In this work, by using the imaginary-time propagation method, we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field. We first examine the effect of in-plane quadrupole field on trapped spinor Bose-Einstein condensate. The numerical results show that Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation. When magnetic field gradient is increased, the vortices around Mermin-Ho vortex display the symmetrical arrangement. For an even larger magnetic field gradient strength, the system only presents the Mermin-Ho vortex because the in-plane quadrupole field can prevent the vortices around Mermin-Ho vortex from occurring. Next, we examine the effect of the rotation on trapped spinor Bose-Einstein condensate. A phase transition from a polar-core vortex to a Mermin-Ho vortex is found through applying a rotational potential, which is caused by the cooperation between the in-plane quadrupole field and the rotation. We further study the combined effects of spin exchange interaction and density-density interaction. The results confirm that in the presence of the quadrupole field both spin exchange interaction and density-density interaction, acting as controllable parameters, can control the number of the vortices around Mermin-Ho vortex. The corresponding number of the vortices shows step behavior with increasing the ratio between spin exchange interaction and density-density interaction, which behaves as hexagon, pentagon, square and triangle. It is found that two types of topology structures, i.e., the hyperbolic meron and half-skyrmion, can occur in the present system. These vortex structures can be realized via time-of-flight absorption imaging technique. Our results not only provide an opportunity to investigate the exotic vortex structures and the corresponding phase transitions in a controlled platform, but also lay the foundation for the study of topological defect subjected to gauge field and dipolar interaction in future.

The ground states and pseudospin textures of rotating two-component Bose–Einstein condensates trapped in harmonic plus quartic potential**Project supported by the National Natural Science Foundation of China (Grant Nos. 91430109 and 11404198), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20111401110004), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2014011005-3).

The ground states of two-component miscible Bose–Einstein condensates (BECs) confined in a rotating annular trap are obtained by using the Thomas–Fermi (TF) approximation method. The ground state density distribution of the condensates experiences a transition from a disc shape to an annulus shape either when the angular frequency increases and the width and the center height of the trap are fixed, or when the width and the center height of the trap increase and the angular frequency is fixed. Meantime the numerical solutions of the ground states of the trapped two-component miscible BECs with the same condition are obtained by using imaginary-time propagation method. They are in good agreement with the solutions obtained by the TF approximation method. The ground states of the trapped two-component immiscible BECs are also given by using the imaginary-time propagation method. Furthermore, by introducing a normalized complex-valued spinor, three kinds of pseudospin textures of the BECs, i.e., giant skyrmion, coaxial double-annulus skyrmion, and coaxial three-annulus skyrmion, are found.

Unconventional Bose-Einstein condensation in a system with two species of bosons in thep-orbital bands in an optical lattice

In the context of Gross-Pitaevskii theory, we investigate the unconventional Bose-Einstein condensations in the two-species mixture with $p$-wave symmetry in the second band of a bipartite optical lattice. A new imaginary-time propagation method is developed to numerically determine the $p$-orbital condensation. Different from the single-species case, the two-species boson mixture exhibits two non-equivalent complex condensates in the intraspecies-interaction-dominating regime, exhibiting the vortex-antivortex lattice configuration in the charge and spin channels, respectively. When the interspecies interaction is tuned across the SU(2) invariant point, the system undergoes a quantum phase transition toward a checkerboard-like spin density wave state with a real-valued condensate wavefunction. The influence of lattice asymmetry on the quantum phase transition is addressed. Finally, we present a phase-sensitive measurement scheme for experimentally detecting the UBEC in our model.

Optimal nonlinear coherent mode transitions in Bose-Einstein condensates utilizing spatiotemporal controls

Bose-Einstein condensates (BECs) offer the potential to examine quantum behavior at large length and time scales, as well as forming promising candidates for quantum technology applications. Thus, the manipulation of BECs using control fields is a topic of prime interest. We consider BECs in the mean field model of the Gross-Pitaevskii equation (GPE), which contains linear and nonlinear features, both of which are subject to control. In this work we report successful optimal control simulations of a one dimensional GPE by modulating the linear and nonlinear terms to stimulate transitions into excited coherent modes. The linear and nonlinear controls are allowed to freely vary over space and time to seek their optimal forms. The determination of the excited coherent modes targeted for optimization is numerically performed through an adaptive imaginary time propagation method. Numerical simulations are performed for optimal control of mode-to-mode transitions between the ground coherent mode and excited modes of a BEC trapped in a harmonic well. The results show greater than 99% success for nearly all trials utilizing reasonable initial guesses for the controls, and analysis of the optimal controls reveals primarily direct transitions between initial and target modes. The success of using solely the nonlinearity term as a control opens up further research toward exploring novel control mechanisms inaccessible to linear Schr\"odinger-type systems.

Analysis of Partition Functions for Metallocenes: Ferrocene, Ruthenocene, and Osmocene.

We present a calculation of the torsional potential of the three metallocenes of the iron group, that is, ferrocene, ruthenocene, and osmocene, calculated with the GAUSSIAN program suite. Both a variational method (through computation of the exact energy levels) and our Chebyshev imaginary time propagation method are used to calculate the hindered rotation partition function, demonstrating the efficiency of the Chebyshev scheme. The transition from a semirigid through a hindered rotor to the free rotor regime is demonstrated, and the effect of the hindered rotation (as opposed to a harmonic) treatment on the thermodynamics of metallocenes is demonstrated.

Modelling of quantum confinement in optical nanostructures

An efficient sensitivity analysis approach for quantum nanostructures is proposed. The imaginary time propagation method (ITP) is utilized to solve the time dependent Schrödinger equation (TDSE). Using this method, an extraction of all the modes and their sensitivity with respect to all the design parameters have been performed with minimal computational effort. The sensitivity analysis is done using the adjoint variable method (AVM) and results are comparable to those obtained using central finite difference method (CFD) applied directly on the response level.

Ground state of a two-component dipolar Bose-Einstein condensate confined in a coupled annular potential

By using the imaginary-time propagation method, we study the ground state structure of a two-component dipolar Bose-Einstein condensate confined in a coupled annular potential. The effects of contact and dipole-dipole interaction on the ground state density distribution of such a system are investigated in detail. The results show that the dipole-dipole interaction, acting as a new degree of freedom, can be used to obtain the desired ground state phases, and to control the phase transition between different ground states.

Vortex dynamics in cubic-quintic Bose-Einstein condensates

We study vortex dynamics in a trapped Bose-Einstein condensate with tunable two- and three-body interactions. The dynamics is governed by two-dimensional cubic-quintic Gross-Pitaevskii equation. A time-dependent variational method has been used to obtain critical rotational frequency and surface mode frequency analytically and are compared with numerical simulation results. An imaginary time propagation method and Crank-Nicolson scheme for discretization have been used for numerical simulation. The numerically calculated average value of the angular momentum per particle shows very clearly its dynamical relation with the time development of the vortex formation. The rotational frequency dependence of the variation of average value of the angular momentum per particle with time shows that vortices form much faster in time for higher rotational frequency. Similarly, the vortex forms much faster in time with an increase of the strength of the repulsive three-body interaction. The simulation of the vortex lattice formation in the condensate shows that the presence of the three-body interactions does not alter the vortex lattice pattern but it helps in the shape deformations of the condensate thereby leading to vortex lattice formation. Likewise, the three-body interactions enable the vortex lattice formation in Bose-Einstein condensates even with attractive two-body interactions and in purely quintic BEC.

Multidimensional Supersymmetric Quantum Mechanics: A Scalar Hamiltonian Approach to Excited States by the Imaginary Time Propagation Method

Supersymmetric quantum mechanics (SUSY-QM) is shown to provide a novel approach to the construction of the initial states for the imaginary time propagation method to determine the first and second excited state energies and wave functions for a two-dimensional system. In addition, we show that all calculations are carried out in sector one and none are performed with the tensor sector two Hamiltonian. Through our tensorial approach to multidimensional supersymmetric quantum mechanics, we utilize the correspondence between the eigenstates of the sector one and two Hamiltonians to construct appropriate initial sector one states from sector two states for the imaginary time propagation method. The imaginary time version of the time-dependent Schrödinger equation is integrated to obtain the first and second excited state energies and wave functions using the split operator method for a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. The computational results indicate that we can obtain the first two excited state energies and wave functions even when a quantum system does not exhibit any symmetry. Moreover, instead of dealing with the increasing computational complexity resulting from computations in the tensor sector two Hamiltonian, this study presents a new supersymmetric approach to calculations of accurate excited state energies and wave functions by directly using the scalar sector one Hamiltonian.

Imaginary time propagation code for large-scale two-dimensional eigenvalue problems in magnetic fields

We present a code for solving the single-particle, time-independent Schrödinger equation in two dimensions. Our program utilizes the imaginary time propagation (ITP) algorithm, and it includes the most recent developments in the ITP method: the arbitrary order operator factorization and the exact inclusion of a (possibly very strong) magnetic field. Our program is able to solve thousands of eigenstates of a two-dimensional quantum system in reasonable time with commonly available hardware. The main motivation behind our work is to allow the study of highly excited states and energy spectra of two-dimensional quantum dots and billiard systems with a single versatile code, e.g., in quantum chaos research. In our implementation we emphasize a modern and easily extensible design, simple and user-friendly interfaces, and an open-source development philosophy. Program summaryProgram title: itp2dCatalogue identifier: AENR_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENR_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License version 3No. of lines in distributed program, including test data, etc.: 11310No. of bytes in distributed program, including test data, etc.: 97720Distribution format: tar.gzProgramming language: C++ and Python.Computer: Tested on x86 and x86-64 architectures.Operating system: Tested under Linux with the g++ compiler. Any POSIX-compliant OS with a C++ compiler and the required external routines should suffice.Has the code been vectorised or parallelized?: Yes, with OpenMP.RAM: 1 MB or more, depending on system size.Classification: 7.3.External routines: FFTW3 (http://www.fftw.org), CBLAS (http://netlib.org/blas), LAPACK (http://www.netlib.org/lapack), HDF5 (http://www.hdfgroup.org/HDF5), OpenMP (http://openmp.org), TCLAP (http://tclap.sourceforge.net), Python (http://python.org), Google Test (http://code.google.com/p/googletest/)Nature of problem: Numerical calculation of the lowest energy solutions (up to a few thousand, depending on available memory), of a single-particle, time-independent Schrdinger equation in two dimensions with or without a homogeneous magnetic field.Solution method: Imaginary time propagation (also known as the diffusion algorithm), with arbitrary even order factorization of the imaginary time evolution operatorAdditional comments: Please see the README file distributed with the program for more information. The source code of our program is also available at https: //bitbucket.org/luukko/itp2d.Running time: Seconds to hours

Numerical simulation of electronic structure, linear and third order nonlinear optical absorption coefficients in the polymeric nanoheterostructure using imaginary time propagation method

In the effective mass approximation, the charge density, the potential profile, the wavefunctions, and the corresponding energy states for zinc sulfide spherical quantum dot have been self-consistently solved, which is embedded in the polymeric media (PVP) by the imaginary time propagation method. Moreover, for the first time, the oscillator strength, linear, third order nonlinear, and total optical absorption coefficients for the transition between intersubbands 1s-1p, 1p-1d, 1d-1f, 1f-1g, 1g-1h, and 1h-1i are investigated with respect to photon energy and the saturation strength has been calculated for every transition too. Calculated result shows that the oscillator strength and the magnitude of optical absorption coefficients are enhanced for transition between intersubbands with higher energies. Also, calculations show that the total absorption coefficient depends on the strength of incident optical intensity and relaxation time.