Antun Balaz

Hybrid OpenMP/MPI programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

We present hybrid OpenMP/MPI (Open Multi-Processing/Message Passing Interface) parallelized versions of earlier published C programs (Vudragovic et al. 2012) for calculating both stationary and non-stationary solutions of the time-dependent Gross–Pitaevskii (GP) equation in three spatial dimensions. The GP equation describes the properties of dilute Bose–Einstein condensates at ultra-cold temperatures. Hybrid versions of programs use the same algorithms as the C ones, involving real- and imaginary-time propagation based on a split-step Crank–Nicolson method, but consider only a fully-anisotropic three-dimensional GP equation, where algorithmic complexity for large grid sizes necessitates parallelization in order to reduce execution time and/or memory requirements per node. Since distributed memory approach is required to address the latter, we combine MPI programming paradigm with existing OpenMP codes, thus creating fully flexible parallelism within a combined distributed/shared memory model, suitable for different modern computer architectures. The two presented C/OpenMP/MPI programs for real- and imaginary-time propagation are optimized and accompanied by a customizable makefile. We present typical scalability results for the provided OpenMP/MPI codes and demonstrate almost linear speedup until inter-process communication time starts to dominate over calculation time per iteration. Such a scalability study is necessary for large grid sizes in order to determine optimal number of MPI nodes and OpenMP threads per node.

Systematically accelerated convergence of path integrals.

We present a new analytical method that systematically improves the convergence of path integrals of a generic N-fold discretized theory. Using it we calculate the effective actions S(p) for p< or =9, which lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit as 1/N(p). We checked this derived speedup in convergence by performing Monte Carlo simulations on several different models.

OpenMP Fortran and C programs for solving the time-dependent Gross-Pitaevskii equation in an anisotropic trap

We present new version of previously published Fortran and C programs for solving the Gross–Pitaevskii equation for a Bose–Einstein condensate with contact interaction in one, two and three spatial dimensions in imaginary and real time, yielding both stationary and non-stationary solutions. To reduce the execution time on multicore processors, new versions of parallelized programs are developed using Open Multi-Processing (OpenMP) interface. The input in the previous versions of programs was the mathematical quantity nonlinearity for dimensionless form of Gross–Pitaevskii equation, whereas in the present programs the inputs are quantities of experimental interest, such as, number of atoms, scattering length, oscillator length for the trap, etc. New output files for some integrated one- and two-dimensional densities of experimental interest are given. We also present speedup test results for the new programs.

Recursive Schrödinger equation approach to faster converging path integrals.

By recursively solving the underlying Schrödinger equation, we set up an efficient systematic approach for deriving analytic expressions for discretized effective actions. With this, we obtain discrete short-time propagators for both one and many particles in arbitrary dimension to orders that have not been accessible before. They can be used to substantially speed up numerical Monte Carlo calculations of path integrals, as well as for setting up an alternative analytical approximation scheme for energy spectra, density of states, and other statistical properties of quantum systems.

OpenMP, OpenMP/MPI, and CUDA/MPI C programs for solving the time-dependent dipolar Gross–Pitaevskii equation

We present new versions of the previously published C and CUDA programs for solving the dipolar Gross–Pitaevskii equation in one, two, and three spatial dimensions, which calculate stationary and non-stationary solutions by propagation in imaginary or real time. Presented programs are improved and parallelized versions of previous programs, divided into three packages according to the type of parallelization. First package contains improved and threaded version of sequential C programs using OpenMP. Second package additionally parallelizes three-dimensional variants of the OpenMP programs using MPI, allowing them to be run on distributed-memory systems. Finally, previous three-dimensional CUDA-parallelized programs are further parallelized using MPI, similarly as the OpenMP programs. We also present speedup test results obtained using new versions of programs in comparison with the previous sequential C and parallel CUDA programs. The improvements to the sequential version yield a speedup of 1.1–1.9, depending on the program. OpenMP parallelization yields further speedup of 2–12 on a 16-core workstation, while OpenMP/MPI version demonstrates a speedup of 11.5–16.5 on a computer cluster with 32 nodes used. CUDA/MPI version shows a speedup of 9–10 on a computer cluster with 32 nodes.

Systematic speedup of path integrals of a generic N-fold discretized theory

We present and discuss a detailed derivation of an analytical method that systematically improves the convergence of path integrals of a generic N-fold discretized theory. We develop an explicit procedure for calculating a set of effective actions S{sup (p)}, for p=1,2,3,... which have the property that they lead to the same continuum amplitudes as the starting action, but converge to that continuum limit ever faster. Discretized amplitudes calculated using the p-level effective action differ from the continuum limit by a term of order 1/N{sup p}. We obtain explicit expressions for the effective actions for levels p{<=}9. We end by analyzing the speedup of Monte Carlo simulations of two different models: an anharmonic oscillator with quartic coupling and a particle in a modified Poeschl-Teller potential.

Properties of quantum systems via diagonalization of transition amplitudes. II. Systematic improvements of short-time propagation.

In this paper, building on a previous analysis [I. Vidanović, A. Bogojević, and A. Belić, preceding paper, Phys. Rev. E 80, 066705 (2009)] of exact diagonalization of the space-discretized evolution operator for the study of properties of nonrelativistic quantum systems, we present a substantial improvement to this method. We apply recently introduced effective action approach for obtaining short-time expansion of the propagator up to very high orders to calculate matrix elements of space-discretized evolution operator. This improves by many orders of magnitude previously used approximations for discretized matrix elements and allows us to numerically obtain large numbers of accurate energy eigenvalues and eigenstates using numerical diagonalization. We illustrate this approach on several one- and two-dimensional models. The quality of numerically calculated higher-order eigenstates is assessed by comparison with semiclassical cumulative density of states.

Analytical and numerical study of dirty bosons in a quasi-one-dimensional harmonic trap

The time-independent Gross-Pitaevskii equation describing a quasi one-dimensional Bose-Einstein-condensed gas in a harmonic trapping potential with an additional delta-correlated disorder potential at zero temperature is numerically solved for the condensate wave function, and disorder ensemble averages are evaluated. In particular, we analyze quantitatively the emergence of mini-condensates in the local minima of the random potential, which occurs for weak disorder preferentially at the border of the condensate, while for intermediate disorder strength this happens in the trap center. In view of a more detailed physical understanding of this phenomenon, we extend a quite recent non-perturbative approach towards the weakly interacting dirty boson problem, which relies on the Hartree-Fock theory and is worked out on the basis of the replica method, from the homogeneous case to a harmonic confinement. In the weak disorder regime we apply the Thomas-Fermi approximation, while in the intermediate disorder regime we use a variational ansatz in order to describe analytically the numerically observed redistribution of the fragmented mini-condensates with increasing disorder strength.

Asymptotic properties of path integral ideals

We introduce and analyze an interesting quantity, the path integral ideal, governing the flow of generic discrete theories to the continuum limit and greatly increasing their convergence. The said flow is classified according to the degree of divergence of the potential at spatial infinity. Studying the asymptotic behavior of path integral ideals we isolate the dominant terms in the effective potential that determine the behavior of a generic theory for large discrete time steps.

Conditions for order and chaos in the dynamics of a trapped Bose-Einstein condensate in coordinate and energy space

Abstract We investigate numerically conditions for order and chaos in the dynamics of an interacting Bose-Einstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size. Graphical abstract