Aleksandar Bogojevic

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.

CUDA programs for solving the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross–Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank–Nicolson method as in the previous version (Kishor Kumar et al., 2015), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12–25, depending on the program and input size.

Ultra-fast converging path-integral approach for rotating ideal Bose–Einstein condensates

A recently developed efficient recursive approach for analytically calculating the short-time evolution of the one-particle propagator to extremely high orders is applied here for numerically studying the thermodynamical and dynamical properties of a rotating ideal Bose gas of 87 Rb atoms in an anharmonic trap. At first, the one-particle energy spectrum of the system is obtained by diagonalizing the discretized short-time propagator. Using this, many-boson properties such as the condensation temperature, the ground-state occupancy, density profiles, and time-of-flight absorption pictures are calculated for varying rotation frequencies. The obtained results improve previous semiclassical calculations, in particular for smaller particle numbers. Furthermore, we find that typical time scales for a free expansion are increased by an order of magnitude for the delicate regime of both critical and overcritical rotation.

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.

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.

Fast converging path integrals for time-dependent potentials: I. Recursive calculation of short-time expansion of the propagator

We calculate the short-time expansion of the propagator for a general many-body quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte-Carlo simulations for path integrals, or to set up various analytic approximation techniques to study dynamic properties of quantum systems in timedependent potentials. The analytically derived results are numerically verified by treating several simple one-dimensional models.

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.

Fast convergence of path integrals for many-body systems

Abstract We generalize a recently developed method for accelerated Monte Carlo calculation of path integrals to the physically relevant case of generic many-body systems. This is done by developing an analytic procedure for constructing a hierarchy of effective actions leading to improvements in convergence of N -fold discretized many-body path integral expressions from 1 / N to 1 / N p for generic p . In this Letter we present explicit solutions within this hierarchy up to level p = 5 . Using this we calculate the low lying energy levels of a two particle model with quartic interactions for several values of coupling and demonstrate agreement with analytical results governing the increase in efficiency of the new method. The applicability of the developed scheme is further extended to the calculation of energy expectation values through the construction of associated energy estimators exhibiting the same speedup in convergence.

Fast converging path integrals for time-dependent potentials: II. Generalization to many-body systems and real-time formalism

Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.

Properties of quantum systems via diagonalization of transition amplitudes. I. Discretization effects.

We analyze the method for calculation of properties of nonrelativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors associated with space discretization. Approaches using direct diagonalization of real-space discretized Hamiltonians lead to polynomial errors in discretization spacing Delta . Here we show that the method based on the diagonalization of the short-time evolution operators leads to substantially smaller discretization errors, vanishing exponentially with 1/Delta(2). As a result, the presented calculation scheme is particularly well suited for numerical studies of few-body quantum systems. The analytically derived discretization errors estimates are numerically shown to hold for several models. In the follow up paper [I. Vidanović, A. Bogojević, A. Balaz, and A. Belić, Phys. Rev. E 80, 066706 (2009)] we present and analyze substantial improvements that result from the merger of this approach with the recently introduced effective-action scheme for high-precision calculation of short-time propagation.