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Journal of Physics Conference Series

Monte Carlo simulations for quantum harmonic and anharmonic oscillators within epistemically restricted phase-space representation

Muhammad Jauhar Kholili, Siska A. Rifianti, Eny Latifah, Ahmad R. T. Nugraha

Published Jun 1, 2022• 2 citations
Publisher

Abstract

Abstract We investigate Monte Carlo simulations within the epistemically-restricted phase-space (ERPS) representation of quantum mechanics for harmonic and anharmonic oscillators. Compared to the variational Monte Carlo (VMC) simulations in the conventional representation, the calculation time in the ER phase-space representation is quite competitive. We also numerically show that the distribution function of the global random variable on the order of Planck’s constant in the ER phase-space representation does not significantly affect the energy accuracy, yet a particular choice of the distribution function may give a better calculation time. This work gives an alternative approach to the Monte Carlo simulations beyond those in the conventional representation of quantum mechanics.

DOI
10.1088/1742-6596/2243/1/012084

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Abstract We investigate Monte Carlo simulations within the epistemically-restricted phase-space (ERPS) representation of quantum mechanics for harmonic and anharmonic oscillators. Compared to the variational Monte Carlo (VMC) simulations in the conventional representation, the calculation time in the ER phase-space representation is quite competitive. We also numerically show that the distribution function of the global random variable on the order of Planck’s constant in the ER phase-space representation does not significantly affect the energy accuracy, yet a particular choice of the distribution function may give a better calculation time. This work gives an alternative approach to the Monte Carlo simulations beyond those in the conventional representation of quantum mechanics.
We develop an unconventional Monte Carlo simulation within the phase-space formulation of quantum mechanics taking into account a global-nonseparable random variable whose strength is epistemically restricted on the order of Planck’s constant. Compared to the ordinary variational Monte simulation, we find that energy expectation values of simple quantum systems such like harmonic oscillators can be calculated by our method with a reasonable accuracy and competitive calculation time at a large sampling N of random numbers (e.g., at N > 105 samples). We also discuss the comparison of simulations performed in different computing resources.
Abstract The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice on which calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger’s equation. The theoretical and computational framework presented here introduces undergraduates to the path integral formulations of quantum mechanics in real time and the partition function in statistical mechanics in imaginary time. The example of the anharmonic oscillator helps to build an intuition about the MCMC method of evaluating the partition function, which can then be used to solve other problems in physics and beyond.
Because the Schrodinger equation cannot be solved analytically for systems larger than hydrogen, approximate solution methods must be applied to study systems of chemical interest. Quantum Monte Carlo is one of these methods. Quantum Monte Carlo simulations involve the diffusion of walkers through configuration space and necessitate the use of one of two imaginary time propagators, 2nd or 4th order. The choice of the propagator implemented depends on a balance between accuracy and efficiency. Obtaining accurate solutions is the primary objective; once this is accomplished, finding ways to make the simulations efficient is the next intention. Using a model system of a multidimensional harmonic oscillator, this study intends to compare and contrast both 2nd and 4th order propagators' diminished accuracy as the dimensionality of the system is increased. Two phases of the propagation are sampled to determine accuracy: a single propagation step and the asymptotic limit. To test the accuracy after one step, we start a simulation by placing walkers at the origin, moving them forward one time step in imaginary time, creating normalized histograms of their positions and comparing these with the known result for the imaginary time propagation of the delta function centered at x=0. The convergence of the simulation is determined by observing the change in distribution from the known asymptotic limit distribution. This is done by distributing the walkers in the known asymptotic distribution and observing the change in average weight upon further propagation. A converged distribution’s average weight will not change.
The quantum harmonic oscillator represents the most common fundamental building block to compute thermal properties of virtually any dielectric crystalline material at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo (PIMC) methods provide a powerful numerical tool to stochastically determine thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance the calculation of thermal conductivity from PIMC, however, has turned out to be substantially more difficult. Although correlation functions of current operators can be determined by PIMC in the form of analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real-time response functions. This task not only relies strongly on the accuracy of the PIMC data, but also introduces noticeable dependence on the underlying model used for the inversion problem. Here, we address both difficulties with great care. In particular, we first devise improved estimators for current correlations which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with more recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies, establishing solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids.
We investigate the phase-space representation of quantum mechanics as a platform to perform Monte Carlo simulation of the non-interacting Jaynes-Cummings-Rabi (JCR) model involving a two-level hydrogen atom as a qubit and a quantum har-monic oscillator as a photonic cavity. Our algorithm starts by defining a trial wave function for solving the JCR model and then implement it on the “epistemically-restricted” phase-space representation, known as Budiyono-Rochrlich theory, with a Monte-Carlo sampling for the phase-space distribution, which is useful to obtain the average energy of our JCR model. We find that the numerical results are close to the analytical ones for every separate system in the JCR model, i.e., the hydrogen and the oscillator, calculated independently. Combining the two components in the non-interacting JCR model, the calculated energy values for the ground and the first excited states are then found to be —0.91 a.u. and 0.41 a.u. with corresponding standard errors of 9.12×10−8 a.u. and 1.95×10−8 a.u., respectively. We expect that the method presented in this paper can be an alternative approach to solving various qubit-photon systems.