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Physical Review A

Monte Carlo simulation of real-time quantum processes

A. Duncan, Robert D. Mawhinney, P. Stehle

Published Oct 1, 1989• 1 citation
Publisher

Abstract

It is shown that Monte Carlo methods can be fruitfully applied to the computation of quantum processes in the real-time (Minkowski) regime. The method is first explained in the context of a simple quantum-mechanical model, then applied to the problem of the quantized electromagnetic field interacting with a large number of discrete two-state systems (the problem of superradiance). Numerical results previously obtained by operator methods are reproduced in this framework and the calculations extended to much larger systems.

DOI
10.1103/physreva.40.4004

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It is shown that Monte Carlo methods can be fruitfully applied to the computation of quantum processes in the real-time (Minkowski) regime. The method is first explained in the context of a simple quantum-mechanical model, then applied to the problem of the quantized electromagnetic field interacting with a large number of discrete two-state systems (the problem of superradiance). Numerical results previously obtained by operator methods are reproduced in this framework and the calculations extended to much larger systems.
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The exact solution of quantum mechanical many-body problems is only possible for few particles. Therefore, numerical methods were developed in the fields of quantum physics and quantum chemistry for larger particle numbers. Configuration Interaction (CI) methods or the No-Core Shell Model (NCSM) allow ab initio calculations for light and intermediate-mass nuclei, without resorting to phenomenology. An extension of the NCSM is the Importance-Truncated No-Core Shell Model, which uses an a priori selection of the most important basis states. The importance truncation was first developed and applied in quantum chemistry in the 1970s and latter successfully applied to models of light and intermediate mass nuclei. Other numerical methods for calculations for ultra-cold fermionic many-body systems are the Fixed-Node Diffusion Monte Carlo method (FN-DMC) and the stochastic variational approach with Correlated Gaussian basis functions (CG). There are also such method as the Coupled-Cluster method, Green’s Function Monte Carlo (GFMC) method, et cetera, used for calculation of many-body systems. In this thesis, we adopt the IT-NCSM for the calculation of ultra-cold Fermi gases at unitarity. Ultracold gases are dilute, strongly correlated systems, in which the average interparticle distance is much larger than the range of the interaction. Therefore, the detailed radial dependence of the potential is not resolved, and the potential can be replaced by an effective contact interaction. At low energy, s-wave scattering dominates and the interaction can be described by the s-wave scattering length. If the scattering length is small and negative, Cooper-pairs are formed in the Bardeen-Cooper-Schrieffer (BCS) regime. If the scattering length is small and positive, these Cooper-pairs become strongly bound molecules in a Bose-Einstein-Condensate (BEC). In between (for large scattering lengths) is the unitary limit with universal properties. Calculations of the energy spectra (ground-state and first excited-state) have so far only been performed for up to five particles using CI or NCSM methods or for up to six particles using the CG method. Calculations with larger particle numbers have only been performed with Monte Carlo methods and only for the ground state of up to 30 particles. We extend ab initio calculations of the energy spectra of ultra-cold Fermi gases at unitarity for up to 20 particle using the IT-NCSM. For our calculations we use different interactions: an effective interaction introduced by Alhassid, Bertsch and Fang and an interaction constructed using an effective field theory (EFT) approach. Furthermore, we use a Gauss-shaped potential as it is also used for the calculations of ultra-cold Fermi gases at unitarity using CG or FN-DMC methods. Although more effort must be invested to make the Gauss-shaped potential suitable for IT-NCSM calculations of ultra-cold Fermi gases, calculations with the other two interactions yield ground-state energies which agree excellently with the results obtained using the FN-DMC method. The IT-NCSM extends the range of the NCSM and permits the calculation of energy spectra (ground-state and excited-state energies) of fermionic systems with large particle numbers for which previously only the ground states could be calculated using Monte Carlo methods.