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arXiv:1203.4878 · Annals of Physics

Grassmann phase space theory and the Jaynes–Cummings model

Bryan J. Dalton, B. M. Garraway, John Jeffers, S. M. Barnett

Published Apr 6, 2013• 11 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2013.03.010

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Phase space distribution functional methods utilising c-number and Grassmann fields for bosons and fermions respectively are being developed for theoretical treatments of degenerate quantum gases with large fermionic and bosonic atom numbers. The Jaynes-Cummings model where a single mode bosonic system and a two mode fermionic system are coupled provides a simple testbed for phase space methods involving Grassmann variables. The Fokker-Planck equation for the distribution function is shown to be equivalent to six coupled equations for c-number bosonic functions. Numerical solutions enable well-known effects such as collapses/revivals of the atomic population difference to be demonstrated.
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.