Dalton, B.J., Garraway, B.M., Jeffers, J. and Barnett, S.M. (2013) Grassmann phase space theory and the Jaynes–Cummings model. Annals of Physics, 334, pp. 100141. (doi:10.1016/j.aop.2013.03.010)

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Abstract
The Jaynes–Cummings model of a twolevel atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an nphoton Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherent state. The observation of revivals for Rydberg atoms in a highQ microwave cavity is key experimental evidence for quantisation of the EM field. Theoretical treatments of the Jaynes–Cummings model based on expanding the state vector in terms of products of atomic and nphoton states and deriving coupled equations for the amplitudes are a wellknown and simple method for determining the effects.
In quantum optics however, the behaviour of the bosonic quantum EM field is often treated using phase space methods, where the bosonic mode annihilation and creation operators are represented by cnumber phase space variables, with the density operator represented by a distribution function of these variables. Fokker–Planck equations for the distribution function are obtained, and either used directly to determine quantities of experimental interest or used to develop cnumber Langevin equations for stochastic versions of the phase space variables from which experimental quantities are obtained as stochastic averages. Phase space methods have also been developed to include atomic systems, with the atomic spin operators being represented by cnumber phase space variables, and distribution functions involving these variables and those for any bosonic modes being shown to satisfy Fokker–Planck equations from which cnumber Langevin equations are often developed. However, atomic spin operators satisfy the standard angular momentum commutation rules rather than the commutation rules for bosonic annihilation and creation operators, and are in fact second order combinations of fermionic annihilation and creation operators. Though phase space methods in which the fermionic operators are represented directly by cnumber phase space variables have not been successful, the anticommutation rules for these operators suggest the possibility of using Grassmann variables—which have similar anticommutation properties. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of phase space methods in quantum optics to treat fermionic systems by representing fermionic annihilation and creation operators directly by Grassmann phase space variables is rather rare. This paper shows that phase space methods using a positive P type distribution function involving both cnumber variables (for the cavity mode) and Grassmann variables (for the TLA) can be used to treat the Jaynes–Cummings model. Although it is a Grassmann function, the distribution function is equivalent to six cnumber functions of the two bosonic variables. Experimental quantities are given as bosonic phase space integrals involving the six functions. A Fokker–Planck equation involving both left and right Grassmann differentiations can be obtained for the distribution function, and is equivalent to six coupled equations for the six cnumber functions. The approach used involves choosing the canonical form of the (nonunique) positive P distribution function, in which the correspondence rules for the bosonic operators are nonstandard and hence the Fokker–Planck equation is also unusual. Initial conditions, such as those above for initially uncorrelated states, are discussed and used to determine the initial distribution function. Transformations to new bosonic variables rotating at the cavity frequency enable the six coupled equations for the new cnumber functions–that are also equivalent to the canonical Grassmann distribution function–to be solved analytically, based on an ansatz from an earlier paper by Stenholm. It is then shown that the distribution function is exactly the same as that determined from the wellknown solution based on coupled amplitude equations. In quantum–atom optics theories for many atom bosonic and fermionic systems are needed. With large atom numbers, treatments must often take into account many quantum modes—especially for fermions. Generalisations of phase space distribution functions of phase space variables for a few modes to phase space distribution functionals of field functions (which represent the field operators, cnumber fields for bosons, Grassmann fields for fermions) are now being developed for large systems. For the fermionic case, the treatment of the simple two mode problem represented by the Jaynes–Cummings model is a useful test case for the future development of phase space Grassmann distribution functional methods for fermionic applications in quantum–atom optics.Item Type:  Articles 

Status:  Published 
Refereed:  Yes 
Glasgow Author(s) Enlighten ID:  Barnett, Professor Stephen 
Authors:  Dalton, B.J., Garraway, B.M., Jeffers, J., and Barnett, S.M. 
College/School:  College of Science and Engineering > School of Physics and Astronomy 
Journal Name:  Annals of Physics 
Publisher:  Elsevier 
ISSN:  00034916 
ISSN (Online):  1096035X 
Copyright Holders:  Copyright © 2013 Elsevier Ltd. 
First Published:  First published in Annals of Physics 334:100141 
Publisher Policy:  Reproduced in accordance with the copyright policy of the publisher 
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