Grassmann Variables and the Jaynes-cummings Model

1 0.1 Abstract The Jaynes-Cummings model of a two-level 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 n-photon 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 high-Q microwave cavity is key experimental evidence for quantization of the EM field. Theoretical treatments of the Jaynes-Cummings model based on expanding the state vector in terms on products of atomic and n-photon states and deriving coupled equations for the amplitudes are a well-known 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 c-number 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 c-number 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 c–number phase space variables, and distribution functions involving these variables and those for any bosonic modes being shown to satisfy Fokker-Planck equations from which c-number 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 fermion annihilation and creation operators. Though phase space methods in which the fermion operators are represented directly by c-number phase space variables have not been successful , the anti-commutation rules for these operators suggests the possibility of using Grassmann variables-which have similar anti-commutation properties. However, in spite of the seminal work by Cahill and Glauber and a few applications , the use of phase space methods in quantum …