The wigner distribution function for systems of bosons or fermions

GERALD J. IAFRATE, H. L. Grubin, David K. Ferry

Sumiyoshi Abe, Norikazu Suzuki

We construct the Wigner distribution function for a coupled boson-fermion system. This function is regarded as a certain ``superfield'' defined on the extended phase space, the points of which are labeled by the anticommuting canonical c variables together with the ordinary ones. In this approach, the variables of a fermion and a boson are treated on a completely equal footing, and the phase-space representation is fully realized. We apply the formalism to the kinetic theory of the optical Dicke model with a two-level atom, and show how systematically a set of the generalized Fokker-Planck equations, which describes the effective dynamics of the radiation field, is derived from a single superfieldlike equation.

Brandon P. van Zyl

We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a 'macroscopic' number of particles, and finite-temperatures.

M. Revzen

Lorenzo Galleani, Leon W. Cohen

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Abstract This chapter presents two examples of applying phase space methods in systems where large numbers of modes may be involved, such as in fermion cases with large particle numbers. Functional Fokker–Planck equations (FFPEs) are derived directly using the field operator correspondence rules. The boson application is to colliding bosons in a trap potential, such as for Bose–Einstein condensates. FFPEs and Ito stochastic field equations (SFEs) are derived for both positive P and Wigner distribution functional cases, and expressions obtained for quantum correlation functions. The fermion application is to colliding spin-1/2 fermions in a trap potential, such as an optical lattice. FFPEs and Ito SFEs are derived for the unnormalised B distribution functional. Solutions to the Ito SFE are presented for the untrapped and optical-lattice cases. For the untrapped case with two fermions, evaluation of short-time Fock state coherences and populations gives the same result as obtained in a previous chapter.