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Introduction

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Published Nov 13, 2014• 0 citations
Publisher

Abstract

Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.

DOI
10.1093/acprof:oso/9780199562749.003.0001

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•Nov 13, 2014•0 citations
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.
Abstract Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. As its title suggests, this book presents the phase space theory approach to treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single-particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation and creation operators, the dynamical equation for the density operator determines a Fokker–Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, the phase space variables are replaced by time-dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker–Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is then treated, the density operator being represented by a distribution functional of field functions which replace field annihilation and creation operators, the distribution functional satisfying a functional Fokker–Planck equation, etc. A novel feature of this book is that the phase space variables for fermions are Grassmann variables, not c-numbers. However, the book shows that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions.
Abstract This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. Specific forms are obtained for master equation dynamics, relating the drift vector and diffusion matrix elements in the FPE to coefficients specifying terms in the Hamiltonian and relaxation operators. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by mode annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functions, these being differentiations and or multiplications by phase space variables (left or right for fermions). Derivations are carried out via changes to characteristic functions or to canonical forms of the density operator. The correct successive application of correspondence rules for fermions is treated, and alternative forms obtained for bosons to correspondence rules and the FPE.
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.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.
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.
Abstract In this chapter, Langevin equations (or Ito stochastic differential equations, SDEs) are derived that are equivalent to Fokker–Planck equations for bosons and fermions. The approach involves replacing modal phase space variables by stochastic phase space variables satisfying Ito SDEs containing c-number Wiener stochastic variables, as well as functions of stochastic phase variables. It is shown that if the Ito SDE quantities are related to drift and diffusion terms in the FPE, then the time dependences of phase space averages of arbitrary functions equal those for stochastic averages of these functions. Using Takagi factorisation, Ito SDE quantities satisfying the required relationship are found, for both bosons and fermions. The Wiener variable numbers differ—2n for bosons, 2 n2 for fermions. For fermion unnormalised B distributions, both the drift and the diffusion Ito SDE terms are linear in the stochastic phase variables. SDEs are applied to fluctuations and quantum correlation functions.