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arXiv:physics/9808029 · Physical Review A

Density operators for fermions

Kevin E. Cahill, Roy J. Glauber

Published Feb 1, 1999• 163 citations
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Abstract

The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.

DOI
10.1103/physreva.59.1538

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The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.
A fermionic analog of a parametric amplifier is used to describe the joint quantum state of the two interacting fermionic modes. Based on a two-mode generalization of the time-dependent density operator, time evolution of the fermionic density operator is determined in terms of its two-mode Wigner and PPP function. It is shown that the equation of motion of the Wigner function corresponds to a fermionic analog of Liouville's equation. The equilibrium density operator for fermionic fields developed by Cahill and Glauber is thus extended to a dynamical context to show that the mathematical structures of both the correlation functions and the weight factors closely resemble their bosonic counterpart. It has been shown that the fermionic correlation functions are marked by a characteristic upper bound due to Fermi statistics, which can be verified in the matter wave counterpart of photon down-conversion experiments.
The quantum dissipation of a fermionic oscillator in an environment of fermionic oscillators is considered. Since fermions anticommute, their eigenvalues are anticommuting numbers. Based on an expansion of the reduced density operator in terms of fermionic coherent states which make use of these anticommuting numbers or Grassmann variables, a Fokker-Planck equation for the associated quasiprobability distribution function is derived. The density operator approach to fermionic fields as introduced by Cahill and Glauber [Phys. Rev. A 59, 1538 (1999)] thus can be extended to the nonequilibrium domain.
The Glauber-Sudarshan P-representation is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. For simplicity only one type of boson or fermion will be considered.
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.
Whereas boson coherent states provide an elegant, intuitive and useful representation, we show that the desirable features of boson coherent states do not carry over very well to fermion fields unless one is prepared to use exotic approaches such as Grassmann fields. Specifically, we identify four appealing properties of boson coherent states (eigenstate of annihilation operator, displaced vacuum state, preservation of product states under linear coupling, and factorization of correlators) and show that fermion coherent states, and approximations to fermion coherent states, defined over the complex field, do not behave well for any of these four criteria.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation functions in terms of phase space functional integrals, where the field functions—complex fields for bosons, Grassmann fields for fermions—replace field annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functionals, with distinct fields for annihilation and creation field operators; however, symmetrically ordered (Wigner type) distribution functionals are also considered, along with unnormalised B distribution functionals, which lead to phase space functional integrals for Fock position state populations and coherences. Characteristic functionals are first defined and then shown to be related to distribution functionals via phase space functional integrals. The existence of the distribution functionals is demonstrated from previous results for equivalent distribution functions by using mode expansions.