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arXiv:1810.04396 · Physical Review A

Combining spatiotemporal and particle-number degrees of freedom

Filippus Stefanus Roux

Published Oct 22, 2018• 13 citations
arXivHTMLPDF

Abstract

Quadrature bases that incorporate spatiotemporal degrees of freedom are derived as eigenstates of momentum-dependent quadrature operators. The resulting bases are shown to be orthogonal for both the particle-number and the spatiotemporal degrees of freedom. Using functional integration, we also demonstrate the completeness of these quadrature bases.

DOI
10.1103/physreva.98.043841

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Quadrature bases that incorporate spatiotemporal degrees of freedom are derived as eigenstates of momentum-dependent quadrature operators. The resulting bases are shown to be orthogonal for both the particle-number and the spatiotemporal degrees of freedom. Using functional integration, we also demonstrate the completeness of these quadrature bases.
Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. To demonstrate the approach, we compute examples of Wigner functionals for a few well-known states and operators.
The definition of quantum states in terms of tagged vector spaces is generalized to incorporate the spatiotemporal and spin degrees of freedom. Considering a tagged vector space where the index space is a function space, representing the additional degrees of freedom, we obtained axioms for the tags that include a completeness condition expressed in terms of a functional integral with an abstract functional integration measure. Using these axioms, we derive a generating functional for the moments of this functional integration measure. These moments are then used to evaluate the functional integrals of Gaussian functionals, leading to expressions in accordance with those obtained as generalizations of equivalent integrals over a finite number of integration variables. For a Gaussian functional used as a probability distributions, we show that its moments, obtained with this functional integration measure, satisfy Carleman's condition, indicating that the measure is unique.
Abstract A Grassmann functional phase space is formulated for the definition of fermionic Wigner functionals by identifying suitable fermionic operators that are analogues to boson quadrature operators. Instead of the Majorana operators, we use operators that are defined with relative spin transformations between the ladder operators. The eigenstates of these operators are shown to provide orthogonal bases, provided that the dual space is defined with the incorporation of a spin transformation. These bases then serve as quadrature bases in terms of which Wigner functionals are defined in a way equivalent to the bosonic case. As an application, we consider a two-level fermion system.