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New Journal of Physics

Reply to Comment on ‘Wigner function for a particle in an infinite lattice’

Margarida Hinarejos, Armando Pérez, Mari Carmen Bañuls

Published Jun 13, 2013• 1 citation
Publisher

Abstract

In a recent paper (2012 New J. Phys. 14 103009), we proposed a definition of the Wigner function for a particle on an infinite lattice. Here we argue that the criticism to our work raised by Bizarro is not substantial and does not invalidate our proposal.
DOI
10.1088/1367-2630/15/6/068002

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The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.

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It is pointed out that in a recent paper (2012 New J. Phys. 14 103009) in which a Wigner function for a particle in an infinite lattice (a system described by an unbounded discrete coordinate and its conjugate angle-like momentum) has been introduced, no reference is made to previous, pioneering work on discrete Wigner distributions (more precisely, on the rotational Wigner function for a system described by a rotation angle and its unbounded discrete-conjugate angular momentum). Not only has the problem addressed in essence been solved for a long time (the discrete coordinate and angle-like conjugate momentum are the perfect dual of the rotation angle and discrete-conjugate angular momentum), but the solution advanced only in some distorted manner obeys two of the fundamental properties of a Wigner distribution (that, when integrated over one period of the momentum variable, it should yield the correct marginal distribution on the discrete position variable, and that it should be invariant with respect to translation).
The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys.14 103009) is extended here to include an internal degree of freedom as spin.This extension is made by introducing a Wigner matrix.The formalism is developed to account for dynamical processes, with or without decoherence.We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited.Discrete processes are also discussed.Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.
The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.
Sadeghi et al. [Phys. Rev. A 82, 012102 (2010)] presented a new nonclassicality indicator which works properly for Husimi and Rivier distribution functions as well as the Wigner distribution function. Some objections on the Sadeghi nonclassicality indicator are presented by Kenfack (preceding Comment, ``Nonclassicality indicator for the real phase-space distribution functions'', Phys. Rev. A 93, 036101 (2016). In this Reply, some misconceptions on Sadeghi's nonclassicality indicator are discussed and clarified.