Wigner formalism for a particle on an infinite lattice: dynamics and spin

Abstract

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.

Quantum simulation of quantum relativistic diffusion via quantum walks

“…A particularity of the present work is thus to introduce the noise on the in- (2)), we are left with the two standard error channels [59,73] that we have introduced on the coin (see Eqs. (51) and (52)): the populations' difference r 3 decays exponentially with a rate γ 2 , and the real (resp. imaginary) part of the coherences, r 1 (resp.…”

“…The problem of nding macroscopic models of relativistic non-quantum diffusion was rst considered in the 1940s by Landau and Eckhart [7,8]. It was rst revisited by Cattaneo [9], who suggested to model relativistic diffusion through the telegraph equation, and whose work, coupled with the Grad expansion technique [10] (see also reference [52] in reference [11]), laid the basis of the so-called extended thermodynamics theories [12]. All models produced by these efforts present serious dif culties, which range from non-causality and instability (see references [6,7,49] in reference [11]) to experimental refutation 9 (see references [58,59] in reference [11]).…”

“…The connections between DTQWs and lattice gauge theories have also been explored [47][48][49][50], and Wigner functions for DTQWs have been proposed in references [51][52][53]. A crucial feature of DTQWs is that they are intrinsically causal, i.e., information propagates, at most, at a nite velocity c = 1, which is why DTQWs are a priori especially suited to model quantum relativistic diffusions.…”

Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment

“…In the same way, the negativity of the Wigner function is also perceived, which is commonly associated with some kind of nonclassical behavior, although some care must be taken when using this interpretation of the negativity because it involves the spin variable also. This issue is associated with the definition of the Wigner function for finite dimensional Hilbert space [ 83 , 84 , 85 , 86 ].…”

Relativistic Wigner Function for Quantum Walks

“…If one follows the standard procedure adopted for Dirac fermions [36][37][38], one should first build a relativistic Wigner function for DTQWs and then describe the phase-space dynamics by the transport equation obeyed by that function. Until now, the only Wigner functions that have been considered for DTQWs [39][40][41][42] are non relativistic [43]. Thus, these function and the equation they obey do not coincide, at the continuous limit, with the usual Wigner function and phase-space transport equation for Dirac fermions.…”