Optical simulation of the free Dirac equation

Abstract

We present a classical optics simulation of the one-dimensional Dirac equation for a free particle. Positive and negative energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle. Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator-known as Zitterbewegung (ZB). We are also able to measure the so called mean-position operator which is a single-particle observable and presents no oscillations. Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.PACS numbers:

Tachyonic Dirac Equation Revisited

“…Let us clarify this aspect further. To better understand the algebraical structure of Matrix (13), let us break it down into the sum of four matrices:…”

“…In physics and quantum chemistry, the Dirac equation and the Schrodinger equation together form the main foundation. In its non-linear version [5][6][7], the Dirac equation is used in condensed matter physics [8][9][10] and quantum optics [11][12][13][14]. The Dirac equation has also been generalised for curved space-time in order to study the behaviour of fermions in gravitational fields [15][16][17].…”

Tachyonic Dirac Equation Revisited

“…where ̂ is the position operator in momentum space, is the non-Hermitian operator (13) and is any of the tachyonic wavefunctions (4). The commutator [ ,̂] returns the vector whose components are the three Dirac matrices .…”

Effect of zitterbewegung on the propagation of wave packets in ABC-stacked multilayer graphene: an analytical and computational approach

“…in ultracold atoms [3,4], semiconductors [5][6][7][8][9][10], carbon nanotubes [11], topological insulators [12], crystalline solids [13,14] and other systems [15][16][17][18]. Although ZBW was theoretically found using a quantum simulation of the Dirac equation for trapped ions [19], Bose-Einstein condensates [20][21][22] and, most recently, an optical simulation [23], up to now, no direct experimental observations have been carried out. The reason is that the Dirac equation predicts ZBW with amplitude of the order of the Compton wavelength (10 −2 Å) and a frequency of ω ZB ≈ 10 21 Hz, which are not accessible with current experimental techniques.…”