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Classical simulation of relativistic quantum mechanics in periodic optical structures

Stefano Longhi

Published Jan 1, 2011• 34 citations
OpenAlex

Abstract

Spatial and/or temporal propagation of light waves in periodic optical structures offers a rather unique possibility to realize in a purely classical setting the optical analogues of a wide variety of quantum phenomena rooted in relativistic wave equations. In this work a brief overview of a few optical analogues of relativistic quantum phenomena, based on either spatial light transport in engineered photonic lattices or on temporal pulse propagation in Bragg grating structures, is presented. Examples include spatial and temporal photonic analogues of the Zitterbewegung of a relativistic electron, Klein tunneling, vacuum decay and pair-production, the Dirac oscillator, the relativistic Kronig-Penney model, and optical realizations of non-Hermitian extensions of relativistic wave equations.

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Spatial and/or temporal propagation of light waves in periodic optical structures offers a rather unique possibility to realize in a purely classical setting the optical analogues of a wide variety of quantum phenomena rooted in relativistic wave equations. In this work a brief overview of a few optical analogues of relativistic quantum phenomena, based on either spatial light transport in engineered photonic lattices or on temporal pulse propagation in Bragg grating structures, is presented. Examples include spatial and temporal photonic analogues of the Zitterbewegung of a relativistic electron, Klein tunneling, vacuum decay and pair-production, the Dirac oscillator, the relativistic Kronig-Penney model, and optical realizations of non-Hermitian extensions of relativistic wave equations.
Zitterbewegung (ZB) is a phenomenon in relativistic quantum systems where the electron wave packet exhibits a trembling or oscillating behavior during its motion, caused by its interaction or coupling with the negative energy state. To directly observe ZB in electronic systems is difficult, due to the challenges associated with the small amplitude of the motion which is of the order of Compton wavelength. Photonic systems offer an alternative paradigm. We exploit the concept of pseudo parity-time (pseudo ) symmetry to study ZB in non-Hermitian quantum systems implemented as an experimentally feasible optical waveguide array. In particular, the non-Hermitian Hamiltonian is realized through evanescent coupling among the waveguides to form a one-dimensional lattice with periodic modulations in gain and loss along the guiding direction. As the modulation frequency is changed, we obtain a number of phenomena including periodically suppressed ZB trembling, spatial energy localization, and Hermitian-like ZB oscillations. We calculate phase diagrams indicating the emergence of different types of dynamical behaviors of the relativistic non-Hermitian quantum system in an experimentally justified parameter space. We provide numerical results and a physical analysis to explain the distinct dynamical behaviors revealed by the phase diagrams. Our findings provide a deeper understanding of both the relativistic ZB phenomenon and non-Hermitian pseudo- systems, with potential applications in controlling/harnessing light propagation in waveguide-based optical systems.
Abstract Engineered photonic waveguides have provided in the past decade an extremely rich laboratory tool to visualize with optical waves the classic analogues of a wide variety of coherent quantum phenomena encountered in atomic, molecular or condensed‐matter physics. As compared to quantum systems, optics offers the rather unique advantage of a direct mapping of the wave function evolution in coordinate space by simple fluorescence imaging or scanning tunneling optical microscopy techniques. In this contribution recent theoretical and experimental advances in the field of quantum‐optical analogies are reviewed. Special attention is devoted to some relevant optical analogies based on the use of curved photonic structures, including: coherent destruction of tunneling in driven bistable potentials; coherent population transfer and adiabatic passage in laser‐driven multilevel atomic systems; quantum decay control and Zeno dynamics; electronic Bloch oscillations and Zener tunneling, Anderson localization and dynamic localization in crystalline potentials.
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. 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.
A classical analog simulator of the two-site Hubbard-Holstein model, describing the dynamics of two correlated electrons coupled with local phonons, is proposed based on light transport in engineered optical waveguide arrays. Our photonic analog simulator enables us to map the temporal dynamics of the quantum system in Fock space into spatial propagation of classical light waves in the evanescently coupled waveguides of the array. In particular, in the strong correlation regime the periodic temporal dynamics, related to the excitation of Holstein polarons with equal energy spacing, can be visualized as a periodic self-imaging phenomenon of the light beam along the waveguide array and explained in terms of generalized Bloch oscillations of a single particle in a semi-infinite inhomogeneous tight-binding lattice.