Zitterbewegung of optical pulses in nonlinear frequency conversion

Abstract

Pulse walk-off in the process of sum frequency generation in a nonlinear χ (2) crystal is shown to be responsible for pulse jittering which is reminiscent to the Zitterbewegung (trembling motion) of a relativistic freely moving Dirac particle. An analytical expression for the pulse center of mass trajectory is derived in the nopump-depletion limit, and numerical examples of Zitterbewegung are presented for sum frequency generation in periodically-poled lithium niobate. The proposed quantumoptical analogy indicates that frequency conversion in nonlinear optics could provide an experimentally accessible simulator of the Dirac equation.

Relativistic quantum mechanics with trapped ions

“…Many other proposals and some experiments, in a wide variety of systems, have recently appeared, as the simulation of Zitterbewegung in semiconductor quantum wells [16] or in graphene [17,18], Klein paradox in graphene [19], Dirac oscillator in a trapped ion [20], Zitterbewegung and Dirac physics with ultracold atoms [21,22,23,24,25,26], Klein paradox with atomic ensembles [27], optical Zitterbewegung in metamaterials [28,29], delocalization of relativistic Dirac particles in cold atoms [30], photon wave function and Zitterbewegung [31], similarity of electron's Zitterbewegung to the Adler-Bell-Jackiw anomaly in QED and its manifestation in graphene [32], photonic analog of Zitterbewegung in binary waveguide arrays [33], Zitterbewegung theory in multiband Hamiltonians [34], classical Zitterbewegung in reduced plasma dynamics [35], Zitterbewegung analogs in nonlinear frequency conversion [36], experimental realization of an optical analog for relativistic quantum mechanics in an optical superlattice [37], relation between parity and Zitterbewegung and proposed simulation in trapped ions [38], Zitterbewegung in a magnetic field and proposal for trapped ion simulation [39], Wilson fermions and axion electrodynamics in optical lattices [40], the Schwinger effect for a possible implementation with atoms in optical lattices [41], Dirac equation for cold atoms in artificial curved spacetimes [42], a theoretical analysis of cold atom simulation of interacting relativistic quantum field theories [43], or an analysis of the photonic simulation of the quark model [44]. For a review of Zitterbewegung of electrons in semiconductors, see Ref.…”

Classical simulation of relativistic quantum mechanics in periodic optical structures

“…A second example of a photonic analogue of ZB is provided by the process frequency conversion of optical pulses in a nonlinear quadratic medium arising from group velocity mismatch [42]. Let us consider the propagation of three optical pulses at carrier frequencies ω 1 , ω 2 and ω 3 = ω 1 + ω 2 in a nonlinear quadratic medium and in presence of group velocity mismatch.…”

“…Photonic analogues of Dirac-type equations have been also theoretically proposed for light propagation in certain triangular or honeycomb photonic crystals, which mimic conical singularity of energy bands of graphene [31,32,33], as well as in metamaterials [34], optical superlattices [35,36,37,38], Bragg gratings [39,40,41], and nonlinear quadratic media [42]. Such studies have motivated extended investigations on the properties of 'photonic graphene' [31,33,43] and to the proposals of photonic analogues of relativistic phenomena like Zitterbewegung [32,34,35,42], Klein tunneling [33,34,36,39], decay of the quantum vacuum and pair production [37], the Dirac oscillator [40], and the relativistic versions of the Kronig-Penney model and surface Tamm states [41]. Noticeably, the introduction of gain and/or loss regions in the optical medium can be exploited to realize in a classical setting certain non-Hermitian relativistic models proposed in the context of non-Hermitian quantum mechanics and quantum field theories [44,45].…”

Different models of gravitating Dirac fermions in optical lattices

“…For instance, the celebrated Haldane model [25] recently experimentally demonstrated with ultracold atoms in an optically shaken brickwall lattice [26] can be interpreted as a realization of lattice Dirac Hamiltonian without doubling due to the breaking of the chiral symmetry [27]. Furthermore, systems governed by the Dirac Hamiltonian display also anomalous Hall conductivity [28][29][30][31][32] and puzzling properties like Klein tunneling [33] and zitterbewegung [34,35], phenomena that are accessible preferably or uniquely with graphene [36][37][38] (or graphene like compounds, see [39]) or artificially engineered systems as in ultracold neutral atoms [40][41][42][43][44][45], trapped ions [46][47][48][49][50], photons [51][52][53], conductor quantum wells [54], and circuit QED [55,56].…”