The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

Haddad, L. H.; Carr, Lincoln D.

doi:10.1016/j.physd.2009.02.001

Cited by 109 publications

(126 citation statements)

References 46 publications

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“…The nonlinear Dirac (NLD) equation in 1 + 1 dimensions [1] has a long history and has emerged as a useful model in many physical systems such as extended particles [2][3][4], the gap solitons in nonlinear optics [5], light solitons in waveguide arrays and experimental realization of an optical analog for relativistic quantum mechanics [6][7][8], BoseEinstein condensates in honeycomb optical lattices [9], phenomenological models of quantum chromodynamics [10], as well as matter influencing the evolution of the universe in cosmology [11]. Further, the multi-component BEC order parameter has an exact spinor structure and serves as the bosonic analog to the relativistic electrons in graphene.…”

Section: Introductionmentioning

confidence: 99%

Approximate analytic solutions to coupled nonlinear Dirac equations

2017

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We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalarscalar self interactions g 2 1 2 (ψψ) 2 + g 2 2 2 (φφ) 2 + g 2 3 (ψψ)(φφ) as well as vector-vector interactions of the form g 2 1 2 (ψγµψ)(ψγ µ ψ) + g 2 2 2 (φγµφ)(φγ µ φ) + g 2 3 (ψγµψ)(φγ µ φ). Writing the two components of the assumed solitary wave solution of these equation in the form ψ = e −iω 1 t {R1 cos θ, R1 sin θ}, φ = e −iω 2 t {R2 cos η, R2 sin η}, and assuming that θ(x), η(x) have the same functional form they had when g3=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 2 3 /g 2 2 and g 2 3 /g 2 1 . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.

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Section: Introductionmentioning

confidence: 99%

Approximate analytic solutions to coupled nonlinear Dirac equations

2017

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“…The first study of nonlinear dynamics in honeycomb lattices was conducted in [13], demonstrating gap solitons, which had no overlap with Bloch modes residing at the vicinity to the Dirac points. Later studies of nonlinear dynamics in honeycomb lattices were a generalization of Dirac approximation [25,26], where a nonlinear version of the massless Dirac equation have been studied.Here, we study the nonlinear dynamics of waves in honeycomb lattices, where initial wave-packets are comprised of Bloch waves from the vicinity of the Dirac points. Such wavepackets are very well described by the massless Dirac equation.…”

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confidence: 99%

Breakdown of Dirac dynamics in honeycomb lattices due to nonlinear interactions

et al. 2010

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We study the dynamics of coherent waves in nonlinear honeycomb lattices and show that nonlinearity breaks down the Dirac dynamics. As an example, we demonstrate that even a weak nonlinearity has major qualitative effects one of the hallmarks of honeycomb lattices: conical diffraction.Under linear conditions, a circular input wave-packet associated with the Dirac point evolves into a ring, but even a weak nonlinearity alters the evolution such that the emerging beam possesses triangular symmetry, and populates Bloch modes outside of the Dirac cone region. Our results are presented in the context of optics, but we propose a scheme to observe equivalent phenomena in Bose-Einstein condensates. 1The past decade has witnessed considerable interest in honeycomb lattices in many fields, ranging from carbon nanotubes [1] and graphene [2][3][4] in condensed matter, cold atoms in honeycomb optical lattices [5][6][7][8], and electromagnetic waves in honeycomb photonic crystals [9][10][11][12] and photonic lattices (waveguide arrays) [13][14][15]. The unique features of the honeycomb lattice result from its unusual band structure, displaying two intersecting bands. The vicinity of the intersection points is conical, and excitations residing at the close vicinity of the intersection points obey the massless Dirac equation: a relativistic wave equation for massless spin-half particles. As a consequence of this unusual linear dispersion and the additional degrees of freedom (two sites in each unit cell, referred as pseudo-spin), a variety of exciting phenomena have been obtained: room temperature quantum Hall effect in graphene [16], conical diffraction in honeycomb photonic lattices [13], negative magnetoresistance which implies anti-localization [17][18][19], Klein tunneling [15,[20][21][22], and Zitterbewegung [23,24] to name a few. The dynamics of Dirac-like excitations in honeycomb lattices has been well studied when the propagation equation is linear (interaction-free). However, the dynamics can also be nonlinear, as happens in the optical domain due to light-matter interactions, and for atomic Bose-Einstein condensates (BEC) due to pair-wise scattering. In either of these, the nonlinear dynamics has received little attention in the context of honeycomb lattices. The first study of nonlinear dynamics in honeycomb lattices was conducted in [13], demonstrating gap solitons, which had no overlap with Bloch modes residing at the vicinity to the Dirac points. Later studies of nonlinear dynamics in honeycomb lattices were a generalization of Dirac approximation [25,26], where a nonlinear version of the massless Dirac equation have been studied.Here, we study the nonlinear dynamics of waves in honeycomb lattices, where initial wave-packets are comprised of Bloch waves from the vicinity of the Dirac points. Such wavepackets are very well described by the massless Dirac equation. However, even the presence of a fairly weak nonlinearity drives the waves away from the vicinity of the Dirac points, hence the nonlinearity brea...

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“…Also, the interplay of non-linearities and disorder, which has proven to host a variety of interesting phenomena in non-relativistic systems 19,50,55 , is yet to be investigated in the relativistic context. A commonly considered non-linear Dirac equation contains an additional interaction term of third order in the spinor 56 , which can be implemented by a Kerr non-linearity in waveguide lattices 57 .…”

Section: Discussionmentioning

confidence: 99%

The random mass Dirac model and long-range correlations on an integrated optical platform

et al. 2013

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Long-range correlation-the non-local interdependence of distant events-is a crucial feature in many natural and artificial environments. In the context of solid state physics, impurity spins in doped spin chains and ladders with antiferromagnetic interaction are a prominent manifestation of this phenomenon, which is the physical origin of the unusual magnetic and thermodynamic properties of these materials. It turns out that such systems are described by a one-dimensional Dirac equation for a relativistic fermion with random mass. Here we present an optical configuration, which implements this one-dimensional random mass Dirac equation on a chip. On this platform, we provide a miniaturized optical test-bed for the physics of Dirac fermions with variable mass, as well as of antiferromagnetic spin systems. Moreover, our data suggest the occurence of long-range correlations in an integrated optical device, despite the exclusively short-ranged interactions between the constituting channels.

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The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

Cited by 109 publications

References 46 publications

Approximate analytic solutions to coupled nonlinear Dirac equations

Approximate analytic solutions to coupled nonlinear Dirac equations

Breakdown of Dirac dynamics in honeycomb lattices due to nonlinear interactions

The random mass Dirac model and long-range correlations on an integrated optical platform

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