The nonlinear Dirac equation for Bose–Einstein condensates (BECs) in honeycomb optical lattices gives rise to relativistic multi-component bright and dark soliton solutions. Using the relativistic linear stability equations, the relativistic generalization of the Boguliubov-de Gennes equations, we compute soliton lifetimes against quantum fluctuations and classify the different excitation types. For a BEC of atoms, we find that our soliton solutions are stable on time scales relevant to experiments. Excitations in the bulk region far from the core of a soliton and bound states in the core are classified as either spin waves or as a Nambu–Goldstone mode. Thus, solitons are topologically distinct pseudospin- domain walls between polarized regions of . Numerical analysis in the presence of a harmonic trap potential reveals a discrete spectrum reflecting the number of bright soliton peaks or dark soliton notches in the condensate background. For each quantized mode the chemical potential versus nonlinearity exhibits two distinct power law regimes corresponding to the free-particle (weakly nonlinear) and soliton (strongly nonlinear) limits.
The nonlinear Dirac equation for Bose–Einstein condensates (BECs) in honeycomb optical lattices gives rise to relativistic multi-component bright and dark soliton solutions. Using the relativistic linear stability equations, the relativistic generalization of the Boguliubov-de Gennes equations, we compute soliton lifetimes against quantum fluctuations and classify the different excitation types. For a BEC of atoms, we find that our soliton solutions are stable on time scales relevant to experiments. Excitations in the bulk region far from the core of a soliton and bound states in the core are classified as either spin waves or as a Nambu–Goldstone mode. Thus, solitons are topologically distinct pseudospin- domain walls between polarized regions of . Numerical analysis in the presence of a harmonic trap potential reveals a discrete spectrum reflecting the number of bright soliton peaks or dark soliton notches in the condensate background. For each quantized mode the chemical potential versus nonlinearity exhibits two distinct power law regimes corresponding to the free-particle (weakly nonlinear) and soliton (strongly nonlinear) limits.
Abstract. We present a thorough analysis of soliton solutions to the quasi-one-dimensional zigzag and armchair nonlinear Dirac equation (NLDE) for a Bose-Einstein condensate in a honeycomb lattice. The two types of NLDEs correspond to quasi-one-dimensional reductions in two independent directions, in direct analogy to the narrowest of graphene nanoribbons. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-one-dimensional NLDE by finding fixed points, delineating the various regions in solution space, and through an invariance relation which we obtain as a first integral of the NLDE. For both the zigzag and armchair geometries we obtain spatially oscillating multi-soliton as well as asymptotically flat single soliton solutions using five different methods: by direct integration; an invariance relation; parametric transformation; a series expansion; and by numerical shooting. By tuning the ratio of the chemical potential to the nonlinearity for a fixed value of the Dirac spinor kinetic energy, we can obtain both bright and dark solitons over a nonzero background. The density contrast between the dark (bright) soliton notch (peak) and the background is about 2/3 (3/1). For both soliton types, we compute the discrete spectra for several spatially quantized states in a harmonic potential. We interpret our solitons as topologically protected domain walls in a quasi-one-dimensional system which separate distinct regions of pseudospin-1/2 with Sz = ±1/2. By solving the relativistic linear stability equations we obtain the low-energy spectrum for excitations in the bulk region far from the soliton core and for bound states in the core and find that excitations occur as spin waves and as a Nambu-Goldstone mode. For a Bose-Einstein condensate of 87Rb atoms, we find that our soliton solutions are stable on time scales relevant to experiments. ar
We present the theoretical and mathematical foundations of stability analysis for a Bose–Einstein condensate (BEC) at Dirac points of a honeycomb optical lattice. The combination of s -wave scattering for bosons and lattice interaction places constraints on the mean-field description, and hence on vortex configurations in the Bloch-envelope function near the Dirac point. A full derivation of the relativistic linear stability equations (RLSE) is presented by two independent methods to ensure veracity of our results. Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex solutions of the nonlinear Dirac equation, which include Anderson–Toulouse skyrmions with lifetime s. Beyond vortex stabilities the RLSE provide insight into the character of collective superfluid excitations, which we find to encode several established theories of physics. In particular, the RLSE reduce to the Andreev equations, in the nonrelativistic and semiclassical limits, the Majorana equation, inside vortex cores, and the Dirac–Bogoliubov–de Gennes equations, when nearest-neighbor interactions are included. Furthermore, by tuning a mass gap, relative strengths of various spinor couplings, for the small and large quasiparticle momentum regimes, we obtain weak-strong Bardeen–Cooper–Schrieffer superconductivity, as well as fundamental wave equations such as Schrödinger, Dirac, Klein–Gordon, and Bogoliubov–de Gennes equations. Our results apply equally to a strongly spin–orbit coupled BEC in which the Laplacian contribution can be neglected.
Bose-Einstein condensates (BECs) are a macroscopic state of quantum matter at temperatures nine orders of magnitude colder than outer space. BECs can be trapped in crystals of light: when the crystal takes the form of a honeycomb optical lattice, Dirac physics arises at the Brillouin zone edges, just like in graphene. The weakly interacting quantum gas comprising the BEC generates an effective nonlinearity, giving rise to the nonlinear Dirac equation (NLDE) [1], as we will show. This equation, together with the relativistic linear stability equations (RLSE) [2], comprise a relativistic generalization of the famous nonlinear Schrodinger equation and Boguliubov-de Gennes equations familiar from, for example, propagation of light in Kerr nonlinear media. We present a zoo of exotic vortex [3], skyrmion, and soliton [4,5] solutions to the NLDE+RLSE system, and explain the practical realization of these concepts in present experiments on BECs in the optical equivalent of 2D graphene and 1D armchair and zigzag graphene nanoribbons.
We propose a detailed experimental procedure for preparing relativistic vortices, governed by the nonlinear Dirac equation, in a two-dimensional Bose-Einstein condensate (BEC) in a honeycomb optical lattice. Our setup contains Dirac points, in direct analogy to graphene. We determine a range of practical values for all relevant physical parameters needed to realize relativistic vortices in a BEC of 87Rb atoms. Seven distinct vortex types, including Anderson-Toulouse and Mermin-Ho skyrmion textures and half-quantum vortices, are obtained, and their discrete spectra and stability properties are calculated in a weak harmonic trap. We predict that most vortices are stable, with a lifetime between 1 and 10 s.
In this thesis, we study the theory of ultracold atoms in two-dimensional (2D) optical
lattices, focusing in particular on honeycomb lattices, the same lattice geometry
as in graphene. We study the case of a Bose-Einstein condensate (BEC) located at the
Dirac points of the reciprocal honeycomb lattice, whose mean-field theory is described
by the nonlinear Dirac equation (NLDE), analogous to the nonlinear Schr¨odinger
equation (NLSE) for ordinary unconstrained BECs in three-dimensions (3D). Physically,
the NLDE describes relativistic quasi-particles which travel at speeds 10 orders
of magnitude slower than the speed of light, a feature which allows access to relativistic
phenomena in the laboratory. We derive the NLDE in coordinate space and
recover the same chiral structure and linear dispersion as for electrons in graphene,
but from an intuitive microscopic lattice perspective. Symmetries of the NLDE are
discussed in detail and compared with those of NLDEs found in the particle physics
and mathematics literature. We determine the low-energy theory by deriving, then
solving the relativistic linear stability equations (RLSE). These are the relativistic
analogs of the Bogoliubov-de Gennes equations (BdGE) and describe quasi-particle
fluctuations and their associated energy eigenvalues for a BEC near the Dirac point,
or Brillouin zone edge, of a honeycomb lattice.
Foundational issues regarding the NLDE are explored, in order to better understand
both the context of the NLDE in relation to the NLSE as well as in terms of
spin statistics. We give a microscopic physical explanation for the transition from
fundamental bosons to Dirac spinors. Similar to graphene, there is a Berry phase
associated with rotations in the honeycomb lattice BEC, and we explain this feature
in relation to the spin-statistics theorem. We explore reductions of the NLDE to the
cubic NLSE with additional correction terms, and study symmetry breaking in this
model in addition to soliton and vortex solutions. By including all Dirac points of
the honeycomb lattice, we find that the reduced NLDE maps to the nonlinear sigma
model. Lagrangian and energy functional approaches are treated, which provide useful
insights into the NLDE.
To place the NLDE on solid experimental ground, we define all relevant physical
parameters and show how they relate back to those of ordinary BECs. We do this
by deriving the parameter renormalizations which occur under dimensional reduction
from 3D to 2D, in addition to the effect of the periodic honeycomb lattice. All of
the constraints and approximations needed to observe the NLDE physics are delineated,
and a consistent range of values is determined for all of our parameters. To
realize NLDE physics along with relativistic vortex excitations, we propose a multistep
process using a spin-dependent lattice potential to turn off and on a mass gap,
Bragg scattering to transfer condensed atoms to the Dirac points, and Gaussian and
Laguerre-Gaussian laser beams to excite the vortices. The combined use of a spindependent
lattice and Bragg scattering allows for the transfer of atoms to a zero-group
velocity state at the Dirac point, resulting in a metastable non-equilibrium BEC removed
from the lattice ground state.
Solitons in the NLDE are realized by tightening the harmonic trap in one of the
planar directions, which produces either an armchair or zigzig pattern in the remaining
spatial degree of freedom of the honeycomb lattice, similar to the geometry of
graphene nanoribbons. We call the resulting (1+1)-dimensional NLDEs the armchair
NLDE and zigzag NLDE, respectively; their solutions are isomorphic under a complex
Pauli rotation. We obtain, by purely analytical means, an extended array of bright
solitons. In addition, using a numerical shooting method we obtain the ground state
and excited states for a gray line soliton in the presence of a weak harmonic potential.
Confinement along the direction of the line soliton leads to spatially quantized states,
and the resulting spectra for chemical potential versus particle interaction are computed. Another important consequence of spatial confinement is the appearance of
Klein-tunneling in regions where the potential becomes large. We give a detailed explanation
of how Klein-tunneling occurs in the NLDE, and contrast this with systems
described by the NLSE.
We find that quantized vortices occur in the full (2 + 1)-dimensional NLDE in
any of seven possible types, distinguished by different combinations of phase winding
number and asymptotic radial forms for each of the spinor components. We
obtain analytical and numerical topological and non-topological vortex solutions for
arbitrary phase winding number, which include skyrmion and half-quantum vortices,
both characterized by a nontrivial pseudospin structure. In the case of unit phase
winding, we obtain a singly wound vortex in one spinor component, and a soliton in
the other component residing at the core of the vortex. These solutions are analogous
to the coreless vortices studied in non-relativistic spinor BECs governed by the
well-known vector NLSE. Similar to the case of solitons, we study our vortices in the
presence of a radial confining potential, and determine the resulting spectra for the
ground state and radial excited states for all of our vortices.
We have extended our study of localized solutions to the general case of a non-zero
mass gap in the NLDE, which can be implemented in the honeycomb lattice using
a variety of methods, but most readily by breaking the degeneracy between the A
and B hexagonal sublattices of the honeycomb lattice. We derive a general method
for translating solitons and vortices, embedded in the continuous spectrum, into the
gap region, i.e., a mapping from embedded to gap-solitons. The presence of a gap in
the spectrum allows for more general massive NLDEs. In particular, we uncover a
mapping from a subspace of the NLDE to the massive Thirring model, an extensively
studied integrable model, thus revealing a subspace of the NLDE possessing enhanced
symmetry.
The first order corrections to mean-field theory are obtained via the RLSE. We
derive the RLSE from first principles, by considering quantum fluctuations at each
lattice site, then diagonalizing the resulting Hamiltonian and imposing the tightbinding
and long-wavelength limits. We analyze the low-energy structure for the case
of a uniform BEC by solving the RLSE to obtain quasi-particle coherence factors and
frequencies, and find that quasi-particle emission has a distinctive Cherenkov directional
and momentum signature when the BEC is displaced from the Dirac point. To
add formal rigor to our results, we present a thorough analysis of Wannier expansions
of the condensate wavefunction and quasi-particle states at the lattice level, and
show that our results depend, to lowest order, on the quality of phase coherence from
site-to-site and on a well defined local particle density. In the same vein, we study
the mapping of local rotation, quantum, and unitary operators in the honeycomb lattice
from the lattice scale to the continuum limit, and obtain the result that discrete
rotation matrices acting on the lattice map to SU(2) operations in the continuum
limit. Application of the RLSE to soliton and vortex states yields spatial structure
for quasi-particle functions, highly localized around the soliton peaks, or dips, and
near the vortex cores. The RLSE reveal Nambu-Goldstone modes associated with
symmetry breaking. We find that, for the case of a BEC of 87Rb atoms, most solitons
and vortices are stable over the lifetime of the BEC.
The full many-body Hamiltonian for bosons near the Dirac points of the honeycomb
lattice is derived in detail in both the linear and quadratic momentum approximations
using nearest neighbor hopping for both derivations. The linear part of the
Hamiltonian describes the full quantum mechanical theory for excitations closest to
the Dirac points, and is identical to the Hamiltonian for massless Dirac fermions.
The next order correction describes quantum excitations with quadratic dispersion
associated with bending of the Dirac cone away from the Dirac points.
This dissertation focuses on the properties of nonlinear waves in Bose-Einstein condensates (BECs). The fundamental model here is the nonlinear Schrodinger equation, the so-called Gross-Pitaevskii (GP) equation, which is a mean-field description of BECs. The systematic analysis begins by considering the dark-bright (DB)-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component BECs. The interaction between two DB solitons in a homogeneous condensate and at the presence of the trap are both considered. Our analytical approximation relies in a Hamiltonian perturbation theory, which leads to an equation of motion of the centers of DB-soliton interacting pairs. Employing this equation, we demonstrate the existence of robust DB-soliton molecules, in the form of stationary two- and three-DB-soliton states. Also the equilibrium distance of the constituent solitons and the corresponding oscillation frequencies are found semianalytically, where the latter corresponds to the characteristic anomalous modes' eigenfrequencies that we numerically computed via a so called Bogoliubov-de Gennes (BdG) linearization analysis. Those studies are discussed in Chapter 2. Then, we extend our studies to the dynamics of dark-bright (DB) solitons in binary BECs at finite temperature using a system of two-coupled dissipative GPs. We show that the effect of the bright soliton is to partially stabilize dark solitons against temperature-induced dissipation, thus providing longer lifetimes in Chapter 3. Furthermore, the dark-dark (DD) solitons as a prototypical coherent structure that emerges in two-component BECs are studied and are connected to dark-bright (DB) solitons via SO(2) rotation. We obtained their beating frequency and their frequency of oscillation inside a parabolic trap. They are identified as exact periodic orbits in the Manakov limit of equal inter- and intra- species nonlinearity strengths with and without the trap and we showcase the persistence of such states upon weak deviations from this limit. Also we investigated in detail the effect of the deviation from the Manakov case by considering different from unity scattering length ratios in Chapter 4. Next, we revisited Hamiltonian eigenvalue problems that typically arise in the linearization around a stationary state of a Hamiltonian nonlinear PDE. Also we presented a overview of the known facts for the eigenvalue counts of the corresponding unstable spectra. In particular, we focused on a straightforward plan to implement finite-dimensional techniques for locating this spectrum via the singular points of the meromorphic Krein Matrix and illustrated the value of the approach by considering realistic problems for recently observed experimentally multivortex and multisoliton solutions in atomic Bose-Einstein condensates in Chapter 5. In the two dimensional scenario, we also examine the stability and dynamics of vortices under the effect of dissipation used as a simplified model for the inclusion of the effect of finite temperatures in atomic BECs, which enables an analytical prediction that can be compared directly to numerical results in Chapter 6. In all the above studies, our analytical prediction from the equation of motion are in good agreement with the numerical results from the BdG analysis.