THE NONLINEAR DIRAC EQUATION IN BOSE-EINSTEIN CONDENSATES
Laith H. Haddad
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.
•Jan 2, 2016•6 citations