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arXiv:1502.05621 · New Journal of Physics

The nonlinear dirac equation in Bose–Einstein condensates: vortex solutions and spectra in a weak harmonic trap

L H Haddad, Lincoln D. Carr

Published Oct 29, 2015• 18 citations
arXivHTMLPDF

Abstract

We analyze the vortex solution space of the ( 2 + 1 ) -dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin–Ho and Anderson–Toulouse vortices for vortex winding ℓ = 1. For ℓ ⩾ 2 we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of ℓ the non-topological solutions include bright ring-vortices which explicitly demonstrate the confining effects of the Dirac operator. We arrive at solutions through an asymptotic Bessel series, algebraic closed-forms, and using standard numerical shooting methods. By including a harmonic potential to simulate a finite trap we compute the discrete spectra associated with radially quantized modes. We demonstrate the continuous spectral mapping between the vortex and free particle limits for all of our solutions.

DOI
10.1088/1367-2630/17/11/113011

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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.

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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

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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. 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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. 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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. 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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

Vortex lattice of a Bose-Einstein condensate in a rotating anisotropic trap

Mehmet Özgür Oktel

We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.

arXiv:cond-mat/0309613•Physical Review A•Feb 26, 2004•52 citations
We analyze the vortex solution space of the ( 2 + 1 ) -dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin–Ho and Anderson–Toulouse vortices for vortex winding ℓ = 1. For ℓ ⩾ 2 we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of ℓ the non-topological solutions include bright ring-vortices which explicitly demonstrate the confining effects of the Dirac operator. We arrive at solutions through an asymptotic Bessel series, algebraic closed-forms, and using standard numerical shooting methods. By including a harmonic potential to simulate a finite trap we compute the discrete spectra associated with radially quantized modes. We demonstrate the continuous spectral mapping between the vortex and free particle limits for all of our solutions.
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^{87}\mathrm{Rb}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.
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.
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
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.
We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.