Honeycomb optical lattices with harmonic confinement

Block, Jens Kusk; Nygaard, Nicolai

doi:10.1103/physreva.81.053421

Cited by 17 publications

(17 citation statements)

References 41 publications

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“…It is also interesting to note that equations (35), (36), (39) and (40) have a Schrödinger-like form for a particle with zero total energy subject to the potential functions Q, R, S and T. For soliton solutions the potential functions develop minima which support stable localized bound states. It is also important to note that equations (35), (36), (39) and (40) must be solved self consistently since the potentials in equations (37), (38), (41) and (42) depend on the eigenvalues Ẽ. In general, the RLSE allow for scattering states and bound states in the uʼs and vʼs.…”

Section: Bound State Fluctuations Of the Soliton Corementioning

confidence: 99%

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The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

Haddad

Carr

2015

New J. Phys.

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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 Rb 87atoms, 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-1 2 domain walls between polarized regions of S 1 2 z = ± . 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.

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Section: Bound State Fluctuations Of the Soliton Corementioning

confidence: 99%

“…where we have used the fact that the square modulus of a quasi-particle amplitude is much smaller than that of the condensate wavefunction. Next, we solve the decoupled equations (35), (36), (39) and (40) using approximate forms for the dark soliton spinor components…”

Section: Bound State Fluctuations Of the Soliton Corementioning

confidence: 99%

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

Haddad

Carr

2015

New J. Phys.

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“…Our own recent work has placed the NLDE in the context of a Bose-Einstein condensate (BEC) [29]. Significantly, our particular form of the NLDE has opened up research in other fields of physics [30][31][32][33][34][35][36][37][38]. For the NLDE in a BEC, the relativistic structure arises naturally as bosons propagate in a shallow periodic honeycomb lattice potential, and yields a rich soliton landscape which we explore in detail in this Article.…”

Section: Introductionmentioning

confidence: 99%

“…Combining equations(31) and(32) determines the envelope and plane wave fractions f e and f p equation(30) to give the full expression for the dispersion…”

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confidence: 99%

The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices

2015

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We present a thorough analysis of soliton solutions to the quasi-one-dimensional (quasi-1D) nonlinear Dirac equation (NLDE) for a Bose-Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-1D reduction of the honeycomb lattice along the zigzag direction, in direct analogy to graphene nanoribbons. Excitations in the remaining large direction of the lattice correspond to the linear subbands in the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-1D 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. We obtain spatially oscillating multi-soliton solutions 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 energy-momentum tensor, we can obtain both bright and dark solitons over a nonzero density background.

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“…Indeed, in the situation when one of the hoping parameter J is larger than the two others J, one observe, for J = 2.5J fig. 2(a), the presence of a dip in the order parameter around the position corresponding to half-filling, where the density depicts a kink, similarly to the non-interactive case [10,24]; for higher hoping imbalance, it deepens, eventually leading, around J = 3.25J fig. 2(b), to a vanishing order parameter, i.e., splitting the superfluid in two (independent) components.…”

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confidence: 99%

Pairing properties of cold fermions in a honeycomb lattice

Grémaud¹

2012

EPL

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-The pairing properties of ultracold fermions, with an attractive interaction, loaded in a honeycomb (graphene-like) optical lattice are studied in a mean-field approach. We emphasize, in the presence of a harmonic trap, the unambiguous signatures of the linear dispersion relation of the band structure around half-filling (i.e. the massless Dirac fermions) in the local order parameter, in particular in the situations of either imbalance hoping parameters or imbalance populations. It can also be observed in the system response to external perturbation, for instance by measuring the pair destruction rate when modulating the optical lattice depth. Going beyond the mean-field level, we estimate the critical temperature for the "condensation" of the preformed pairs.Since its first experimental observation, graphene has attracted considerable attention due to its interest in fundamental physics as well as for potential applications [1,2]. In particular, the energy band spectrum depicts "conical points" where the valence and conduction bands are connected, and the Fermi energy at half-filling is therefore only made of points. Around these points, the energy varies proportionally to the modulus of the wave-vector and the excitations (holes or particles) of the system are equivalent to ultra-relativistic (massless) Dirac fermions since their dispersion relation is linear [3]. In the presence of interaction, the vanishing density of states at the Fermi energy leads to extremely rich physics. Contrary to the square lattice, where the nesting of the Fermi surface generally leads to ordered phases even for arbitrarily small interaction strengths, the Fermi-Hubbard model on the honeycomb lattice depicts two quantum phases transitions at half-filling [4]: first, for weak interaction, one observes a semi-metallic behavior with Dirac-like excitations [5][6][7] similar to the non-interacting situation; for an interaction strength U ≈ 3.5J, the system enters into a spin-liquid phase, i.e. a Mott-Insulator (charge gap) without longrange magnetic ordering [8,9]; eventually, for larger interaction strength U ≈ 4.3J, anti-ferromagnetic order sets in.

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Honeycomb optical lattices with harmonic confinement

Cited by 17 publications

References 41 publications

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices

Pairing properties of cold fermions in a honeycomb lattice

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