Weak Lattice Research Articles

Controlling directed transport of matter-wave solitons using the ratchet effect

We demonstrate that directed transport of bright solitons formed in a quasi-one-dimensional Bose-Einstein condensate can be reliably controlled by tailoring a weak optical lattice potential, biharmonic in both space and time, in accordance with the degree of symmetry breaking mechanism. By considering the regime where matter-wave solitons are narrow compared to the lattice period, (i) we propose an analytical estimate for the dependence of the directed soliton current on the biharmonic potential parameters that is in good agreement with numerical experiments, and (ii) we show that the dependence of the directed soliton current on the number of atoms is a consequence of the ratchet universality.

Hypergraphs induced by algebras of fixed type

A characterization of the weak subalgebra lattice of a partial algebra of a fixed type is a natural algebraic problem. In Pióro (2000, 2002) [13,15] we have shown that this algebraic problem is equivalent to the following hypergraph question, interesting in itself: When can edges of a hypergraph be directed to form a partial algebra of a fixed type (equivalently, to form a directed hypergraph of a fixed type)? This problem will be solved in the present paper.

Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schrödinger lattices

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schrödinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to the existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy.

Structural and electrical properties of quantum wells with nanoscale InAs inserts in In y Al1 − y As/In x Ga1 − x As heterostructures on InP substrates

A complex study of the effect ofintroduction of nanoscale InAs inserts of different thicknesses into an In0.53Ga0.47As quantum well on the electrical properties and structural features of In0.50Al0.50As/In0.53Ga0.47As/In0.50Al0.50As nanoheterostructures with bilateral δ-Si doping grown on InP substrates has been performed. The layers of nanoheterostructures with a weak lattice mismatch are found to be equally (cube-on-cube) oriented. The introduction of a nanoscale InAs insert leads to an increase in mobility. At an insert thickness of about 1.8 nm, the effect of increasing mobility is saturated due to structural deterioration. The segregation of the second (apparently, wurtzite) phase is revealed; this process, as well as the formation of other defects in the nanoheterostructure layers, is due to local strains caused by variations of the indium content in the layers.

Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons

Quantum many-body systems can have phase transitions even at zero temperature; fluctuations arising from Heisenberg's uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system's Hamiltonian changes across a finite critical value. A well-known example is the Mott-Hubbard quantum phase transition from a superfluid to an insulating phase, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect--the sine-Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator--in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine-Gordon model. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose-Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.

A Property of the Weak Subalgebra Lattice for Algebras with Some Non-Equalities

Let A be a locally finite total algebra of finite type such that ai for every operation , elements an and . We show that the weak subalgebra lattice of A uniquely determines its (strong) subalgebra lattice. More precisely, for any algebra B of the same finite type, if the weak subalgebra lattices of A and B are isomorphic, then their subalgebra lattices are also isomorphic. Moreover, B is also total and locally finite.

Stability of matter-wave solitons in optical lattices

We consider localized states of both single- and two-component Bose-Einstein condensates (BECs) confined in a potential resulting from the superposition of linear and nonlinear optical lattices and make use of Vakhitov-Kolokolov criterion to investigate the effect of nonlinear lattice on the stability of the soliton solutions in the linear optical lattice (LOL). For the single-component case we show that a weak nonlinear lattice has very little effect on the stability of such solitons while sufficiently strong nonlinear optical lattice (NOL) squeezes them to produce narrow bound states. For two-component condensates we find that when the strength of the NOL (γ 1) is less than that of the LOL (V 0) a relatively weak intra-atomic interaction (IAI) has little effect on the stability of the component solitons. This is true for both attractive and repulsive IAI. A strong attractive IAI, however, squeezes the BEC solitons while a similar repulsive IAI makes the component solitons wider. For γ 1 > V 0, only a strong attractive IAI squeezes the BEC solitons but the squeezing effect is less prominent than that found for γ 1 < V 0. We make useful checks on the results of our semianalytical stability analysis by solving the appropriate Gross-Pitaevskii equations numerically.

Stability of Bose-Einstein condensates in two-dimensional optical lattices

Both Landau instability and dynamical instability of Bose-Einstein condensates in moving two-dimensional optical lattices are investigated numerically and analytically. Phase diagrams for both instabilities are obtained numerically for different system parameters. These phase diagrams show that the Landau instability does not depend on direction for weak lattices while the dynamic instability is direction dependent. These features are explained analytically.

Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials

We study the behavior of an ultracold atomic gas of bosons in a bichromatic lattice, where the weaker lattice is used as a source of disorder. We numerically solve a discretized mean-field equation, which generalizes the one-dimensional Aubry-Andr\`e model for particles in a quasiperiodic potential by including the interaction between atoms. We compare the results for commensurate and incommensurate lattices. We investigate the role of the initial shape of the wave packet as well as the interplay between two competing effects of the interaction, namely, self-trapping and delocalization. Our calculations show that, if the condensate initially occupies a single lattice site, the dynamics of the interacting gas is dominated by self-trapping in a wide range of parameters even for weak interaction. Conversely, if the diffusion starts from a Gaussian wave packet, self-trapping is significantly suppressed and the destruction of localization by interaction is more easily observable.

Characteristic triangles of closure operators with applications in general algebra

Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such “lattices with closure” is the so-called characteristic triangle, a certain substructure of the square L2. For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias Hamiltonian groups) are precisely those with modular weak congruence lattices; similar results are obtained for other classes of algebras.

Bose–Einstein Condensates in a One-Dimensional Optical Lattice: from Superfluidity to Number-Squeezed States

We study the phase coherence property of Bose−Einstein condensates confined in a one-dimensional optical lattice formed by a standing-wave laser field. The lattice depth is determined using a method of Kapitza–Dirac scattering between a condensate and a short pulse lattice potential. Condensates are then adiabatically loaded into the optical lattice. The phase coherence property of the confined condensates is reflected by the interference patterns of the expanded atomic cloud released from the optical lattice. For weak lattice, nearly all of the atoms stay in a superfluid state. However, as the lattice depth is increased, the phase coherence of the whole condensate sample is gradually lost, which confirms that the sub-condensates in each lattice well have evolved into number-squeezed states.

Interaction of (1+1)-dimensional dipole solitons in optical lattice

The interaction between two (1+1)-dimensional dipole solitons in a weak modulation lattice is studied numerically. Two unstable dipole solitons can form the bound state, and their propagation can be stabilized due to their mutual interaction. The propagation patterns with the interaction are dependent on the interval, relative phase, and input energy of the two dipole solitons. The two poles of the dipole soliton exchanges their energy after the interaction. Under the proper conditions, the interaction force between the two dipole solitons exhibits some properties similar to those between two molecules.

Charged defects in soft semiconductors and their influence on organic photovoltaics

Charged defects play an important, but often unrecognized, role in organic semiconductors and in photovoltaic cells made from them. Thermodynamic considerations show that the weak lattice forces characteristic of soft semiconductors inevitably lead to a high concentration of defects, some of which will be charged. The defect density in actual materials is discussed along with the different types of defects and their influence on the electrical properties and chemical stability. Charged defects act as dopants and thus may sometimes be beneficial, although low defect materials with purposely added dopants should be superior.

Discrete chaotic states of a Bose-Einstein condensate

We examine spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For a low laser barrier, chaotic regions of the parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high-barrier case, bounded perturbed solutions that describe a set of discrete chaotic states are constructed for discrete barrier heights and magic numbers of condensed atoms. Chaotic density profiles are exhibited numerically for the lowest quantum number, and analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.

Dynamics of Matter-Wave Solitons in a Ratchet Potential

We study the dynamics of bright solitons formed in a Bose-Einstein condensate with attractive atomic interactions perturbed by a weak bichromatic optical lattice potential. The lattice depth is a biperiodic function of time with a zero mean, which realizes a flashing ratchet for matter-wave solitons. We find that the average velocity of a soliton and the soliton current induced by the ratchet depend on the number of atoms in the soliton. As a consequence, soliton transport can be induced through scattering of different solitons. In the regime when matter-wave solitons are narrow compared to the lattice period the dynamics is well described by the effective Hamiltonian theory.

Sound speed of a Bose-Einstein condensate in an optical lattice

The speed of sound of a Bose-Einstein condensate in an optical lattice is studied both analytically and numerically in all spatial three dimensions. Our investigation shows that the sound speed depends strongly on the strength of the lattice. In the one-dimensional case, the speed of sound falls monotonically with increasing lattice strength. The dependence on lattice strength becomes much richer in two and three dimensions. In the two-dimensional case, when the interaction is weak, the sound speed first increases and then decreases as the lattice strength increases. For the three-dimensional lattice, the sound speed can even oscillate with the lattice strength. These rich behaviors can be understood in terms of compressibility and effective mass. Our analytical results in the limit of weak lattices also offer an interesting perspective to help with our understanding: they show that the lattice component perpendicular to the sound propagation increases the sound speed while the lattice component parallel to the propagation decreases the sound speed. The various dependences of the sound speed on the lattice strength are the result of this competition.

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included.

Competition between attractive and repulsive interactions in two-component Bose-Einstein condensates trapped in an optical lattice

We consider effects of interspecies attraction on two-component gap solitons (GSs) in the binary BEC with intraspecies repulsion, trapped in the one-dimensional optical lattice (OL). Systematic simulations of the coupled Gross-Pitaevskii equations corroborate an assumption that, because the effective mass of GSs is negative, the interspecies attraction may split the two-component soliton. Two critical values, ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$, of the OL strength $(\ensuremath{\kappa})$ are identified. Two-species GSs with fully overlapping wave functions are stable in strong lattices $(\ensuremath{\kappa}&gt;{\ensuremath{\kappa}}_{1})$. In an intermediate region, ${\ensuremath{\kappa}}_{1}&gt;\ensuremath{\kappa}&gt;{\ensuremath{\kappa}}_{2}$, the soliton splits into a double-humped state with separated components. Finally, in weak lattices $(\ensuremath{\kappa}&lt;{\ensuremath{\kappa}}_{2})$, the splitting generates a pair of freely moving single-species GSs. We present and explain the dependence of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ on the number of atoms (total norm), and on the relative strength of the competing interspecies attraction and intraspecies repulsion. The splitting of asymmetric solitons, with unequal norms of the two species, is briefly considered too. It is found and explained that the splitting threshold grows with the increase of the asymmetry.

Phase Diagram for a Bose-Einstein Condensate Moving in an Optical Lattice

The stability of superfluid currents in a system of ultracold bosons was studied using a moving optical lattice. Superfluid currents in a very weak lattice become unstable when their momentum exceeds 0.5 recoil momentum. Superfluidity vanishes already for zero momentum as the lattice deep reaches the Mott insulator (MI) phase transition. We study the phase diagram for the disappearance of superfluidity as a function of momentum and lattice depth between these two limits. Our phase boundary extrapolates to the critical lattice depth for the superfluid-to-MI transition with 2% precision. When a one-dimensional gas was loaded into a moving optical lattice a sudden broadening of the transition between stable and unstable phases was observed.

Slowly moving matter-wave gap soliton propagation in weak random optical lattices

We systematically investigate slowly moving matter-wave gap soliton propagation in weak random optical lattices. With the weak randomness, an effective-particle theory is constructed to show that the motion of a gap soliton is similar to a particle moving in random potentials. Based on the effective-particle theory, the effects of the randomness on gap solitons are obtained and the trajectories of gap solitons are well predicted. Moreover, the general laws that describe the movement depending on the weak randomness are obtained. We find that with an increase of the random strength, the ensemble-average velocity reduces slowly and the reflection probability becomes larger. The theoretical results based on the effective-particle theory are confirmed by the numerical simulations based on the Gross-Pitaevskii equation.