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Discrete breathers for a discrete nonlinear Schrodinger ring coupled to a central site
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-6708-1560
2016 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 93, no 1, 012219- p.Article in journal (Refereed) PublishedText
Abstract [en]

We examine the existence and properties of certain discrete breathers for a discrete nonlinear Schrodinger model where all but one site are placed in a ring and coupled to the additional central site. The discrete breathers we focus on are stationary solutions mainly localized on one or a few of the ring sites and possibly also the central site. By numerical methods, we trace out and study the continuous families the discrete breathers belong to. Our main result is the discovery of a split bifurcation at a critical value of the coupling between neighboring ring sites. Below this critical value, families form closed loops in a certain parameter space, implying that discrete breathers with and without central-site occupation belong to the same family. Above the split bifurcation the families split up into several separate ones, which bifurcate with solutions with constant ring amplitudes. For symmetry reasons, the families have different properties below the split bifurcation for even and odd numbers of sites. It is also determined under which conditions the discrete breathers are linearly stable. The dynamics of some simpler initial conditions that approximate the discrete breathers are also studied and the parameter regimes where the dynamics remain localized close to the initially excited ring site are related to the linear stability of the exact discrete breathers.

Place, publisher, year, edition, pages
AMER PHYSICAL SOC , 2016. Vol. 93, no 1, 012219- p.
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:liu:diva-125683DOI: 10.1103/PhysRevE.93.012219ISI: 000369333600003PubMedID: 26871085OAI: oai:DiVA.org:liu-125683DiVA: diva2:908209
Available from: 2016-03-01 Created: 2016-02-29 Last updated: 2016-08-22
In thesis
1. Theoretical studies of Bose-Hubbard and discrete nonlinear Schrödinger models: Localization, vortices, and quantum-classical correspondence
Open this publication in new window or tab >>Theoretical studies of Bose-Hubbard and discrete nonlinear Schrödinger models: Localization, vortices, and quantum-classical correspondence
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is mainly concerned with theoretical studies of two types of models:  quantum mechanical Bose-Hubbard models and (semi-)classical discrete nonlinear Schrödinger (DNLS) models.

Bose-Hubbard models have in the last few decades been widely used to describe Bose-Einstein condensates placed in periodic optical potentials, a hot research topic with promising future applications within quantum computations and quantum simulations. The Bose-Hubbard model, in its simplest form, describes the competition between tunneling of particles between neighboring potential wells (`sites') and their on-site interactions (can be either repulsive or attractive). We will also consider extensions of the basic models, with additional interactions and tunneling processes.

While Bose-Hubbard models describe the behavior of a collection of particles in a lattice, the DNLS description is in terms of a classical field on each site. DNLS models can also be applicable for Bose-Einstein condensates in periodic potentials, but in the limit of many bosons per site, where quantum fluctuations are negligible and a description in terms of average values is valid. The particle interactions of the Bose-Hubbard models become  nonlinearities in the DNLS models, so that the DNLS model, in its simplest form, describes a competition between on-site nonlinearity and tunneling to neighboring sites. DNLS models are however also applicable for several other physical systems, most notably for nonlinear waveguide arrays, another rapidly evolving research field.

The research presented in this thesis can be roughly divided into two parts:

1) We have studied certain families of solutions to the DNLS model.

First, we have considered charge flipping vortices in DNLS trimers and hexamers. Vortices represent a rotational flow of energy, and a charge flipping vortex is one where the rotational direction (repeatedly) changes. We have found that charge flipping vortices indeed exist in these systems, and that they belong to continuous families of solutions located between two stationary solutions.

Second, we have studied discrete breathers, which are spatially localized and time-periodic solutions, in a DNLS models with the geometry of a ring coupled to an additional, central site. We found under which parameter values these solutions exist, and also studied the properties of their continuous solution families. We found that these families undergo different bifurcations, and that, for example, the discrete breathers which have a peak on one and two (neighboring) sites, respectively, belong to the same family below a critical value of the ring-to-central-site coupling, but to separate families for values above it.

2) Since Bose-Hubbard models can be approximated with DNLS models in the limit of a large number of bosons per site, we studied signatures of certain classical solutions and structures of DNLS models in the corresponding Bose-Hubbard models.

These studies have partly focused on quantum lattice compactons. The corresponding classical lattice compactons are solutions to an extended DNLS model, and consist of a cluster of excited sites, with the rest of the sites exactly zero (generally localized solutions have nonzero `tails'). We find that only one-site classical lattice compactons remain compact for the Bose-Hubbard model, while for several-site classical compactons there are nonzero probabilities to find particles spread out over more sites in the quantum model. We have furthermore studied the dynamics, with emphasize on mobility, of quantum states that correspond to the classical lattice compactons. The main result is that it indeed is possible to see signatures of the  classical compactons' good mobility, but that it is then necessary to give the quantum state a `hard kick' (corresponding to a large phase gradient). Otherwise, the time scales for quantum fluctuations and for the compacton to travel one site become of the same order.

We have also studied the quantum signatures of a certain type of instability (oscillatory) which a specific solution to the DNLS trimer experiences in a parameter regime. We have been able to identify signatures in the quantum energy spectrum, where in the unstable parameter regime the relevant eigenstates undergo many avoided crossings, giving a strong mixing between the eigenstates. We also introduced several measures, which either drop or increase significantly in the regime of instability.

Finally, we have studied quantum signatures of the charge flipping vortices mentioned above, and found several such, for example when considering the correlation of currents between different sites.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1775
National Category
Physical Sciences
Identifiers
urn:nbn:se:liu:diva-129564 (URN)10.3384/diss.diva-129564 (DOI)978-91-7685-735-9 (Print) (ISBN)
Public defence
2016-09-02, Hörsal Planck, Fysikhuset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2016-08-22 Created: 2016-06-21 Last updated: 2016-08-22Bibliographically approved

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