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Theoretical studies of Bose-Hubbard and discrete nonlinear Schrödinger models: Localization, vortices, and quantum-classical correspondence
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, Faculty of Science & Engineering.
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: urn:nbn:se:liu:diva-129564DOI: 10.3384/diss.diva-129564ISBN: 978-91-7685-735-9 (Print)OAI: oai:DiVA.org:liu-129564DiVA: diva2:954403
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
List of papers
1. Exact localized eigenstates for an extended Bose-Hubbard model with pair-correlated hopping
Open this publication in new window or tab >>Exact localized eigenstates for an extended Bose-Hubbard model with pair-correlated hopping
2012 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 1, 016603(R)- p.Article in journal (Refereed) Published
Abstract [en]

We show that a Bose-Hubbard model extended with pair-correlated hopping has exact eigenstates, quantum lattice compactons, with complete single-site localization. These appear at parameter values where the one-particle tunneling is exactly canceled by nonlocal pair correlations, and correspond in a classical limit to compact solutions of an extended discrete nonlinear Schrödinger model. Classical compactons at other parameter values, as well as multisite compactons, generically get delocalized by quantum effects, but strong localization appears asymptotically for increasing particle number.

Place, publisher, year, edition, pages
American Physical Society, 2012
National Category
Atom and Molecular Physics and Optics
Identifiers
urn:nbn:se:liu:diva-73926 (URN)10.1103/PhysRevA.85.011603 (DOI)000298861100001 ()
Note
Funding agencies|Swedish Research Council||Available from: 2012-01-16 Created: 2012-01-16 Last updated: 2016-08-22
2. Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer
Open this publication in new window or tab >>Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer
2012 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, Vol. 86, no 1, 016214- p.Article in journal (Refereed) Published
Abstract [en]

We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration and search for quantum signatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopf bifurcations for the "single-depleted-well" family of stationary states in the discrete nonlinear Schrodinger equation. In the regimes of classical stability, single quantum eigenstates with properties analogous to those of the classical stationary states can be identified already for rather small particle numbers. On the other hand, in the instability regime the interaction with other eigenstates through avoided crossings leads to strong mixing, and no single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resulting from initial conditions taken as perturbed quantum eigenstates and SU(3) coherent states, respectively, in a quantum-semiclassical transitional regime of 10-100 particles. While the perturbed quantum eigenstates do not show a classical-like behavior in the instability regime, a coherent state behaves analogously to a perturbed classical stationary state, and exhibits initially resonant oscillations with oscillation frequencies well described by classical internal-mode oscillations.

Place, publisher, year, edition, pages
American Physical Society, 2012
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-79982 (URN)10.1103/PhysRevE.86.016214 (DOI)000306470900001 ()
Note

Funding Agencies|Swedish Research Council||Swedish Institute||

Available from: 2012-08-17 Created: 2012-08-17 Last updated: 2016-08-22
3. Quantum dynamics of lattice states with compact support in an extended Bose-Hubbard model
Open this publication in new window or tab >>Quantum dynamics of lattice states with compact support in an extended Bose-Hubbard model
2013 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 88, no 3, 033605- p.Article in journal (Refereed) Published
Abstract [en]

We study the dynamical properties, with special emphasis on mobility, of quantum lattice compactons (QLCs) in a one-dimensional Bose-Hubbard model extended with pair-correlated hopping. These are quantum counterparts of classical lattice compactons (localized solutions with exact zero amplitude outside a given region) of an extended discrete nonlinear Schrödinger equation, which can be derived in the classical limit from the extended Bose-Hubbard model. While an exact one-site QLC eigenstate corresponds to a classical one-site compacton, the compact support of classical several-site compactons is destroyed by quantum fluctuations. We show that it is possible to reproduce the stability exchange regions of the one-site and two-site localized solutions in the classical model with properly chosen quantum states. Quantum dynamical simulations are performed for two different types of initial conditions: “localized ground states” which are localized wave packets built from superpositions of compactonlike eigenstates, and SU(4) coherent states corresponding to classical two-site compactons. Clear signatures of the mobility of classical lattice compactons are seen, but this crucially depends on the magnitude of the applied phase gradient. For small phase gradients, which classically correspond to a slow coherent motion, the quantum time scale is of the same order as the time scale of the translational motion, and the classical mobility is therefore destroyed by quantum fluctuations. For a large phase instead, corresponding to fast classical motion, the time scales separate so that a mobile, localized, coherent quantum state can be translated many sites for particle numbers already of the order of 10.

Place, publisher, year, edition, pages
American Physical Society, 2013
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-98144 (URN)10.1103/PhysRevA.88.033605 (DOI)000323942100007 ()
Note

Funding Agencies|Swedish Research Council||

Available from: 2013-09-30 Created: 2013-09-30 Last updated: 2016-08-22
4. Charge flipping vortices in the discrete nonlinear Schrodinger trimer and hexamer
Open this publication in new window or tab >>Charge flipping vortices in the discrete nonlinear Schrodinger trimer and hexamer
2015 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, no 2, 022910- p.Article in journal (Refereed) Published
Abstract [en]

We examine the existence and properties of charge flipping vortices (CFVs), vortices which periodically flip the topological charge, in three-site (trimer) and six-site (hexamer) discrete nonlinear Schrodinger lattices. We demonstrate numerically that CFVs exist as exact quasiperiodic solutions in continuous families which connect two different stationary solutions without topological charge, and that it is possible to interpret the dynamics of certain CFVs as the result of perturbations of these stationary solutions. The CFVs are calculated with high numerical accuracy and we may therefore accurately determine many of their properties, such as their energy and linear stability, and the CFVs are found to be stable over large parameter regimes. We also show that, like in earlier studies for lattices with a multiple of four sites, trimer and hexamer CFVs can be obtained by perturbing stationary constant amplitude vortices with certain linear eigenmodes. However, in contrast to the former case where the perturbation could be infinitesimal, the magnitude of the perturbations for trimers and hexamers must overcome a quite large threshold value. These CFVs may be interpreted as exact quasiperiodic CFVs, with a small perturbation applied. The concept of a charge flipping energy barrier is introduced and discussed.

Place, publisher, year, edition, pages
American Physical Society, 2015
National Category
Physical Sciences
Identifiers
urn:nbn:se:liu:diva-117256 (URN)10.1103/PhysRevE.91.022910 (DOI)000351205700005 ()
Note

Funding Agencies|Swedish Research Council

Available from: 2015-04-22 Created: 2015-04-21 Last updated: 2016-08-22
5. Discrete breathers for a discrete nonlinear Schrodinger ring coupled to a central site
Open this publication in new window or tab >>Discrete breathers for a discrete nonlinear Schrodinger ring coupled to a central site
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) Published
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
National Category
Physical Sciences
Identifiers
urn:nbn:se:liu:diva-125683 (URN)10.1103/PhysRevE.93.012219 (DOI)000369333600003 ()26871085 (PubMedID)
Available from: 2016-03-01 Created: 2016-02-29 Last updated: 2016-08-22

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