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Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.ORCID iD: 0000-0001-6708-1560
Linköping University, Department of Physics, Chemistry and Biology. Linköping University, The Institute of Technology.
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. Vol. 86, no 1, 016214- p.
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-79982DOI: 10.1103/PhysRevE.86.016214ISI: 000306470900001OAI: diva2:544950

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

Available from: 2012-08-17 Created: 2012-08-17 Last updated: 2016-08-22
In thesis
1. Comparisons between classical and quantum mechanical nonlinear lattice models
Open this publication in new window or tab >>Comparisons between classical and quantum mechanical nonlinear lattice models
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved.

The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models.

Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise.

In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 48 p.
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1648
Discrete Breathers; Bose-Einstein-Condensation; Instabilities; Oscillatory Instabilities; Classical-Quantum correspondence, Bose-Hubbard model; DNLS; discrete nonlinear Schrödinger equation; nonlinear; nonlinear lattice models; compacton
National Category
Other Physics Topics Atom and Molecular Physics and Optics
urn:nbn:se:liu:diva-105817 (URN)10.3384/lic.diva-105817 (DOI)978-91-7519-375-5 (print) (ISBN)
2014-04-24, 13:00 (English)
Available from: 2014-04-14 Created: 2014-04-08 Last updated: 2014-04-14Bibliographically approved
2. 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
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1775
National Category
Physical Sciences
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)
Available from: 2016-08-22 Created: 2016-06-21 Last updated: 2016-08-22Bibliographically approved

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