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Comparisons between classical and quantum mechanical nonlinear lattice models
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
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.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1648
Keyword [en]
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
Identifiers
URN: urn:nbn:se:liu:diva-105817DOI: 10.3384/lic.diva-105817ISBN: 978-91-7519-375-5 (print)OAI: oai:DiVA.org:liu-105817DiVA: diva2:710947
Presentation
2014-04-24, 13:00 (English)
Opponent
Supervisors
Available from: 2014-04-14 Created: 2014-04-08 Last updated: 2014-04-14Bibliographically 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: 2017-12-08
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, E-ISSN 1550-2376, 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: 2017-12-07
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: 2017-12-06

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