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arXiv:cond-mat/0507631 · Physical Review A

Quantum dynamics of localized excitations in a symmetric trimer molecule

Ricardo A. Pinto, Sergej Flach A

Published Feb 24, 2006• 19 citations
arXivHTMLPDF

Abstract

We study the time evolution of localized (local bond) excitations in a symmetric quantum trimer molecule. We relate the dynamical properties of localized excitations such as their spectral intensity and their temporal evolution (survival probability and tunneling of bosons) to their degree of overlap with quantum tunneling pair states. We report on the existence of degeneracy points in the trimer eigenvalue spectrum for specific values of parameters due to avoided crossings between tunneling pair states and additional states. The tunneling of localized excitations which overlap with these degenerate states is suppressed at all times. As a result local bond excitations may be strongly localized forever, similar to their classical counterparts.

DOI
10.1103/physreva.73.022717

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We study the time evolution of localized (local bond) excitations in a symmetric quantum trimer molecule. We relate the dynamical properties of localized excitations such as their spectral intensity and their temporal evolution (survival probability and tunneling of bosons) to their degree of overlap with quantum tunneling pair states. We report on the existence of degeneracy points in the trimer eigenvalue spectrum for specific values of parameters due to avoided crossings between tunneling pair states and additional states. The tunneling of localized excitations which overlap with these degenerate states is suppressed at all times. As a result local bond excitations may be strongly localized forever, similar to their classical counterparts.
We study the quantum dynamics of the Bose-Hubbard model on a ladder formed by two rings coupled by the tunneling effect. By implementing the Bogoliubov approximation scheme, we prove that, despite the presence of the inter-ring coupling term, the Hamiltonian decouples in many independent sub-Hamiltonians ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ associated with momentum-mode pairs $\ifmmode\pm\else\textpm\fi{}k$. Each sub-Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ is then shown to be part of a specific dynamical algebra. The properties of the latter allow us to perform the diagonalization process, to find the energy spectrum and the conserved quantities of the model, and to derive the time evolution of important physical observables. We then apply this solution scheme to the simplest possible closed ladder, the double trimer. After observing that the excitations of the system are weakly populated vortices, we explore the corresponding dynamics by varying the initial conditions and the model parameters. Finally, we show that the inter-ring tunneling determines a spectral collapse when approaching the border of the dynamical-stability region.
We present analytical relations for the quantum evolution of the number difference of bosons between the two sites of a double-well potential, by using perturbative results for small tunneling amplitudes in the two-mode approximation. Results are obtained for two different initial conditions: completely localized states and coherent spin states. In the former case both the short and the long time behavior is investigated and the characteristic Bohr frequencies of the energy spectrum are determined. In the latter case we calculate the short time-scale evolution of the number difference. The analytical expressions compare favorably with direct numerical solutions of the quantum problem. Finally, we discuss the corresponding Gross-Pitaevskii (i.e., mean-field) dynamics and we point out the differences between the full quantum evolution and the mean-field evolution.
We investigate quantum tunneling in a translation invariant chain of particles. The particles interact harmonically with their nearest neighbors except for one bond, which is anharmonic. It is described by a symmetric double-well potential. In the first step, we show how the anharmonic coordinate can be separated from the normal modes. This yields a Lagrangian which has been used to study quantum dissipation. Elimination of the normal modes leads to a nonlocal action of Caldeira-Leggett type. If the anharmonic bond defect is in the bulk, one arrives at Ohmic damping, i.e., there is a transition of a delocalized bond state to a localized one if the elastic constant exceeds a critical value Ccrit{C}_{\text{crit}}Ccrit​. The latter depends on the masses of the bond defect. Super-Ohmic damping occurs if the bond defect is in the site MMM at a finite distance from one of the chain ends. If the observation time TTT is smaller than a characteristic time ${\ensuremath{\tau}}_{M}\ensuremath{\sim}M$, depending on the location MMM of the defect, the behavior is similar to the bulk situation. However, for $T⪢{\ensuremath{\tau}}_{M}$ tunneling is never suppressed.
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites.In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies.Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer.These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis.In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories.Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites. In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies. Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer. These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis. In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories. Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
The phenomenon of tunneling is investigated for a symmetric double-well potential perturbed by a monochromatic driving force. The analysis is based on a numerical treatment of the quantum map that propagates the system over one period of the external force, and of the spectrum of its eigenphases (quasienergies). The variety in the quasienergy spectrum, such as exact and avoided crossings, leads to novel forms of coherent tunneling. In particular, for specific parameter values of the driving force, we find almost complete localization of the wave packet in one of the wells.