This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.

Jan 1, 20160 citations

Bifurcations in resonance widths of an open Bose-Hubbard dimer

Moritz Hiller, Tsampikos Kottos, Alexander Ossipov

We investigate the structure of resonance widths of a Bose-Hubbard dimer with intersite hopping amplitude $k$ , which is coupled to a continuum at one of the sites with strength $\ensuremath{\gamma}$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\ensuremath{\chi}$ the resonances undergo consequent bifurcations. For $\ensuremath{\Lambda}=k∕\ensuremath{\gamma}\ensuremath{\geqslant}0.5$, the bifurcation points follow a scaling law ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{m}\ensuremath{\equiv}{\ensuremath{\chi}}_{m}N∕k={f}_{\ensuremath{\Lambda}}(m\ensuremath{-}0.5∕\ensuremath{\Lambda})$, where $N$ is the number of bosons. For the function ${f}_{\ensuremath{\Lambda}}$ two different $\ensuremath{\Lambda}$ dependences are found around the minimum and maximum bifurcation points.We investigate the structure of resonance widths of a Bose-Hubbard dimer with intersite hopping amplitude $k$, which is coupled to a continuum at one of the sites with strength $\ensuremath{\gamma}$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\ensuremath{\chi}$ the resonances undergo consequent bifurcations. For $\ensuremath{\Lambda}=k∕\ensuremath{\gamma}\ensuremath{\geqslant}0.5$, the bifurcation points follow a scaling law ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{m}\ensuremath{\equiv}{\ensuremath{\chi}}_{m}N∕k={f}_{\ensuremath{\Lambda}}(m\ensuremath{-}0.5∕\ensuremath{\Lambda})$, where $N$ is the number of bosons. For the function ${f}_{\ensuremath{\Lambda}}$ two different $\ensuremath{\Lambda}$ dependences are found around the minimum and maximum bifurcation points.

arXiv:cond-mat/0602626Physical Review AJun 29, 200653 citations

We investigate the structure of resonance widths of a Bose-Hubbard dimer with intersite hopping amplitude

k

, which is coupled to a continuum at one of the sites with strength $\ensuremath{\gamma}$. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term $\ensuremath{\chi}$ the resonances undergo consequent bifurcations. For $\ensuremath{\Lambda}=k∕\ensuremath{\gamma}\ensuremath{\geqslant}0.5$, the bifurcation points follow a scaling law ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{m}\ensuremath{\equiv}{\ensuremath{\chi}}_{m}N∕k={f}_{\ensuremath{\Lambda}}(m\ensuremath{-}0.5∕\ensuremath{\Lambda})$, where

N

is the number of bosons. For the function ${f}_{\ensuremath{\Lambda}}$ two different $\ensuremath{\Lambda}$ dependences are found around the minimum and maximum bifurcation points.