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arXiv:cond-mat/0602626 · Physical Review A

Bifurcations in resonance widths of an open Bose-Hubbard dimer

Moritz Hiller, Tsampikos Kottos, Alexander Ossipov

Published Jun 29, 2006• 53 citations
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Abstract

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.

DOI
10.1103/physreva.73.063625

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We investigate the structure of resonance widths of a Bose-Hubbard dimer with intersite hopping amplitude kkk, which is coupled to a continuum at one of the sites with strength \ensuremathγ\ensuremath{\gamma}\ensuremathγ. Using an effective non-Hermitian Hamiltonian formalism, we show that by varying the on-site interaction term \ensuremathχ\ensuremath{\chi}\ensuremathχ the resonances undergo consequent bifurcations. For \ensuremathΛ=k∕\ensuremathγ\ensuremath⩾0.5\ensuremath{\Lambda}=k∕\ensuremath{\gamma}\ensuremath{\geqslant}0.5\ensuremathΛ=k∕\ensuremathγ\ensuremath⩾0.5, the bifurcation points follow a scaling law \ensuremathχ\ifmmode~\else˜\fim\ensuremath≡\ensuremathχmN∕k=f\ensuremathΛ(m\ensuremath−0.5∕\ensuremathΛ){\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{m}\ensuremath{\equiv}{\ensuremath{\chi}}_{m}N∕k={f}_{\ensuremath{\Lambda}}(m\ensuremath{-}0.5∕\ensuremath{\Lambda})\ensuremathχ\ifmmode~\else˜\fi​m​\ensuremath≡\ensuremathχm​N∕k=f\ensuremathΛ​(m\ensuremath−0.5∕\ensuremathΛ), where NNN is the number of bosons. For the function f\ensuremathΛ{f}_{\ensuremath{\Lambda}}f\ensuremathΛ​ two different \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ dependences are found around the minimum and maximum bifurcation points.
Abstract We consider the open two-site Bose–Hubbard dimer, a well-known quantum mechanical model that has been realised recently for photons in two coupled photonic crystal nanocavities. The system is described by a Lindblad master equation which, for large numbers of photons, gives rise to a limiting semiclassical model in the form of a four-dimensional vector field. From the situation where both sites trap the same amount of photons under symmetric pumping, one encounters a transition that involves symmetry breaking, the creation of periodic oscillations and multistability as the pump strength is increased. We show that the associated one-parameter bifurcation diagram of the semiclassical model captures the essence of statistical properties of computed quantum trajectories as the pump strength is increased. Even for small numbers of photons, the fingerprint of the semiclassical bifurcations can be recognised reliably in observables of quantum trajectories.
The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number kkk, each subspace corresponding to a system of only kkk particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its kkk-particle subspaces. In particular, this Hamiltonian, when restricted to one of the kkk-particle subspaces, is exactly the spin projection operator along the xxx-axis, where the number of particles kkk in the dimer system yields the projection matrix for spin quantum number s=k/2s=k/2s=k/2. Thus, a new method for computing the eigenvalues and eigenvectors of the xxx-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schrödinger cat states in the two-site setting.
We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.
In this thesis, we study interacting bosons on lattices consisting of a few sites. The main focus is on their response to external driving fields, their transport and decay properties. We address fundamental questions of quantum classical correspondence and investigate the implications of interactions in the transport properties of (ultra-)cold atom devices. Quantum mechanically, we describe bosons on lattices using a parametric Bose-Hubbard Hamiltonian while the corresponding classical limit is described by the discrete nonlinear Schrödinger equation.In the first part of this thesis we focus on the chaotic regime of the Bose-Hubbard Hamiltonian. We investigate the parametric change of eigenstates as the coupling strength between neighboring sites is varied, using perturbative and semiclassical methods. We then turn to the quantum evolution of driven interacting bosonic systems and study the resulting quantum dynamics and the irreversibility (fidelity) of the quantum motion.In the second part we initiate the study of quantum pumping/stirring in Bose-Einstein condensates with the aim to identify optimal pumping cycles and propose such a device in order to probe the interatomic interactions. Finally, in the last part we focus on the interplay of intrinsic dynamics with coupling to the continuum and investigate the structure of the resonance widths of a Bose-Hubbard dimer which is coupled to the continuum at one of the sites using an effective non-Hermitian Hamiltonian formalism.
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.
Motivated by the role that spectral properties play for the dynamical evolution of a quantum many-body system, we investigate the level spacing statistics of the extended Bose–Hubbard model. In particular, we focus on the distribution of the ratio of adjacent level spacings, useful at large interactions, to distinguish between chaotic and non-chaotic regimes. After revisiting the bare Bose–Hubbard model, we study the effect of two different perturbations: next-nearest-neighbor hopping and nearest-neighbor interaction. The system size dependence is investigated together with the effect of the proximity to integrable points or lines. Lastly, we discuss the consequences of a cutoff in the number of onsite bosons on the level statistics.