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arXiv:1705.02115 · Physical Review A

Quantum dynamics of bosons in a two-ring ladder: Dynamical algebra, vortexlike excitations, and currents

Andrea Richaud, Vittorio Penna

Published Jul 18, 2017• 31 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.96.013620

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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 H\ifmmode^\elseˆ\fik{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}H\ifmmode^\elseˆ\fi​k​ associated with momentum-mode pairs \ifmmode±\else\textpm\fik\ifmmode\pm\else\textpm\fi{}k\ifmmode±\else\textpm\fik. Each sub-Hamiltonian H\ifmmode^\elseˆ\fik{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}H\ifmmode^\elseˆ\fi​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 investigate the weak excitations of a system made up of two condensates trapped in a Bose-Hubbard ring and coupled by an interspecies repulsive interaction. Our approach, based on the Bogoliubov approximation scheme, shows that one can reduce the problem Hamiltonian to the sum of sub-Hamiltonians ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$, each one associated to momentum modes $\ifmmode\pm\else\textpm\fi{}k$. Each ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ is then recognized to be an element of a dynamical algebra. This uncommon and remarkable property allows us to present a straightforward diagonalization scheme, to find constants of motion, to highlight the significant microscopic processes, and to compute their time evolution. The proposed solution scheme is applied to a simple but nontrivial closed circuit, the trimer. The dynamics of low-energy excitations, corresponding to weakly populated vortices, is investigated considering different choices of the initial conditions and the angular-momentum transfer between the two condensates is evidenced. Finally, the condition for which the spectral collapse and dynamical instability are observed is derived analytically.
Bosonic two-ring ladders constitute an important class of atomtronic circuits, where coherent current flows not only can offer a new insight into many-body physics, but also can play the role of actual degrees of freedom, and hence allow for a viable implementation of cold-atom based devices and qubit systems. In this work, we exhaustively investigate the ground state properties and the low-lying energy spectrum of two linearly coupled Bose-Hubbard rings. We show that the competition among interactions, intra- and inter-ring hopping processes gives place to a rather rich physical scenario, where Mott-like states and (different kinds of) superfluid-like states emerge. The latter ones depend also on the (in)commensurate filling of the atoms. Our analysis, carried out within a simple analytical framework and by means of the exact numerical diagonalization of the system Hamiltonian, provides one with a rather complete characterization of the static properties of the two-ring ladder, including, but not limited to, coherence, fragmentation, correlations, and entanglement. We complement our investigation by studying how these indicators depend on the commensurability of the total number of bosons with respect to the total number of sites and show that the two stacked rings are always entangled for an odd number of atoms.
We consider a system of weakly interacting bosons confined on a planar double lattice ring subjected to two artificial gauge fields. This system is known to display three phases: the Meissner phase, where the flow of particles is carried at the edges of the system without transverse current; a vortex phase, characterized by nonzero transverse current; and a biased-ladder phase, characterized by an imbalance of the population of the two rings. We use the Bogoliubov approximation to determine the excitation spectrum in the three phases, the dynamic structure factor, and the quantum fluctuation corrections to the first-order correlation function. Our analysis reveals supersolid features as well as Josephson modes, corresponding to out-of-phase modes of the finite ring.
Abstract Bosonic two-ring ladders constitute an important class of atomtronic circuits, where coherent current flows not only can offer a new insight into many-body physics, but also can play the role of actual degrees of freedom, and hence allow for a viable implementation of cold-atom based devices and qubit systems. In this work, we exhaustively investigate the ground state properties and the low-lying energy spectrum of two linearly coupled Bose–Hubbard rings. We show that the competition among interactions, intra- and inter-ring hopping processes gives place to a rather rich physical scenario, where Mott-like states and (different kinds of) superfluid-like states emerge. The latter ones depend also on the (in)commensurate filling of the atoms. Our analysis, carried out within a simple analytical framework and by means of the exact numerical diagonalization of the system Hamiltonian, provides one with a rather complete characterization of the static properties of the two-ring ladder, including, but not limited to, coherence, fragmentation, correlations, and entanglement. We complement our investigation by studying how these indicators depend on the commensurability of the total number of bosons with respect to the total number of sites and show that the two stacked rings are always entangled for an odd number of atoms.
We determine the low-energy spectrum and the eigenstates for a two-bosonic mode nonlinear model by applying the In\"{o}n\"{u}-Wigner contraction method to the Hamiltonian algebra. This model is known to well represent a Bose-Einstein condensate rotating in a thin torus endowed with two angular-momentum modes as well as a condensate in a double-well potential characterized by two space modes. We consider such a model in the presence of both an attractive and a repulsive boson interaction and investigate regimes corresponding to different values of the inter-mode tunneling parameter. We show that the results ensuing from our approach are in many cases extremely satisfactory. To this end we compare our results with the ground state obtained both numerically and within a standard semiclassical approximation based on su(2) coherent states.
We investigate the dynamics of a two-component Bose-Einstein condensate trapped in two coupled rings in ladder configuration. First, we derive from the two-component Gross-Pitaevskii equation a few state model for the angular momentum modes. Then, we analyse the stability of the stationary states by solving the Bogoliubov-de Gennes equations. Also, we study the dynamics of the system for an arbitrary population imbalance between the rings, characterizing the transition from Josephson oscillations to quantum self-trapping. We also derive the self-trapping condition for the bosonic mixture, finding simultaneous self-trapping of the two components.