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arXiv:1707.09934 · Physical Review A

Two-species boson mixture on a ring: A group-theoretic approach to the quantum dynamics of low-energy excitations

Vittorio Penna, Andrea Richaud

Published Nov 28, 2017• 32 citations
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Abstract

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.

DOI
10.1103/physreva.96.053631

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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 H\ifmmode^\elseˆ\fik{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}H\ifmmode^\elseˆ\fi​k​, each one associated to momentum modes \ifmmode±\else\textpm\fik\ifmmode\pm\else\textpm\fi{}k\ifmmode±\else\textpm\fik. Each H\ifmmode^\elseˆ\fik{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}H\ifmmode^\elseˆ\fi​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.
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 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 determine the low-energy spectrum and the eigenstates for a two-bosonic mode nonlinear model by applying the Inönü-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.
A theory that describes the dynamic behavior of a system containing an arbitrary number of different interacting Bose species is developed. The excited eigenstates of the system are generated by the superposition of plane-wave creation- and annihilation-operator products ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}=\ensuremath{\Sigma}{\ensuremath{\gamma}}^{}\ensuremath{\Sigma}{p}^{}g(p,k,\ensuremath{\gamma}){{a}_{p+k}}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\gamma}){a}_{p}(\ensuremath{\gamma})$, where the ccc-number coefficients $g(p,k,\ensuremath{\gamma})$ are determined by solving the equation of motion for ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}$ in an approximate way. The sum over $\ensuremath{\gamma}$ extends over all different species of the system. The equation of motion is solved by a method similar to that originally derived by Pines and Bohm for the electron gas. Results are obtained for the energy spectrum of the elementary excitations in the system. In particular, explicit results are obtained for the energy spectrum of a two-species system---first for a weakly coupled gas, and secondly for the low-momentum collective modes of a strongly coupled fluid. The low-momentum excitations of the strongly coupled fluid are completely analogous to the "zero sound" modes commonly associated with Fermi liquids. The theory is then applied to a dilute mixture of He3{\mathrm{He}}^{3}He3 in He4{\mathrm{He}}^{4}He4 and the phonon velocity is determined as a function of He3{\mathrm{He}}^{3}He3 concentration. The probability per unit time $P(\mathbf{k},{\ensuremath{\omega}}_{k})$ that a particle transfers momentum k and energy ${\ensuremath{\omega}}_{k}$ to the system by scattering is also calculated in terms of the dynamic form factor ${S}_{\ensuremath{\gamma}}(\mathbf{k},\ensuremath{\omega})$.
We investigate the stability and dynamics of the orbital angular momentum modes of a repulsive Bose-Einstein condensate trapped in two tunnel-coupled rings in a stack configuration. Within mean-field theory, we derive a two-state model for the system in the case in which we populate both rings equally with a single orbital angular momentum mode and include small perturbations in other modes. Analyzing the fixed-point solutions of the model and the associated classical Hamiltonian, we characterize the destabilization of the stationary states and the subsequent dynamics. By populating a single orbital angular momentum mode with an arbitrary population imbalance between the rings, we derive analytically the boundary between the regimes of Josephson oscillations and macroscopic quantum self-trapping and study numerically the stability of these solutions.