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arXiv:0906.5229 · Journal of Physics B Atomic Molecular and Optical Physics

Excitation spectrum of Mott shells in optical lattices

Dirk-Sören Lühmann, Kai Bongs, Daniela Pfannkuche

Published Jun 24, 2009• 6 citations
arXivHTMLPDF

Abstract

We theoretically study the excitation spectrum of confined macroscopic optical lattices in the Mott-insulating limit. For large systems, a fast numerical method is proposed to calculate the ground state filling and excitation energies. We introduce many-particle on-site energies capturing multi-band effects and discuss tunnelling on a perturbative level using an effectively restricted Hilbert space. Results for small one-dimensional lattices obtained by this method are in good agreement with the exact multi-band diagonalization of the Hamiltonian. Spectral properties associated with the formation of regions with constant filling, so-called Mott shells, are investigated and interfaces between the shells with strong particle fluctuations are characterized by gapless local excitations.

DOI
10.1088/0953-4075/42/14/145305

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We theoretically study the excitation spectrum of confined macroscopic optical lattices in the Mott-insulating limit. For large systems, a fast numerical method is proposed to calculate the ground state filling and excitation energies. We introduce many-particle on-site energies capturing multi-band effects and discuss tunnelling on a perturbative level using an effectively restricted Hilbert space. Results for small one-dimensional lattices obtained by this method are in good agreement with the exact multi-band diagonalization of the Hamiltonian. Spectral properties associated with the formation of regions with constant filling, so-called Mott shells, are investigated and interfaces between the shells with strong particle fluctuations are characterized by gapless local excitations.
We investigate the energy spectrum of ground state and quasi-particle excitation spectrum of hard-core bosons, which behave very much like spinless noninteracting fermions, in optical lattices by means of the perturbation expansion and Bogoliubov approach. The results show that the energy spectrum has a single band structure, and the energy is lower near zero momentum; the excitation spectrum gives corresponding energy gap, and the system is in Mott-insulating state at Tonks limit. The analytic result of energy spectrum is in good agreement with that calculated in terms of Green's function at strong correlation limit.
We study bosonic atoms in optical honeycomb lattices with anisotropic tunneling and find dimerized Mott insulator (MI) phases with fractional filling. These incompressible insulating phases are characterized by an interaction-driven localization of particles in respect to the individual dimers and large local particle-number fluctuations within the dimers. We calculate the ground-state phase diagrams and the excitation spectra using an accurate cluster mean-field method. The cluster treatment enables us to probe the fundamental excitations of the dimerized MI where the excitation gap is dominated by the intra-dimer tunneling amplitude. This allows the distinction from normal Mott insulating phases gapped by the on-site interaction. In addition, we present analytical results for the phase diagram derived by a higher-order strong-coupling perturbative expansion approach. By computing finite lattices with large diameters the influence of a harmonic confinement is discussed in detail. It is shown that a large fraction of atoms forms the dimerized MI under experimental conditions. The necessary anisotropic tunneling can be realized either by periodic driving of the optical lattice or by engineering directly a dimerized lattice potential. The dimers can be mapped to their antisymmetric states creating a lattice with coupled p -orbitals.
The density-matrix renormalization-group (DMRG) method is used to investigate optical excitations in the Mott insulating phase of a one-dimensional extended Hubbard model. The linear optical conductivity is calculated using the dynamical DMRG method and the nature of the lowest optically excited states is investigated using a symmetrized DMRG approach. The numerical calculations agree perfectly with field-theoretical predictions for a small Mott gap and analytical results for a large Mott gap obtained with a strong-coupling analysis. It is shown that four types of optical excitations exist in this Mott insulator: pairs of unbound charge excitations, excitons, excitonic strings, and charge-density-wave (CDW) droplets. Each type of excitation dominates the low-energy optical spectrum in some region of the interaction parameter space and corresponds to distinct spectral features: a continuum starting at the Mott gap (unbound charge excitations), a single peak or several isolated peaks below the Mott gap (excitons and excitonic strings, respectively), and a continuum below the Mott gap (CDW droplets).
The one-dimensional Bose-Hubbard model at a unit filling factor is studied by means of a very high-order symbolic perturbative expansion. Analytical expressions are derived for the ground-state quantities such as energy per site, variance of on-site occupation, and correlation functions: $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j+r}⟩$ and $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{j+r}⟩$. These findings are compared to numerics and good agreement is found in the Mott insulator phase. Our results provide analytical approximations to important observables in the Mott phase, and are also of direct relevance to future experiments with ultracold atomic gases placed in optical lattices. We also discuss the symmetry of the Bose-Hubbard model associated with the sign change of the tunneling coupling.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
We study Bragg spectroscopy of ultracold atoms in one-dimensional optical lattices as a method for probing the excitation spectrum in the Mott-insulator phase, in particular the one-particle-hole excitation band. Within the framework of perturbation theory we obtain an analytical expression for the dynamic structure factor and use it to calculate the imparted energy which was shown to be a relevant observable in recent experiments. We test the accuracy of our approximations by comparing them with numerically exact solutions of the Bose-Hubbard model in restricted cases and establish the limits of validity of our linear-response analysis. Finally we show that when the system is deep in the Mott-insulator regime, its response to the Bragg perturbation is temperature dependent. We suggest that this dependence might be used as a tool to probe temperatures of order of the Mott gap.