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arXiv:2202.07535 · SciPost Physics

Time-dependent variational Monte Carlo study of the dynamic response of bosons in an optical lattice

Mathias Gartner, Ferran Mazzanti, Robert E. Zillich

Published Aug 18, 2022• 6 citations
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Abstract

We study the dynamics of a one-dimensional Bose gas at unit filling in both shallow and deep optical lattices and obtain the dynamic structure factor \bm{S(k,\omega)} 𝐒 ( 𝐤 , 𝛚 ) by monitoring the linear response to a weak probe pulse. We introduce a new procedure, based on the time-dependent variational Monte Carlo method (tVMC), which allows to evolve the system in real time, using as a variational model a Jastrow-Feenberg wave function that includes pair correlations. Comparison with exact diagonalization results of \bm{S(k,\omega)} 𝐒 ( 𝐤 , 𝛚 ) obtained on a lattice in the Bose-Hubbard limit shows good agreement of the dispersion relation for sufficiently deep optical lattices, while for shallow lattices we observe the influence of higher Bloch bands. We also investigate non-linear response to strong pulses. From the power spectrum of the density fluctuations we obtain the excitation spectrum, albeit broadened, by higher harmonic generation after a strong pulse with a single low wave number. As a remarkable feature of our simulations we furthermore demonstrate that the full excitation spectrum can be retrieved from the power spectrum of the density fluctuations due to the stochastic noise inherent in any Monte Carlo method, without applying an actual perturbation.

DOI
10.21468/scipostphys.13.2.025

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We study the dynamics of a one-dimensional Bose gas at unit filling in both shallow and deep optical lattices and obtain the dynamic structure factor \bm{S(k,\omega)} 𝐒 ( 𝐤 , 𝛚 ) by monitoring the linear response to a weak probe pulse. We introduce a new procedure, based on the time-dependent variational Monte Carlo method (tVMC), which allows to evolve the system in real time, using as a variational model a Jastrow-Feenberg wave function that includes pair correlations. Comparison with exact diagonalization results of \bm{S(k,\omega)} 𝐒 ( 𝐤 , 𝛚 ) obtained on a lattice in the Bose-Hubbard limit shows good agreement of the dispersion relation for sufficiently deep optical lattices, while for shallow lattices we observe the influence of higher Bloch bands. We also investigate non-linear response to strong pulses. From the power spectrum of the density fluctuations we obtain the excitation spectrum, albeit broadened, by higher harmonic generation after a strong pulse with a single low wave number. As a remarkable feature of our simulations we furthermore demonstrate that the full excitation spectrum can be retrieved from the power spectrum of the density fluctuations due to the stochastic noise inherent in any Monte Carlo method, without applying an actual perturbation.
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 have used inelastic light scattering to study correlated phases of an array of one-dimensional interacting Bose gases. In the linear response regime, the observed spectra are proportional to the dynamic structure factor. In particular we have investigated the superfluid to Mott insulator crossover loading the one-dimensional gases in an optical lattice and monitoring the appearance of an energy gap due to finite particle-hole excitation energy. We attribute the low frequency side of the spectra to the presence of some superfluid and normal phase fraction between the Mott insulator regions with different fillings produced in the inhomogeneous systems. In the Mott phase we also investigated excitations to higher excited bands of the optical lattice, the spectra obtained in this case being connected to the single particle spectral function. In one-dimensional systems the effect of thermal fluctuations and interactions is enhanced by the reduced dimensionality showing up in the dynamic structure factor. We measured the dynamic structure factor of an array of one-dimensional bosonic gases pointing out the effect of temperature-induced phase fluctuations in reducing the coherence length of the system.
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.
We study Bragg spectroscopy of strongly interacting one-dimensional bosons loaded in an optical lattice plus an additional parabolic potential. We calculate the dynamic structure factor by using Monte Carlo simulations for the Bose-Hubbard Hamiltonian, exact diagonalizations and the results of a recently introduced extended fermionization model. We find that, due to the system's inhomogeneity, the excitation spectrum exhibits a multibranched structure, whose origin is related to the presence of superfluid regions with different densities in the atomic distribution. We thus suggest that Bragg spectroscopy in the linear regime can be used as an experimental tool to unveil the shell structure of alternating Mott insulator and superfluid phases characteristic of trapped bosons.
We investigate the spin-2 chain model corresponding to the small hopping limit of the spin-2 Bose-Hubbard model using density-matrix renormalization-group and time-evolution techniques. We calculate both static correlation functions and the dynamic structure factor. The dynamic structure factor in the dimerized phase differs significantly between parameters near the SU(5)-symmetric point and those deeper in the phase where the dimerization is strong. In the former case, most of the spectral weight is concentrated in a single excitation line, while in the latter case, a broad excitation continuum shows up. For the trimerized phase, we find gapless excitations at momenta $k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3$ in agreement with previous results, although the visibility of these excitations in the dynamic spin response depends strongly on the specific parameters. We also consider parameters for specific atoms which may be relevant for future optical-lattice experiments.
We calculate the dynamic structure factor $S(q,\ensuremath{\omega})$ of a one-dimensional (1D) interacting Bose gas confined in a harmonic trap. The effective interaction depends on the strength of the confinement enforcing the (1D) motion of atoms; interaction may be further enhanced by superimposing an optical lattice on the trap potential. In the compressible state, we find that the smooth variation in the gas density around the trap center leads to softening of the singular behavior of $S(q,\ensuremath{\omega})$ at the first Lieb excitation mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response $\overline{S}(q,\ensuremath{\omega})$ remains a nonanalytic function of qqq and $\ensuremath{\omega}$ at the first Lieb excitation mode in the limit of weak trap confinement. The exponent of the power-law nonanalyticity is modified due to the inhomogeneity in a universal way and thus bears unambiguously the information about the (homogeneous) Lieb-Liniger model. A strong optical lattice causes formation of Mott phases. Deep in the Mott regime, we predict a semicircular peak in $S(q,\ensuremath{\omega})$ centered at the on-site repulsion energy, $\ensuremath{\omega}=U$. Similar peaks of smaller amplitudes exist at multiples of UUU as well. We explain the suppression of the dynamic response with entering into the Mott regime, observed recently by Cl\'ement et al. [Phys. Rev. Lett. 102, 155301 (2009)], based on an fff-sum rule for the Bose-Hubbard model.