Research ArticlePHYSICS

Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose-Hubbard system

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Science Advances  22 Dec 2017:
Vol. 3, no. 12, e1701513
DOI: 10.1126/sciadv.1701513
  • Fig. 1 Engineered dissipation of inelastic two-body collision.

    (A) Schematic of the introduced inelastic two-body collision. When there are singly and doubly occupied sites in the lattice (top), the atoms in the doubly occupied sites are converted into the molecules by applying the PA laser (middle) and then escape from the lattice because of the high kinetic energy given by the dissociation (bottom). (B) Inelastic collision coefficient βPA as a function of the intensity of the PA laser. The dashed line indicates the linear fit to the low intensity data with a slope of 2.10(7) × 10−11 cm3 s−1/(W cm−2), which well agrees with the theoretical estimation of 2.12 × 10−11 cm3 s−1/(W cm−2) (42). Note that a saturating behavior is observed at the highest intensity, the behavior of which is reported in other experiments performed in a harmonic trap (4346). The inset shows time evolution of the remaining atom number in a 3D optical lattice for the measurement of the inelastic collision rate ΓPA. The lattice depth is set to V0 = 14 ER. a.u., arbitrary units. (C) Inelastic collision rate ΓPA dependence of the two-body loss rate κ for atoms initially prepared in a Mott insulating state with singly occupied sites. The values of κ are determined by fitting of the two-body loss function N(t) = N(0)/(1 + κt) to the data (27). The scales on the right in (B) and the top in (C) indicate the dimensionless dissipation strength γ.

  • Fig. 2 Atom loss and condensate fraction.

    (A) Numerical calculation of the atom number per site Embedded Image (left) and the condensate fraction Embedded Image (right) based on the dissipative Bose-Hubbard model with the Gutzwiller approximation. The time sequence of the lattice depth and the strength of the dissipation are set to be almost identical to those in the experiments shown in (B) (see also fig. S11). (B) Atom number diagram. The experimental data of the atom number are shown as the gray dots as a function of the final lattice depth for various strengths of dissipation and are interpolated. The white triangles show the lattice depths at which the atom loss sets in, determined from the analysis in (C). The numbers ① to ④ correspond to the dissipation strengths for which the atom number changes are plotted in (C). (C) Temporal change of the atom number during a ramp-down sequence for four representative strengths of the dissipation. The atom number is normalized by the initial atom number at the lattice depth of V0 = 20 ER. Blue lines are double linear fits to extract the onset of the atom loss, which are shown as dotted lines.

  • Fig. 3 Coherence properties across the Mott insulator to superfluid crossover.

    (A) TOF absorption image. The images are taken with different final lattice depths and strengths of the dissipation and averaged over 20 shots at each parameter. (B) Visibility of the interference peak of the images. (C) Width of the density distribution. The width is the full width at half maximum obtained by the Gaussian fitting. The insets in (B) and (C) show the values varying the dissipation strength in the fixed lattice depth of 8 ER.

  • Fig. 4 Dynamics after turning off the dissipation.

    (A) Experimental sequence for the observation of the dynamics after turning off the dissipation. After ramping up the lattice to V0 = 20 ER to prepare the Mott insulator state, we ramped down the lattice to the final lattice depth V0 = s ER by applying the PA laser. The ramp-down speed is −2 ER/ms, and the ramp-down time is tramp = (20 − s)/2 ms. After ramping down the lattice, we turned off the PA laser and held the lattice for thold. (B) Time evolution of TOF image after turning off the dissipation. The hold time after turning off the PA laser is shown at the bottom right of each image. (C) Time evolution of the visibility and the width. (D) Lattice depth dependence of the visibility and the width with 0- and 4-ms hold time. (E) Atom number after turning off the dissipation. The atom number is normalized by the initial atom number at thold = 0 ms.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/12/e1701513/DC1

    section S1. Derivation of the dissipative Bose-Hubbard model

    section S2. Loss dynamics from the Mott insulating state with double filling

    section S3. Details of the theoretical analyses using the Gutzwiller variational approach

    section S4. Unexpectedly large atom loss for strong intensity of PA laser

    fig. S1. Time evolution of the normalized atom density Formula for Formula = 2.

    fig. S2. Measurement of the one-body molecular loss Formula.

    fig. S3. On-site interaction U and the hopping energy J as a function of the lattice depth.

    fig. S4. Time sequence of the atom loss measurement from the Mott insulating state with unit filling.

    fig. S5. Time evolution of the atom density Formula for Formula.

    fig. S6. Loss rate κ as a function of the dissipation strength γ.

    fig. S7. Time evolution of ρ3,3.

    fig. S8. Formula and Formula as a function of γ.

    fig. S9. Time evolution of the amplitude of the superfluid order parameter and its growth rate.

    fig. S10. Contour plot of the growth rate, Formula.

    fig. S11. Time sequence for the dynamical melting of the Mott insulating state with unit filling.

    fig. S12. Atom density Formula as a function of the instantaneous value of the lattice depth V0/ER.

    fig. S13. Condensate fraction Formula as a function of the instantaneous value of the lattice depth V0/ER.

    References (4759)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Derivation of the dissipative Bose-Hubbard model
    • section S2. Loss dynamics from the Mott insulating state with double filling
    • section S3. Details of the theoretical analyses using the Gutzwiller variational approach
    • section S4. Unexpectedly large atom loss for strong intensity of PA laser
    • fig. S1. Time evolution of the normalized atom density 〈nˆA〉(t) for 〈nˆA〉(0) = 2.
    • fig. S2. Measurement of the one-body molecular loss ΓPA.
    • fig. S3. On-site interaction U and the hopping energy J as a function of the lattice depth.
    • fig. S4. Time sequence of the atom loss measurement from the Mott insulating state with unit filling.
    • fig. S5. Time evolution of the atom density 〈nˆA〉(t) for 〈nˆA〉 (0) = 1.
    • fig. S6. Loss rate κ as a function of the dissipation strength γ.
    • fig. S7. Time evolution of ρ3,3.
    • fig. S8. ρmax3,3 and ΓPA × ρmax3,3 as a function of γ.
    • fig. S9. Time evolution of the amplitude of the superfluid order parameter and its growth rate.
    • fig. S10. Contour plot of the growth rate, G=d/dt ln |ψ|2.
    • fig. S11. Time sequence for the dynamical melting of the Mott insulating state with unit filling.
    • fig. S12. Atom density 〈nˆA〉 as a function of the instantaneous value of the lattice depth V0/ER.
    • fig. S13. Condensate fraction |ψ|2/ 〈nˆA〉 as a function of the instantaneous value of the lattice depth V0/ER.
    • References (47–59)

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