Research ArticleQUANTUM PHYSICS

Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice

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Science Advances  20 Nov 2015:
Vol. 1, no. 10, e1500854
DOI: 10.1126/sciadv.1500854
  • Fig. 1 Optical Lieb lattice.

    (A) Lieb lattice. A unit cell is indicated by the green square. (B) Tight-binding energy band structure of the Lieb lattice. (C) Experimental realization of the Lieb lattice. Black arrows indicate polarizations of the lattice beams. (D) Lattice potential for (slong, sshort, sdiag) = (8, 8, 9.5) at φx = φz = 0 and ψ = π/2. (E) Band structures of the optical Lieb lattice at (slong, sshort, sdiag) = (8, 8, 9.5) (red dashed), (34, 34, 37.4) (solid black).

  • Fig. 2 Coherent band transfer.

    (A) Principle of the transferring method. (B) Absorption images reveal the coherent oscillations between the |B〉 + |C〉 and |B〉 − |C〉 states. In the upper left image, the first three Brillouin zones are displayed by white, green, and red lines, respectively. (C) Oscillating behavior of the band population during phase imprinting in the absence of lattice confinement along the y direction (blue circles) and with lattice confinement −Vycos2(2kLy) (red squares). Solid lines are the fit results using the single-particle solution of the Schrödinger equations (see main text). Error bars denote SD of three independent measurements.

  • Fig. 3 Lifetime of atoms in the flat band.

    (A) Absorption images for the lifetime measurement of the 2nd band with three different hold times, taken after 14-ms time of flight. The diagonal lattice depth is sdiag = 9.5. The first three Brillouin zones are indicated by the white dashed lines. In the top image, the areas used to evaluate the lifetime of a condensate (τc) are also displayed with the red squares. (B) Decay of the flat band at (slong, sshort) = (8, 8) and variable sdiag. Solid lines are the fit results with double-exponential curves. Error bars denote the SD of three independent measurements. (C) Lifetime of the flat band. τ1,2 are the fast and slow decay time obtained from the data shown in (B), respectively. τc is the e−1 lifetime of condensates. Error bars represent fitting error.

  • Fig. 4 Tunneling dynamics in the Lieb lattice.

    (A) Demonstrating the measurement of sublattice occupancy. Here, sublattice mapping technique is applied to atoms loaded into (left) ((slong(x),slong(z)), sshort, sdiag) = ((8,8), 8, 0), (middle) ((2,8), 8, 19), and (right) ((8,2), 8, 19), corresponding to atoms in A, B, and C sites, respectively. (B) Measured tunneling dynamics of |+〉 and |−〉 initial states in the Lieb lattice with (slong, sshort, sdiag) = (8, 8, 9.5). Solid lines are the fits to the experimental data with damped sinusoidal oscillation (for |+〉) and double exponentials (for |−〉). Inset shows dynamics of the |−〉 state for longer hold times. Error bars denote SD. Illustration of tunneling process for each initial state is also shown on the right-hand side. (C) Frequencies of coherent intersite oscillations. Solid lines are the calculated band gap between the 1st and 2nd (red) and the 1st and 3rd bands (blue). Error bars denote fitting error. (D) Bending flat band. Dynamics of the |−〉 state in the presence of imbalance Δslong = slong(x)slong(z) shows restoration of coherent dynamics. Error bars denote SD.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/10/e1500854/DC1

    Section S1. Calibration of the relative phase.

    Section S2. Tight-binding model for the optical Lieb lattice.

    Section S3. Momentum distributions in coherent band transfer.

    Section S4. Effect of interactions on inter-sublattice oscillations of a BEC.

    Fig. S1. Phase dependence of a time-of-flight signal.

    Fig. S2. Tunneling parameters in the optical Lieb lattice.

    Fig. S3. Wannier functions of the optical Lieb lattice.

    Fig. S4. Momentum space observation of coherent band transfer.

    Fig. S5. Density dependence of oscillation frequency.

    Table S1. Initial conditions for the inter-sublattice oscillations.

    Reference (38)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Calibration of the relative phase.
    • Section S2. Tight-binding model for the optical Lieb lattice.
    • Section S3. Momentum distributions in coherent band transfer.
    • Section S4. Effect of interactions on inter-sublattice oscillations of a BEC.
    • Fig. S1. Phase dependence of a time-of-flight signal.
    • Fig. S2. Tunneling parameters in the optical Lieb lattice.
    • Fig. S3. Wannier functions of the optical Lieb lattice.
    • Fig. S4. Momentum space observation of coherent band transfer.
    • Fig. S5. Density dependence of oscillation frequency.
    • Table S1. Initial conditions for the inter-sublattice oscillations.
    • Reference (38)

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