Probing topology by “heating”: Quantized circular dichroism in ultracold atoms

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Science Advances  18 Aug 2017:
Vol. 3, no. 8, e1701207
DOI: 10.1126/sciadv.1701207
  • Fig. 1 Topology through heating.

    (A) A 2D Fermi gas is initially prepared in the lowest band (LB) of a lattice, with Chern number νLB, and it is then subjected to a circular time-periodic modulation (Eq. 3). (B) The rate Γ associated with the depletion of the populated band is found to depend on the orientation of the drive, Γ+ ≠ Γ, whenever the LB is characterized by a nontrivial Chern number νLB ≠ 0. (C) Integrating the differential rate over a relevant drive-frequency range, ΔΓint = ∫dω(Γ+ − Γ)/2, leads to a quantized result, ΔΓint/Asyst = (νLB/2)E2, where E is the strength of the drive and Asyst is the system’s area (Eq. 1).

  • Fig. 2 Transition matrix elements for the driven two-band Haldane model with 104 lattice sites and PBC.

    Specifically, the inset shows the matrix elements Embedded Image, as defined in Eq. 7, for all possible transitions, ω = [ε1(k) − ε0(k)]/ℏ; the main plot shows the averaged values Embedded Image, defined within each interval of width Δω = 0.1J/ℏ. The model parameters are set such that Δgap ≈ 2J, where J is the nearest-neighbor hopping amplitude; the strength of the drive (Eq. 3) is E = 0.001J/d, where d is the lattice spacing. The DIR (Eq. 8) obtained from this numerical data yields ΔΓint(ℏ2/AsystE2) ≈ −1.00, in agreement with Eq. 1 and the theoretical prediction νLB = −1. The matrix elements Embedded Image are expressed in units of J2/ℏ, whereas the frequency ω is given in units of J/ℏ.

  • Fig. 3 Local Chern marker C(rj) in a 2D lattice with boundaries (OBC) realizing the Haldane model.

    Far from the boundaries, the marker is C(rj) ≈ −1, in agreement with the Chern number of the populated band νLB = −1. Close to the edges, the local marker is very large and positive (see the zoom shown in inset) such that the total contribution of the edges exactly cancels the bulk contribution, Embedded Image: The DIR in Eq. 17 vanishes in a system with boundaries. Here, d is the lattice spacing.

  • Fig. 4 Depletion rates for the trap-release protocol and the related Chern number measurement.

    (A) Depletion rates Γ±(ω) extracted from a numerical simulation of the circularly shaken Haldane model with OBC. The edge-state contribution has been annihilated by initially confining the cloud in a disc of radius r = 20d and then releasing it in a larger lattice of size 120 × 120, after which the heating protocol (circular drive) was applied; other system parameters are the same as in Fig. 2. The rates Γ±l), which are expressed in units of J/ℏ, were obtained by measuring the number of excited particles after a time t = 4ℏ/J for fixed values of ωl separated by Δω = 0.05J/ℏ. (B) Approximate value for the Chern number of the populated band Embedded Image, as extracted from the numerical rates and Eq. 1, and represented as a function of the step Δω used to sample the drive frequencies; note that the area Asyst entering Eq. 1 corresponds to the initial area of the cloud in the trap release protocol. A satisfactory measure is reached when the frequency sampling accurately probes the resonant peaks; we find Δω ≲ 0.5J/ℏ (that is, at least 20 different values within the proper frequency range) for an observation time t = 4ℏ/J. The saturation value Embedded Image is limited by the fraction of particles populating the upper band, after abruptly removing the confinement, and can be improved by further increasing the initial radius r (or by softening the trap release).

Supplementary Materials

  • Supplementary Materials

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    • fig. S1. Depletion rates Γ±(ω) as a function of the drive frequency ω for the driven two-band Haldane model with 104 lattice sites and PBC.

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