Research ArticleQuantum Mechanics

Observation of prethermalization in long-range interacting spin chains

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Science Advances  25 Aug 2017:
Vol. 3, no. 8, e1700672
DOI: 10.1126/sciadv.1700672
  • Fig. 1 Emergent double-well potential.

    Calculated spin excitation probability after a quantum quench. (A) The spin chain starts with a single-spin excitation on the left end in an effective double-well potential, Ueff, whose barrier height is determined by the range α of the interactions. The black dots represent the positions of the ions, whereas the red dots represent the simulated spin excitation probabilities. (B) For short-range interactions (α = 1.33), we map the system to a particle in a 1D square well, where the excitation becomes symmetrically distributed across the chain as predicted by the GGE, Embedded Image. (D) However, for long-range interactions (α = 0.55), there is an emergent double-well potential, which prevents the efficient transfer of the spin, and the excitation location retains memory of the initial state, in contrast to Embedded Image. (C) The double-well gives rise to near-degenerate eigenstates as α is decreased, as seen in the calculated energy difference between all pairs of eigenstates versus α. The height of the plot quantifies the off-diagonal density matrix elements of the initial state.

  • Fig. 2 Measured location of spin excitation.

    The average position of the spin excitation, 〈C〉, is plotted for an initial excitation on the left (dark blue unfilled squares) and right (dark red unfilled circles) along with their cumulative time average (blue and red filled circles and squares) for short-range (α = 1.33) and long-range (α = 0.55) interactions with initial states with one spin excitation (top panels) and two spin excitations (bottom panels). The cumulative time average of the deviation of the postselected individual spin projections from the GGE, Embedded Image, is also plotted. For short-range interactions, the spins quickly thermalize to a value predicted by the GGE, but for the long-range interacting system, a new type of prethermal state emerges. Error bars are statistical, calculated on the basis of quantum projection noise and calibrated detection errors (38).

  • Fig. 3 Long-time dynamics.

    Comparison between a numerical simulation (red line; with laser power fluctuations included) of the transverse-field Ising model and experimental data (black dots) for single (A) and double (B) initial spin excitations. The inset shows good agreement between experiment and numerics for 〈C〉. For both the single and double spin flip cases, the prethermal state persists for much longer than the experimental time scale and eventually relaxes to the thermal state. Error bars, 1 SD.

  • Fig. 4 Scaling to larger system size.

    Time evolution (light blue) and cumulative time average (orange) of 〈C〉 with false color images of the 22-ion chain, where the brightness of each ion is determined by the value of Embedded Image. The ions fluoresce during detection when in the | ↑ 〉z state (top image). We initialize the spins with a single spin excitation on the left end (middle image). After evolving for 36 Jmax, the spin excitation is delocalized, but its average position remains on the left half of the chain (bottom image). Effective range of the interaction is α ≈ 0.9. Error bars, 1 SD.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/8/e1700672/DC1

    Experimental noise sources and their influence on the thermalization dynamics

    Measuring the spin-spin coupling matrix

    Justification for postselection

    The spin-boson mapping and the GGE

    Single-particle properties of H0

    Discussion

    fig. S1. We directly measure the spin-spin coupling matrix with seven ions for both short-range (left matrix) and long-range (right matrix) interactions and see if it is symmetric.

    fig. S2. Numerical calculation to illustrate the origin of the double-well potential.

    Reference (39)

  • Supplementary Materials

    This PDF file includes:

    • Experimental noise sources and their influence on the thermalization dynamics.
    • Measuring the spin-spin coupling matrix
    • Justification for postselection
    • The spin-boson mapping and the GGE
    • Single-particle properties of H0
    • Discussion
    • fig. S1. We directly measure the spin-spin coupling matrix with seven ions for both short-range (left matrix) and long-range (right matrix) interactions and see if it is symmetric.
    • fig. S2. Numerical calculation to illustrate the origin of the double-well potential.
    • Reference (39)

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