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High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock

William R. Milner, Lingfeng Yan, Ross B. Hutson, Christian Sanner, and Jun Ye
Phys. Rev. A 107, 063313 – Published 26 June 2023
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  • INTRODUCTION
  • HIGH-INTENSITY IMAGING
  • IN SITU IMAGING CHARACTERIZATION
  • BAND INSULATOR DEMONSTRATION
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We report on the observation of a high-density, band insulating state in a three-dimensional optical lattice clock. Filled with a nuclear-spin-polarized degenerate Fermi gas of Sr87, the three-dimensional (3D) lattice has one atom per site in the ground motional state, thus guarding against frequency shifts due to contact interactions. At this high density where the average distance between atoms is comparable to the probe wavelength, conventional imaging techniques at saturation intensity suffer from large systematic errors. To spatially probe frequency shifts in the clock and measure thermodynamic properties of this system, accurate imaging techniques at high optical depths are required. Using a combination of highly saturated fluorescence and absorption imaging, we confirm the density distribution in our 3D optical lattice in agreement with a single spin band insulating state. Combining our clock platform with this high filling fraction opens the door to studying new classes of long-lived, many-body states arising from dipolar interactions.

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    • Received 9 January 2023
    • Revised 9 March 2023
    • Accepted 31 May 2023

    DOI:https://doi.org/10.1103/PhysRevA.107.063313

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Atomic, optical & lattice clocks
    Atomic, Molecular & OpticalQuantum Information, Science & Technology

    Authors & Affiliations

    William R. Milner*, Lingfeng Yan, Ross B. Hutson, Christian Sanner, and Jun Ye†

    • JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA

    • *william.milner@colorado.edu
    • †ye@jila.colorado.edu

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    Vol. 107, Iss. 6 — June 2023

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    Images

    • Figure 1
      Figure 1

      Schematic of our clock platform. Vertical and horizontal imaging systems with numerical apertures of 0.2 and 0.1 respectively provide measurements of the two-dimensional density distribution ñ. Accounting for the lattice spacing a=407 nm, ña2 is determined from highly saturated absorption imaging. To mitigate imaging errors, the atoms are highly saturated and each scatters photons with a maximum rate of Γ/2. Measurements from our high-resolution imaging system integrated along gravity are presented in panel (a), where the density distribution is extracted for thermodynamic modeling. Images from the horizontal imaging system in panel (b) are just used to determine our atom cloud aspect ratio for our inverse Abel transform.

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    • Figure 2
      Figure 2

      A comparison of high-intensity fluorescence and standard absorption imaging (I≈Isat) at optical depths exceeding 200 in our highly degenerate Fermi gas is shown. in situ absorption imaging at low intensity yields strikingly erroneous measurements at high density. The calculated two-dimensional Fermi gas distribution according to our experimental parameters is shared for comparison in qualitative agreement.

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    • Figure 3
      Figure 3

      (a) Calibration method for in situ fluorescence detection using atom counts from time-of-flight absorption imaging. Collected photon counts from both the vertical and horizontal imaging systems are plotted, with solid and dashed lines representing fits to the horizontal and vertical measurements respectively. Inset: Collected photon count with the vertical imaging system as a function of I/Isat at 1−µs pulse duration. (b) Peak column density as a function of fluorescence pulse duration. Measurements are normalized by 1.9×1011atoms/cm2, the column density at the shortest pulse duration of 500 ns. Images at 500 ns and 2µs in the inset are plotted for comparison. The error bars denote the standard error of the mean.

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    • Figure 4
      Figure 4

      (a) The three-dimensional density distribution and the corresponding lattice filling fraction are determined from the in situ absorption image in Fig. 1 and the use of an inverse Abel transformation. (b) A line cut along z=0 and y=0 provides the data points in circles. Errorbars are both the statistical uncertainty of the Abel transformation and atom number uncertainty added in quadrature. We start with a prediction based on thermodynamic calculation, using independently measured values for the entropy per particle, atom number, and harmonic confinement. The best fit to the data results in a 10% reduction of the measured aspect ratio ωy/ωx and 5% reduction of the predicted entropy per particle. The red line captures this fit, with entropy-per-particle uncertainty in the shaded band. The blue dashed line is a fit to the Gaussian in qualitative disagreement with na3.

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    • Figure 5
      Figure 5

      (a) Integrated counts from the images in Fig. 3 of the main text along the x axis as a function of pulse duration. The total counts at each pulse duration are plotted in panel (b), normalized by the counts at 500 ns. Given the detected photon count increases linearly with pulse duration, we observe minimal atom loss or molecular formation over the full 2−µs range. The inset shows the Gaussian rms width of the cloud as a function of pulse duration.

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    • Figure 6
      Figure 6

      SNR comparison between absorption and fluorescence imaging. The relevant imaging parameters from the main figures of the paper are used for this calculation. For absorption imaging the atom count variance scales inversely proportional with intensity in the nonsaturated limit I≪Isat, and proportional with intensity in the high-saturation limit. The variance is for both imaging methods proportional to 1/τ. In the fully saturated regime (and assuming no technical noise) the normalized variance for fluorescence imaging is independent of atomic column density. To avoid imaging defects at the high densities used in clock operation, I/Isat>50 was used in all imaging measurements. The black dashed line indicates the intensity used for our inverse Abel measurements.

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    • Figure 7
      Figure 7

      Readout noise calibration. A π pulse on our optical clock transition is used so pe≈1 and Vpe=R2Ct¯2+C. We use four pulse durations between 5 and 20µs to vary Ct. We fit R=100.2±24.6 and C=2.73×10−6±1.02×10−6.

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    • Figure 8
      Figure 8

      aQPN calibration. The atoms in our optical lattice are placed in a superposition of the ground and clock states with a π/2 pulse so pe≈0.5 for these measurements and Vpe is fit to Eq. (E9). We determine aQPN=1.72±0.16.

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