DOI: 10.5061/dryad.qv9s4mwq5
title: Data from: Coherent evolution of superexchange interaction in seconds long optical clock spectroscopy

description: Measurement science now connects strongly with engineering of quantum
 coherence, many-body states, and entanglement. To scale up the performance
 of an atomic clock using a degenerate Fermi gas loaded in a
 three-dimensional optical lattice, we must understand complex many-body
 Hamiltonians to ensure meaningful gains for metrological applications. In
 this work, we use a highly filled Sr 3D lattice to study the effect of a
 tunable Fermi-Hubbard Hamiltonian. The clock laser introduces a spin-orbit
 coupling spiral phase and breaks the isotropy of superexchange
 interactions, changing the Heisenberg spin model into one exhibiting
 XXZ-type spin anisotropy. By tuning the lattice confinement and applying
 imaging spectroscopy we map out favorable atomic coherence regimes. With
 weak transverse confinement, both s- and p-wave interactions contribute to
 decoherence and atom loss, and their contributions can be balanced. At
 deep transverse confinement, we directly observe coherent superexchange
 interactions, tunable via on-site interaction and site-to-site energy
 shift, on the clock Ramsey fringe contrast over timescales of multiple
 seconds. This study provides a groundwork for using a 3D optical lattice
 clock to probe quantum magnetism and spin entanglement.
description: We use absorption imaging of ultracold atoms to detect the
 relative populations of ground and clock atoms. These populations are used
 to determine the excitation fraction. We relate the excitation fraction to
 various physics effects as detailed in the manuscript. 
description: ## ‘Coherent evolution of superexchange interaction in seconds long
 optical clock spectroscopy’ dataset   ## Description of the data and file
 structure All experimental data presented in “Coherent evolution of
 superexchange interaction in seconds long optical clock spectroscopy” in
 both the main text and supplementary materials is provided. Each folder
 provided in ‘paper_data.zip’ is labelled by figure name and contains the
 data for each figure. Here we go folder-by-folder and specifically
 annotate each dataset.   ### Fig 2: **(B)** `plot_fig_2b.py` is the Python
 script to plot the data in Fig.2(B). Left panel: ‘dark_times’ are the
 Ramsey interferometer dark times in seconds.  ‘C_’ are the Ramsey fringe
 contrasts. A detailed explanation of Ramsey spectroscopy is provided in
 the main text of our manuscript. Our method of determining these contrasts
 is in the section ‘Ellipse fitting analysis’ in the Supplemental
 materials. ‘C_errs’ are one standard deviation error bars for the ‘C_’
 measurements. Upper right panel: `P1_2s.csv` are the excitation fractions
 in region P1 for 2 second Ramsey spectroscopy dark time. Details of
 regions P1 and P2 are provided in ‘Ellipse fitting analysis’ of the
 manuscript. ‘P2_2s.csv’ are the excitation fractions in region P2 for 2
 second Ramsey spectroscopy dark time. Bottom right panel: `P1_16s.csv` is
 the excitation fractions in region P1 for 16 second Ramsey spectroscopy
 dark time. `P2_16s.csv` is the excitation fractions in region P2 for 16
 second Ramsey spectroscopy dark time. **(C)** `plot_fig_2c.py` is the
 python script to plot the data in Fig.2(C). ‘Fig_2c.csv’ is the data for
 Fig.2(C). Each grid point corresponds to a quality factor Q as defined in
 the caption of Fig. 2.  The grid corresponds to the transverse and
 vertical optical lattice confinement depths detailed in the paper figure.
 ### Fig3: **(A)** ‘fig3a_measurement_resubmission.csv’. Left column are
 measurement dark times in units of second. Middle column are Ramsey fringe
 contrasts. Right column are one standard deviation error bars for Ramsey
 fringe contrasts. ‘fig3a_theory_resubmission’. First column are dark times
 for theoretical simulation in units of second. Second column are
 theoretically calculated Ramsey contrasts using the numerically solved
 many-body Hamiltonian described in the Supplementary materials. Third and
 fourth column are error bands stemming from the uncertainty on the
 temperature and T2. **(B)** ‘fig3b_measurement_resubmission.csv’. Left
 column are measurement dark times in units of second. Middle column are
 Ramsey fringe contrasts. Right column are one standard deviation error
 bars for Ramsey fringe contrasts. ‘fig3b_theory_resubmission’. First
 column are dark times for theoretical simulation in units of second.
 Second column are theoretically calculated Ramsey contrasts using the
 numerically solved many-body Hamiltonian described in the Supplementary
 materials. Third and fourth column are error bands stemming from the
 uncertainty on the temperature and T2. **(C)** ‘fig3c_measurement.csv’.
 Left column are perpendicular trap depths in units of the lattice photon
 recoil energy. Middle column is the measured oscillation frequency in Hz.
 Right column is the one standard deviation frequency error in the
 oscillation frequency. ‘fig3c_2site_model.csv’. Left column are
 perpendicular trap depths in units of the lattice photon recoil energy.
 Right column are the modelled oscillation frequencies as explained in the
 ‘Heuristic averaging of superexchange dynamics’ section of the
 supplemental materials. ‘fig3c_full_sim.csv’. Left column are
 perpendicular trap depths in units of the lattice photon recoil energy.
 Middle column are the fit results obtained from the full theoretical
 simulations used in panel (A) and (B) analysis. Right column are one
 standard deviation error. **(D)** ‘fig3d_measurement.csv’. First column
 are the scaled dark times for Ramsey spectroscopy measurements.
 Sequentially in each column from left to right, we scan the transverse
 confinement and provide the Ramsey fringe contrast and the one standard
 deviation uncertainty in contrast. The top row of each column provides a
 label for additional clarity. ‘fig3d_simulation.csv’ First column are
 rescaled dark times. Second column are modelled contrast decay using the
 theoretical model described in the ‘General averaging’ section of the
 supplemental materials. ### Fig4: **(B)** ‘plot_fig_4b.py’ is the python
 script to plot the data in Fig.4(B). ‘Pulse_0.csv’, ‘Pulse_1.csv’,
 ‘Pulse_2.csv’, ‘Pulse_3.csv’ are the Ramsey spectroscopy data with tunable
 clock pulse areas as described in the caption of Fig. 4. First row are
 dark times in units of seconds. Second row are Ramsey fringe contrasts.
 Third row are one standard deviation uncertainty in contrast. **(D)**
 ‘plot_fig_4d.py’ is the python script to plot the data in Fig.4(D).
 ‘-24.csv’ to ‘+12.csv’ are experimental positions of atoms in units of
 microns. First row are dark times in units of seconds. Second row are
 Ramsey fringe contrasts. Third row are one standard deviation uncertainty
 in contrast. Data are offset in contrast for clarity when plotting.
 **(E)** ‘fig4e_measurement.csv’ Left column are experimental positions of
 atoms in units of microns. Middle column is the measured oscillation
 frequency in Hz. Right column is the one standard deviation frequency
 error in the oscillation frequency. ‘fig4e_simulation.csv’ Left column are
 positions of atoms in units of microns. Middle column are oscillation
 frequencies from heuristic superexchange simulation. Right column are one
 standard deviation in oscillation frequencies. ### S1: **(A)**
 ‘plot_fig_S1A.py’ is the python script to plot the data in Fig.S1(A).
 ‘dark_times_S1_a.csv’ are the dark times in units of seconds. ‘N_S1_a.csv’
 are the atom numbers as a function of dark time. **(B)** ‘plot_fig_S1B.py’
 is the python script to plot the data in Fig.S1(B). ‘dark_times_S1_b.csv’
 are the dark times in units of seconds. ‘Ns_S1_b.csv’ are the atom numbers
 as a function of dark time. ‘Ns_errs_S1_b.csv’ are the one standard
 deviation errors in atom numbers as a function of dark time. ### S2: The
 ‘x’ and ‘y’ data for each panel in Fig. S2 are provided. The last three
 numbers in each measurement (e.g. ‘404010’ ) refer to the transverse and
 vertical trap depths respectively in units of the lattice photon recoil
 energy. As seen in Fig. S2, there is a small anharmonic correction and the
 corrected trap depths are provided in the manuscript. ‘x_data_.csv’ refers
 to the clock laser frequency in Hz.  ‘y_data_.csv’ refers to the
 excitation fraction of the atomic ensemble. ### S3:
 ‘fig_s3_measurement.csv’ Left column are perpendicular trap depths in
 units of the lattice photon recoil energy. Middle column is the measured
 oscillation frequency in Hz. Right column is the one standard deviation
 frequency error in the oscillation frequency. ‘fig_s3_model.csv’ Left
 column are perpendicular trap depths in units of the lattice photon recoil
 energy. Right column are the modelled oscillation frequency in Hz. ### S4:
 ‘plot_fig_s4.py’ is the python script to plot the data in Fig. S4.
 ’14.csv’ is the measurement at 14 px radius. Top row are the dark times in
 units of seconds. Second row are Ramsey fringe contrasts. Third row are
 one standard deviation uncertainty in contrast. ’48.csv’ is the
 measurement at 48 px radius. Top row are the dark times in units of
 seconds. Second row are Ramsey fringe contrasts. Third row are one
 standard deviation uncertainty in contrast. ### S5: In panels a-e, we vary
 the perpendicular trap depth. For a given trap depth (i.e. 19.6 Er), we
 theoretically model Ramsey fringe contrast decay as a function of
 temperature. (e.g. 19.6 Er, 200 nK): ‘datContTempV19d6T200.csv’. Left
 column are the dark times in units of seconds. Right column are Ramsey
 fringe contrasts. These are overlayed with experimental measurements (e.g.
 19.6 Er, 200 nK): ‘datContTempV19d6experiment.csv’. Left column are the
 dark times in units of seconds. Middle column are Ramsey fringe contrasts.
 Right column are Ramsey fringe contrast error bars (one standard
 deviation). ### S6: **(B)** ‘plot_fig_s6.py’ is the python script to plot
 the data in Fig. S6(B). ‘Vz_X.csv’ are coherence time measurements where X
 is the vertical trap depth in units of the lattice photon recoil energy.
 First row is the trap depth. Second row is the coherence time in units of
 seconds. Third row is the coherence time error bar (one standard
 deviation). ‘plot_fig_s6_inset.py’ is the python script to plot the data
 in Fig. S6(B) inset. ‘Vz_NX.csv’ are atom lifetime measurements where X is
 the vertical trap depth in units of the lattice photon recoil energy.
 First row is the trap depth. Second row is the atom lifetime in units of
 seconds. Third row is the atom lifetime error bar (one standard
 deviation). **(C)** ‘plot_fig_S6c_1.py’ is the python script to plot the
 upper left figure in in Fig. S6(C). ‘Vz_perp_X.csv’ are coherence time
 measurements where X is the perpendicular trap depth in units of the
 lattice photon recoil energy. First row is the trap depth. Second row is
 the coherence time in units of seconds. Third row is the coherence time
 error bar (one standard deviation). ‘plot_fig_S6c_2.py’ is the python
 script to plot the upper right figure. ‘Vz_perp_X.csv’ are coherence time
 measurements where X is the perpendicular trap depth in units of the
 lattice photon recoil energy. First row is the trap depth. Second row is
 the coherence time in units of seconds. Third row is the coherence time
 error bar (one standard deviation). ‘plot_fig_S6c_3.py’ is the python
 script to plot the lower left figure. ‘Vz_perp_X.csv’ are atom lifetime
 measurements where X is the perpendicular trap depth in units of the
 lattice photon recoil energy. First row is the trap depth. Second row is
 the atom lifetime in units of seconds. Third row is the atom lifetime
 error bar (one standard deviation). ‘plot_fig_S6c_4.py’ is the python
 script to plot the lower right figure. ‘Vz_perp_X.csv’ are atom lifetime
 measurements where X is the perpendicular trap depth in units of the
 lattice photon recoil energy. First row is the trap depth. Second row is
 the atom lifetime in units of seconds. Third row is the atom lifetime
 error bar (one standard deviation). ## Access information The link to this
 data will be provided in the citations section of our published
 manuscript. All questions may be directed to the paper’s corresponding
 authors. ## Code/Software * Theory code - A Mathematica notebook is
 provided containing sample code used to numerically solve the full
 many-body Hamiltonian in Fig 3. * Data code - Camera images are processed
 using ‘image_processing/absorption imaging.py’ code. We use this
 absorption imaging procedure to extract the atom number and excitation
 fraction values in the provided data.