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Impacts of random filling on spin squeezing via Rydberg dressing in optical clocks

Jacques Van Damme, Xin Zheng, Mark Saffman, Maxim G. Vavilov, and Shimon Kolkowitz
Phys. Rev. A 103, 023106 – Published 11 February 2021
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  • INTRODUCTION
  • SPIN SQUEEZING VIA RYDBERG DRESSING
  • ONE-DIMENSIONAL LATTICE SQUEEZING
  • LATTICE CLOCK STABILITY COMPARISON
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We analyze spin squeezing via Rydberg dressing in optical lattice clocks with random fractional filling. We compare the achievable clock stability in different lattice geometries, including unity-filled tweezer clock arrays and fractionally filled lattice clocks with varying dimensionality. We provide practical considerations and useful tools in the form of approximate analytical expressions and fitting functions to aid in the experimental implementation of Rydberg-dressed spin squeezing. We demonstrate that spin squeezing via Rydberg dressing in one-, two-, and three-dimensional optical lattices can provide significant improvements in stability in the presence of random fractional filling.

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    • Received 9 October 2020
    • Accepted 21 January 2021

    DOI:https://doi.org/10.1103/PhysRevA.103.023106

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Atomic, optical & lattice clocksSqueezing of quantum noise
    1. Physical Systems
    Rydberg atoms & molecules
    Atomic, Molecular & OpticalQuantum Information, Science & Technology

    Authors & Affiliations

    Jacques Van Damme, Xin Zheng, Mark Saffman, Maxim G. Vavilov, and Shimon Kolkowitz*

    • Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

    • *kolkowitz@wisc.edu

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    Vol. 103, Iss. 2 — February 2021

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    Images

    • Figure 1
      Figure 1

      (a) Strontium energy level diagram. The clock laser drives the transition between |g〉 and |e〉 with Rabi frequency Ωc. The squeezing laser drives the Rydberg transition between |e〉 and |r〉 far off-resonance with Rabi frequency Ωr. (b) Spin-echo sequence of rotation (with the clock transition) and squeezing intervals (with the Rydberg transition) to perform one-axis squeezing of the coherent spin state |g〉⊗N=|−1/2〉⊗N. (c) Bottom view of the Bloch sphere illustrating the squeezing of the binomial distribution uncertainty of the coherent spin state towards reduced uncertainty ΔĴ⊥ along the perpendicular direction at angle θmin. [(d) and (e)] Fully or partially filled one-dimensional lattice with the dressed Rydberg interaction potential given by Eq. (1) for the central atom (solid line) and the corresponding Heaviside approximation (dashed line).

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    • Figure 2
      Figure 2

      (a) The minimal squeezing parameter ξmin2 averaged over 200 randomly filled one-dimensional lattice configurations (M=103) at each lattice site filling probability Pfilling∈{0.1,0.2,⋯,1} with Rydberg interaction radius Rc/a=9. The error bars indicate the minimal and maximal value from the simulations. (b) Corresponding scatter plot of minimal squeezing ξmin2 plotted versus the actual filling fraction N/1000 loaded into the lattice. The additional solid lines correspond to fully filled lattices with their lattice constant scaled to the average gap size of a randomly filled lattice using the exact interaction potential (green), and with the approximated Heaviside interaction potential (light blue). The dashed lines represent special case limits for fractionally filled lattices: the case where the interaction radius is infinite (gold) and the case when there are no empty sites between the atoms (purple).

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    • Figure 3
      Figure 3

      (a) The angle of minimal uncertainty θmin as illustrated in Fig. 1 plotted for one-dimensional lattices of M=1000 sites, randomly filled with Pfilling=50%. The red line fits to the average 〈θmin〉 of the random lattice configurations as a function of the Rydberg interaction radius Rc/a. (b) The squeezing ξ2 obtained in these lattices, where the filled circle data points are the results of simulations with τopt and θmin optimized for each individual random lattice configuration, while the cross data point simulations used a fixed τ and θ derived from the fitting functions for all the random lattice configurations.

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    • Figure 4
      Figure 4

      Simulation results of clock frequency stability as a function of the number of atoms in the lattice. This plot compares the performance of clocks with fully filled tweezer arrays or partially filled lattices (Pfilling=50%) of different dimensionality (1D, 2D, 3D). The dashed fitting function is derived in Appendix pp3 and predicts the performance of lattice sizes that are no longer practical to fully simulate. The dotted vertical black line illustrates the rough bound on the number of tweezer traps imposed by current experimentally practical laser powers.

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    • Figure 5
      Figure 5

      The squeezing parameter ξmin2 and the figure of merit τ̃/τopt as a function of detuning |Δ|/Ωr from the Rydberg transition in a one-dimensional fully filled lattice of M=1000 sites with Rydberg interaction radius Rc/a=9.

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    • Figure 6
      Figure 6

      The squeezing parameter ξmin2 in a one-dimensional fully filled lattice calculated with the solution Eq. (4). We plot the impact of using the exact Rydberg interaction potential Vij of Eq. (B1) with different Förster defects δ∈{10MHz,100MHz,1.26GHz} to the van der Waals approximated potential Eq. (1) and the Heaviside solution.

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    • Figure 7
      Figure 7

      The optimal squeezing time τopt in a one-dimensional fully filled lattice calculated with the solution Eq. (4). We plot the impact of using the exact Rydberg interaction potential Vij of Eq. (B1) with different Förster defects δ∈{10MHz,100MHz,1.26GHz} to the van der Waals approximated potential Eq. (1) and the Heaviside solution.

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    • Figure 8
      Figure 8

      Förster defect of the S13 series in the bosonic isotopes of Sr with nuclear spin zero. Calculations assume coupling to P0,1,23 (blue, yellow, green dots) and a spin weighted average (red dots).

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    • Figure 9
      Figure 9

      The squeezing parameter ξmin2 as a function of the interaction radius Rc/a in a fully filled one-dimensional lattice of size M.

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    • Figure 10
      Figure 10

      (a) The optimal squeezing time τopt for 100 randomly filled one-dimensional lattice configurations (M=103) at lattice site filling probability Pfilling=0.5 with Rydberg interaction radius Rc/a=9. We compare the actual interaction potential with the Heaviside approximation. (b) Corresponding scatter plot of minimal squeezing ξmin2 plotted versus the actual filling fraction N/1000 loaded into the lattice.

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    • Figure 11
      Figure 11

      The squeezing parameter ξmin2 in random fractionally filled one-dimensional lattices of size M=103 simulated for different Rydberg interaction radii Rc together with the corresponding fitting function from Eq. (C1).

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    • Figure 12
      Figure 12

      The optimal squeezing time τopt in random fractionally filled one-dimensional lattices of size M=103 simulated for different Rydberg interaction radii Rc together with the corresponding fitting function from Eq. (C3).

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