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Atom-light entanglement for precise field sensing in the optical domain

D. Barberena, R. J. Lewis-Swan, J. K. Thompson, and A. M. Rey
Phys. Rev. A 102, 052615 – Published 20 November 2020
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  • INTRODUCTION
  • ATOM-LIGHT QUANTUM SENSOR
  • ENGINEERING THE DISPERSIVE INTERACTION
  • EFFECTS OF DISSIPATION
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Macroscopic arrays of cold atoms trapped in optical cavities can reach the strong atom-light collective coupling regime thanks to the simultaneous interactions of the cavity mode with the atomic ensemble. In a recent work [R. J. Lewis-Swan, D. Barberena, J. A. Muniz, J. R. K. Cline, D. Young, J. K. Thompson, and A. M. Rey, Phys. Rev. Lett. 124, 193602 (2020)] we reported a protocol that takes advantage of the strong and collective atom-light interactions in cavity QED systems for precise electric-field sensing in the optical domain. We showed that it can provide between 10 and 20 dB of metrological gain over the standard quantum limit in current cavity QED experiments operating with long-lived alkaline-earth atoms. Here, we give a more in depth discussion of the protocol using both exact analytical calculations and numerical simulations, and describe the precise conditions under which the predicted enhancement holds after thoroughly accounting for both photon loss and spontaneous emission, natural decoherence mechanisms in current experiments. The analysis presented here not only serves to benchmark the protocol and its utility in cavity QED arrays but also sets the conditions required for its applicability in other experimental platforms such as arrays of trapped ions.

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    • Received 8 June 2020
    • Accepted 6 October 2020

    DOI:https://doi.org/10.1103/PhysRevA.102.052615

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Atoms, ions, & molecules in cavitiesCavity quantum electrodynamicsCollective effects in quantum opticsQuantum metrologyQuantum sensing
    1. Physical Systems
    Quantum cavities
    1. Techniques
    Tavis-Cummings model
    Atomic, Molecular & Optical

    Authors & Affiliations

    D. Barberena1,2,*, R. J. Lewis-Swan1,2, J. K. Thompson1, and A. M. Rey1,2

    • 1Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
    • 2Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

    • *diego.barberena@colorado.edu

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    Vol. 102, Iss. 5 — November 2020

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    Images

    • Figure 1
      Figure 1

      Wigner function of an analogous bosonic generalized cat state |ψB〉∝∑mz=−N/2N/2cmz|αe−iωmzt〉 with ωmz=χmz at various evolution times t. Sensitivity to displacements increases from panel (a) to panel (c) as it relates to the smallest scale structure observed in the Wigner function.

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    • Figure 2
      Figure 2

      Preparation of the generalized cat state |ψcatSB〉 and interferometric protocol. (i) The cavity is injected with a coherent field α and the collective spin is fully polarized along x̂ (blue circles). (ii) Fluctuations in the spin projection combined with the dispersive interaction drive a rotation of the initial bosonic coherent state into a superposition at angles θm∼χmzτ. Conversely, the large cavity occupation rotates the collective Bloch vector by ϕ1∼χ|α|2τ about ẑ. (iii) The cavity field is coherently displaced by β (red circles). The spin degree of freedom is unaffected. (iv) By reversing the sign of the dispersive interaction the initial rotations are undone. If β≠0 the final cavity state (red circles) does not return to the original coherent state. Similarly, the collective spin rotates back under the evolution by ϕ2∼−χ|αe−iχSzτ+β|τ about ẑ, leading to an overall rotation ϕtot=ϕ1+ϕ2∼−2χαβτcos(χSzτ) relative to the initial state along x̂.

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    • Figure 3
      Figure 3

      (a) Dependence of the measurement observable on β for fixed χτ=0.1: 〈Ŝy〉(2τ) (red, oscillatory) and 〈X̂〉(2τ) (blue, linear). Shaded regions indicate rms fluctuations due to quantum noise, i.e., 〈[ΔŜy(2τ)]2〉 and 〈[ΔX̂(2τ)]2〉. The period of oscillations in 〈Ŝy〉(2τ) is enhanced by the amplitude α, allowing a more precise inference of β. (b) Comparison of attainable metrological gain relative to the SQL as a function of interaction time τ for N=51 and α=15. The short-time approximation (upper line, solid blue) is given by Eq. (14), while the exact result (lower solid red line) is given by Eq. (13).

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    • Figure 4
      Figure 4

      Evolution of 〈Ŝz〉 predicted by the complete Tavis-Cummings model (solid blue), Eq. (17), for the initial state |N/2z〉|α〉 with N=40 and α=40. The decay envelope of the oscillations is compared to that predicted by the effective dispersive interaction Eq. (27) (dashed black). In the inset we show that the frequency of Rabi oscillations is also captured correctly when including the small correction in the first line of Eq. (27) (red dots). Though not noticeable, the discrepancy between the exact evolution and the approximation is about 4%, which is consistent with N/α2=0.025.

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    • Figure 5
      Figure 5

      Cràmer-Rao bound, 1/FQ, on metrological gain with respect to the SQL, independently calculated using the Tavis-Cummings model (solid red) and effective dispersive interaction Eq. (27) (dashed black). Calculations are for the initial state |N/2z〉|α〉 with N=40 and α=40.

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    • Figure 6
      Figure 6

      (a) Evolution of QFI with interaction time. Numerical evaluation of FQ using Eq. (47) (dashed black) is compared to approximate analytic expressions Eq. (49) (lower solid, blue) and Eq. (50) (upper solid, red) for N=1000, α=100N, and χαN/κ=73≫1. (b) Optimal QFI as a function of χαN/κ for N=1000 and α=100N. We compare the numerical optimization of Eq. (47) (dashed black) to the approximate analytic expression Eq. (51) (solid red). The gray horizontal line indicates the optimal QFI for κ=0, which is attained for very large values of χαN/κ. The vertical lines mark the region delimited by Eq. (52), where our results for (FQ)opt are expected to work.

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    • Figure 7
      Figure 7

      Metrological gain as a function of interaction time τ (in μs) when photon leakage from the cavity is accounted for: χα=g=2π×11kHz and N=106. We compare two cavity decay rates: κ/2π=15kHz (dashed blue) and κ/2π=150kHz (solid orange).

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    • Figure 8
      Figure 8

      Comparison of metrological gain using the time-reversal protocol and M̂=Ŝy (dashed dark blue) to that predicted from the Cràmer-Rao bound (solid red). Calculations are for N=106, χα=g=2π×11kHz, and κ/2π=150kHz.

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    • Figure 9
      Figure 9

      Optimal sensitivity as a ratio of the characteristic interaction scale and cavity decoherence rate, χNα/κ for different N. The inset shows (δβ)opt2 as a function of χα/κ, emphasizing that, overall, larger N is better. When χαN/κ≳1, there is enhanced sensitivity which scales like N−1/3. Conversely the protocol does not beat the SQL for χαN/κ≲1.

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    • Figure 10
      Figure 10

      Sensitivity as a function of τ2 for χα=g=2π×11kHz, N=106, and κ/2π=150kHz and fixed τ1=85 ns. Note that the best gain is obtained for τ2 slightly larger than τ1.

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    • Figure 11
      Figure 11

      Sensitivity as a function of time using the resonant protocol for χα=g=2π×11kHz, γ/2π=7.5kHz, and N=102 (lower, dot-dashed red), 104 (middle, dashed blue), and 106 (upper, solid orange). Note that the optimum occurs always at the same time.

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    • Figure 12
      Figure 12

      Robustness to detection noise as a function of measurement basis M̂=Ŝφ=cos(φ)Ŝx+sin(φ)Ŝy. Detection noise of σdet=N/4 is included for the case with spontaneous emission [γ/(2π)=7.5 kHz, upper solid line, in blue], and with photon loss [κ/(2π)=150 kHz, lower solid line, in red], leading to an optimal sensitivity for φ=π/2. For comparison we also plot the relevant results for σdet=0 (dashed lines). The inset shows scaling of sensitivity with σdet for φ=π/2. Dashed lines in this case represent the σdet=0 result. All calculations are for N=106, t=85ns, and other parameters used in Refs. [19, 56].

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