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Analytical and numerical study of subradiance-only collective decay from atomic ensembles

Anirudh Yadav and D. D. Yavuz
Phys. Rev. A 110, 023709 – Published 5 August 2024
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  • INTRODUCTION
  • EXCHANGE HAMILTONIAN: THE SCHRÖDINGER…
  • ANALYTICAL STUDY OF INITIAL-TIME…
  • CONCLUSIONS AND DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We analytically and numerically study collective emission from an ensemble of atoms in the weak-excitation regime with the initial condition of uniform excitation of the ensemble. We show that under certain conditions, subradiance at the later stages of the collective decay does not necessarily need to be accompanied by early-time superradiance. We analyze the conditions where such subradiance-only decay occurs for ordered and disordered ensembles in one-, two-, and three-dimensional spatial configurations.

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    • Received 15 March 2024
    • Revised 13 June 2024
    • Accepted 8 July 2024

    DOI:https://doi.org/10.1103/PhysRevA.110.023709

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Collective effects in quantum opticsLight-matter interactionLong-range interactionsQuantum description of light-matter interactionSpontaneous emission
    Atomic, Molecular & OpticalQuantum Information, Science & Technology

    Authors & Affiliations

    Anirudh Yadav and D. D. Yavuz

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    Vol. 110, Iss. 2 — August 2024

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    Images

    • Figure 1
      Figure 1

      Four spatial configurations for the atomic sample that we consider in this paper. The plots on the top left and right show ordered one-dimensional and two-dimensional arrays with fixed spacing s between the atoms. The plots on the bottom left and right show disordered two-dimensional and three-dimensional ensembles; the locations of the atoms are uniform random variables along each axis (see text for details).

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    • Figure 2
      Figure 2

      A visual cartoon schematic for the condition of observing subradiance-only decay. The solid blue line in both plots is the time evolution of Φ(t). The dashed green line (again in both plots) is the time derivative Φ′(0), whose slope is the initial collective decay rate, while the dotted red line is the uncorrelated rate. The plot on the left shows the case where Φ′(0) is lower than the uncorrelated decay rate of −mΓ/2, hence showing early-time superradiance. The plot on the right shows subradiance-only decay since the initial tangent Φ′(0) is greater than −mΓ/2.

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    • Figure 3
      Figure 3

      Plot of I1 vs s/λa. In the regions where I1(s/λa) is negative, the condition of Eq. (17) is satisfied, and the decay is subradiance only. This happens between the roots of the function, I1(s/λa)=0 and when s/λa is an integer.

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    • Figure 4
      Figure 4

      Four different regions that are seen by the atom at a position of (ux,uy). The calculation of the quantity I2′(s/λa) requires a summation over all the atoms in each of these regions, A,B,C, and D. In the limit of a large number of atoms, N→∞, regions A,B,C, and D become identical (see text for details).

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    • Figure 5
      Figure 5

      Plot of I2 as a function of s/λa. In the regions where I2(s/λa) is negative, the decay is subradiance only. The plot also illustrates the positions of its singularities. The function is defined everywhere except the points in the set {u2+v2:(u,v)∈Z2} marked by dashed radial lines.

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    • Figure 6
      Figure 6

      Graph of I2R/(N2−N) versus L for d=50 nm and λa=780 nm for a 2D disordered ensemble of atoms. The dependence of I2R on L changes sign after roughly 5λa and remains negative for the rest of the domain. This suggests that the system will experience early-time superradiance decay until a domain size of ∼5λa. For larger domain sizes, decay will be subradiance only. We pick four specific points, which are indicated by vertical lines, and verify the collective decay behavior through direct numerical simulation in Fig. 7.

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    • Figure 7
      Figure 7

      The numerically calculated expectation value of Φ over 1000 iterations for 25 atoms for four specific cases: L=2λa,3λa,6.1λa, and 7.1λa. These four selected cases are shown by the vertical lines in Fig. 6. Since the uncorrelated decay rate of ln∥ϕ(t)∥ is −Γ/2, it becomes the dashed zero line in this plot. All the other decay processes can be understood as deviations from this line, as illustrated in Fig. 2. In agreement with the predictions of Fig. 6, L=2λa and L=3λa display early-time superradiance, while L=6.1λa and L=7.1λa show subradiance-only decay.

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    • Figure 8
      Figure 8

      Similar to Fig. 7, the numerically calculated expectation value of Φ over 100 iterations for 250 atoms for the same four cases of L=2λa,3λa,6.1λa, and 7.1λa. The results are again in agreement with the predictions of Fig. 6.

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    • Figure 9
      Figure 9

      Graph of I3R/(N2−N) versus L for d=50 nm and λa=780 nm for a 3D disordered ensemble of atoms. The function shows oscillatory behavior, and in the regions where I3R/(N2−N)<0, the decay is subradiance only. We pick four specific points, which are indicated by vertical lines, and verify the collective decay behavior through direct numerical simulation in Figs. 10 and 11.

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    • Figure 10
      Figure 10

      The numerically calculated expectation value of Φ over 1000 iterations for 25 atoms for four specific cases that are shown by the vertical lines in Fig. 9. In agreement with the predictions of Fig. 8, L=2.5λa and L=3.5λa display early-time superradiance, while L=6λa and L=7λa show subradiance-only decay.

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    • Figure 11
      Figure 11

      Similar to Fig. 10, the numerically calculated expectation value of Φ over 100 iterations for 250 atoms for the same four cases that are shown by the vertical lines in Fig. 9. The results are again in agreement with the predictions of Fig. 9.

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