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Integrability, multifractality, and two-photon dynamics in disordered Tavis-Cummings models

Agnieszka Wierzchucka, Francesco Piazza, and Pieter W. Claeys
Phys. Rev. A 109, 033716 – Published 19 March 2024
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  • INTRODUCTION
  • THE HOMOGENEOUS TAVIS-CUMMINGS…
  • DISORDERED COUPLINGS
  • RELAXATION TIMES FOR PERMUTATION…
  • DISORDERED BARE ENERGIES
  • SEMILOCALIZATION AND MULTIFRACTALITY
  • DYNAMICS AND PHOTON BUNCHING
  • CONCLUSION AND DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The Tavis-Cummings model is a paradigmatic central-mode model in which a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties, and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity, and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we establish here the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and we study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates, and it illustrates how these models support a collective spin description despite the presence of disorder.

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    • Received 16 January 2024
    • Accepted 4 March 2024

    DOI:https://doi.org/10.1103/PhysRevA.109.033716

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Cavity quantum electrodynamicsQuantum optics
    1. Physical Systems
    Disordered systemsQuantum cavities
    1. Techniques
    Integrable systemsTavis-Cummings model
    Atomic, Molecular & Optical

    Authors & Affiliations

    Agnieszka Wierzchucka1,2,*, Francesco Piazza1,3, and Pieter W. Claeys1

    • 1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
    • 2Merton College, University of Oxford, Oxford OX1 4JD, United Kingdom
    • 3Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

    • *agnieszka.wierzchucka@merton.ox.ac.uk

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    Issue

    Vol. 109, Iss. 3 — March 2024

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    • Figure 1
      Figure 1

      Eigenspectrum of the clean and disordered Tavis-Cummings Hamiltonian for two excitations. In the absence of disorder, three classes of eigenstates can be observed resulting in dark states (singlet states) and polaritons (doublet and triplet states). For disordered systems (gray lines) the dark states and the doublet polaritons split into bands of states, while the triplet polaritons remain isolated. Parameters: N=40, εi∈[−0.1,0.1], and γi∈[1,2] uniformly distributed for the disordered model, and ε=εi¯ and γ2=γi2¯ for the homogeneous model (a.u.).

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    • Figure 2
      Figure 2

      Graphical illustration of the secular equation. Each intersection between the left-hand side (red line) and the right-hand side (blue line) returns the eigenvalue of a polariton state, leading to two classes of solutions: doublet polaritons (circles) and triplet polaritons (squares). Parameters: Δ=1, ε=0, N=4, and gi2=1,⋯,4 (a.u.).

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    • Figure 3
      Figure 3

      Scaling of the IPR for the three different classes of eigenstates in the different subsection of the Hilbert space for q=0.125. Gray dashed lines indicate best fits ∝Nα. Parameters: Δ=1, εi uniformly distributed in [−0.1,0.1], and gi uniformly distributed in [1,3]/N (a.u.).

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    • Figure 4
      Figure 4

      Illustration of the one-photon wave-function components |ψi|2q for representative polariton (triplet and doublet) and dark (singlet) states. For homogeneous couplings, these components are a smooth function of the corresponding bare energy εi, with different states characterized by the number of poles where the amplitudes scale as |ψi|2q∝1/(εi−Eα)2q. In the presence of disorder in the couplings, this overall structure is preserved and in turn reflected in the IPR. Parameters: N=100, Δ=1, εi uniformly distributed in [−0.1,0.1], and gi uniformly distributed in [1,2]/N (a.u.).

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    • Figure 5
      Figure 5

      Dynamics of the probability of observing n photons in the cavity for an initial state with two photons (a) and an initial state with one photon and a single cavity excitation (b). Parameters: N=40, gi uniformly distributed in [1,3]/N, and εi uniformly distributed in [−0.2,0.2] (a.u.).

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    • Figure 6
      Figure 6

      Time-averaged probability of observing two photons in the cavity for an initial state with n photons. Parameters: Δ=1, gi equally spaced in [1,3]/N, and εi uniformly distributed in [−0.2,0.2] (a.u.).

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    • Figure 7
      Figure 7

      Graphical illustration of the secular equation in the case in which a pair of poles vanishes. Each intersection between the left-hand side (red line) and the right-hand side (blue line) returns the eigenvalue of a polariton state, leading to different classes of solutions. Parameters: Δ=1, ε=0, N=5, and gi2∈[2,4,6,8,32] (a.u.).

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    • Figure 8
      Figure 8

      Scaling of the IPR for the three different classes of eigenstates in the different subsection of the Hilbert space for q=0.125 and an integrable model with disordered bare energies. Gray dashed lines indicate best fits ∝Nα. Parameters: Δ=1, g=2/N, and εi uniformly distributed in [−0.1,0.1] (a.u.).

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    • Figure 9
      Figure 9

      Scaling of the IPR for the polaritons in the different subsections of the Hilbert space for q=0.125 and an integrable model with disordered interaction strengths. Gray dashed lines indicate best fits ∝Nα. Parameters: Δ=1, εi=0, and gi uniformly distributed in [1,3]/N (a.u.).

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