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This work analyzes lifetimes of collective states in dense ensembles of point quasi-stationary atoms using the finite complex spectrum of a two-atom reradiation matrix. The objective of the work is to derive the finite interatomic part of the two-atom reradiation matrix and use its complex spectrum to calculate collective decay rates and lifetimes. The tasks of the work are to formulate the microscopic atom-field model, specify the atomic level scheme and transition dipoles, include the virtual one-photon sector needed in the perturbative derivation, identify the ultraviolet problem of the off-diagonal Green-matrix elements, compare three derivation routes, and diagonalize the two-atom matrix for axial and arbitrary-coordinate geometries. The study uses analytical derivation in the electric-dipole and p·A gauges, direct principal-value analysis of the interatomic Green-matrix elements, parameter differentiation, and numerical diagonalization of the 6×6 reradiation matrix. The main result is a finite complex interaction kernel and numerical curves showing how superradiant and subradiant lifetime branches depend on interatomic separation. MATLAB was used for numerical matrix construction, eigenvalue calculation, lifetime extraction, branch classification, and plotting.

• Contents
• LIST OF ABBREVIATIONS
• INTRODUCTION
• CHAPTER 1. LITERATURE REVIEW
  • 1.1. Cooperative decay and long-lived states
  • 1.2. Lifetime approaches and limitations
  • 1.3. Relevance, objective, and tasks
  • 1.4. Summary of Chapter 1
• CHAPTER 2. MICROSCOPIC MODEL AND FINITE COMPLEX KERNEL
  • 2.1. Atomic-level scheme and lifetime formulation
  • 2.2. Off-diagonal cutoff-sensitive complex kernel
  • 2.3. Method 1: parameter differentiation and finite complex kernel
  • 2.4. Extraction rule for decay rates and lifetimes
  • 2.5. Summary of Chapter 2
• CHAPTER 3. GAUGE COMPARISON AND CUTOFF SENSITIVITY BEFORE LIFETIME EXTRACTION
  • 3.1. Method 2: direct electric-dipole-gauge integration and cutoff sensitivity
  • 3.2. Method 3: p A gauge with the electrostatic dipole-dipole term
  • 3.3. Comparison of the three routes before lifetime extraction
  • 3.4. Summary of Chapter 3
• CHAPTER 4. LIFETIMES OF COLLECTIVE STATES IN THE TWO-ATOM SYSTEM
  • 4.1. Axial two-atom geometry and collective branches
  • 4.2. Eigenvalues, decay rates, and lifetimes
  • 4.3. Direct 6×6 diagonalization for arbitrary two-atom coordinates
  • 4.4. Summary of Chapter 4
• CONCLUSION
• REFERENCES
• APPENDIX
  • A.1. Method 1: differentiated finite kernel
  • A.2. Method 2: direct electric-dipole-gauge integration
  • A.3. Method 3: p A-gauge decomposition
  • A.4. Axial two-atom lifetime calculation
  • A.5. Arbitrary-coordinate lifetime calculation