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This work studies the cooperative Lamb shift in dense ensembles of point quasi-stationary atoms through a microscopic two-atom model. The objective of the work is to derive the finite interatomic part of the two atom reradiation matrix and use it to calculate distance-dependent cooperative frequency shifts. The tasks of the work are to formulate the atom-field Hamiltonian, specify the atomic level scheme and transition dipoles, include the one-photon virtual sector needed in the perturbative derivation, identify the ultraviolet problem of the off diagonal Green-matrix elements responsible for the cooperative Lamb shift, 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 off-diagonal interaction integral, parameter differentiation, and numerical diagonalization of the 6×6 reradiation matrix. The main result is a consistent finite working formula for the off-diagonal interaction kernel and numerical curves showing how cooperative Lamb-shift branches depend on interatomic separation. MATLAB was used for numerical matrix construction, eigenvalue calculation, branch classification, and plotting.

• Contents
• LIST OF ABBREVIATIONS
• INTRODUCTION
• CHAPTER 1. LITERATURE REVIEW
  • 1.1. Dense ensembles and collective shifts
  • 1.2. Cooperative Lamb-shift approaches and limitations
  • 1.3. Relevance, objective, and tasks
  • 1.4. Summary of Chapter 1
• CHAPTER 2. MICROSCOPIC MODEL AND ULTRAVIOLET PROBLEM
  • 2.1. Atomic-level scheme and microscopic formulation
  • 2.2. Off-diagonal cutoff-sensitive kernel
  • 2.3. Method 1: parameter differentiation in the electric-dipole gauge
  • 2.4. Finite kernel and extraction rule for the cooperative shift
  • 2.5. Summary of Chapter 2
• CHAPTER 3. DIRECT CALCULATIONS IN DIFFERENT GAUGES AND CUTOFF SENSITIVITY
  • 3.1. Method 2: direct electric-dipole-gauge calculation and cutoff sensitivity
  • 3.2. Method 3: direct calculation in the p A gauge
  • 3.3. Comparison of the three derivation routes
  • 3.4. Summary of Chapter 3
• CHAPTER 4. COOPERATIVE LAMB SHIFT IN THE TWO-ATOM SYSTEM
  • 4.1. Axial two-atom geometry and projector reduction
  • 4.2. Symmetric and antisymmetric collective eigenvalues
  • 4.3. Direct 6×6 matrix diagonalization and arbitrary two-atom coordinates
  • 4.4. Summary of Chapter 4
• CONCLUSION
• REFERENCES
• APPENDIX
  • A.1. Method 1: parameter-differentiation tensor block
  • A.2. Method 2: direct principal-value quadrature
  • A.3. Method 3: p A-gauge Coulomb and transverse parts
  • A.4. Axial two-atom shift calculation
  • A.5. Arbitrary-coordinate shift calculation