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Theoretical and Mathematical Physics

Green's functions in the dicke model. I. Evolution equation

N. N. Bogolyubov, A. R. Kazaryan, A. M. Kurbatov, V. N. Neskoromnyi

Published Jan 1, 1983• 0 citations
Publisher
DOI
10.1007/bf01017129

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•Oct 13, 2011•22 citations
By using the thermodynamic Green's function method, the authors study the phase transition of a collection of two- as well as three-level atoms interacting with the electromagnetic field and contained within a volume much smaller than the smallest resonance wavelength (Dicke model). They show for the case of three-level atoms the existence of two critical temperatures where a second-order phase transition takes place. The upper critical temperature is determined by the largest of the three coupling constants in the three-level model, whereas the lower critical temperature is a function of the remaining two coupling constants. It is also shown that for a collection of two- as well as three-level atoms, the critical temperature depends upon the number of atoms contained within lambda r 3 ( lambda r is a resonance wavelength) rather than the density as has previously been suggested. Also, using the Green's function method, we show the formal equivalence of the Dicke model in thermodynamic equilibrium with the BCS model of superconductivity.
We study the dynamical properties of the Dicke model for increasing spin length, as the system approaches the limit of a classical spin. First, we describe the emergence of collective excitations above the ground state that converge to the coupled spin-oscillator oscillations found in the classical limit. The corresponding Green functions reveal quantum dynamical signatures close to the superradiant quantum phase transition. Second, we identify signatures of classical quasiperiodic orbits in the quantum time evolution using numerical time propagation of the wave function. The resulting phase-space plots are compared to the classical trajectories. We complete our study with the analysis of individual eigenstates close to the quasiperiodic orbits.
Introduction. 1: Evolution of States of Quantum Systems of Finitely Many Particles. 1. Principal Concepts of Quantum Mechanics. 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles. 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation. Mathematical Supplement I. References. 2: Evolution of States of Infinite Quantum Systems. 4. Bogolyubov Equations for Statistical Operators. 5. Solution of the Bogolyubov Equations. 6. Gibbs Distributions. Mathematical Supplement II. Mathematical Supplement III. References. 3: Thermodynamic Limit. 7. Thermodynamic Limit for Statistical Operators. 8. Statistical Operators in the Case of Quantum Statistics. 9. Bogolyubov's Principle of Weakening of Correlations. Mathematical Supplement IV. References. 4: Mathematical Problems in the Theory of Superconductivity. 10. Frohlich Model. 11. Bogolyubov's Compensation Principle for 'Dangerous' Diagrams. Compensation Equations. 12. Bardeen--Cooper--Schrieffer (BCS) Hamiltonian. 13. Microscopic Theory of Superfluidity. Mathematical Supplement V. References. 6: Green's Functions. 14. Green's Functions. Equations for Green's Functions. 15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity. 16. Green's Functions in the Thermodynamic Limit. References. 6: Exactly Solvable Models. 17. Description of the Hamiltonians of Model Systems. 18. Functional Spaces of Translation--Invariant Function. 19. ModelHamiltonians in the Spaces of Translation Invariant Functions. 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs. 21. Equations for Green's Functions and their Solutions. Mathematical Supplement VI. References. 7: Quasiaverages. Theorem on Singularities of Green's Functions of 1/q2-Type. 22. Quasiaverages. 23. Green's Functions and their Spectral Representations. 24. Theorem on Singularities of Green's Functions of 1/q2-Type. References. Subject Index.