Dynamics of a system of two-level emitters in the Dicke model taking into account small correlations between them

Abstract

On the basis of our previous study of the quantum system of atoms and the electromagnetic field, a system of non-relativistic stationary and non-interacting small one-electron atoms located in a region much smaller in size than the characteristic wavelength of the field is considered. Atoms are considered in the two-level approximation and interact with the field by dipole-electric interaction. We carefully analyze the state space of the atom and the operators in it in the spin s = 1 / 2 formalism, which leads to the Dicke Hamiltonian in terms of the spin formalism.

Non-equilibrium states of the system are investigated using the reduced description method and described by the occupation numbers of photon states η_αk , the degree of excitation of the atoms η₁ , and the value of correlations between them η₂. The basis of our consideration is the Peletminsky–Yatsenko model, the foundations of which and, in particular, the approach to solving the Cauchy problem, based on the notion of effective initial conditions, are thoroughly discussed.

Since the Dicke model deals with the dynamics of a system of particles with spin, the technique of calculating the average values of products of spin operator proposed by Vax, Larkin, and Pikin is developed to a certain extent. Averages in a state with a statistical operator of a system of atoms with given excitations are considered. The obtained results are maximally simplified in the case of spin s = 1 / 2 .

The impossibility of calculations with a quasi-local statistical operator when describing the state of the system with parameters η_αk , η₁, η₂ is overcome with using a somewhat simplified approach of our previous work, in which, instead of a parameter η₂, the system is described by a small deviation δη₂ = η₂ - η₂₀ of the parameter η₂ from its value η₂₀ when describing the system only by the parameters η_αk, η₁. Developed approaches to calculations with spin operators are used to calculate the quasi-equilibrium statistical operator when choosing of the quantities η_αk, η₁ , and δη₂ as reduced description parameters and the right-hand sides of the time equations for these parameters.