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arXiv:2008.11056 · Journal of Physics A Mathematical and Theoretical

The atomic damping basis and the collective decay of interacting two-level atoms

William Alvarez-Giron, Pablo Barberis-Blostein

Published Aug 24, 2020• 4 citations
arXivHTMLPDF

Abstract

Abstract We find analytical solutions to the evolution of interacting two-level atoms when the master equation is symmetric under the permutation of atomic labels. The master equation includes atomic independent dissipation. The method to obtain the solutions is: first, we use the system symmetries to describe the evolution in an operator space whose dimension grows polynomially with the number of atoms. Second, we expand the solutions in a basis composed of eigenvectors of the dissipative part of the master equation that models the independent dissipation of the atoms. This atomic damping basis is an atomic analog to the damping basis used for bosonic fields Briegel and Englert (1993 Phys. Rev. A 47 3311–29). The solutions show that the system decays as a sum of sub- and super-radiant exponential terms.

DOI
10.1088/1751-8121/abb1e2

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Abstract We find analytical solutions to the evolution of interacting two-level atoms when the master equation is symmetric under the permutation of atomic labels. The master equation includes atomic independent dissipation. The method to obtain the solutions is: first, we use the system symmetries to describe the evolution in an operator space whose dimension grows polynomially with the number of atoms. Second, we expand the solutions in a basis composed of eigenvectors of the dissipative part of the master equation that models the independent dissipation of the atoms. This atomic damping basis is an atomic analog to the damping basis used for bosonic fields Briegel and Englert (1993 Phys. Rev. A 47 3311–29). The solutions show that the system decays as a sum of sub- and super-radiant exponential terms.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.
We derive a generalized master equation on the basis of system-reservoir theory where the system is a two-level atom interacting with a superintense classical electromagnetic field within the Born and Markov approximations. We show that the usual assumption of independence of the atom-field coupling and the atomic decay processes breaks down and the relaxation becomes field dependent in the superintense regime. This results in significant deviation from the exponential decay of the Bloch vector components.
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A dissipating system consisting of two strongly interacting subsystems, each coupled to its respective thermal reservoir is considered. The usual method for obtaining a master equation for the reduced density operator of such a system is to assume the factorization of this density operator in the derivation of the irreversible term, retaining the coupling in the reversible term alone. This approach has the serious limitation that it fails to preserve detailed balance in general and yields an incorrect canonical form for the stationary solution of the density operator. A general approach, based on the introduction of a new interaction picture, is presented. This method is illustrated by applying it to a coupled boson system and to a two-level atom interacting with a single mode of the electromagnetic field.
An approximate solution is given for the interaction of a set of AAA two-level atoms with a single mode of a quantized radiation field when the field is damped by a reservoir at zero temperature. Assuming that the average photon number is much larger than the number of atoms, we study the influence of dissipation on the approximate factorization of the density matrix. For realistic values of the parameters, the factorization survives under the presence of field damping. The main effect of dissipation is an exponential decay of the average photon number and a loss of coherence among the different factorized states. The influence of field dissipation on the atomic inversion, purity, and field squeezing are studied.