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Collective Density Matrix of Identical Multi-level Atoms and Application in Superradiance

Yuan Zhang

Published Sep 10, 2019• 0 citations
OpenAlex

Abstract

We present a collective description of density matrix for identical multi-level atoms, which are either excited initially or driven coherently or pumped incoherently. We consider density matrix as expectation value of projection or transition operators in a basis of atom's product states and then identify identical matrix elements with several integers, which specify uniquely the involved transition or projection operators. To remove the redundancy of these identical elements, we treat them as single quantity and derive the equation for this quantity by mapping the transition or projection operators to a single vector specified with the integers in Liouville space and work out the actions of the master equation constitutions (as superoperators) on these vectors. As a result, the number of necessary elements increases polynomially with the number of atoms rather than exponentially as expected if the redundancy is not removed. This numerical advantage is explored to study superradiance from hundreds of two-level atoms. Our simulation identifies the conditions of observing the superradiance with and without dipole involved.

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Collective Description of Density Matrix of Identical Multi-level Atoms for Superradiance

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A collective description of density matrix is presented for identical multi-level atoms, which are either excited initially, driven coherently or pumped incoherently. The density matrix is defined as expectation value of projection or transition operators in a basis of atom's product states. The identical matrix elements are identified with several integers, which specify uniquely the involved operators. To remove the redundancy, these identical elements are treated as single quantity and the equation for this quantity is dervied by mapping the transition or projection operators to a single vector specified with these integers. As a result, the number of computed elements increases polynomially rather than exponentially with the number of atoms. As an example, we carry out exact simulation of hundreds of two-level atoms and demonstrate the different conditions for observing superradiance and superfluorescence.

arXiv:1909.04792•Sep 10, 2019•0 citations

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Exact solution of the superradiance master equation. I. Complete initial excitation

Ching Tsung Lee

The cooperative spontaneous emission from a completely excited system of $N$ identical two-level atoms is studied under the condition that the cooperation number is conserved. The master equation for such a system has been solved analytically. The exact expressions for the probability distribution and the expectation values of various observable physical quantities are presented. Through a power-series expansion of the exact photon-number expectation value and by using properties of Stirling numbers of the second kind, it is shown that our result agrees with that of Haake and Glauber asymptotically. As by-products, some new identities involving binomial coefficients and Stirling numbers of the second kind have also been obtained. The exact solution of the same master equation for arbitrary initial conditions will be presented in a subsequent publication.

Physical Review A•May 1, 1977•16 citations
We present a collective description of density matrix for identical multi-level atoms, which are either excited initially or driven coherently or pumped incoherently. We consider density matrix as expectation value of projection or transition operators in a basis of atom's product states and then identify identical matrix elements with several integers, which specify uniquely the involved transition or projection operators. To remove the redundancy of these identical elements, we treat them as single quantity and derive the equation for this quantity by mapping the transition or projection operators to a single vector specified with the integers in Liouville space and work out the actions of the master equation constitutions (as superoperators) on these vectors. As a result, the number of necessary elements increases polynomially with the number of atoms rather than exponentially as expected if the redundancy is not removed. This numerical advantage is explored to study superradiance from hundreds of two-level atoms. Our simulation identifies the conditions of observing the superradiance with and without dipole involved.
A collective description of density matrix is presented for identical multi-level atoms, which are either excited initially, driven coherently or pumped incoherently. The density matrix is defined as expectation value of projection or transition operators in a basis of atom's product states. The identical matrix elements are identified with several integers, which specify uniquely the involved operators. To remove the redundancy, these identical elements are treated as single quantity and the equation for this quantity is dervied by mapping the transition or projection operators to a single vector specified with these integers. As a result, the number of computed elements increases polynomially rather than exponentially with the number of atoms. As an example, we carry out exact simulation of hundreds of two-level atoms and demonstrate the different conditions for observing superradiance and superfluorescence.
We simulate the superradiant dynamics of ensembles of atoms in the presence of collective and individual atomic decay processes. We apply the Monte-Carlo wave-function method and identify quantum jumps in a reduced Dicke state basis, which reflects the permutation symmetry of the system. While the number of density matrix elements in the Dicke representation increases polynomially with atom number, the quantum jump dynamics populates only a single Dicke state at the time and thus efficient simulations can be carried out for tens of thousands of atoms. The superradiant pulses from initially excited atoms agree quantitatively with recent experimental results of strontium atoms but rapid atom loss in these experiments does not permit steady-state superradiance. By introducing an incident flux of new atoms to maintain a large average atom number, our theoretical calculations predict lasing with a millihertz linewidth despite rapid atom number fluctuations.
In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with NNN resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.
We investigate properties of the cooperative emission of radiation from NNN identical two-level atoms by using, for the collective atomic operators, the Holstein-Primakoff transformation to boson operators. In the basis of the Glauber states, a weight function is associated with the atomic density operator. The equation of motion for the weight function is derived from the master equation and approximate solutions are given in the two limit cases of superradiance and diffusion. The moments of the radiation intensity are calculated and are found to be in close agreement with previous works.
The cooperative spontaneous emission from a system of NNN identical two-level atoms, excited arbitrarily and interacting with a single mode of radiation, is studied by using the Bonifacio-Schwendimann-Haake approach. The evolution of the probability distribution of the atomic system is considered as a Markov process described by a time-evolution matrix (TEM). The exact analytic expressions for the elements of this TEM are obtained by solving the superradiance master equation. It is proved that the TEM satisfies the normalization condition and the Chapman-Kolmogorov equation as required. The exact general formulas for the expectation values of various physical observables, expressed in terms of an arbitrary initial distribution, are obtained. Finally, these results are applied to a specific example, the superradiant initial state.
The cooperative spontaneous emission from a completely excited system of NNN identical two-level atoms is studied under the condition that the cooperation number is conserved. The master equation for such a system has been solved analytically. The exact expressions for the probability distribution and the expectation values of various observable physical quantities are presented. Through a power-series expansion of the exact photon-number expectation value and by using properties of Stirling numbers of the second kind, it is shown that our result agrees with that of Haake and Glauber asymptotically. As by-products, some new identities involving binomial coefficients and Stirling numbers of the second kind have also been obtained. The exact solution of the same master equation for arbitrary initial conditions will be presented in a subsequent publication.