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arXiv:1909.04792

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

Yuan Zhang

Published Sep 10, 2019• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1909.04792

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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.
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 density-matrix approach is developed to provide a theoretical description of the intensity, angular distribution, and polarization of superradiative emission from an ensemble of many-electron atomic systems. The many-electron atomic systems are described as cooperatively interacting by means of forces that can be long range. Particular emphasis is given to the coherent excitation of the collective atomic-ensemble states, which may be produced by incident laser radiation. The initial excitation and spontaneous emission processes may be described as independent. Both frequency-domain and time-domain formulations of the density-matrix approach are developed. The collective atomic-ensemble states are specified in a detailed hyperfine representation, corresponding to successively coupling the individual hyperfine angular momenta F pertaining to the many-electron atoms. A less detailed fine-structure angular-momentum representation may also be used. In the density-operator approach, account can be taken of the coherent excitation of a particular subspace of the initial atomic-ensemble states. For a comprehensive and unified development of time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations, a reduced-density-matrix (quantum-open-systems) approach is introduced. The non-equilibrium atomic-ensemble-state kinetics and the homogeneous spectral-line shapes can thereby be systematically and self-consistently determined. The collective atomic-ensemble states may be obtained using a variety of different methods. These states can be determined using a dressed-state approach, in which the required states are calculated in the presence of an electromagnetic field.
The notion of atomic coherent states is extended to the multilevel case. Since the representation based on coherent states is convenient in treating collective interactions of atoms with photons, and since many optical processes involve atoms of three or more levels, it is expected that this extension will play a role in the theory of such processes as cascade superfluorescence and superradiant lasing. Like their bosonic counterparts, atomic coherent states may be used to define a holomorphic representation for atomic states and operators. We discuss this in detail and give examples to illustrate applications.
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.
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.
While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.