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Physical Review A

N-atom optical Bloch equations: A microscopic theory of quantum optics

Robert Goodell Brown, Mikael Ciftan

Published Sep 1, 1989• 11 citations
Publisher

Abstract

We present a microscopic optical-Bloch-equation model for N two-level atoms being driven by a monochromatic radiation field. The model is obtained from the equations of motion that result from the microscopic Hamiltonian by introducing at a certain stage an Ansatz that accounts for spontaneous emission and bath dephasing in a unitary manner. The equations of motion are a numerically tractable set of coupled c-number differential equations with a discrete, stochastic term deduced from the quantum-mechanical equations of motion. We apply these equations to various model problems and show that they correctly describe such varied optical phenomena as superradiance, photon echoes, and absorptive bistability. In particular, they yield an interesting microscopic picture of the dynamical phase transition that occurs in the latter example. We conclude that this unified approach has advantages that make it useful for studying a number of problems of current interest.

DOI
10.1103/physreva.40.3080

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Physical Review A•Sep 1, 1973•20 citations
We present a microscopic optical-Bloch-equation model for N two-level atoms being driven by a monochromatic radiation field. The model is obtained from the equations of motion that result from the microscopic Hamiltonian by introducing at a certain stage an Ansatz that accounts for spontaneous emission and bath dephasing in a unitary manner. The equations of motion are a numerically tractable set of coupled c-number differential equations with a discrete, stochastic term deduced from the quantum-mechanical equations of motion. We apply these equations to various model problems and show that they correctly describe such varied optical phenomena as superradiance, photon echoes, and absorptive bistability. In particular, they yield an interesting microscopic picture of the dynamical phase transition that occurs in the latter example. We conclude that this unified approach has advantages that make it useful for studying a number of problems of current interest.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.
A set of coupled non-Markovian equations of motion is derived for the Green's function of a driven two-level atom. These equations can be expressed in terms of the density-matrix elements only in the Markovian limit, in which case the set reduces to optical Bloch equations. The non-Markovian effect arises from correlations between successive photon emissions, an effect which is neglected in the existing equations of motion. The resulting correction is of the order of ${\ensuremath{\gamma}}_{0}$/${\ensuremath{\Omega}}_{0}$ in the strong-field region (${\ensuremath{\gamma}}_{0}$${\ensuremath{\Omega}}_{0}$) and it vanishes in the weak-field region, where ${\ensuremath{\gamma}}_{0}$ is the Einstein A coefficient and ${\ensuremath{\Omega}}_{0}$ is the Rabi frequency.
The analysis in Chapter 3 used a phenomenological form of the permittivity to describe active materials. A proper understanding of optical amplification requires a quantum-mechanical approach for describing the interaction of light with atoms of an active medium [1]. However, even a relatively simple atom such as hydrogen or helium allows so many energy transitions that its full description is intractable even with modern computing machinery [2, 3]. The only solution is to look for idealized models that contain the most essential features of a realistic system. The semiclassical two-level-atom model has proven to be quite successful in this respect [4]. Even though a real atom has infinitely many energy levels, two energy levels whose energy difference nearly matches the photon energy suffice to understand the interaction dynamics when the atom interacts with nearly monochromatic radiation. Moreover, if the optical field contains a sufficiently large number of photons (> 100), it can be treated classically using a set of optical Bloch equations. In this chapter, we learn the underlying physical concepts behind the optical Bloch equations. We apply these equations in subsequent chapters to actual optical amplifiers and show that they can be solved analytically under certain conditions to provide a realistic description of optical amplifiers. It is essential to have a thorough understanding of the concept of a quantum state [5]. To effectively use the modern machinery of quantum mechanics, physical states need to be represented as vectors in so-called Hilbert space [6].
Generalized single-atom Maxwell-Bloch equations for optically dense media are derived taking into account noncooperative radiative atom-atom interactions. Applying a Gaussian approximation and formally eliminating the degrees of freedom of the quantized radiation field and of all but a probe atom leads to an effective time-evolution operator for the probe atom. The mean coherent amplitude of the local field seen by the atom is shown to be given by the classical Lorentz-Lorenz relation. The second-order correlations of the field lead to terms that describe relaxation or pump processes and level shifts due to multiple scattering or reabsorption of spontaneously emitted photons. In the Markov limit a nonlinear and nonlocal single-atom density matrix equation is derived. To illustrate the effects of the quantum corrections, we discuss amplified spontaneous emission and radiation trapping in a dense ensemble of initially inverted two-level atoms and the effects of radiative interactions on intrinsic optical bistability in coherently driven systems.
Theory of the interaction between atom and radiation field: three pictures in quantum mechanics description two-level atom and the optical Bloch equation quantized description of the radiation field Dicke Hamiltonian and Jaynes - Cummings model quantum theory of a small system coupling to a reservoir. The quantum properties of light: coherence of light squeezed states of light resonance fluorescence superfluorescence optical bistability effects of virtual photon processes. Quantum properties of the atomic behaviour under the interaction of the radiation field: collapses and revivals of the atomic populations squeezing effects of the atomic operators coherent trapping of the atomic population quantum characteristics of a two-atom system under the interaction of radiation field autoionization of the atom in a laser field motion of the atom in a laser field laser cooling.
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.