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Physical Review A

Generalization of the Maxwell-Bloch equations to the case of strong atom-field coupling

Olga A. Kocharovskaya, Shi-Yao Zhu, Marlan O. Scully, Paul Mandel, Yevgeny V. Radeonychev

Published Jun 1, 1994• 47 citations
Publisher

Abstract

We derive the generalized set of Maxwell-Bloch equations which takes into account the dependence of relaxation coefficients on the amplitude and frequency of the coherent field. This has interesting implications in many problems in quantum optics and laser physics, e.g., the problem of lasing without inversion due to a strong coherent generating field.

DOI
10.1103/physreva.49.4928

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We derive the generalized set of Maxwell-Bloch equations which takes into account the dependence of relaxation coefficients on the amplitude and frequency of the coherent field. This has interesting implications in many problems in quantum optics and laser physics, e.g., the problem of lasing without inversion due to a strong coherent generating field.
We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.
The linear and nonlinear properties of propagation of an electromagnetic field through a medium may be derived by the effective elimination of the medium from the coupled Maxwell-Bloch equations. In this paper we suggest, on the other hand, the elimination of the field from the Maxwell-Bloch equations as a means to derive a nonlinear master equation for, e.g., an atomic sample in a laser field. This presents a straightforward and physically intuitive derivation of equations for ``nonlinear atom optics,'' obtained recently by other approaches.
A generalized Bloch-Maxwell formulation for laser-field-coupled semiconductors is derived from a two-band model which includes direct Coulomb interactions. The momentum-dependent, microscopic, electron-hole equations of motion in the time-dependent Hartree-Fock approximation and neglecting interband exchange interactions form the starting point for the formulation. A self-consistent set of coupled equations in four dynamical variables for the medium, together with the electric-field amplitude coupled through the Maxwell wave equation in a semiclassical approximation, are obtained to lowest order in the coherent Coulomb exchange interaction and the density-of-states distribution. Intrinsic optical bistability is predicted in a steady state due to a carrier density-dependent redshift of the band edge due explicitly to coherent Coulomb exchange. Integration of the dynamical equations for conditions which correspond to the ultrafast time regime exhibit intrinsic adiabatic inversion, adiabatic following, anomalous Rabi cycling, and unique, as well as fast, optical switching, all of which depend upon the coherent Coulomb exchange interaction.
Based on a model of strong coupling for two electrons that are localized inside a quantum dot characterized by only two levels of size quantization, a set of equations is derived, which allows one to describe coherent transient processes in an ensemble of isolated dots. Here one considered the case of ultrashort durations of electromagnetic field pulses, when the processes of polarization and population relaxation can be neglected. The resulting set of equations is a generalization of the Maxwell-Bloch equations, which are used in studying the processes occurring under the action of ultrashort electromagnetic pulses. The model obtained is reduced to the model of two-level atoms only if radiation is circularly polarized.
generalised model operator bloch equations are derived for some models in the field of quantum optics. these include the behaviour of a single 2-level atom coupled with a broadband squeezed vacuum (phase - sensitive) reservior outside the familiar rotating wave approximation the bistable behaviour in a ring cavity configuration subject to an average atomic coherent field as part of its Environment tal reservior in case the average atomic inversion exhibits transient and steady oscillatory behaviour while in case a further non linear structure shows in the characteristic input state relation due to field dependent relaxation processes
The optical Bloch equations, which give the time evolution of the elements of the density matrix of an atomic system subject to radiation, are generalized so that they can be applied when transitions between pairs of states can proceed by more than one stimulated route. The case considered is that for which the time scale of interest in the problem is long compared with that set by the differences in detuning of the radiation fields stimulating via the different routes. It is shown that the Bloch equations then reduce to the standard form of linear differential equations with constant coefficients. The theory is applied to a two-state system driven by two lasers with different intensities and frequencies and to a three-state Λ-system with one laser driving one transition and two driving the second. It is also shown that the theory reproduces well the observed response of a cold 40Ca+ ion when subject to a single laser frequency driving the 4S1/2–4P1/2 transition and a laser with two strong sidebands driving 3D3/2–4P1/2.