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Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences

Closed equations of motion for expectation values of collective operators for spontaneous emission in the presence of external driving fields

JOSIP SEKE

Published Dec 18, 1984• 1 citation
Publisher

Abstract

Abstract By using a modified Robertson projection technique, developed by the author in previous work, three exact closed equations of motion for expectation values of collective atomic operators for spontaneous emission in the presence of a classical driving field are derived.
DOI
10.1098/rsta.1984.0137

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In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of NNN identical two-level atoms. Three different observation levels of the atomic system are considered. First, exact closed integrodifferential equations for the expectation values of each observation level are obtained. Second, making the standard approximations---Born, Markov, and long-time approximations, we find that these equations reduce to closed differential equations for the expectation values. The quality of the Born approximation for different observation levels is also discussed. Moreover, in the obtained equations no kind of rotating-wave approximation was made.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
We consider the radiation from a system of NNN identical two-level atoms coupled to a continuum of quantized em modes, and possibly, to an external driving field near resonance. The atoms can be distributed over a region large in comparison to the resonant wavelength, but smaller than the spontaneous pulse length. Radiation rates and correlation functions are expressed in terms of expectation values of time-dependent atomic operators, which are shown to satisfy coupled first-order differential equations involving similar atomic operators and the initial radiation operators. The corresponding equations for the expectation values simplify considerably if no driving field is present. Similar results are derived for a model in which each atom is replaced by a harmonic oscillator.
We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and applies equally well to distinguishable and indistinguishable atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find closed-form formulas for a number of relevant states (Gaussian states, Fock states, and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion.