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Physical Review Letters

Stationary two-level atomic inversion in a quantized cavity field

Craig M. Savage

Published May 2, 1988• 67 citations
Publisher

Abstract

A quantum mechanical analysis of a two-level atom in a coherently driven optical cavity is shown to predict steady-state atomic population inversion. Semi-classically this is forbidden because of the factorization of operator-product expectation values. The full quantum theory is much richer and different field states may be correlated with different atomic states. One consequence is that damping of the cavity field allows atomic polarization to be transferred from higher to lower field states and thus steady-state inversion becomes possible.

DOI
10.1103/physrevlett.60.1828

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International Letters of Chemistry Physics and Astronomy•Jul 1, 2015•1 citation
A quantum mechanical analysis of a two-level atom in a coherently driven optical cavity is shown to predict steady-state atomic population inversion. Semi-classically this is forbidden because of the factorization of operator-product expectation values. The full quantum theory is much richer and different field states may be correlated with different atomic states. One consequence is that damping of the cavity field allows atomic polarization to be transferred from higher to lower field states and thus steady-state inversion becomes possible.
A two-level atom illuminated by a laser may be driven into a population-inverted steady state if it is coupled to a cavity. This is a result of the quantum nature of the electromagnetic field and is forbidden by the semiclassical theory. We numerically and analytically analyze this inversion. We find the maximum possible inversion and determine the rate of approach to the steady state. The quantized cavity modifies the electromagnetic vacuum seen by the atom; this connects our work to that on cavity-enhanced spontaneous emission and on dynamical line narrowing. Experimental signatures of the inversion and potential experimental difficulties are considered. For example, neither the presence of many atoms in the mode nor many modes in the cavity destroys the inversion.
Placing an optical cavity around a two-level atom modifies the electromagnetic vacuum to which it couples. Effects such as inhibited and enhanced spontaneous emission are known to result. Recently one of us predicted a new nonclassical effect of vacuum engineering.1 A two-level transition may be laser driven into an inverted steady state. Semiclassical theory forbids this fully quantum mechanical result. Unlike squeezing and antibunching, this inversion is not a property of the linearized quantum fluctuations. We present numerical and analytical results which clarify the physics of the inversion. The maximum inversion possible is determined. We attempt to identify obstacles to the experimental realization of steady-state inversion. Multiple atoms and multiple cavity modes do not destroy the effect. A significant difficulty appears to be localizing the atoms at the cavity beamwaist until they reach steady state. The inversion may be observed via fluorescence changes as the cavity is blocked and unblocked or via the presence of a fluoresence maximum as a function of the driving field intensity.
We investigate properties of a two-level atom coupled to a single cavity mode, and driven by a strong, in general off-resonance, external field. The cavity field is tuned to resonance with a Mollow sideband. We show that the steady-state atomic population may be highly (nearly totally) inverted, a phenomenon forbidden in semiclassical theory, and that a strong enhancement of the resonance fluorescence results. This system is suggested as an alternative approach to create the atomic inversion necessary for laser or maser generation without using additional atomic levels. We also investigate properties of the cavity field, derive an analytical expression for the photon number distribution in the good cavity or strong-coupling limit, and discuss the conditions under which this system acts as a single-atom dressed-state laser.
The quantum mechanical master equation for a single two-level atom in a single-mode optical cavity is numerically solved in both the quantum and the semiclassical limits. The quantum limit of few cavity photons shows semiclassically forbidden behaviour such as steady state two-level population inversion. Qualitatively new fluorescent spectra, having sidebands broadened by the cavity interaction, also occur. The quantum theory of the single-atom laser with injected signal is presented. At the interface between its quantum and semiclassical dynamics we elucidate the signature of semiclassical limit cycles.
The interaction of a two-level atom with a single mode of the radiation field is examined. The cavity mode is damped by coupling to a reservoir in a squeezed state, and the atom is also driven by an external laser field. A master equation for the atomic density operator in the bad-cavity limit is derived. The effect of the squeezing in the stationary population inversion and in the resonance spectrum of the light emitted by the atom into the background modes is studied. The results are compared with the numerical solution of the full master equation for the atom-plus-cavity mode-density operator.
A mathematical model was developed to obtain the population inversion of the atoms in laser field in a laser cavity by considering the electric field in the optical cavity and the atomic states of the medium to be quantized. The master equation of the density operator of the laser field was studied analytically and numerically. Using coherent states, the Fokker-Plank equation for the phase space density for the laser field was solved analytically for the time dependent and steady state situations. The laser field above the threshold can be represented by a randomly phased mixture of coherent states. As the pump parameter increases in the laser process, the phase density becomes narrower, tending toward a delta function, creating a coherence laser field.