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Physical Review A

Macroscopic dark periods without a metastable state

Gerhard C. Hegerfeldt, Martin Bodo Plenio

Published Jul 1, 1992• 85 citations
Publisher

Abstract

The well-known three-level system of Dehmelt [Bull. Am. Phys. Soc. 20, 60 (1975)] exhibits macroscopic dark periods because one of the upper levels is metastable. In this paper we show that one can also have the same phenomenon for a three-level-atom system with both upper states rapidly decaying to the ground state, provided (a) their level separation is very small and (b) their dipole moments for the ground-state transition are parallel. For suitably chosen laser parameters we also exhibit an exact nonradiating solution of the complete time development for atom plus radiation field. This leads to the surprising effect that the atom, after emission of a number of photons, will stop to fluoresce although still irradiated by the laser. The mean number of photons emitted before this switching off of the atom is calculated, as well as the mean duration of the dark periods for the general case.

DOI
10.1103/physreva.46.373

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The well-known three-level system of Dehmelt [Bull. Am. Phys. Soc. 20, 60 (1975)] exhibits macroscopic dark periods because one of the upper levels is metastable. In this paper we show that one can also have the same phenomenon for a three-level-atom system with both upper states rapidly decaying to the ground state, provided (a) their level separation is very small and (b) their dipole moments for the ground-state transition are parallel. For suitably chosen laser parameters we also exhibit an exact nonradiating solution of the complete time development for atom plus radiation field. This leads to the surprising effect that the atom, after emission of a number of photons, will stop to fluoresce although still irradiated by the laser. The mean number of photons emitted before this switching off of the atom is calculated, as well as the mean duration of the dark periods for the general case.
Advances in trap technology have made feasible experiments on single, or a few, atoms and ions. Photon counts from a single atom may exhibit new features that are washed out in ensembles. Wilser and I recently proposed a theoretical approach based on repeated gedanken measurements, which was then also independently proposed by Dalibard et al. and by Carmichael. The optical Bloch equations are not used but rather follow as a consequence. The approach is applied to a three-level V-system with small separation of the upper levels having parallel dipole moments. Macroscopic dark periods are predicted for such a laser-driven system, although none of the levels is metastable as in the Dehmelt system. The reason is a coherence effect. For particular values the dark period may become arbitrarily long. A similar result holds for A-systems. This explains the observed dark resonances in ensembles of such systems by increasingly long dark periods of individual atoms. The approach is then applied to atoms driven by incoherent light, and nonrate equations with off-diagonal terms are derived. It is shown that these may lead to coherence effects.
We consider a three-level atomic system driven strongly on one transition and weakly on the other. The excited state on the weak transition is assumed to be metastable. We give an analysis of the fluorescence from the strong transition in terms of the elementary probability density p[0{p}_{[0}p[0​,t)(t1{\mathrm{t}}_{1}t1​,t2{t}_{2}t2​,..., tn{t}_{n}tn​) which gives us the probability density that exactly n photons are emitted at times t1{t}_{1}t1​,t2{t}_{2}t2​,...,tn{t}_{n}tn​ by the atom in the time interval [0,t). We show that p[0{p}_{[0}p[0​,t) essentially factorizes into products of conditional densities c\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\tau}) that, given a photon is emitted at time zero, the next photon emission occurs at time \ensuremath{\tau}. This enables a simulation of the individual photon emissions to be given which shows directly the existence of prolonged dark windows in the fluorescence corresponding to the shelving of the electron in the metastable state or ``quantum jumps.''
We present a microscopic model for the so-called quantum jumps in three-level atoms with one transition between two levels being strongly driven and the third level being metastable. Our model is based on calculating the elementary probability density P(0,t)(t1, … , t n ) which gives the probability density that exactly n photons are emitted at times t1, … , t n by the atom in the time interval (0, t) on the strong transition. This enables us to calculate the conditional probabilities c ˜ ( τ ) that given a photon has emitted at τ = 0 the next photon emission occurs at time τ. This quantity is shown to be the sum of two exponentials which for sufficient differetial in the time scales describe the periods of light and darkness. Using this conditional probability it is possible to numerically simulate the individual photon emissions.
Mathematical modeling of the population dynamics is performed for states of a three-level system (atom) with a V-type configuration transforming a light pulse. It is assumed that the excited eigenstates of the atom are degenerate and coupled by coherent interaction, one of the states being radiating (radiative), while the other state is nonradiating (“dark”). The population dynamics of atomic states is described on the basis of numerical solutions of equations for the matrix elements of the density operator. The dependence of the efficiency of population of the atomic dark state from the values of the parameters of an irradiation pulse and from the ratio of the period of population oscillations of excited atomic states (caused by their coherent interaction) to the lifetime of the atomic radiating state is determined. Typical examples of the time dependence of the population of states of the atom considered are presented for the cases of irradiation by a short (as compared to the lifetime of the radiating state) sinusoidal light pulse and by a long rectangular light pulse with the resonance carrier frequency.
The quantum Zeno effect (QZE) predicts a slow-down of the time development of a system under rapidly repeated ideal measurements, and experimentally this was tested for an ensemble of atoms using short laser pulses for non-selective state measurements. Here we consider such pulses for selective measurements on a single system. Each probe pulse will cause a burst of fluorescence or no fluorescence. If the probe pulses were strictly ideal measurements, the QZE would predict periods of fluorescence bursts alternating with periods of no fluorescence (light and dark periods) which would become longer and longer with increasing frequency of the measurements. The non-ideal character of the measurements is taken into account by incorporating the laser pulses in the interaction, and this is used to determine the corrections to the ideal case. In the limit, when the time between the laser pulses goes to zero, no freezing occurs but instead we show convergence to the familiar macroscopic light and dark periods of the continuously driven Dehmelt system. An experiment of this type should be feasible for a single atom or ion in a trap
The temporal behavior of a single quantum system may contain information, for instance, about macroscopic dark periods in electron shelving, which is washed out or even lost in ensemble averages. The same is true in principle for spectral information. For example, for an atom with light and dark periods one might try to spectrally analyze the light emitted in a light period of given length or, more generally, the light emitted into one solid angle while monitoring the temporal emission in another solid angle. Considering different emission behaviors one may get different spectra, i.e., spectra that are dependent on the conditions imposed. In this paper, such questions, in particular the notion of spectrum in a light period, are studied. To this end we generalize the recent quantum jump approach to include resonance-fluorescence spectra of single atoms. We explicitly show that the theory developed in this paper easily generalizes, e.g., to absorption spectra. We apply the theory to a V system that has one metastable level and that may exhibit intermittent fluorescence when driven by two lasers. We show that the spectrum of resonance fluorescence of the complete ensemble differs substantially from that observed in a light period. A physical explanation of this behavior is presented. \textcopyright{} 1996 The American Physical Society.