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Physical Review A

Interrupted fluorescence, quantum jumps, and wave-function collapse

David Thomas Pegg, P. L. Knight

Published Jun 1, 1988• 38 citations
Publisher

Abstract

The phenomenon of antibunching provides important support for the physical principle of the collapse of the wave function following a detection of a photon. In this paper we show that interrupted fluorescence in a three-level, V-configuration atom, with one transition strongly driven and one weakly driven, can be regarded as evidence supporting the complementary principle of ``collapse by nondetection.'' Use of this principle leads to the immediate deduction of remarkably simple general expressions for the average bright and dark periods, which allows these quantities to be calculated for both coherent and incoherent excitation without the need to solve the density-matrix equations, or even to find their long-term steady-state solutions. Thus the concept of collapse by nondetection appears to be not only valid but is also useful for predicting physically observable phenomena.

DOI
10.1103/physreva.37.4303

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The phenomenon of antibunching provides important support for the physical principle of the collapse of the wave function following a detection of a photon. In this paper we show that interrupted fluorescence in a three-level, V-configuration atom, with one transition strongly driven and one weakly driven, can be regarded as evidence supporting the complementary principle of ``collapse by nondetection.'' Use of this principle leads to the immediate deduction of remarkably simple general expressions for the average bright and dark periods, which allows these quantities to be calculated for both coherent and incoherent excitation without the need to solve the density-matrix equations, or even to find their long-term steady-state solutions. Thus the concept of collapse by nondetection appears to be not only valid but is also useful for predicting physically observable phenomena.
For pt.I see Phys. Rev. A, vol.37, p.4303 (1988). The principle of wavefunction collapse by non-detection of photons leads to simple but rigorous expressions for the average dark and bright periods, T D and T B , of intermittent quantum jump fluorescence from a V-configuration three-level atom. Here further consequences of this principle are explored, including: the derivation of some new correlation functions which exhibit the physical characteristics of extended dark and bright periods; the expressions for T D and T B when one transition is driven coherently and the other incoherently; and a general expression for the characteristic time to be sure that a quantum jump has occurred for any type of driving field.
Correlations between the emission of photons from an excited three-level atom with one transition more strongly driven than the other are studied. Antibunching, anticorrelation, and superbunching are exhibited in the four possible configurations: ``vee'' (V), ``lambda'' (\ensuremath{\Lambda}), and two cascades with the upper or lower transition strongly driven. We pay particular attention to the projection of the atom into particular states by the measurement of fluorescence, and its significance for detecting quantum jumps in the weak transition by observing the fluorescence from the strong transition. For incoherent excitation we find that cross antibunching (the anticorrelation between emissions of different types of photons) manifests itself as prolonged periods of darkness. For the V and the cascade configuration with a strongly driven lower transition, these periods herald the emission of a weak-transition photon; for the other cascade and the \ensuremath{\Lambda} configuration the periods are heralded by the emission of a weak-transition photon. For on-resonance coherent excitation, dark periods are extremely infrequent for the former two configurations, and are extremely long or infinite for the latter two.
Wave function collapse has been a contentious concept in quantum mechanics for a considerable time. Here we show examples of how the concept can be used to advantage in predicting the statistical results of three experiments in atomic physics and quantum optics: photon antibunching, single-photon phase difference states and interrupted single-atom fluorescence. We examine the question of whether or not collapse is 'really' a physical process, and discuss the consequences of simply omitting it but including the observer as a part of the overall system governed by the laws of quantum mechanics. The resulting entangled world does not appear to be inconsistent with experience.
One of the experiments made possible by the trapping and cooling of a single atom is to observe the quantum jumps to a metastable state in the atom by monitoring the resonance fluorescence of a strong transition in which at least one of the states is coupled to the metastable state. When the atomic electron moves to the metastable state, the fluorescence from the strongly driven transition disappears. When the electron drops back into the strongly driven transition, the fluorescence abruptly returns. Statistical analysis of these random interruptions to the fluorescence signal implies photon antibunching in the emission of the weak photon.
Time correlations in the fluorescence of a multilevel atom are studied in terms of the quantum statistics which describe the emission of the next photon. The next photon equations are substantially simpler than the a photon equations (or so-called optical Bloch equations) yet they contain phase information lost in the averaging procedure that yields the a photon equations. Employing the next photon equations we present a quantum formalism in which the time development of the wave function of the atom is dramatically modified by the observation of a dark period. This ``collapse'' of the quantum state due to measurements with a null result (such as the failure to record events in a photodetector) is the cause of intermittent atomic fluorescence even when the exciting field is arbitrarily coherent. The next photon and a photon formalisms are contrasted and applied to photon antibunching. The phenomenon of photon bunching which characterizes sideband correlations is also calculated. Observable effects particular to the coherent intermittency are emphasized.
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.''