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Journal of the Optical Society of America

Time-dependent spectra

P. Stehle

Published Jul 1, 1979• 17 citations
Publisher

Abstract

A definition of a time-dependent spectrum is proposed as an alternative to that given by Eberly and Wódkiewicz. The alternative definition excludes most of the measuring apparatus from the system to which the spectrum pertains, and is based directly on the quantum-mechanical transition probability. It is closely related to that used by Renaud et al. in the case of resonance fluorescence, and gives similar results in this case.

DOI
10.1364/josa.69.001043

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A definition of a time-dependent spectrum is proposed as an alternative to that given by Eberly and Wódkiewicz. The alternative definition excludes most of the measuring apparatus from the system to which the spectrum pertains, and is based directly on the quantum-mechanical transition probability. It is closely related to that used by Renaud et al. in the case of resonance fluorescence, and gives similar results in this case.
The definition of a time-dependent spectrum registered by an idealized spectrometer responding to a time-varying electromagnetic field as proposed by Eberly and Wodkiewicz and subsequently applied to the spectrum of laser-induced fluorescence by Eberly, Kunasz, and Wodkiewicz is here extended to allow a stochastically fluctuating (interruption model) environment: we provide an algorithm for numerical determination of the time-dependent fluorescence spectrum of an atom subject to excitation by an intense noisy laser and interruptive relaxation.
Extending the concept of the time-dependent physical spectrum introduced by Eberly and Wodkiewicz (1977) the authors present a formula for the n-fold time-resolved correlation spectrum depending on n time and n frequency arguments. The corresponding measuring device consists of n coupled photodetectors with frequency resolving equipment in front of each. As an application the authors consider the resonance fluorescence radiation of a two-level atom irradiated monochromatically. Under certain conditions the authors derive a simple expression for the n-fold detection probability of photons at time t i (i=1, . . ., n) with frequencies belonging to the central peak or to one of the sidebands of the fluorescence triplet.
Using the concept of the time-dependent physical spectrum introduced by Eberly and Wodkiewicz (1977) the authors develop the theoretical basis of time-resolved correlation spectroscopy. The corresponding measuring device consists of two coupled photodetectors with frequency resolving equipment in front of each. A formula is presented for the time-resolved correlation spectrum depending on two time and two frequency arguments. It may be used to gain insight into the time correlation of photons with given frequencies. The authors apply the theory to the resonance fluorescence radiation of a two-level atom irradiated monochromatically. It is shown that in the case of a weak irradiation with large detuning the photons are emitted in a definite time order into the two sidebands of the fluorescence triplet. Under strong resonant irradiation the photons show up bunching (antibunching) behaviour if they belong to different (equal) sidebands. In addition to these results which are already known the authors present new results concerning intermediate situations.
So far in our development of quantum mechanics, we have given very little consideration to problems involving the time dependence of the wave function. This is rather surprising as in classical mechanics it is time-dependent phenomenai.e., dynamics rather than statics-that command most attention. Moreover, most experimental observations necessarily involve some change in the quantity being observed, so we might expect that the successful prediction of experimental results will require a detailed understanding of the way in which a system changes with time. In fact we have already made some implicit assumptions about time dependence and the principal reason why we have got so far without discussing it in detail is that many observed quantum phenomena are associated with sudden discontinuous changes between otherwise stable states. Thus, most of our information concerning the energy levels of atoms has been obtained from measurements of the frequencies of electromagnetic radiation emitted or absorbed as the atom undergoes a transition from one energy eigenstate to another, assuming the correctness of the formula E = }ω, but not considering the mechanism of the transition in any detail. Assumptions concerning time dependence are also implicit in the quantum theory of measurement which refers to the probability of obtaining a particular result following a measurement performed on a system in a given state; thus the state vector of a spinhalf particle changes from being an eigenvector Sˆ z to being one of Sˆ x following a measurement of the latter property using an appropriately oriented Stern-Gerlach apparatus. However, as was pointed out in Chapter 7, and will be discussed further in Chapters 15 and 16, the collapse of the wave function associated with a measurement is not a direct consequence of the time-dependent Schro¨dinger equation.
Time-dependent features of the fluorescence radiation from a strongly driven three-level atom in a Raman configuration are studied applying the Eberly-W\'odkiewicz counting-rate definition of the spectrum. Using our earlier formalism, where we assume that one or both of the driving fields are intense, we obtain analytical expressions for the time-dependent fluorescence spectra. We have also studied the influence on the transient spectrum of the initial preparation of the atom in three different dressed states.