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Physical Review A

Emission rate of a multilevel atom and its quantum beats

I. C. Khoo, Joseph H. Eberly

Published Dec 1, 1976• 24 citations
Publisher

Abstract

Using the Heisenberg operator approach, we have calculated the emission rate of a multilevel atom whose transitions involve both upper- and lower-state splittings. The emission rate is first calculated by a frequently used power-flow expression and is shown to exhibit lower-state beats. This power-flow expression requires care in its interpretation because it does not account for the measurement process. We then derive an alternative, detection-oriented emission rate and show that lower-state beats are not observable in a photodetection experiment.

DOI
10.1103/physreva.14.2174

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Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.

•Jan 1, 1974•0 citations
Using the Heisenberg operator approach, we have calculated the emission rate of a multilevel atom whose transitions involve both upper- and lower-state splittings. The emission rate is first calculated by a frequently used power-flow expression and is shown to exhibit lower-state beats. This power-flow expression requires care in its interpretation because it does not account for the measurement process. We then derive an alternative, detection-oriented emission rate and show that lower-state beats are not observable in a photodetection experiment.
The quantum electrodynamics of two atoms in free space, separated by many wavelengths and interacting with one another via the coupling of their dipole moments to the electromagnetic field, is studied using the Heisenberg-operator approach. The problem is formulated in such a way that one of the atoms, which has two levels, plays the role of a detector; the other atom, whose levels are arbitrary but do not involve cascadelike transitions, plays the role of the source. Several aspects of the interaction are considered, including: a review of photodetection theory, the theory of an alternative detection method that uses an excited-state detector, and a study of quantum beats. These dynamical studies of emission and detection provide new insights into the observation of quantum beats in the transition rate of a detector. We show that ground-state beats are not observed by a detector atom making an upward transition, but that ground-state beats are, in principle, observable if the detector atom makes a downward transition.
Spontaneous emission from a multilevel atom is studied from the viewpoint of quantum beats. The master equation describing the atomic dynamics is shown to have some parameters that oscillate at the beat frequencies, and because of this a Pauli type of master equation for the diagonal elements of the density matrix is not obtained. As a result, the transition rates in the atomic system show the characteristic beats. The relation of the rate of change of the energy of the atomic system to the total power radiated in the far zone is also established. The normally and antinormally ordered correlation functions of the spontaneously emitted radiation are calculated under the approximation that the level widths are much smaller than the beat frequencies. The contribution arising from the interference of the free field and the source terms to antinormally ordered correlations is carefully evaluated. The nature of beats present in normally and antinormally ordered correlations is discussed, and their detection from the viewpoint of causing transitions in another atom is considered. The change in the nature of beats due to atomic collective effects is studied. Finally, it is shown that because of quantum effects, intensity correlations do not show any beats.
Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field EEE, and the existence of a spectral component at the beat frequency in E2{E}^{2}E2. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.