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Physical Review A

How to reset an atom after a photon detection: Applications to photon-counting processes

Gerhard C. Hegerfeldt

Published Jan 1, 1993• 158 citations
Publisher

Abstract

The question is studied as to which state a radiating atom has to be reset to if a photon is detected and absorbed. This is important for applications of a recently proposed approach to photon counts from a radiating atom which is based upon repeated gedanken measurements and associated reductions as a technical tool. The general reset density matrix is determined. It may depend on the atomic state shortly before the photon detection, and may thus be time dependent. It may also have off-diagonal terms which may lead to interesting coherence effects. It is shown that the repeated measurements together with the reset matrix lead to the optical Bloch equations for a general N-level atom. The general photon-counting distribution for an N-level radiating atom is derived and the axiomatic continuous-measurement theory of Srinivas and Davies [Opt. Acta 28, 981 (1981); 29, 235 (1982)] for quantum-counting processes is shown to follow in the present framework from ordinary quantum mechanics.

DOI
10.1103/physreva.47.449

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The question is studied as to which state a radiating atom has to be reset to if a photon is detected and absorbed. This is important for applications of a recently proposed approach to photon counts from a radiating atom which is based upon repeated gedanken measurements and associated reductions as a technical tool. The general reset density matrix is determined. It may depend on the atomic state shortly before the photon detection, and may thus be time dependent. It may also have off-diagonal terms which may lead to interesting coherence effects. It is shown that the repeated measurements together with the reset matrix lead to the optical Bloch equations for a general N-level atom. The general photon-counting distribution for an N-level radiating atom is derived and the axiomatic continuous-measurement theory of Srinivas and Davies [Opt. Acta 28, 981 (1981); 29, 235 (1982)] for quantum-counting processes is shown to follow in the present framework from ordinary quantum mechanics.
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.
The concept of photodetection as a continuous quantum measurement introduced by Srinivas and Davies is extended to the detection of resonance fluorescence. This is a new approach to an old subject. This is also a new development in continuous photodetection in the sense that a fermion rather than the traditional boson field is the subject of monitoring. The superoperators for the no-count and the one-count processes are postulated, following the examples of Srinivas and Davies. The probability for the m-count process is then derived on the basis of these postulates. The first two factorial moments of the photon-number distribution are calculated, which are then used to evaluate the non-classical effects in resonance fluorescence. A parameter slightly different from Mandel's is found to be more suitable indicator of photon antibunching in the transient state.
This paper proposes a microscopic model for the continuous measurement of photon numbers where the continuous measurement is simulated by a sequence of infinitesimal processes. Each process is composed of a unitary evolution for the coupled system, which consists of the field and the measuring apparatus, followed by a projection according to the readout of the measuring apparatus. The proposed model consists of two-level atoms that are coupled one by one with an optical cavity field via the electric-dipole interaction. Each atomic level is then read out after the interaction, causing a nonunitary state reduction of the field. In particular, it is shown that this microscopic model leads to the Srinivas-Davies model that has been postulated for the photodetection process.
The counting statistics give insight into the properties of quantum states of light and other quantum states of matter such as ultracold atoms or electrons. The theoretical description of photon counting was derived in the 1960s and was extended to massive particles more recently. Typically, the interaction between each particle and the detector is assumed to be limited to short time intervals, and the probability of counting particles in one interval is independent of the measurements in previous intervals. There has been some effort to describe particle counting as a continuous measurement, where the detector and the field to be counted interact continuously. However, the formalism based on continuous measurements does not provide a formula applicable to general time- and space-dependent fields. In our work, we derive a fully time- and space-dependent description of the counting process for linear quantum many-body systems, taking into account the back-action of the detector on the field. We apply our formalism to an expanding Bose-Einstein condensate of ultracold atoms, and show that it describes the process correctly, whereas the standard approach gives unphysical results in some limits. The example illustrates that, in certain situations, the back-action of the detector cannot be neglected and has to be included in the description.
A quantum-mechanical theory of photon detection is presented which takes fully into account the attenuation of the field due to the detection process. The time evolution of the joint quantum state of the detector and the field is found, to all orders in perturbation theory. Formulas are derived for the probabilities of absorbing specified numbers of photons within a given time interval, and for the correlations in the positions and times at which absorptions take place. The results are free from the inconsistencies which arise in the conventional theory of photon detection, and remain valid even when the field becomes appreciably attenuated during the experiment. It is found that an initially coherent field state remains coherent during its interaction with a given detector, and that its amplitude becomes attenuated by an amount which is completely independent of the number of counts which the detector records. This independence is shown to be a simple consequence of the Poisson quantum-number distribution of coherent fields. The counting statistics for arbitrary fields are expressed in a way which shows explicitly the relationship between the attenuation of the field and the absorption of quanta by the detector. The analysis is performed by first treating the case in which the field is confined within a homogeneous detecting medium throughout the experiment, and then generalizing to the case in which the field spends a limited amount of time in a spatially localized detecting region. The detector is assumed to consist of harmonic oscillators, which are shown to represent a suitable formal model for the absorption of radiation by large numbers of conventional detecting atoms.
This paper presents a general theory for a continuous quantum-nondemolition measurement of photon number. This theory treats a time-distributed measurement as a sequence of measurements in which at most one photon can be detected in an infinitesimal time, and shows that the average number of photons remaining in the measured field increases when a photon is detected and decreases when no photon is detected. The state of the measured system evolves nonunitarily and reduces continuously to a nunber state whose eigenvalue is uniquely determined by the average rate of photodetection and whose probability distribution coincides with the initial photon-number distribution. Applying the general theory to typical quantum states---coherent, thermal, and squeezed states---shows that the continuous-state reduction towards a number state depends strongly on the initial photon statistics. Despite the nonunitarity of state evolution, an initially pure state keeps its purity: the initial density operator becomes diagonalized only if the readout information is discarded.