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Journal of the Physical Society of Japan

Decay of an Excited Atom under Imperfect Photon Counting Measurement

Kazuki Koshino, Akira Shimizu

Published Jan 1, 2003• 0 citations
Publisher

Abstract

We investigate the decay of an excited two-level atom under a situation where the emitted photon is detected by the conventional photon counting measurement. We discuss an imperfect measurement with finite probability of failing in counting the emitted photon. By analyzing the extended quantum system including the measurement apparatus, we have calculated the survival probability of the excited atom, as well as the response of the apparatus and the errors in the measurement. The decay rate under the measurement is obtained in a simple form. It is shown that not only the response time of the detector but also the measurement error play crucial roles in modification of the decay rate. The conditions to observe the quantum Zeno effect and the anti-Zeno effect are also clarified for the case of Lorentzian form factor.

DOI
10.1143/jpsjs.72sc.77

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We investigate the decay of an excited two-level atom under a situation where the emitted photon is detected by the conventional photon counting measurement. We discuss an imperfect measurement with finite probability of failing in counting the emitted photon. By analyzing the extended quantum system including the measurement apparatus, we have calculated the survival probability of the excited atom, as well as the response of the apparatus and the errors in the measurement. The decay rate under the measurement is obtained in a simple form. It is shown that not only the response time of the detector but also the measurement error play crucial roles in modification of the decay rate. The conditions to observe the quantum Zeno effect and the anti-Zeno effect are also clarified for the case of Lorentzian form factor.
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Abstract We study the measurement-induced enhancement of the spontaneous decay for a two-level subsystem, where measurements are treated as couplings between the excited state and an auxiliary state rather than the von Neumann's wave function reduction. The photon radiated in a fast decay of the atom, from the auxiliary state to the excited state, triggers a quasi-measurement, as opposed to a projection measurement. Our use of the term “quasi-measurement” refers to a “coupling-based measurement”. Such frequent quasi-measurements result in an exponential decay of the survival probability of atomic initial state with a photon emission following each quasi-measurement. Our calculations show that the effective decay rate is of the same form as the one based on projection measurements. The survival probability of the atomic initial state obtained by tracing over all the photon states is equivalent to that of the atomic initial state with a photon emission following each quasi-measurement.
It was predicted that frequently repeated measurements on an unstable state may alter the decay rate of the state. This is called the quantum Zeno effect (QZE) or the anti-Zeno effect (AZE), depending on whether the decay is suppressed or enhanced. In conventional theories of the QZE and AZE, effects of measurements are simply described by the projection postulate, assuming that each measurement is an instantaneous and ideal one. However, real measurements are not instantaneous and ideal. For the QZE and AZE by such general measurements, interesting and surprising features have recently been revealed, which we review in this article. The results are based on the quantum measurement theory, which is also reviewed briefly. As a typical model, we consider a continuous measurement of the decay of an excited atom by a photodetector that detects a photon emitted from the atom upon decay. This measurement is an indirect negative-result one, for which the curiosity of the QZE and AZE is emphasized. It is shown that the form factor is renormalized as a backaction of the measurement, through which the decay dynamics is modified. In a special case of the flat response, where the detector responds to every photon mode with an identical response time, results of the conventional theories are reproduced qualitatively. However, drastic differences emerge in general cases where the detector responds only to limited photon modes. For example, against predictions of the conventional theories, the QZE or AZE may take place even for states that exactly follow the exponential decay law. We also discuss relation to the cavity quantum electrodynamics.
We study the influence of a detector on the decay law of a quantum state whose undisturbed survival probability is purely exponential. In particular, we consider a detector with a finite energy band of detection, i.e., it interacts only with decay products having an energy within a certain range of values. In one case, we assume that the detector performs many repeated measurements at short time intervals in all of which a collapse of the wave function occurs (bang-bang or pulsed-type measurements). In the second case, we assume a continuous measurement, which preserves unitarity. We confirm the slowing down of the decay in presence of a measuring apparatus, the quantum Zeno effect, but the outcomes of the detector are in general qualitatively and quantitatively different in the two cases. In turn, this implies that the so-called Schulman relation (the equivalence of pulsed and continuous measurements) does not hold in this case and that it is in principle possible to experimentally access how a certain detector performs a measurement.