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arXiv:1405.6882 · Physical Review A

Pulsed and continuous measurements of exponentially decaying systems

Francesco Giacosa, Giuseppe Pagliara

Published Nov 11, 2014• 11 citations

Abstract

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.

DOI: 10.1103/physreva.90.052107

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After reviewing the description of an unstable state in the framework of nonrelativistic Quantum Mechanics (QM) and relativistic Quantum Field Theory (QFT), we consider the effect of pulsed, ideal measurements repeated at equal time intervals on the lifetime of an unstable system. In particular, we investigate the case in which the `bare' survival probability is an exact exponential (a very good approximation in both QM and QFT), but the measurement apparatus can detect the decay products only in a certain energy range. We show that the Quantum Zeno Effect can occur in this framework as well.

After reviewing the description of an unstable state in the framework of nonrelativistic Quantum Mechanics (QM) and relativistic Quantum Field Theory (QFT), we consider the effect of pulsed, ideal measurements repeated at equal time intervals on the lifetime of an unstable system. In particular, we investigate the case in which the ‘bare’ survival probability is an exact exponential (a very good approximation in both QM and QFT), but the measurement apparatus can detect the decay products only in a certain energy range. We show that the Quantum Zeno Effect can occur in this framework as well.

After reviewing the description of an unstable state in the framework of nonrelativistic Quantum Mechanics (QM) and relativistic Quantum Field Theory (QFT), we consider the effect of pulsed, ideal measurements repeated at equal time intervals on the lifetime of an unstable system. In particular, we investigate the case in which the ‘bare’ survival probability is an exact exponential (a very good approximation in both QM and QFT), but the measurement apparatus can detect the decay products only in a certain energy range. We show that the Quantum Zeno Effect can occur in this framework as well.

The quantum Zeno effect--suppression of decay by frequent measurements--was believed to occur only when the response of the detector is so quick that the initial tiny deviation from the exponential decay law is detectable. However, we show that it can occur even for exactly exponentially decaying systems, for which this condition is never satisfied, by considering a realistic case where the detector has a finite energy band of detection. The conventional theories correspond to the limit of an infinite bandwidth. This implies that the Zeno effect occurs more widely than expected thus far.

In this paper measurement is defined via an observation operator with simple, purely discrete spectrum. When the quantum Zeno effect holds, a continuously measured quantum process with fixed initial state ‘‘freezes’’ in that state. Such states are called ‘‘regular’’ states. If the final state is also fixed, then movement from the initial state to the final state is forced to occur, whether or not the initial state is regular. This forced movement is studied here in the context of a discrete state model which contains a distinguished nonregular state which mediates all transitions. In this model the forced movement between any two states satisfies a ‘‘least action’’ principle, wherein action is identified with change of state transitions. For completeness, the model is also studied when only the initial state is fixed under continuous measurement. In that case the distinguished state exhibits exact exponential decay at all times. The model is of interest independently of issues related to continuous measurement inasmuch as it is a ‘‘discrete’’ approximation to a ‘‘continuous’’ decay model. More precisely, the distinguished state exhibits exact exponential decay on a finite time interval which expands without bound as the discrete decay products densely approach the continuum. This unexpected result provides a striking confirmation of Fermi’s ‘‘golden rule.’’