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arXiv:quant-ph/0001020 · Physical Review A

Influence of measurement on the lifetime and the linewidth of unstable systems

B. Elattari, S. A. Gurvitz

Published Aug 8, 2000• 48 citations
arXivHTMLPDF

Abstract

We investigate the quantum Zeno effect in the case of electron tunneling out of a quantum dot in the presence of continuous monitoring by a detector. It is shown that the Schr\"odinger equation for the whole system can be reduced to Bloch-type rate equations describing the combined time development of the detector and the measured system. Using these equations we find that continuous measurement of the unstable system does not affect its exponential decay to a reservoir with a constant density of states. The width of the energy distribution of the tunneling electron, however, is not equal to the inverse lifetime---it increases due to the decoherence generated by the detector. We extend the analysis to the case of a reservoir described by an energy-dependent density of states, and we show that continuous measurement of such quantum systems affects both the exponential decay rate and the energy distribution. The decay does not always slow down, but might be accelerated. The energy distribution of the tunneling electron may reveal the lines invisible before the measurement.

DOI
10.1103/physreva.62.032102

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We investigate the quantum Zeno effect in the case of electron tunneling out of a quantum dot in the presence of continuous monitoring by a detector. It is shown that the Schr"odinger equation for the whole system can be reduced to Bloch-type rate equations describing the combined time development of the detector and the measured system. Using these equations we find that continuous measurement of the unstable system does not affect its exponential decay to a reservoir with a constant density of states. The width of the energy distribution of the tunneling electron, however, is not equal to the inverse lifetime---it increases due to the decoherence generated by the detector. We extend the analysis to the case of a reservoir described by an energy-dependent density of states, and we show that continuous measurement of such quantum systems affects both the exponential decay rate and the energy distribution. The decay does not always slow down, but might be accelerated. The energy distribution of the tunneling electron may reveal the lines invisible before the measurement.
We investigated the electron tunneling out of a quantum dot in the presence of continuous monitoring by a detector. It is shown that the Schrodinger equation for the whole system can be reduced to new Bloch-type rate equations describing the time development of the detector and the measured system at once. Using these equations we find that the continuous measurement of the unstable system does not affect its exponential decay, exp(-gammat), contrary to expectations based on the quantum Zeno effect. However, the width of the energy distribution of the tunneling electron is no longer gamma, but increases due to the decoherence, generated by the detector.
We study the dynamics of the measurement process in quantum-dot systems, where a particular state out of coherent superposition is observed. The ballistic point contact placed near one of the dots is taken as a noninvasive detector. We demonstrate that the measurement process is fully described by the Bloch-type equations applied to the whole system. These equations clearly reproduce the collapse of the density matrix into the statistical mixture in the course of the measurement process. The corresponding dephasing width is uniquely defined. We show that the continuous observation of one of the states in a coherent superposition may accelerate decay from this state---in contradiction with rapidly repeated observations, which slow down the transitions between quantum states (the quantum Zeno effect).
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.
We consider non‐selective continuous measurements of a particle tunneling to a reservoir of finite band‐width (Λ). The particle is continuously monitored by frequent projective measurements (“quantum trajectory”), separated by a time‐interval τ. A simple analytical expression for the decay rate has been obtained. For Markovian reservoirs ( ), no effect of the measurements is found. Otherwise for a finite Λ, the decay rate always depends on the measurement time τ. This result is compared with alternative calculations, with no intermediate measurements, but when the measurement device is included in the Schrödinger evolution. We found that the detector affects the system by the decoherence rate ( ), related to the detector's signal. Although both treatments are different, the final results become very close for . This τ corresponds to the minimal time for which the detector's signal can be distinguished by an “observer”. This indicates a fundamental role of information in quantum motion and can be used for the extension of the quantum trajectory method for non‐Markovian environments.
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.