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arXiv:quant-ph/9908054 · Physical Review Letters

Effect of Measurement on the Decay Rate of a Quantum System

B. Elattari, S. A. Gurvitz

Published Mar 6, 2000• 45 citations
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Abstract

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.

DOI
10.1103/physrevlett.84.2047

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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 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 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 consider a qubit interacting with its environment and continuously monitored by a detector represented by a point contact. Bloch-type equations describing the entire system of the qubit, the environment, and the detector are derived. Using these equations we evaluate the detector current and its noise spectrum in terms of the decoherence and relaxation rates of the qubit. Simple expressions are obtained that show how these quantities can be accurately measured. We demonstrate that due to interaction with the environment, the measurement can never localize a qubit even for infinite decoherence rate.
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 have investigated how the decay dynamics of an unstable quantum system is affected by a false measurement, where the decay is monitored by detecting a decay product but the active energy band of the detector does not match the energy of the decay product. It is shown that, although such a measurement is ineffective and has almost no effect if the detector response is slow, the detectability of decay is increased and the decay is accelerated considerably if the response is fast. This is due to the new decay channel, which is generated as a counteraction of measurement.
We consider the continuous measurement of a double quantum dot by a weakly coupled detector (tunnel point contact nearby). While the conventional approach describes the gradual system decoherence due to the measurement, we study the situation when the detector output is explicitly recorded that leads to the opposite effect: gradual purification of the double-dot density matrix. The nonlinear Langevin equation is derived for the random evolution of the density matrix which is reflected and caused by the stochastic detector output. Gradual collapse, gradual purification, and the quantum Zeno effect are naturally described by the equation. We also discuss the possible experiments to confirm the theory.