The change of dynamics in a quantum system under frequent or continuous observation, known as the quantum Zeno effect, is generally derived from the projection or reduction of the wave-packet hypothesis that is the central postulate in the theory of quantum measurements. The only experiment in which the Zeno effect has yet been clearly demonstrated, though, allows no conclusion on the necessity or validity of the projection postulate. This is shown by calculating, in detail, the outcome of the experiment on the basis of the standard three-level Bloch equations. These equations follow from the quantum theory of irreversible processes with no additional assumptions necessary, such as which part of the system serves as measuring apparatus or how efficient the measurement would be.
The change of dynamics in a quantum system under frequent or continuous observation, known as the quantum Zeno effect, is generally derived from the projection or reduction of the wave-packet hypothesis that is the central postulate in the theory of quantum measurements. The only experiment in which the Zeno effect has yet been clearly demonstrated, though, allows no conclusion on the necessity or validity of the projection postulate. This is shown by calculating, in detail, the outcome of the experiment on the basis of the standard three-level Bloch equations. These equations follow from the quantum theory of irreversible processes with no additional assumptions necessary, such as which part of the system serves as measuring apparatus or how efficient the measurement would be.
A purely dynamical explanation for the quantum Zeno effect is proposed. It is argued that a quantum system undergoes a quantum-Zeno-type dynamics as a consequence of a particular type of evolution involving a series of frequent spectral decompositions. The role of quantum measurements and of the ``collapse of the wave function'' is investigated and it is clarified that, provided a final observation is performed, the dynamical quantum Zeno effect can be obtained without making use of von Neumann's projection postulate. The meaning of infinitely frequent measurements is critically discussed and it is argued that it should be regarded as a mathematical idealization, impossible to realize from a physical point of view.
This is a theoretical thesis in the area of quantum measurement theory. Due to the extensive breadth of this field we choose to narrow our focus to examine a particular problem - the quantum Zeno effect (defined below). Quantum measurement theory is introduced in Chap. 1, using the terminology of effects and operations. This approach allows an operational definition of such terms as a state vector, an ensemble, and a measurement device (for instance), and a consideration of interactions between quantum systems and inaccurate measurement devices. We further introduce the quantum trajectories approach to consider the evolution of an individual quantum system subject to measurement. The quantum Zeno effect is introduced in Chap. 2. Any quantum treatment of a measurement interaction must consider the measurement backaction onto the measured system and this backaction will disrupt the free evolution of the system. The quantum Zeno effect occurs in the strong measurement limit where the measurement backaction totally freezes the evolution of the system, thus rendering the measurement useless. The effect is introduced via projective measurements of two level systems subject to measurement of level populations. At this stage we are able to discuss the main questions addressed by this thesis, and present its structure in Chap. 2. We then develop a new measurement model for the interaction between a system and a measurement device in Chap. 3. Our motivation in doing this is to better model the usual laboratory meter, and in our approach the meter dynamics are such that it relaxes towards an appropriate readout of the system parameter of interest. The irreducible quantum noise of the meter introduces fluctuations that drive the stochastic dynamical collapse of the system wavefunction. In our model, the measured system dynamics (if treated selectively) are described by a stochastic, nonlinear Schroedinger equation. A double well system subject to position measurement provides a natural first application for this model. This is done in Chap. 4 where we monitor the coherent tunnelling of a particle from one well to the other. The advantage afforded by considering this system is that it displays differing regimes where the measurement observable (position) is approximated as possessing either, respectively, a continuous or a discrete eigenvalue structure. Thus, we use this one model to explore the quantum Zeno effect in both measurement regimes. The above treatment is of a theoretical measurement model. In Chap. 5 we turn to consider a recent experimental test of the quantum Zeno effect which examined the dynamics of a two level atom subject to pulsed measurements of atomic level populations. We treat a slightly modified experiment in a fully continuous measurement regime. By first unravelling the optical Bloch equations, and second, using the quantum trajectories approach we demonstrate the existence of certain measurement regimes where there is a quantum Zeno effect, and other regimes where no measurement of the atomic populations is being effected at all. Through these results we demonstrate the importance of making a full analysis of the system-detector interaction before any conclusions can be made. In the remainder of the thesis we propose further possible tests of the quantum Zeno effect. In Chap. 6 the evolution of a Rydberg atom exchanging one photon with a single cavity mode subject to measurement is examined. The measurement is made by monitoring the photon number occupancy of the cavity mode using a beam of Rydberg atoms configured so as to perform phase sensitive detection. In the limit of frequent monitoring we show that the free oscillation of the atomic inversion is disrupted, and the atom is trapped close to its initial state. This is the quantum Zeno effect. In Chap. 7 we realize the Zeno effect on two possible systems. We consider first, a two level Jaynes-Cumming atom interacting with a cavity mode, and second, two electromagnetic modes configured as a multi-level parametric frequency converter. These systems interact with another cavity mode via a quadratic coupling system based on four wave mixing, and constructed to be a quantum nondemolition measurement of the photon number. This mode is damped to the environment thus effecting a measurement of the system populations. Again we show that this interaction, can manifest the quantum Zeno effect. Our explicit modelling of the system-detector interaction enables us to show how the effect depends on the resolution time of the detector. Finally, we consider a proposed measurement of the square of the quadrature phase of an electromagnetic mode in Chap. 8. Here, a three mode interaction mediated by a second order nonlinear susceptibility is considered. One mode, the pump, is prepared in a feedback generated photon number state to give insight into the role of pump noise. The other two modes are treated as an angular momentum system, and we show that photon counting on the two mode rotation system effects the above mentioned measurement. In addition, this measurement provides a direct measure of the second order squeezing of the signal. With that we finish our investigation of the quantum Zeno effect using the techniques of quantum measurement theory. However, in the epilogue [Chap. 9] we note that no thesis in quantum measurement theory would be complete without some consideration of the ``meaning attributed to the theory. In the epilogue we take a novel historical approach and examine the method by which metaphysical theories are formed to draw conclusions regarding quantum metaphysics.
The projection postulate has been used to predict a slow-down of the time evolution of the state of a system under rapidly repeated measurements, and ultimately a freezing of the state. To test this so-called quantum Zeno effect an experiment was performed by Itano et al. [Phys. Rev. A 41, 2295 (1990)] in which an atomic-level measurement was realized by means of a short laser pulse. The relevance of the results has given rise to controversies in the literature. In particular, the projection postulate and its applicability in this experiment have been cast into doubt. In this paper we show analytically that for a wide range of parameters, such a short laser pulse acts as an effective level measurement to which the usual projection postulate applies with high accuracy. The corrections to the ideal reductions and their accumulation over n pulses are calculated. Our conclusion is that the projection postulate is an excellent pragmatic tool for a quick and simple understanding of the slow-down of time evolution in experiments of this type. However, corrections have to be included, and an actual freezing does not seem possible because of the finite duration of measurements. \textcopyright{} 1996 The American Physical Society.
Itano and co-workers [Wayne M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990)] have recently reported the experimental verification of the quantum Zeno effect, which is the inhibition of a quantum transition by frequent measurements. In this article, we offer an alternative interpretation of the quantum Zeno effect. We show that an analysis of the dynamics of the full three-level system gives the same result. There is no need to assume explicitly that the wave function has collapsed, nor even to assume that an ideal measurement has been made. In addition, we differentiate between what has been referred to as the quantum Zeno effect and what has been termed the quantum Zeno paradox. The former is the inhibition of induced transitions, and the latter is the, as yet experimentally unobserved, inhibition of spontaneous decay. Our interpretation, which emphasizes the ``measurement''-induced interruption of atomic-state coherences as the cause of inhibited quantum transitions, suggests a resolution to the quantum Zeno paradox. The theoretical limit of continuous observation is discussed.
Quantum Zeno effect is conventionally interpreted by the assumption of the wave-packet collapse, in which does not involve the duration of measurement. However, we predict duration τm of each measurement will appear in quantum Zeno effect by a dynamical approach. Moreover, there exists a model-free critical measurement time, which quantum Zeno effect does not occur when τm takes some special values. In order to give these predictions, we first present a description of quantum Zeno effect in the Heisenberg picture, which is based on the expectation value of an observable and its fluctuation. Then we present a general proof for quantum Zeno effect in the Heisenberg picture, which is independent of the concrete systems. Finally, we calculate the average population and relative fluctuation after N successive measurements in XX model, which agrees with our prediction about the critical measurement time.