Conditions which may lead to a freezing of the motion of a system under continuous observation (the so-called "Zeno paradox" or "watchdog effect") are examined. The measurement process is treated phenomenologically by the usual wave-packet reduction as well as in a more realistic way by including the measuring apparatus. For this purpose a model for an ideal measurement process is employed, following an example given by von Neumann. The resulting behavior varies between complete freezing and a mere suppression of interference terms and constant transition rates as represented by a master equation (rate equation). The most familiar example of the latter is Fermi's golden rule, with integration leading to exponential decay. Reviewing and extending the derivation of the Pauli master equation, the conditions leading to constant transition rates are discussed. The importance of the interaction with the natural environment for establishing a master equation is emphasized. Some consequences for the derivation of macroscopic equations of motion and for the physical foundations of superselection rules are pointed out.
Conditions which may lead to a freezing of the motion of a system under continuous observation (the so-called "Zeno paradox" or "watchdog effect") are examined. The measurement process is treated phenomenologically by the usual wave-packet reduction as well as in a more realistic way by including the measuring apparatus. For this purpose a model for an ideal measurement process is employed, following an example given by von Neumann. The resulting behavior varies between complete freezing and a mere suppression of interference terms and constant transition rates as represented by a master equation (rate equation). The most familiar example of the latter is Fermi's golden rule, with integration leading to exponential decay. Reviewing and extending the derivation of the Pauli master equation, the conditions leading to constant transition rates are discussed. The importance of the interaction with the natural environment for establishing a master equation is emphasized. Some consequences for the derivation of macroscopic equations of motion and for the physical foundations of superselection rules are pointed out.
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.’’
We modify the theory of the Quantum Zeno Effect to make it consistent with the postulates of quantum mechanics. This modification allows one, throughout a sequence of observations of an excited system, to address the nature of the observable and thereby to distinguish survival from non-decay, which is necessary whenever excited states are degenerate. As a consequence, one can determine which types of measurements can possibly inhibit the exponential decay of the system. We find that continuous monitoring taken as the limit of a sequence of ideal measurements will only inhibit decay in special cases, such as in well-controlled experiments. Uncontrolled monitoring of an unstable system, however, can cause exponentially decreasing non-decay probability at all times. Furthermore, calculating the decay rate for a general sequence of observations leads to a straightforward derivation of Fermi's Golden Rule, that avoids many of the conceptual difficulties normally encountered. When multiple decay channels are available, the derivation reveals how the total decay rate naturally partitions into a sum of the decay rates for the various channels, in agreement with observations. Continuous and unavoidable monitoring of an excited system by an uncontrolled environment may therefore be a mechanism by which to explain the exponential decay law.
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.
The evolution of a quantum system undergoing very frequent measurements takes place in a proper subspace of the total Hilbert space (quantum Zeno effect). When the measuring apparatus is included in the quantum description, the Zeno effect becomes a pure consequence of the dynamics. We show that for continuous measurement processes the quantum Zeno evolution derives from an adiabatic theorem. The system is forced to evolve in a set of orthogonal subspaces of the total Hilbert space and a dynamical superselection rule arises. The dynamical properties of this evolution are investigated and several examples are considered.
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 evolution of a quantum system undergoing very frequent measurements takes place in a proper subspace of the total Hilbert space (quantum Zeno effect). When the measuring apparatus is included in the quantum description, the Zeno effect becomes a pure consequence of the dynamics. We show that for continuous measurement processes the quantum Zeno evolution derives from an adiabatic theorem. The system is forced to evolve in a set of orthogonal subspaces of the total Hilbert space and a dynamical superselection rule arises. The dynamical properties of this evolution are investigated and several examples are considered. 1