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.’’
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 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.
If one continuously measures a decaying system, the system will appear to never decay that was called quantum Zeno effect. The continuous measurement is defined by a sequence of measurements whose time interval t between measurements approaches zero. Later many works chose the time interval t as finite (and greater than the Zeno time) which corresponds to making equal spaced measurements over a discrete time interval. With the discrete variable formulism one can derive the so-called Anti-Zeno effect. Our study is trying to contrast the results between continuous time interval measurement versus discrete time interval measurement. We demonstrate that we can obtain so-called “Zeno” and “Anti-Zeno” in a classical system if we apply the definition of non-ideal measurement.
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.
We develop a general framework for characterizing the response of an evolving quantum system to repetitive quantum measurements. Modeling each evolution-measurement cycle as a quantum channel induced by an effective Liouvillian generator, we find that the Liouvillian spectral gap determines the measurement-induced decay rate. We analyze how the spectral gap responds to the measurement frequency, and define a quantum Zeno response as a decrease in the gap with increasing measurement frequency, and an anti-Zeno response as the opposite. We illustrate this criterion for both discrete-time and continuous-time quantum measurements. In an exactly solvable discrete-time qubit model, the exceptional-point spectral coalescence or spectral crossings mark the transition between Zeno and anti-Zeno responses, which can be experimentally distinguished from the long-time decay envelope of the survival probability. In a continuous-time superconducting-qubit defect model, the same transition manifests as smooth extrema of the spectral gap. Our results establish a universal spectral criterion for measurement-induced decay, offering a practical route to identify and manipulate these effects in generic quantum systems.
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.
Abstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resu