We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region---this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [Phys. Rev. A 88, 022128 (2013)], demonstrating the suppression of quantum-mechanical reflection by decoherence.
We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region---this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the \ensuremathℏ\ensuremath→0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [Phys. Rev. A 88, 022128 (2013)], demonstrating the suppression of quantum-mechanical reflection by decoherence.
In quantum mechanics an incoming particle wave packet with sufficient energy will undergo both transmission and reflection when encountering a barrier of lower energy, but in classical mechanics there is no reflection, only transmission. In this paper we seek to explain the disappearance of quantum-mechanical reflection in the quasiclassical limit, using the standard machinery of decoherence through environmental interaction. We consider two models. In the first, the incoming particle is classicalized by coupling to an environment and modeled using a standard master equation of Lindblad form with Lindblad operator proportional to position (the simplest version of quantum Brownian motion). We find, however, that suppression of reflection is achieved only for environmental interaction so strong that large fluctuations in momentum are generated, which blurs the distinction between incoming and reflected wave packets. This negative conclusion also holds for a complex potential which has similar implications for attempts to understand the suppression of the Zeno effect using the same mechanism (discussed in more detail elsewhere). A different master equation with Lindblad operator proportional to momentum is shown to be successful in suppressing reflection without large fluctuations but such a master equation is unphysical. We consider a second model in which the barrier is modeled quantum mechanically by a massive target particle coupled to an environment to maintain it in a quasiclassical state. This avoids the fluctuations problem since the incoming particle is not coupled to the environment directly. We find that reflection is significantly suppressed as long as the decoherence time scale of the target particle is much smaller than certain characteristic scattering time scales of the incoming particle, or equivalently, as long as the velocity fluctuations in the target are larger than the velocity of the incoming particle.
The quantum Zeno effect arises due to frequent observations. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely what is the probability of a system, remaining always in a particular subspace. This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases the quantum Zeno effect is still present.
The quantum Zeno effect arises due to frequent observation. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely: "what is the probability of a system, remaining always in a particular subspace". This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, and in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases, the quantum Zeno effect is still present.
The quantum Zeno effect is a distinctive phenomenon in quantum mechanics, describing the nontrivial effect of frequent projective measurements on hindering the evolution of a quantum system. However, when subjected to environmental noise, the quantum system may dissipate, and the quantum Zeno effect no longer works. This research starts from the physical mechanism for the decay of the quantum Zeno effect in the presence of noise and investigates the effect of coherent quantum controls on mitigating the decrease of the survival probability that the system stays in the initial state induced by the noise. We derive the decay rate of the survival probability with and without coherent quantum controls in general, and show that when the frequency of the projective measurements is large but finite, proper coherent controls by sufficiently strong Hamiltonians can be designed to decrease the decay rate of the survival probability. A two-level quantum system suffering from typical unitary and nonunitary noise is then considered to demonstrate the effect of the proposed coherent quantum control scheme in protecting the quantum Zeno effect against the noise. The decay rate of the survival probability is obtained in the presence of noise, and the control Hamiltonian is further optimized analytically to minimize the decay rate by a variational approach. The evolution paths of the quantum system with the optimal coherent controls are illustrated numerically for different scenarios to explicitly show how the coherent control scheme works in lowering the decay of survival probability.
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/E, where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/ E , where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.