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Scientific Reports

The quantum Zeno and anti-Zeno effects with non-selective projective measurements

Mehwish Majeed, Adam Zaman Chaudhry

Published Oct 1, 2018• 13 citations
Publisher

Abstract

In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective projective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such nonselective measurements. Importantly, we show that there is a difference between using non-selective projective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective projective measurements can differ very significantly as compared to the repeated selective measurement scenario.

DOI
10.1038/s41598-018-33181-9

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The quantum Zeno and anti-Zeno effects with non-selective measurements

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In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective projective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such nonselective measurements. Importantly, we show that there is a difference between using non-selective projective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective projective measurements can differ very significantly as compared to the repeated selective measurement scenario.
In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such non-selective measurements. Importantly, we show that there is a difference between using non-selective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems, and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective measurements can differ very significantly as compared to the repeated selective measurement scenario.
A general treatment of the quantum Zeno and anti-Zeno effects is presented which is valid for an arbitrary system-environment model in the weak system-environment coupling regime. It is shown that the effective lifetime of a quantum state that is subjected to repeated projective measurements depends on the overlap of the spectral density of the environment and a generalized `filter function' which in turn depends on the system state that is repeatedly being prepared, the measurement interval, the system parameters, the system-environment coupling, and the environment correlation function. This general framework is then used to study explicitly the Zeno to anti-Zeno crossover behavior for the spin-boson model where a single two-level system is coupled to a bosonic environment. It is shown that our framework reproduces results for the usual population decay case as well as for the pure dephasing model, while at the same time allowing us to study the Zeno to anti-Zeno transition when both decay and dephasing take place. These results are then extended to many two-level systems coupled collectively to the bosonic environment to show that the distinction between the Zeno and anti-Zeno effects becomes more pronounced as the number of two-level systems increases.
Repeated measurements can slow down (the quantum Zeno effect) or speed up (the quantum anti-Zeno effect) the temporal evolution of a quantum system. In this paper, a general treatment of the quantum Zeno and anti-Zeno effects is presented which is valid for an arbitrary system-environment model in the weak system-environment coupling regime. It is shown that the effective lifetime of a quantum state that is subjected to repeated projective measurements depends on the overlap of the spectral density of the environment and a generalized 'filter function'. This filter function depends on the system-environment Hamiltonian, the state of the environment, and the measurement being performed. Our general framework is then used to study explicitly the Zeno to anti-Zeno crossover behaviour for the spin-boson model where a single two-level system is coupled to a bosonic environment. It is possible to not only reproduce results for the usual population decay case as well as for the pure dephasing model, but to also study the regime where both decay and dephasing take place. These results are then extended to many two-level systems coupled collectively to the bosonic environment to further illustrate the importance of the correct evaluation of the effective decay rate.
Measurements in quantum mechanics can not only effectively freeze the quantum system (the quantum Zeno effect) but also accelerate the time evolution of the system (the quantum anti-Zeno effect). In studies of these effects, a quantum state is prepared repeatedly by projecting the quantum state onto the initial state. In this paper, we repeatedly prepare the initial quantum state in a different manner. Instead of only performing projective measurements, we allow unitary operations to be performed, on a very short time-scale, after each measurement. We can then repeatedly prepare the initial state by performing some projective measurement and then, after each measurement, we perform a suitable unitary operation to end up with the same initial state as before. Our objective is to find the projective measurements that minimize the effective decay rate of the quantum state. We find such optimal measurements and the corresponding decay rates for a variety of system-environment models such as the pure dephasing model and the spin-boson model. We find that there can be considerable differences between this optimized effective decay rate and the usual decay rate obtained by repeatedly projecting onto the initial state. In particular, the Zeno and anti-Zeno regimes can be considerably modified.
The fact that repeated projective measurements can slow down (the Zeno effect) or speed up (the anti-Zeno effect) quantum evolution is well-known. However, to date, studies of these effects focus on quantum systems that are weakly interacting with their environment. In this paper, we investigate what happens to a quantum system under the action of repeated measurements if the quantum system is strongly interacting with its environment. We consider as the quantum system a single two-level system coupled strongly to a collection of harmonic oscillators. A so-called polaron transformation is then used to make the problem in the strong system-environment coupling regime tractable. We find that the strong coupling case exhibits quantitative and qualitative differences as compared with the weak coupling case. In particular, the effective decay rate does not depend linearly on the spectral density of the environment. This then means that, in the strong coupling regime that we investigate, increasing the system-environment coupling strength can actually decrease the effective decay rate. We also consider a collection of two-level atoms coupled strongly with a common environment. In this case, we find that there are further differences between the weak and strong coupling cases since the two-level atoms can now indirectly interact with one another due to the common environment.
Quantum Zeno and anti-Zeno effects on pure dephasing are studied using exactly solvable microscopic models. The crossover between these two opposite effects is investigated. The case of a single two-level system undergoing dephasing is already different from the previously studied population-decay problem, even without taking into account any back action from the environment. For many two-level systems interacting with a common environment, multiple transitions between Zeno and anti-Zeno regimes are predicted. Finally, if the system-environment coupling strength is not weak, we show that the nontrivial evolution of the environment between measurements can considerably alter the quantum Zeno and anti-Zeno effects. This is qualitatively different from population-decay models where the environment state is reset to equilibrium after each measurement.