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

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.

Quantum Zeno effect appears in stages

“…In this regime, as a result of repeated measurements, the system state is mostly frozen next to one of the measurement eigenstates, yet rarely performs quantum jumps between them. The crossover between coherent oscillations and the Zeno regime is controlled by the frequency of the measurement and has been extensively explored both theoretically [29][30][31][32][33][34] and experimentally [35][36][37][38][39]. Beyond projective measurements, the onset of the Zeno regime is richer [40,41], and quantum jumps appear as part of continuous stochastic dynamics.…”

The quantum Zeno and anti-Zeno effects with driving fields in the weak and strong coupling regimes

“…We also note that there are different ways to define the survival probability and hence the decay rate, as well as different ways of identifying the Zeno and anti-Zeno regimes. For example, one can also look at the history of measurements 74 , 75 when calculating the survival probability 34 . Similarly, we identify the Zeno and anti-Zeno regimes by looking at when the decay rate is an increasing function (the Zeno regime) or a decreasing function (the anti-Zeno regime) 46 ; an alternative approach is to compare the measurement modified decay rate with the decay rate without measurement 76 .…”

Quantum Zeno effect appears in stages

“…Therefore, it is more convenient to use f γ (θ) in the first form form in Eq. (28). At the same time, the integrals for the norm and the boundary condition at θ = π are more conveniently calculated using the expressions in Eq.…”

“…In this regime, as a result of repeated measurements, the system state is mostly frozen next to one of the measurement eigenstates, yet rarely performs quantum jumps between them. The crossover between coherent oscillations and the Zeno regime is controlled by the frequency of the measurement and has been extensively explored both theoretically [23][24][25][26][27][28] and experimentally [29][30][31][32][33]. Beyond projective measurements, the onset of the Zeno regime is richer [34,35], and quantum jumps appear as part of continuous stochastic dynamics.…”