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Physical Review A

Zeno and anti-Zeno effects in two-level systems

Alfredo Luis

Published Jun 30, 2003• 22 citations
Publisher

Abstract

We examine the appearance of Zeno and anti-Zeno effects when the unitary evolution of a two-level system is observed. We demonstrate that, contrary to previous claims, the anti-Zeno effect can occur in two-level systems so that the observation can enhance the transition probability. Moreover, for nonresonant couplings we show that the Zeno and anti-Zeno effects always appear sequentially in the evolution of the same system: there is Zeno effect in the short-time regime and anti-Zeno effect in the long-time regime.

DOI
10.1103/physreva.67.062113

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We examine the appearance of Zeno and anti-Zeno effects when the unitary evolution of a two-level system is observed. We demonstrate that, contrary to previous claims, the anti-Zeno effect can occur in two-level systems so that the observation can enhance the transition probability. Moreover, for nonresonant couplings we show that the Zeno and anti-Zeno effects always appear sequentially in the evolution of the same system: there is Zeno effect in the short-time regime and anti-Zeno effect in the long-time regime.
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.
The effect of the anti-rotating terms on the short-time evolution and the quantum Zeno (QZE) and anti-Zeno (AQZE) effects is studied for a two-level system coupled to a bosonic environment. A unitary transformation and perturbation theory are used to obtain the electron self-energy, energy shift and the enhanced QZE or the AQZE, simultaneously. The calculated Zeno time depends on the atomic transition frequency sensitively. When the atomic transition frequency is smaller than the central frequency of the spectrum of boson environment, the Zeno time is prolonged and the anti-rotating terms enhance the QZE; when it is larger than that the Zeno time is reduced and the anti-rotating terms enhance the AQZE. PACS numbers: 03.65.Xp; 42.50.Ct; 03.65.Yz 1 The quantum Zeno effect (QZE) and anti-Zeno effects (AQZE) have been widely discussed for decades theoretically [** * early papers and recent papers] and recently experimentally [***the paper of Ref. B and others]. In an unstable quantum twolevel (multi-level) system, frequently measurement can reduce or accelerate the decay
Repeated measurements in quantum mechanics can freeze (the quantum Zeno effect) or enhance (the quantum anti-Zeno effect) the time-evolution of a quantum system. In this paper, we present a general treatment of the quantum Zeno and anti-Zeno effects for arbitrary driven open quantum systems, assuming only that the system-environment coupling is weak. In particular, we obtain a general expression for the effective decay rate of a two-level system subjected to arbitrary driving fields as well as periodic measurements. We demonstrate that the driving fields change the decay rate, and hence the quantum Zeno and anti-Zeno behavior, both qualitatively and quantitatively. We also extend our results to systems consisting of more than one two-level system, as well as a two-level system strongly coupled to an environment of harmonic oscillators, to further illustrate the non-trivial effect of the driving fields on the quantum Zeno and anti-Zeno effects.
We examine the appearance of Zeno and anti-Zeno effects (Misra and Sudarshan 1977 J. Math. Phys. 18 756; Kofman and Kurizki 2000 Nature 405 546; Kofman and Kurizki 1996 Phys. Rev. A 54 3750; Facchi et al 2001 Phys. Rev. Lett. 86 2699) in the entanglement dynamics of two qubits off-resonantly coupled to the same lossy cavity when the unitary evolution of the system is interrupted by repeated projective measurements. We describe in detail these quantum effects by comparing the measurement-induced coarse-grained dynamics to the entanglement evolution in the absence of measurements in several scenarios (Francica et al 2009 Phys. Rev. A 79 032310). In particular, we examine the strong and weak coupling regimes, the role of the relative coupling strengths between the two qubits and the reservoir and the effect of the detuning from the main cavity frequency. We show that the anti-Zeno effect can occur in the entanglement dynamics when the qubit frequencies are detuned from the main reservoir frequency. Furthermore, we find that Zeno and anti-Zeno effects can even appear sequentially many times as a function of the interval between the measurements. Finally, we show that, in the off-resonant regime, we can preserve the entanglement using the quantum Zeno effect more efficiently than in the resonant limit (Maniscalco et al 2008 Phys. Rev. Lett. 100 090503), even if, in this case, no sub-radiant state exists.
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.
In this work, we study the decay behavior of a two-level system under the competing influence of a dissipative environment and repetitive measurements. The sign of the second derivative of the environmental spectral density function with respect to the system transition frequency is found to be a sufficient condition to distinguish between the quantum Zeno (negative) and the anti-Zeno (positive) effects raised by the measurements. We check our criterion for practical measurement intervals, which are larger than the conceptual Zeno time, in various environments. In particular, with the Lorentzian spectrum, the quantum Zeno and anti-Zeno phenomena are found to emerge respectively in the near-resonant and off-resonant cases. For the interacting spectra of hydrogenlike atoms, the quantum Zeno effect usually occurs and the anti-Zeno effect can rarely occur unless the transition frequency is close to the cut-off frequency. With a power-law spectrum, we find that sub-Ohmic and super-Ohmic environments lead to the quantum Zeno and anti-Zeno effects, respectively.