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Physical Review A

Anti-Zeno-effect recovery and Lamb-shift modification in modified vacuum

Da‐Wei Wang, Li‐Gang Wang, Zhenghong Li, Shi‐Yao Zhu

Published Oct 1, 2009• 13 citations
Publisher

Abstract

The influence of the modification of the free vacuum on the quantum Zeno and anti-Zeno effects, as well as the Lamb (energy) shift of a multilevel atom, is studied. A unitary transformation method has been used to include the counter-rotating terms and to obtain the ground state of the whole Hamiltonian. The anti-Zeno effect is recovered if the modified density of states (DOS) has a dip near the transition frequency. The modification of DOS by a dip or a peak will change the Lamb shift compared with that in the free vacuum. Together with our unitary transformation method, there are three different methods to obtain the change in the Lamb shift. Using our method we investigate the change in the Lamb shift and obtain a formula for the change, which is the same as obtained with the method of Louisell [Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] but different from the method of Cohen-Tannoudji et al. [Atom-Photon Interactions: Basic Processes and Applications (Wiley, New York, 1992), pp. 317 and 417]. The correctness of the Lamb shift formulas obtained by the three methods can be easily testified experimentally.

DOI
10.1103/physreva.80.042101

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The influence of the modification of the free vacuum on the quantum Zeno and anti-Zeno effects, as well as the Lamb (energy) shift of a multilevel atom, is studied. A unitary transformation method has been used to include the counter-rotating terms and to obtain the ground state of the whole Hamiltonian. The anti-Zeno effect is recovered if the modified density of states (DOS) has a dip near the transition frequency. The modification of DOS by a dip or a peak will change the Lamb shift compared with that in the free vacuum. Together with our unitary transformation method, there are three different methods to obtain the change in the Lamb shift. Using our method we investigate the change in the Lamb shift and obtain a formula for the change, which is the same as obtained with the method of Louisell [Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] but different from the method of Cohen-Tannoudji et al. [Atom-Photon Interactions: Basic Processes and Applications (Wiley, New York, 1992), pp. 317 and 417]. The correctness of the Lamb shift formulas obtained by the three methods can be easily testified experimentally.
A unitary transformation method is used to investigate the dynamic evolution of two multilevel atoms, in the basis of symmetric and antisymmetric states, with one atom being initially prepared in the first excited state and the other in the ground state. The unitary transformation guarantees that our calculations are based on the ground state of the atom-field system and the self-energy is subtracted at the beginning. The total Lamb shifts of the symmetric and antisymmetric states are divided into transformed shift and dynamic shift. The transformed shift is due to emitting and reabsorbing of virtual photons, by a single atom (nondynamic single atomic shift) and between the two atoms (quasi-static shift). The dynamic shift is due to the emitting and reabsorbing of real photons, by a single atom (dynamic single atomic shift) and between the two atoms (dynamic interatomic shift). The emitting and reabsorbing of virtual and real photons between the two atoms result in the interatomic shift, which does not exist for the one-atom case. The spectra at the long-time limit are calculated. If the distance between the two atoms is shorter than or comparable to the wavelength, the strong coupling between the two atoms splits the spectrum into two peaks, one from the symmetric state and the other from the antisymmetric state. The origin of the red or blue shifts for the symmetric and antisymmetric states mainly lies in the negative or positive interaction energy between the two atoms. In the investigation of the short time evolution, we find the modification of the effective density of states by the interaction between two atoms can modulate the quantum Zeno and quantum anti-Zeno effects in the decays of the symmetric and antisymmetric states.
The spontaneous decay of a multilevel atom interacting with the electromagnetic field in free space is investigated with a unitary transformation method, which is introduced in order to include all rotating and counter-rotating terms in the Hamiltonian. By using the ground state of the total Hamiltonian, the evolution of the effective decay rate and the energy shift are calculated. When the atomic transition frequency is smaller than the central frequency of the spectrum, the Zeno effect dominates, and if the atomic transition frequency is larger than the central frequency, the anti-Zeno effect will dominate. The time evolution of the energy shift is obtained. The counter-rotating terms lead to a shift toward the low frequency region for the frequency distribution of the emitted photon.
We show that the counterrotating, neglected in the previous studies of the quantum Zeno effect (QZE) in atomic decay, can have a large impact on the short-time evolution. We calculate the electron self-energy, the Lamb shift, and the QZE without making the rotating-wave approximation (RWA) and show that, for hydrogen in free space, the Zeno time is longer by 2 orders of magnitude than that obtained from the RWA. We also show that there is no anti-Zeno effect as the counterrotating terms and rotating terms represent the opposite processes in the higher frequency region. Consequently, the experimental measurement of the QZE may be much easier than what was determined with the RWA results.
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
A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.
The giant atom system is a new paradigm in quantum optics, in which the traditional dipole approximation is not available. In this paper, we construct an artificial giant atom model by coupling a superconducting circuits to a transmission line by two coupling points. In the ultrastrong coupling regime, we show that the Lamb shift of the giant atom, which is induced by the non-negligible counter-rotating atom-waveguide coupling term, will modify its dissipation process. Furthermore, we investigate quantum Zeno and anti-Zeno effect where the size of the giant atom serves as a sensitive controller. Specifically, by comparing the critical measurement interval and the life time of the giant atom, we clarify the condition for the occurring of quantum anti-Zeno effect. We hope our work is useful for the application of giant atom system in the investigation of fundamental problems of quantum mechanics.