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Physical Review A

Effect of the counterrotating-wave terms on the spontaneous emission from a multilevel atom

Zheng-Hong Li, Dawei Wang, Hang Zheng, Shi-Yao Zhu, Muhammad Suhail Zubairy

Published Aug 3, 2009• 43 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.80.023801

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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.
The third level has great influence on the time evolution of the spontaneous decay from the excited state, when the counter-rotating terms are taken into account. The influence in the cascade-type and V\mathsf{V}V-type three-level systems is investigated for two initial states, excited from the ground states of the whole system (atom plus vacuum modes) and from the bare atom. The third level results in the additional virtual photon processes, emitting a photon from one level and reabsorbing the same photon to another level and vice versa. The main influence of the third level is to accelerate the decay, which leads to or enhances the anti-Zeno effect, especially for the initial state excited from the ground state of the whole system. Therefore, the third level cannot be neglected, if the counter-rotating terms are taken into account.
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
The collective spontaneous emission of NNN multilevel atoms is studied in optical vector theory and without applying the rotating-wave approximation. The counter-rotating terms are included using a unitary transformation method. We analyze the decay dynamics starting from two initial conditions, the standard Dicke and timed Dicke states. In addition to the dependence on ensemble volume and density, we also study the effect of the ensemble geometry on cooperative emission for spheres, cubes, and quasi-two-dimensional shapes in different orientations. Finally, time-dependent cooperative spontaneous emission rates are introduced and investigated.
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.
Quantum interference in spontaneous emission from a four-level atom is investigated with the counter-rotating terms and the energy shifts included. The atom has two upper levels which are coupled to a common lower level by the same vacuum modes and are driven by a coherent field to an auxiliary level. The effect of the counter-rotating terms in the coupling through the vacuum modes is taken into account by a unitary transformation method. We show how the quantum interference due to the energy shifts affects the spontaneous emission.