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Journal of Experimental and Theoretical Physics

Induced superradiation for a collection of three-level atoms

R. A. Ismailov, Alexander Ya. Kazakov

Published Oct 1, 2000• 2 citations
Publisher

Abstract

N three-level atoms interact simultaneously with classical and quantum fields, which are in quasiresonance with various atomic transitions. The classical and quantum fields exchange photons by means of the atoms. It is shown that under certain conditions this process is collective. The number of photons in a quantized mode oscillates, and the amplitude of these oscillations is proportional to N 2. The frequency of the oscillations is determined by the frequencies of the classical and external fields.

DOI
10.1134/1.1326961

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Cambridge University Press eBooks•Sep 29, 1995•1 citation
N three-level atoms interact simultaneously with classical and quantum fields, which are in quasiresonance with various atomic transitions. The classical and quantum fields exchange photons by means of the atoms. It is shown that under certain conditions this process is collective. The number of photons in a quantized mode oscillates, and the amplitude of these oscillations is proportional to N 2. The frequency of the oscillations is determined by the frequencies of the classical and external fields.
If a two-level atom is in the two-photon resonance with a quantized mode and simultaneously inter-acts with a quasi-resonance classical field, then a photon exchange is observed in this system between the quantized and classical modes. It is demonstrated that such a physical system can serve as a source of squeezed radiation in the quantized mode. The squeezing can be arbitrarily close to unity, while the radiation amplitude can be relatively large. A situation is discussed when N atoms are in the two-photon resonance with a quantized mode and simultaneously interact with a classical field. The phenomenon of exponential superradiation is described when the number of photons in the quantized mode exponentially depends on the number N of atoms.
We systematically study the interaction between two quantized optical fields and a cyclic atomic ensemble driven by a classic optical field. This so-called atomic cyclic ensemble consists of three-level atoms with $\ensuremath{\Delta}$-type transitions due to the symmetry breaking, which can also be implemented in the superconducting quantum circuit by Yu-xi Liu et al. [Phys. Rev. Lett. 95, 087001 (2005)]. We explore the dynamic mechanisms to creating the quantum entanglements among photon states, and between photons and atomic collective excitations by the coherent manipulation of the atom-photon system. It is shown that the quantum information can be completely transferred from one quantized optical mode to another, and the quantum information carried by the two quantized optical fields can be stored in the collective modes of this atomic ensemble by adiabatically controlling the classic field Rabi frequencies.
An atomic electron, in the vacuum of photons, can emit and reabsorb photons of any frequency. Such processes give rise to well known radiative corrections (Lamb—shift, g — 2, natural width of excited states...). We review in this paper similar effects which can be induced by the interaction of an atomic electron with an incident electromagnetic wave, and we try to get some physical insight in radiative processes by comparing these two types of « spontaneous » and « induced » effects. Optical pumping methods have been very useful for studying how atoms are perturbed by optical or RF photons. The Zeeman sublevels of an atomic ground state are broadened and shifted by a resonant or quasiresonant optical irradiation. The corresponding light-shifts (or « Lamp-shifts » as A. Kastler called them) are due to virtual absorptions and reemissions of incident photons. Another example of an induced effect is the modification of the magnetic properties of an atomic level due to a high frequency non resonant RF irradiation. The Larmor precession is slowed down and becomes anisotropic. The two previous examples correspond to two situations leading to simple calculations. For a resonant or quasiresonant excitation, a single atomic transition needs to be considered. Non perturbative treatments can be given which include the effect of the atom field coupling to all orders. For example, the « dressed atom » approach allows a unified treatment of various effects such as light shifts, Autler-Townes doublets resonance fluorescence triplets... On the other hand, for a high frequency irradiation, all atomic transitions must be included. It is possible in this case to derive an effective hamiltonian describing, to lowest order in the fine structure constant α, how electronic energy levels are perturbed in the presence of N photons ω. The remaining part of the paper is devoted to the analysis of such a situation. The terms of the effective hamiltonian proportional to the number N of incident photons correspond to the induced effects. They can be fully interpreted in semiclassical terms and they describe how the slow motion of the electron in the external static fields is modified by the high frequency vibration in the incident monochromatic wave. For example, the vibrating electron averages the static potential over a finite length, the angular oscillation of the spin reduces the effective magnetic moment. The remaining terms, which are independent of N, exist even in the vacuum of photons (N = 0) and describe the contribution of the mode ω to spontaneous radiative corrections. It is possible to split them into two parts. The first one has the same structure as the N-dependent terms, with N replaced by 1/2. It describes the corrections due to the vibration of the electron in vacuum fluctuations which have a spectral power density ħω/2 per mode. The second part represents the effect of radiation reaction, i.e. the interaction of the electron with its self field. Such an interpretation is confirmed by a Heisenberg equations approach where the total force acting upon the electron is split into two physical (i.e. hermitian) parts representing respectively the effect of the vacuum « free » field and the effect of the « source » field produced by the electron itself. These considerations are finally applied to the problem of the positive sign of g - 2. The g-factor can be written as 2 ωL/ωc, where ω L is the Larmor frequency of the spin and ωc the cyclotron frequency of the charge for an electron in a weak static field B 0. In the absence of radiative corrections, g = 2, and ω L = ωc. To lowest order in 1/c (order zero), radiation reaction slows down ωc but not ωL (a charge is more coupled to its self field than a magnetic moment). The effect of vacuum fluctuations appears only at order 2 in 1/c and corresponds to relativistic and magnetic effects which slow down both ωL and ωc. Therefore, it appears that, in the non relativistic domain, the main effect is a reduction of ωc by radiation reaction and this explains why 2 ωL/ωc becomes larger than 2. A complete relativistic calculation (to all orders in 1/ c, but to lowest order in α) does not change this conclusion since the contribution of non relativistic modes (ħω < mc2) is predominant in the integral giving g — 2.
A collection of NNN Rydberg atoms and a resonant millimeter-wave cavity are shown to exchange energy back and forth at a rate $2(\frac{d}{\ensuremath{\hbar}}){\mathcal{E}}_{0}\sqrt{N}$, where ddd is the electric dipole matrix element of the atomic transition and E0{\mathcal{E}}_{0}E0​ the "field per photon" in the cavity. This experiment is a demonstration of self-induced Rabi oscillations in a two-level atom system coupled to a single electromagnetic field mode and can also be considered as a very simple illustration of "ringings" in superradiant emission.
In the previous chapter we studied the interaction between a single two-level atom and the electromagnetic field, both when the field is treated classically and when it is quantized. We encountered some interesting phenomena such as Rabi oscillations, the a.c. Stark effect, and photon antibunching, all of which have been observed. These phenomena are essentially single-atom effects, in the sense that either they require a single atom for the effect to be observed, as in the last case, or at least they do not require more than one, although a group of atoms may be used in practice. In this chapter we shall turn to a discussion of some effects that depend in an essential way on the presence of a group of atoms. In some cases we shall find that the group or collective behavior of the atomic system is relatively trivial, in the sense that we can account for the phenomenon by summing the contributions of the individual atoms to the total field, and treating each of them as if it acts almost independently of the others. This is the situation in free induction decay and in the photon echo . In other cases it is essential to include the effect of each atom on all the other atoms, because this modifies the behavior of each in a significant way. These phenomena, such as self-induced transparency and superradiance , are collective effects in a deeper sense. They are sometimes called cooperative effects .