Reduced model for the description of radiation-matter interaction including atomic recoil
J. Javaloyes and G. L. Lippi
Institut Non Line ´aire de Nice, UMR 6618 CNRS, Universite ´ de Nice – Sophia Antipolis, 1361 Route des Lucioles,
F-06560 Valbonne, France
A. Politi
Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, 50125 Firenze, Italy
Received 15 January 2003; published 12 September 2003
We show that a model for the collective atomic recoil laser, previously introduced to include collisions with
an external buffer gas, can be reduced to a single dynamical equation for the probe amplitude. This is the result
of a clever adiabatic elimination of the atomic variables and of the assumption of a negligible effect of the
probe field onto the atomic motion. This reduced model provides a fairly accurate description of the phase
diagram of the original set of equations and allows for the investigation of more realistic regimes, where the
direct simulation of the full model would be otherwise unfeasible. As a result, we find that the onset of a
coherent field can be either described by a second- or first-order transition, the former scenario being observ-
able only below a given temperature. Moreover, the first-order transition is accompanied by an intrinsic optical
bistability region.
DOI: 10.1103/PhysRevA.68.033405 PACS numbers: 42.50.Vk, 05.45.Xt, 05.65.+b, 42.65.Sf
I. INTRODUCTION
The interaction between atoms and the electromagnetic
e.m. radiation is a domain of physics that has attracted
attention for over a century. At the origin of the development
of quantum mechanics, the interpretation of atomic and mo-
lecular spectra and the prediction of their features has been
the object of a large wealth of work e.g., cf. Refs. 1–5.
The problem has been analyzed, in most cases, semiclassi-
cally, but fully quantum mechanical treatments have been
also carried out. Within the semiclassical approach—the
scope of the present work—several authors have developed
ways of treating the interaction between a quantized two-
level or multilevel atom, with a formalism analogous to the
vector description of spin states 6, and the classical mac-
roscopic e.m. field. This way of modeling the interaction is
particularly appealing when the e.m. field is sufficiently
strong to neglect its fluctuations, and whenever the atomic
response to the external field is sought. The number of physi-
cal effects that can be treated in this fashion is particularly
large, and the complete literature cannot be cited here a
good overview of a number of classic effects can be found in
Refs. 7,8. Within this framework, the contribution of this
paper is to take into account aspects that so far have been
treated independently or in a perturbative approach: the ef-
fect of the radiation scattered by the atoms into the global
e.m. field and its feedback on the atoms 9, i.e., taking into
account the atomic motion due to atomic recoil because of
the photon exchange and collisions between atoms.
The rest of this introduction is dedicated to a brief review
of known effects that are going to play a role in the system
we choose to describe. Some of them will be arbitrarily ex-
cluded e.g., Rayleigh or Raman scattering from our discus-
sion, but they are indeed contained in the literature cited, as
well as in the problem that we consider and in the way we
treat it.
The advent of the laser marked a revolution in the study
of light-atom interactions. Work concerning the shape of
atomic emission spectra under different experimental condi-
tions was published very early on 10–12 and applications
to nonlinear effects arising in gas laser amplifiers were con-
sidered 13,14. In Refs. 10–12 it was also shown that, at
the lowest level of semiclassical description, the study of the
response of a two-level atom to the incident radiation re-
quires the inclusion of saturation effects on the optical tran-
sition and the influence of detuning between field and atoms.
Immediately, a concern arises about the strength of the inci-
dent radiation, which modifies the atom’s level structure, in-
troducing the so-called Rabi sidebands or level splitting
15: the absorption and emission spectrum of the atom have
been strongly modified by the interaction. The development
of narrow-linewidth and sufficiently powerful tunable lasers
allowed for experimentally probing, a few years later, these
somewhat surprising predictions. The experiments 16–19
confirmed the existence of sidebands—an object of debate, at
the time—but also showed that the complete picture was
quite more complex.
One of the questions that presented themselves concerned
the measurement of the modifications in the atomic spectral
properties when subject to an intense external field. For this
purpose, it was natural to introduce a second field of variable
frequency and of very weak intensity, which may be scanned
across the frequency range over which the atom reacts, with-
out perturbing in any significant way the atomic line shape: a
weak probe field. Unfortunately, the mathematical treatment
of the problem becomes immediately intractable in closed
form, and approximations have to be introduced 20.
The problem of two independent traveling waves pump
and weak probe interacting with an atomic sample was first
investigated in the 1960s with reference to a multilevel
atomic structure 11,21–23 and a detailed discussion can be
found already in Ref. 24. A subsequent, more general, treat-
ment of the interaction of an ensemble of atoms with qua-
siresonant counterpropagating pump and probe fields, includ-
PHYSICAL REVIEW A 68, 033405 2003
1050-2947/2003/683/03340513/$20.00 ©2003 The American Physical Society 68 033405-1