Cavity QED analog of the harmonic-oscillator probability distribution function
and quantum collapses
Z. Ficek
1,2
and S. Swain
1
1
Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 1NN, Northern Ireland
2
Department of Physics and Centre for Laser Science, The University of Queensland, Brisbane QLD 4072, Australia
Received 10 January 2001; published 16 May 2001
We establish a connection between the simple harmonic oscillator and a two-level atom interacting with
resonant, quantized cavity and strong driving fields, which suggests an experiment to measure the harmonic-
oscillator’s probability distribution function. To achieve this, we calculate the Autler-Townes spectrum by
coupling the system to a third level. We find that there are two different regions of the atomic dynamics
depending on the ratio of the Rabi frequency
c
of the cavity field to that of the Rabi frequency of the
driving field. For
c
and moderate coupling of the transition to the cavity mode the spectral peaks are
composed of multiplets. A quantized dressed-atom approach provides a simple explanation of the spectral
features and shows that the oscillations in the spectral components arise from the oscillations of the population
distribution in the dressed states. The observation of these features would provide evidence for the quantum
nature of the cavity field. The distribution is an analog of the harmonic-oscillator probability distribution
function, and should be experimentally observable. For
c
there is no Autler-Townes splitting and the
spectrum is composed of a single peak located at the frequency of the probe transition. We show that this effect
results from the collapse of the atom to the ground state, which has been predicted by Alsing, Cardimona, and
Carmichael Phys. Rev. A 45, 1793 1992 for a two-level atom in a lossless cavity.
DOI: 10.1103/PhysRevA.63.063815 PACS numbers: 42.50.Ct, 03.65.-w, 32.80.-t
I. INTRODUCTION
With recent successful experiments in the laser cooling
and trapping of a single atom within a single mode of a
microscopic cavity 1, it is now possible to test theoretical
predictions of quantum physics 2 and the cavity quantum
electrodynamics CQED of the strong interaction of atoms
with single quanta of the radiation field. The fundamental
model of the atom-field interaction is the Jaynes-Cummings
model 3 consisting of an excited two-level atom strongly
coupled to a single mode of the radiation field. The model
has been extensively studied and many interesting quantum
effects have been predicted and observed, among the most
well known of which are collapse and revival of the inver-
sion 4, subnatural linewidths 5, fluorescence spectra 6,7,
and nonclassical photon statistics 8. These features result
from the presence of a multiple exchange of photons be-
tween the radiating atom and the cavity mode and occur
when the coupling strengths between the atom and the cavity
mode are larger than the damping rates of the system.
The Jaynes-Cummings model has been extended to in-
clude spontaneous emission, cavity damping, and external
driving fields. Two different configurations of atom driving
have been analyzed. In the first case the external field drives
the cavity mode 8,9, and in the second case the driving
field couples to the atom through an auxiliary field, different
than the cavity mode 7,10. The cases of strong and weak
atom-cavity couplings have been considered. In the case of
the atom driven through an auxiliary mode and weak atom-
cavity coupling the system behaves formally the same as in
free space, but with significantly modified spontaneous-
emission rates. For instance, the fluorescence spectrum of a
strongly driven atom is a triplet, as in free space 11, but
with widely differing linewidths. Depending on the detuning
of the cavity mode from the atomic resonance, the central or
even all three spectral components can be significantly nar-
rowed 12. For strong atom-cavity coupling, each Mollow
triplet component is composed of a multiplet, whose detailed
structure depends on the atom-cavity coupling strength, the
cavity and spontaneous-emission decay rates, and the
photon-number distribution of the cavity field 6,7. More-
over, in the case of the lossless cavity and exact resonance of
the cavity and the driving fields to the atomic transition fre-
quency, the atom can remain in its ground state resulting in
the disappearance of the atomic resonance fluorescence 10.
Recently, considerable interest in the study of the Jaynes-
Cummings model has been devoted to observing the signa-
tures of the discrete nature of field quanta in the atom-cavity
interaction that are sensitive to the presence of single quanta
in the cavity mode. The most recent are experiments on the
detection of quantum Rabi oscillations 13, Fock states of
the radiation field 14, and a quantum phase gate 15. How-
ever, the basic signature of a discrete small number of pho-
tons in the cavity mode is the dependence of the energy
spectrum of the Jaynes-Cummings model on the number of
photons n. The energy spectrum is composed of a single
ground ( n =0) level, and a ladder of doublets separated by
0
. The intradoublet splitting is equal to g n , where
0
is the resonance frequency and g is the atom-cavity coupling
constant. The splitting is characterized by n , the signature
of a discrete number of photons in the cavity mode. The
splitting of the lowest energy doublet ( n =1), called the
vacuum Rabi splitting, has been observed experimentally
16, and a photon correlation spectroscopy technique in-
volving a weak multichromatic field has been proposed to
measure the unequal splitting of the second and third
PHYSICAL REVIEW A, VOLUME 63, 063815
1050-2947/2001/636/06381510/$20.00 ©2001 The American Physical Society 63 063815-1