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We define the fluorescence spectrum into the structured reservoir
modes as a rate at which the mean number of photons
of the reservoir
mode at frequency
changes in time for the steady state
conditions. It is given by

(29) 
Using the Heisenberg equations of motion for the bosonic reservoir
operators, we obtain the following formula for the
fluorescence spectrum emitted into the cavity modes

(30) 
Formula (30) differs from the standard definition of the
resonance fluorescence spectrum, as the Fourier transform of the atomic
correlation function, by the frequency dependent factor
,
which is important here. The standard definition
assumes that the atomic rate is constant.
The equations of motion for the atomic correlation function appearing in (30) can be obtained
from the generalized Bloch equations (22) with the use of the
quantum regression theorem [13]. Taking the Laplace transform
of the evolution equations for the atomic correlation functions with
the appropriate initial conditions, we finally arrive at the following
expression for the Laplace transform of the correlation function
,
which enters the
definition of the resonance fluorescence spectrum
The incoherent part of the spectrum can be calculated from

= 

(32) 
where we have used the expression (6) for the frequency
dependent coupling constant .
We would like to emphasize that the presence of this factor is
necessary when one wish to derive the fluorescence spectrum into the
structured reservoir modes. When the fluorescent light is emitted to
the structureless background modes the traditional definition is
applicable and
can be omitted. This factor is crucial for
``tailored'' reservoirs and/or very strong laser fields.
Figure 3:
Fig. 3. The incoherent part of the fluorescence spectrum
for
,
,
,
(solid line), and
(dashed line).

Figure 4:
Fig. 4. The incoherent part of the fluorescence spectrum
for
,
,
,
.
Exact solution (solid line), no shifts
(dashed line).

However, the expressions (31)
and (32) are quite general and they are applicable for both
strong and weak driving fields and all reservoirs with
sufficiently broad linewidth, which is much broader than the atomic linewidth
to justify the Markovian approximation used to derive the master equation.
Of course, for very strong driving fields, in the secular limit, the
results can be simplified considerably.
To illustrate our results, we have plotted in Fig. 3 and
Fig. 4 the fluorescence spectra
for moderately strong laser fields, for which the principal value
terms (shifts) and the density of
modes of the reservoir play an important role.
We can observe that the structured reservoir with nonflat density of
modes leads to narrowing of the central line and simultaneously
broadening of the sidebands when the cavity is tuned to the central
frequency (Fig. 3). This is the
effect reminiscent of the analogous effect observed for the squeezed
vacuum reservoir [14], and it is related to a possibility of
getting negative values of M_{r} by tuning the Lorentzian
representing the reservoir density of modes to the central line.
We can also observe that including the principal value terms (shifts)
in the derivation of the fluorescence spectrum, i.e. using our
solutions for ``tailored'' reservoirs can lead to the observable asymmetry
in the fluorescence emitted into the cavity modes. This effect is
clearly seen from Fig. 4, where we compare the spectra with
and without the shift terms. For very strong driving fields, in the
secular limit, the shift terms become negligible, and the resonance
fluorescence spectrum becomes symmetric in accordance with the earlier
results [8].
Next: Summary
Up: Twolevel atom in a
Previous: Generalized Bloch equations
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