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Resonance fluorescence

We define the fluorescence spectrum into the structured reservoir modes as a rate at which the mean number of photons $b^{+}(\omega)b(\omega)$ of the reservoir mode at frequency $\omega$ changes in time for the steady state conditions. It is given by
 \begin{displaymath}
{\cal F}(\omega)=\lim_{t\rightarrow\infty}\frac{d}{dt}\langl...
...a,t)b(\omega,t)
+b^{+}(\omega,t)\frac{d}{dt}b(\omega,t)\rangle
\end{displaymath} (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
 \begin{displaymath}
{\cal F}(\omega)=2 K^{2}(\omega)\, {\rm Re}\int_{0}^{\infty}...
...{+}(0)\sigma_{-}(\tau)\rangle e^{i(\omega-\omega_{L})\tau}\, .
\end{displaymath} (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 $K^{2}(\omega)$, 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 $\langle\sigma_{+}(0)\sigma_{-}(\tau)\rangle$, which enters the definition of the resonance fluorescence spectrum
 
F(z) = $\displaystyle {1\over 2zd(z)}\Biggr\{{z\over 2}(1+\langle\sigma_{z}\rangle_{ss})
\left[2(z+2\Gamma)(z+\Gamma+i\Delta')
+\Omega(\Omega+b_{i}+i\Lambda_{r})\right]$  
  + $\displaystyle \langle\sigma_{+}\rangle_{ss}\biggl[-i\,[\Omega(z+a)+\Lambda_{i}
(z+2\Gamma)](z+\Gamma+ M+i\Delta')$  
  + $\displaystyle b_{r}\left[\Omega(\Omega+b_{i}+i\Lambda_{r})+
(z+2\Gamma)(z+\Gamma-M+i\Delta')\right]\biggr]\Biggr\} \, .$ (31)

The incoherent part of the spectrum can be calculated from
 
$\displaystyle {\cal F}_{inc}(\omega)$ = $\displaystyle \frac{\gamma}{\pi}\left(\frac{\omega}{\omega_{A}}
\right)^{3}\eta...
...over
z}\lim_{z\rightarrow 0}zF(z)
\right]_{z=-i(\omega-\omega_{L})}\right\}\, ,$ (32)

where we have used the expression (6) for the frequency dependent coupling constant $K(\omega)$. 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 $K(\omega)$ 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 ${\cal F}_{inc}(\omega )$ for $\Delta =0$, $\Omega /\gamma =15$, $\omega _{c}=\omega _{L}$, $\gamma _{c}/\gamma =10$ (solid line), and $\gamma _{c}/\gamma =10000$ (dashed line).
\resizebox{7.0cm}{!}{\rotatebox{0}{
\includegraphics{fig3.eps}
}}

  
Figure 4: Fig. 4. The incoherent part of the fluorescence spectrum ${\cal F}_{inc}(\omega )$ for $\Delta /\gamma =5$, $\Omega /\gamma =15$, $\omega _{c}=\omega _{L}+\Omega ^{'}$, $\gamma _{c}/\gamma =10$. Exact solution (solid line), no shifts (dashed line).
\resizebox{7.0cm}{!}{\rotatebox{0}{
\includegraphics{fig4.eps}
}}

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 non-flat 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 Mr 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 up previous
Next: Summary Up: Two-level atom in a Previous: Generalized Bloch equations
7th CEWQO
2000-05-22