Generalized Bloch equations

The generalized master equation (17) leads immediately to the generalized Bloch equations describing the evolution of the expectation values of the atomic operators. The Bloch equations can be written in the matrix form as

 

$${{\rm d}\over {\rm d} t}\left(\begin{array}{l} {\langle\sigma_{-}... ...\langle\sigma_{+}(t)\rangle}\\ {\langle\sigma_{z}(t)\rangle}\end{array}\right) {\bf A}\left(\begin{array}{l} {\langle\sigma_{-}(t)\rangle}\\ {\... ...rray}{c}b_{r}-i \Lambda_{i}\\ b_{r}+i \Lambda_{i}\\ -2a\end{array}\right)\, ,$$ = (22)

where the matrix A has the form

 

$$\left(\begin{array}{ccc} i\Delta'-\Gamma&- M&{i\over 2}\Omega\\ ... ...b_{i})+\Lambda_{r}&-i(\Omega+b_{i})+\Lambda_{r}&-2\Gamma \end{array}\right)\, .$$ <I>A = (23)

To make the notation shorter we denote the real part of the complex number Q by Q_r and the imaginary part by Q_i, and we have used the substitutions

 

$$\Gamma \frac{1}{2}(a+2N)\, ,\quad \Lambda=b+2L\, .$$ = (24)

On introducing the Hermitian operators representing the two quadrature components of the atomic dipole moment

 

$$\sigma_{x} {1\over 2}(\sigma_{-}+\sigma_{+})\, ,\quad \sigma_{y}={1\over 2i}(\sigma_{-}-\sigma_{+})\, ,$$ = (25)

the Bloch equations can be rewritten as

 

$${{\rm d}\over {\rm d} t}\left(\begin{array}{l} {\langle\sigma_{x}... ...\langle\sigma_{y}(t)\rangle}\\ {\langle\sigma_{z}(t)\rangle}\end{array}\right) {\bf B}\left(\begin{array}{l} {\langle\sigma_{x}(t)\rangle}\\ {\... ...{1\over 2}\left(\begin{array}{c} b_{r}\\ -\Lambda_{i}\\ -2a\end{array}\right)$$ = (26)

with the matrix <I>B given by

 

$$\left(\begin{array}{ccc} -\Gamma- M_{r}&-\Delta'+ M_{i}&0\\ \Del... ...{1\over2}\Omega\\ \Lambda_{r}&-2(\Omega+b_{i})&-2\Gamma \end{array}\right)\, .$$ <I>B = (27)

From the matrix ${\bf B}$ it is easily seen that two components of the atomic dipole moment, $\langle\sigma_{x}\rangle$ and $\langle\sigma_{y}\rangle$ decay with different rates when M_r is different from zero. This effect is well known for squeezed reservoirs [14], but here it is associated with the modification of the damping rates in very strong fields and/or with the non-flat density of modes of the reservoir. A new feature of the Bloch equations (26) is the presence of the shifts b₀ and $b_{\pm}$, given by (21), which do not appear in other Markovian approaches, but they do appear in the non-Markovian approach [6]. On neglecting the shifts, the Bloch equations (22) are equivalent of the Bloch equations obtained by Kocharovskaya et al. [4].

The steady state solutions of the Bloch equations (26) can be easily obtained for general case. In particular, for very strong laser fields and equal number of photons, at each frequency, $N(\omega)=N_{0}=N_{\pm}$, the steady state solutions for the atomic operators take the approximate form

 

$$\langle\sigma_{x}\rangle_{ss} {\tilde{\Omega}\over 2(1+2N_{0})}\, \frac{(1+\tilde{\Delta})^{2}\... ...^{2}\,a_{+}} {(1+\tilde{\Delta})^{2}\,a_{-}+ (1-\tilde{\Delta})^{2}\,a_{+}}\, ,$$ =

$$\langle\sigma_{y}\rangle_{ss} -{\tilde{\Omega}\over 4\Omega'}\, \frac{2(1-\tilde{\Delta}^{2})\,... ...{+}]\,a_{0}} {(1+\tilde{\Delta})^{2}\,a_{-}+ (1-\tilde{\Delta})^{2}\,a_{+}}\, ,$$ = (28)

$$\langle\sigma_{z}\rangle_{ss} -{\tilde{\Delta}\over 1+2N_{0}}\,\frac{(1+\tilde{\Delta})^{2}\, a... ...{2}\, a_{+}} {(1+\tilde{\Delta})^{2}\,a_{-}+ (1-\tilde{\Delta})^{2}\,a_{+}}\, .$$ =

From solutions (28), it is seen that for thermal reservoir and very strong laser fields the steady state inversion between the atomic states for the specific range of the laser field detuning can be realized, the effect reported in [5] and attributed to $((\omega\pm\Omega^{'})/\omega_{A})^{3}$ factors. Moreover, a nonzero solution for $\langle\sigma_{x}\rangle_{ss}$ component in the resonant case can be found. It is also obvious that placing the atom inside a cavity, where there is a peak in the density of modes at some

  

Figure: Fig. 1. Time dependence of the atomic inversion $\langle\sigma_{z}\rangle(t)$ for $\Omega /\gamma =15$ $\omega _{c}=\omega _{L}-\Omega ^{'}$, $\Delta =-0.4\Omega$, and $\gamma _{c}/\gamma =10$ (solid line), $\gamma _{c}/\gamma =10000$ (dashed line).

characteristic frequency, may increase the values of $\langle\sigma_{x}\rangle_{ss}$ and the steady state atomic inversion [6,12].

For not too strong laser fields the exact solutions have to be used. In

  

Figure: Fig. 2. Time dependence of the atomic inversion $\langle\sigma_{z}\rangle(t)$ for $\Omega /\gamma =15$ $\omega _{c}=\omega _{L}-\Omega ^{'}$, $\gamma _{c}/\gamma =10$, and $\Delta =-0.4\Omega$. Exact solution (solid line), no shifts (dashed line).

this laser field intensity regime it is possible to observe changes of the atomic behavior coming from the nonzero principal value terms in the case of structured reservoirs. In Fig. 1 we have plotted the time dependence of the mean value of the atomic inversion $\langle\sigma_{z}\rangle(t)$. The figure shows the differences, for the moderate laser field intensity $\Omega /\gamma =15$, between the broadband reservoir (dashed line) and the reservoir with the mode structure being a Lorentzian centered at frequency $\omega _{c}=\omega _{L}-\Omega ^{'}$ (solid line). The laser field is detuned by $\Delta =-0.4\Omega$ from the atomic resonance, and we can see that there are significant differences in behavior of $\langle\sigma_{z}\rangle(t)$ for the two cases. When the atom is placed inside a tailored reservoir with narrow bandwidth the Rabi oscillations of the atomic population inversion have larger amplitude than in the case of broadband reservoir and their decay time is longer. In the long time limit there is a considerable amount of population inversion, $\langle\sigma_{z}\rangle_{ss}\approx 0.15$, in the case of narrow bandwidth reservoir, while there is no inversion between atomic states for broadband reservoir. In Fig. 2 we have illustrated the role of the shifts coming from the principal value terms in the atomic evolution by plotting the long time behavior of the atomic inversion: the exact solution (solid line) and the solution with the shifts equal to zero (dashed line). It is clear that the exact solution gives slightly lower atomic inversion in the long time limit, for the parameter values taken in the figure, and whole the trajectory is shifted by the shift terms. The possibility of creating the steady state atomic inversion by tuning the cavity was discussed in the non-Markovian approach by Lewenstein and Mossberg [6]. We can see that also the much simpler Markovian approach presented here, leads to similar effects.