Spectrum at resonance in the Steady State

The general expressions for the spectrum and correlator find a simple analytic solution at resonance, $ \omega_{E1}=\omega_{E2}$, and in the SS under incoherent continuous pump. In what follows, we refer always to such a situation and therefore drop the SS label. The four complex coefficients,

\begin{subequations}\begin{align}l_1+ik_1=&\frac{1}{16 R z_1}\Big\{\Big[2(2z_1+i...
...a_2)-2(R+z_2-i\Gamma_+)\Big]n_\mathrm{12}\Big\}\,, \end{align}\end{subequations}

are defined in terms of the parameters

$\displaystyle a_1=\frac{g^2}{\Gamma_+^2}[4\Gamma_+^2+2P_{E1}(P_{E2}-2\Gamma_+)-P_{E2}\Gamma_1]\,,\quad a_2=\frac{g^2}{\Gamma_+^2}P_{E1}(P_{E1}-\gamma_{E1})\,,$ (4.18)

and the corresponding frequencies and decay rates
\begin{subequations}\begin{align}&\frac{\gamma_1}{2}+i\omega_1=2\Gamma_++iz_1\,,...
...ad &\frac{\gamma_4}{2}+i\omega_4=2\Gamma_+-iz_2\,. \end{align}\end{subequations}

They are all given in terms of two complex parameters:

$\displaystyle z_{1,2}=\sqrt{(D^sg)^2-(\Gamma_+\mp i R)^2}\,.$ (4.20)

$ D^s$ is a real positive quantity, between 0 and $ \sqrt{2}$, given by

$\displaystyle D^s=\frac{\sqrt{(\gamma_{E1}P_{E2}+\gamma_{E2}P_{E1})/2}}{\Gamma_+}$ (4.21)

that tells about the degree of symmetry in the system. For example, it is 1 when all parameters are equal, $ \gamma_{E1}=\gamma_{E2}=P_{E1}=P_{E2}$, and 0 if one of the parameters is much larger than the others. The renormalized coupling,

$\displaystyle G=D^sg\,,$ (4.22)

reaches its maximum when the parameters are such that $ D^s=\sqrt{2}$. The enhancement by $ 2$ is related to cooperative behavior of two coupled modes, similarly to the superradiance of $ 2$ atoms of the Dicke model (Chapter 6) or the renormalization by $ 2$ in the JCM with $ 2$ photons (Chapter 5).

The last and main parameter appearing in the previous expressions is

$\displaystyle R=\sqrt{g^2-(D^s g)^2-\Gamma_-^2}\,.$ (4.23)

This is the true analog of the (half) Rabi frequency, $ R_0$, in the LM [Eq. (3.33)]. We refer to it by $ R$ and not by $ R_0^\mathrm{2TD}$, as it would correspond to the two-time related quantity at resonance, for simplicity of notation. At vanishing pump, the renormalized coupling converges to $ g$, and both $ R$ and $ R_0^\mathrm{1TD}$ converge exactly to $ R_0$ (LM).

The normalized spectra for the direct emission of dot $ 1$ follows from Eq. (4.11) with the coefficients we have just obtained. It is composed of four peaks with positions and broadenings given respectively by the real and imaginary parts of $ z_1$ and $ z_2$. These latter parameters are valid also in the SE case by only setting the pumping rates to zero. Fig. 4.3 is an example of the spectrum $ S_1(\omega)$ we have constructed (in solid black), with its four peaks, each of them a combination of Lorentzian and dispersive parts.

Figure 4.3: Direct emission spectrum from QD 1 (thick solid black line) in the SC regime ( $ \gamma_{E1}=g$ and $ \gamma_{E2}=g/2$) for the SS under vanishing pump ( $ P_{E1}=0.02g$ and $ P_{E2}=0.01g$). The four peaks that compose the spectra are plotted with a thin blue line (lower transitions 1, 4) and with a thin red line (upper transitions 2, 3). The equivalent spectra in the LM is plotted with a dashed purple line for comparison. Due to the low pump, the doubly excited state $ \ket{B}$ is almost empty and the emission from upper transitions very weak (magnified $ \times 30$ to be visible). This is the linear regime where all models for two coupled modes converge.
\includegraphics[width=0.7\linewidth]{chap4/Fig1.eps}

Elena del Valle ©2009-2010-2011-2012.