Mean values

Let us start by introducing the notations that will be useful in the general description of SE and SS emission. The effective broadenings reduce to the decay rates in the SE case but get renormalized by the pumping rate in the SS case:

$\begin{subequations}\begin{align}\Gamma_{a,b}&=\gamma_{a,b}\,,&\text{(SE case)}\... ...ma_{a,b}&=\gamma_{a,b}-P_{a,b}\,.&\text{(SS case)} \end{align}\end{subequations}$

We shall also use thoroughly the combinations:

$\displaystyle \gamma_{\pm}=\frac{\gamma_a\pm\gamma_b}{4}$ and $\displaystyle \quad\Gamma_{\pm}=\frac{\Gamma_a\pm\Gamma_b}{4}\,.$

(3.5)

Thanks to the general relations $\langle C\rangle=\Tr(C\rho)$ and $d\langle C\rangle/dt=\Tr(Cd\rho/dt)=\Tr(C\mathcal{L}\rho)$ , we can obtain from Eq. (3.3) the single-time mean values of interest for this problem, by solving the equation of motion of the coupled system:

$\displaystyle \frac{d\mathbf{u}(t)}{dt}=-\mathbf{M}_0\mathbf{u}(t)+\mathbf{p}$

(3.6)

with

$\begin{subequations}\begin{gather}\mathbf{u}= \begin{pmatrix}n_{a}\\ n_{b}\\ n_{... ...-ig & ig & 0 & i\Delta+2\Gamma_+ \end{pmatrix}\,, \end{gather}\end{subequations}$

where $n_{c}=\langle\ud{c}c\rangle \in\mathbb{R}$ (for

) and $n_{ab}=\langle\ud{a}b\rangle =n_{ba}^*\in\mathbb{C}$ . The SE case corresponds to setting $P_{a,b}=0$ and providing the initial conditions, $\mathbf{u}(0)$ , with:

$\displaystyle n_a^0\equiv n_{a}(0),\quad n_b^0\equiv n_{b}(0),$ and $\displaystyle \quad n_{ab}^0\equiv n_{ab}(0)\,.$

(3.8)

The solution,

$\displaystyle \mathbf{u}^\mathrm{SE}(t)=e^{-\mathbf{M}_0t}\mathbf{u}(0)\,,$

(3.9)

gives the cavity population:

$\displaystyle n_a(t)=e^{-2 \gamma_+ t}\Bigg\{$	$\displaystyle \Big[\cos{(R_\mathrm{r} t)}+\cosh{(R_\mathrm{i} t})\Big]\frac{n_a^0}{2}$
$\displaystyle -$	$\displaystyle \Big[\cos{(R_\mathrm{r}t)}-\cosh{(R_\mathrm{i} t)}\Big]\frac{(\fr... ...+ g^2 n_b^0+ g \Delta \Re{n_{ab}^0}-2g \gamma_- \Im{n_{ab}^0}}{2\vert R\vert^2}$
$\displaystyle +$	$\displaystyle \left[\frac{\sin{(R_\mathrm{r} t)}}{R_\mathrm{r}}+\frac{\sinh{(R_\mathrm{i}t})}{R_\mathrm{i}} \right]\left( g \Im{n_{ab}^0}-\gamma_- n_a^0 \right)$
$\displaystyle +$	$\displaystyle \left[\frac{\sin{(R_\mathrm{r}t)}}{R_\mathrm{r}}-\frac{\sinh{(R_\... ...}+g(\frac{\Delta^2}{4}-\gamma_-^2+g^2) \Im{n_{ab}^0}}{\vert R\vert^2}\Bigg\}\,.$	(3.10)

The expression for follows from $a\leftrightarrow b$ . The crossed mean value that reflects the coherent coupling reads:

	$\displaystyle n_{ab}(t)=e^{-2 \gamma_+ t}\Bigg\{\Big[\cos{(R_\mathrm{r} t)}+\cosh{(R_\mathrm{i} t})\Big]\frac{n_{ab}^0}{2}$
$\displaystyle -$	$\displaystyle \Big[\cos{(R_\mathrm{r} t)}-\cosh{(R_\mathrm{i} t)}\Big]\frac{g(\... ..._b^0-(\frac{\Delta^2}{4}+\gamma_-^2) n_{ab}^0+g^2(n_{ab}^0)^*}{2\vert R\vert^2}$
$\displaystyle +$	$\displaystyle \left[\frac{\sin{(R_\mathrm{r} t)}}{R_\mathrm{r}}+\frac{\sinh{(R_\mathrm{i} t})}{R_\mathrm{i}} \right]\frac{i(\Delta n_{ab}^0-g(n_a^0-n_b^0))}{2}$
$\displaystyle +$	$\displaystyle \left[\frac{\sin{(R_\mathrm{r} t)}}{R_\mathrm{r}}-\frac{\sinh{(R_\mathrm{i} t})}{R_\mathrm{i}} \right]\times$
	$\displaystyle \frac{g(\Delta \gamma_--i(\frac{\Delta}{2}+g^2-\gamma_-^2)) n_a^0... ...^0+i\Delta(\frac{\Delta^2}{4}+\gamma_-^2+g^2) n_{ab}^0}{2\vert R\vert^2}\Bigg\}$	(3.11)

where we have defined the complex (half) Rabi frequency:

$\displaystyle R=\sqrt{g^2-\left(\Gamma_-+i\frac{\Delta}{2}\right)^2}\,,$

(3.12)

that arises as a direct extension of the dissipationless case, Eq. (2.56c). $R_\mathrm{r,i}$ are its real and imaginary parts respectively, $R=R_\mathrm{r}+iR_\mathrm{i}$ . Note that in the SE case $\Gamma_-\rightarrow\gamma_-$ in

, that we define now in general. It is clear from Eqs. (3.10)-(3.11) that $R_\mathrm{r}$ is responsible for the oscillations and $R_\mathrm{i}$ , together with $\gamma_+$ , for the damping.

**Figure 3.1:** Dressed or polariton modes for the LM at resonance and in the absence of pumping, as a function of the effective broadening $\vert\gamma_-\vert$ . Left hand side corresponds to strong coupling and right hand side to weak coupling. The number of shared excitations increases upwards from 0 to 3. All manifolds have proportional splittings that undergo the SC-WC transition simultaneously.
$\includegraphics[width=0.3\linewidth]{chap3/newFigs/boson.ps}$

It is of interest to note that Eqs. (3.10)-(3.11) are reproduced by introducing decay as an imaginary part to the energies in the Heisenberg picture, i.e., substituting $\omega_{a,b}$ by $\omega_{a,b}-i\gamma_{a,b}/2$ and solving directly in a full Hamiltonian picture the operator equations of motion: with . This method goes along the lines of the manifold picture (closely related to the Langevin equations), represented in Fig. 3.1. Although essentially incorrect (as we explained in the previous Chapter), following this method provides the right average quantities, such as the correlator $\langle\ud{a}(t)a(t+\tau)\rangle$ and therefore leads also to the correct expression for the SE spectra. The expressions that we obtain for the four time-dependent operators ( $\ud{a}(t)$ , , $\ud{b}(t)$ , ) solving the Heisenberg (not the Langevin!) equations are all contained in:

$\begin{multline} a(t)=e^{-(\gamma_+-i\frac{\Delta}{2})t}\Big[ e^{iRt} \frac{... ...Rt} \frac{(R-i\gamma_++\frac{\Delta}{2})a(0)+g b(0)}{2R}\Big]\,. \end{multline}$

The operators at zero time,

and

, are the same as in the Schrödinger picture. The hermitian conjugation and exchange $a\leftrightarrow b$ leads to the other three operators. They are all missing the noise contribution and for instance, one can check that $[a(t),\ud{a}(t)]$ decays exponentially instead of remaining equal to one.

On the other hand, the SS case corresponds to setting the time derivative on the left hand side of Eq. (3.6) to zero, and solving the resulting set of linear equations. The solution

$\displaystyle \mathbf{u}^\mathrm{SS}=\mathbf{M}_0^{-1}\mathbf{p}$

(3.13)

explicitly yields:

$\begin{subequations}\begin{align}n_a^\mathrm{SS}&=\frac{g^2\Gamma_+(P_a+P_b)+P_a... ...mma_a\Gamma_b(\Gamma_+^2+(\frac{\Delta}{2})^2)}\,. \end{align}\end{subequations}$

Both photonic and excitonic reduced density matrices are diagonal. They correspond to thermal distributions of particles with the above mean numbers:

$\begin{subequations}\begin{align}&\rho^\mathrm{a}_{n,p}=\sum_{m}\rho_{n,m;\,p,m}... ...(n_b^\mathrm{SS})^m}{(1+n_b^\mathrm{SS})^{m+1}}\,. \end{align}\end{subequations}$

Behind their forbidding appearance, Eqs. (3.15) enjoy a transparent physical meaning, that they inherit from the semi-classical--and therefore intuitive--picture of rate equations. When the coupling strength between the two modes, , vanishes, the solutions are those of thermal equilibrium for and [Eqs. (2.34) and (2.52)]. In the general case where $g\neq0$ , the mean numbers can also be written in the same form:

$\displaystyle n_a^\mathrm{SS}=\frac{P_a^\mathrm{eff}}{\gamma_a^\mathrm{eff}-P_a^\mathrm{eff}}\,,$

(3.16)

(Id. for mode

throughout by exchanging indexes $a\leftrightarrow b$ ), in terms of effective pumping and decay rates:

$\begin{subequations}\begin{align}&P_a^\mathrm{eff}=P_a+\frac{Q_a}{\Gamma_a+\Gamm... ...frac{Q_a}{\Gamma_a+\Gamma_b}(\gamma_a+\gamma_b)\,, \end{align}\end{subequations}$

with

the rate at which mode

exchanges particles with mode

$\displaystyle Q_a=\frac{4(g^\mathrm{eff})^2}{\Gamma_b}\,,$

(3.18)

in terms of the effective coupling strength at nonzero detuning:

$\displaystyle g^\mathrm{eff}=\frac{g}{\sqrt{1+\Big(\frac{\Delta/2}{\Gamma_+} \Big)^2}}\,.$

(3.19)

is a generalization of the Purcell rate $\gamma_a^\mathrm{P}=4g^2/\gamma_b$ , which is the rate at which the population

, [cf. Eq. (3.10)], decays in weak coupling when $\gamma_b,\gamma_a^\mathrm{P}\gg \gamma_a$ . The dimensionless parameter $4(g^\mathrm{eff})^2/(\Gamma_a\Gamma_b)$ is a generalization of the so-called ``cooperativity parameter'' that quantifies the rate of effective coherent exchange in the system as compared to dissipation.

From the point of view of mode , the coupling with mode is both adding particles, contributing to $P_a^\mathrm{eff}$ , and removing them, contributing to $\gamma_a^\mathrm{eff}$ . The total effective decay is:

$\displaystyle \Gamma_a^\mathrm{eff}=\gamma_a^\mathrm{eff}-P_a^\mathrm{eff}=\Gamma_a+Q_a\,.$

(3.20)

Note that the generalized Purcell rate

appears in the same way in both effective parameters in Eqs. (3.18), due to the ``bidirectionality'' of the coupling (the coupling both brings in and removes excitations).

The mean value of the coherence can also be expressed in terms of these quantities:

$\displaystyle n_{ab}^\mathrm{SS}=\frac{2g^\mathrm{eff}}{\Gamma^\mathrm{eff}_a+\Gamma^\mathrm{eff}_b}\frac{\gamma_aP_b-\gamma_bP_a}{\Gamma_a\Gamma_b}\,e^{i\phi}$

(3.21)

where $\phi=\arctan{(\frac{\Gamma_+}{\Delta/2})}$ .

The quantities defined in Eqs. (3.18) and Eq. (3.21) are all positive when $\Gamma_b>0$ () and all negative when $\Gamma_b<0$ (if there exists a solution for the steady state). The conditions for the pumping terms , to yield a physical state (a steady state), are therefore those for which the mean values $n_{a,b}^\mathrm{SS}$ are positive and finite, implying:

$\begin{subequations}\begin{align}\Gamma_+&>0\,,\\ 4(g^\mathrm{eff})^2&>-\Gamma_a\Gamma_b\,. \end{align}\end{subequations}$

The first condition requires that pumps

are not simultaneously larger than their respective decay rates $\gamma_a$ , $\gamma_b$ . The second condition only represents a restriction when one of the effective parameters, either $\Gamma_a$ or $\Gamma_b$ , is negative. Then, it reads explicitly $4(g^\mathrm{eff})^2>\vert\Gamma_a\Gamma_b\vert$ . Note that, out of resonance, the pumping rates appear both in $g^\mathrm{eff}$ and $\Gamma_a$ , $\Gamma_b$ and therefore the explicit range of physical values for them needs to be found self-consistently.

From now on, we shall refer with ``SE'' and ``SS'' to the expressions that apply specifically to the spontaneous emission and to the steady state, respectively, leaving free of index those that are of general validity. In some cases, as for instance in Eq. (3.4), no index is required if it is understood that $P_{a/b}$ are defined and equal to zero in the SE case. For that reason, we shall leave $\Gamma$ free of the SE/SS redundant index.