First order correlation function and power spectrum

We now turn to the luminescence spectrum of the system $S(\omega)$ given by Eq. (2.75). The equations for the two-time correlator $\langle\ud{a}(t)a(t+\tau)\rangle$ follow from the quantum regression formula. The most general set of correlators we can construct is $\{C_{\{mn\mu\nu\}}=\ud{a}^ma^n\ud{b}^\mu b^\nu\}$ , which satisfy Eq. (2.98) for any operator $\Omega_1$ through the general regression matrix for the linear problem given by:

$\begin{subequations}\begin{align}&M_{\substack{mn\mu\nu\\ mn\mu\nu}}=i\omega_a(m... ... M_{\substack{mn\mu\nu\\ mn+1,\mu\nu-1}}=-ig\nu\,. \end{align}\end{subequations}$

However in order to compute $\langle\Omega_1(t)a(t+\tau)\rangle$ , we only need the subset of correlators $\{C_{\{0n0\nu\}}=a^nb^\nu\}$ , which satisfies Eq. (2.98) with a regression matrix

defined only by:

$\begin{subequations}\begin{align}&M_{\substack{n\nu\\ n\nu}}=-i(n\omega_a+\nu\om... ...nu-1}}=M_{\substack{\nu,n\\ \nu-1,n+1}}=-ig\nu\, , \end{align}\end{subequations}$

and zero everywhere else. Furthermore, for the computation of the optical spectrum, it is enough to consider the subset $\{a,b\}$ and $\Omega_1=\ud{a}$ . In Fig. 3.2 we can see a schema of this minimal (and finite) set of correlators (left) and mean values (right), labelled with the indices $\{\eta\}=\{m,n,\mu,\nu\}$ . The coherent (through

) and incoherent (through $P_{a,b}$ ) links between the various correlators, given by the regression matrix, are shown with arrows (see a detailed explanation of the figure in the caption).

**Figure 3.2:** Chain of correlators--indexed by $\{\eta\}=(m,n,\mu,\nu)$ --linked by the Hamiltonian dynamics with pump and decay for two coupled harmonic oscillators. On the left (resp., right), the set $\mathcal{N}_1$ (resp., $\tilde{\mathcal{N}}_1$ ) involved in the equations of the two-time (resp., single-time) correlators. In this and similar figures throughout the manuscript (Figs. 4.2 and 5.11), in green is shown the first manifold, the only one needed to compute the spectrum in the linear model. The equation of motion $\langle \ud{a}(t)C_{\{\eta\}}(t+\tau)\rangle$ with $\eta\in\mathcal{N}_1$ requires for its initial value the correlator $\langle C_{\{\tilde\eta\}}\rangle$ with $\{\tilde\eta\}\in\tilde{\mathcal{N}}_1$ defined from $\{\eta\}=(m,n,\mu,\nu)$ by $\{\tilde\eta\}=(m+1,n,\mu,\nu)$ , as seen on the diagram. The red arrows indicate which elements are linked by the coherent (SC) dynamics, through the coupling strength , while the green/blue arrows show the connections due to the incoherent cavity/QD pumpings. The sense of the arrows indicates which element is ``calling'' which in its equations. The self-coupling of each node to itself is not shown. This is where $\omega_{a,b}$ and $\Gamma_{a,b}$ enter. These links are obtained from the rules in Eq. (3.24), that result in the matrices of regression $\mathbf{M}_1$ and $\mathbf{M}_0$ . Higher manifolds $\mathcal{N}_k$ and $\tilde{\mathcal{N}}_k$ (not plotted), that include higher order correlators, increase their dimension as and , respectively. A manifold is only linked directly to in this model. For example, when computing $g^{(2)}$ in Section 3.6, only manifolds $\tilde{\mathcal{N}}_{k\leq 2}$ will be involved.
$\includegraphics[width=0.8\linewidth]{chap3/manifolds/FigNew-mani.ps}$

Thanks to the linearity of the problem, we obtain a simple equation,

$\displaystyle \frac{\mathbf{v}(t,t+\tau)}{d\tau}=-\mathbf{M}_1\mathbf{v}(t,t+\tau)\,,$

(3.25)

for the two-time correlators

$\displaystyle \mathbf{v}(t,t+\tau)= \begin{pmatrix}\langle\ud{a}(t){a}(t+\tau)\rangle\\ \langle\ud{a}(t)b(t+\tau)\rangle \end{pmatrix}$

(3.26)

where

$\displaystyle \mathbf{M}_1=- \begin{pmatrix}M_{\substack{10\\ 10}} & M_{\substa... ...a+\frac{\Gamma_a}{2} & ig \\ ig & i\omega_b+\frac{\Gamma_b}{2} \end{pmatrix}\,.$

(3.27)

The formal solution follows straightforwardly from $\mathbf{v}(t,t+\tau)=e^{-\mathbf{M}_1\tau}\mathbf{v}(t,t)$ . The initial vector $\mathbf{v}(t,t)$ is that of the mean values that we computed in Sec. 3.2 for the SE and SS case. They can also be found through the quantum regression formula, applied on the set of correlators $\tilde{\mathcal{N}}_1$ (see Fig. 3.2) with $\Omega_1=1$ and the regression matrix $\mathbf{M}_0$ . In terms of these averaged one-time quantities, the correlator of interest reads explicitly (for positive $\tau$ ):

$\begin{multline} \langle\ud{a}(t){a}(t+\tau)\rangle = e^{-\Gamma_+\tau}e^{-i... ...{(R-i\Gamma_-+\frac{\Delta}{2}) n_a(t) + g\, n_{ab}(t)}{2R}\Big] \end{multline}$

with the complex (half) Rabi frequency that we defined in Eq. (3.12). The second correlator found together with Eq. (3.29) is the cross correlation function [defined in Eq. (2.83)] which here reads:

$\begin{multline} \langle\ud{a}(t){b}(t+\tau)\rangle = e^{-\Gamma_+\tau}e^{-i... ...R+i\Gamma_--\frac{\Delta}{2}) n_{abq}(t)-g\, n_a(t)}{2R}\Big]\,. \end{multline}$

Before computing the spectrum of emission, let us look into the Rabi frequency more in detail. Out of resonance, the Rabi frequency is a complex number with both nonzero real, $R=R_\mathrm{r}$ , and imaginary, $R_\mathrm{i}$ , parts. The absolute value of these frequencies can be written as:

$\displaystyle \vert R_\mathrm{r,i}\vert=\frac{1}{\sqrt{2}}\sqrt{\vert R\vert^2\pm(g^2-\Gamma_-^2+\frac{\Delta^2}{4})}\,.$

(3.28)

For parameters $\Gamma_-$ and

which result in SC at resonance ( $g>\vert\Gamma_-\vert$ ), this can be further simplified into

$\displaystyle \vert R_\mathrm{r,i}\vert=\frac{\vert R\vert}{\sqrt{2}}\sqrt{1\pm\sqrt{1-\left(\frac{\Gamma_-\Delta}{\vert R\vert^2}\right)^2}}\,.$

(3.29)

At resonance,

is either pure imaginary (in the WC regime) or real (in the SC regime). For this latter case, it is worth defining a new quantity:

$\displaystyle R_0=R(\Delta=0)=\sqrt{g^2-\Gamma_-^2}\,.$

(3.30)

**Figure 3.3:** Complex Rabi , separated in its real (a) and imaginary (b) parts, as a function of the decoherence parameter $\Gamma_-/g$ for various detunings ( $\Delta/g$ from , up, to 0, bottom, by steps of 0.4). Solid black lines correspond to resonance.
$\includegraphics[width=0.45\linewidth]{chap3/fig3a-Re-Rabi-detuning-decoherence.eps}$ $\includegraphics[width=0.45\linewidth]{chap3/fig3b-Im-Rabi-detuning-decoherence.eps}$

The real and imaginary parts of are plotted in Fig. 3.3 as a function of $\Gamma_-/g$ for various negative detunings. The invariance of under exchange of indexes $a\leftrightarrow b$ results in the property $R(-\Delta,\Gamma_-)=R(\Delta,-\Gamma_-)=R^*(\Delta,\Gamma_-)$ . From this follows the results of $R_\mathrm{r}$ and $R_\mathrm{i}$ for the combinations of $\Delta\neq0$ and $\Gamma_-$ that are not plotted in the figure:

$\begin{subequations}\begin{align}R_\mathrm{r}(-\Delta,\Gamma_-)= R_\mathrm{r}(\D... ...elta,-\Gamma_-)= -R_\mathrm{i}(\Delta,\Gamma_-)\,. \end{align}\end{subequations}$

In the limit of high detuning, $\vert\Delta\vert\gg g$ , $\vert\Gamma_-\vert$ , regardless of WC or SC, the real part becomes independent of the dissipation (decay and pumping), $R_\mathrm{r}\approx\vert\Delta\vert/2$ , and the imaginary part becomes $R_\mathrm{i}\approx\mp\Gamma_{-}$ . We can see in Fig. 3.3 that this sets an upper bound for $R_\mathrm{i}$ :

$\displaystyle \vert R_\mathrm{i}\vert<\vert\Gamma_{-}\vert\,.$

(3.32)

Once again, for the steady state case, we can obtain a range of physical combinations of pumping intensities, , , by ensuring that the correlator of Eq. (3.29) converges to zero when $\tau \rightarrow \infty$ . Here, the condition follows from having a positive total decay rate:

$\displaystyle \Gamma_+-\vert R_\mathrm{i}\vert$

$\displaystyle >$

$\displaystyle 0\,.$

(3.33)

The first consequence of this condition is simply that $\Gamma_+$ must be positive, as we already found with the analysis of the mean values and wrote in Eq. (3.23a). With $\Gamma_+>0$ , the other decay rate appearing in Eq. (3.29) is automatically fulfilled ( $\Gamma_++\vert R_\mathrm{i}\vert>0$ ). On the one hand, if $\Gamma_a$ , $\Gamma_b>0$ , Eq. (3.36) is always true, as we know that $\vert R_\mathrm{i}\vert<\vert\Gamma_-\vert<\Gamma_+$ [from Eq. (3.35)]. This includes the spontaneous emission case where there is no restriction in the parameters. On the other hand, if either $\Gamma_a$ or $\Gamma_b$ is negative, Eq. (3.36) represents a further limitation for the pumping parameters. One can check that it is again exactly equivalent to the condition we already found in Eq. (3.23b). Therefore, the condition that the correlators are well behaved are exactly the same as those that the populations are positive and a physical steady state exists.

Using the result of Eq. (3.29) into the definition of Eq. (2.78), we obtain the formal structure of the emission spectrum:

$\displaystyle S(\omega)=\frac{1}{2}(\mathcal{L}^1+\mathcal{L}^2)-\frac{1}{2}\Im... ...mathcal{L}^1-\mathcal{L}^2)-\frac{1}{2}\Re\{W\}(\mathcal{A}^1-\mathcal{A}^2)\,,$

(3.34)

with $\mathcal{L}(\omega)$ and $\mathcal{A}(\omega)$ being the Lorentzian and dispersive functions whose features (position and broadening) are entirely specified by the complex Rabi frequency [Eq. (3.12)], $\Gamma_+$ [Eq. (3.5)] and the detuning $\Delta$ :

$\begin{subequations}\begin{align}\mathcal{L}^{\substack{1,2}}(\omega)=&\frac{1}{... ...\omega_a-\frac{\Delta}{2}\mp R_\mathrm{r}}))^2}\,. \end{align}\end{subequations}$

We also introduced the weight

, a complex dimensionless coefficient given by

$\displaystyle W=\frac{\Gamma_-+i(\frac{\Delta}2+gD)}{R}\,,$

(3.36)

that we define in terms of still another dimensionless parameter,

$\displaystyle D=\frac{\displaystyle\int_0^\infty\langle\ud{a}b\rangle(t)\,dt}{\displaystyle\int_0^\infty\langle\ud{a}a\rangle(t)\,dt}\,.$

(3.37)

Written in this form, Eqs. (3.37)-(3.40) assume a transparent physical meaning with a clear origin for each term. The spectrum consists of two peaks (that we label 1 and 2), as is well known qualitatively for the SC regime. These are composed of a Lorentzian $\mathcal{L}$ and a dispersive $\mathcal{A}$ part. We already introduced the Lorentzian as the fundamental lineshape for free particles with a lifetime, and in the expression above, it inherits most of how the dissipation gets distributed in the coupled system, including the so-called subnatural linewidth averaging that sees the broadening at resonance below the cavity mode width, as pointed out by Carmichael et al. (1989). The dispersive part originates from the dissipative coupling as in the Lorentz (driven) oscillator. It stems from the existence of resonant eigenenergies (polaritons or dressed modes) in the system that overlap in energy and interfere. The overlapping is due to decoherence that impedes the Hamiltonian polaritons of Eq. (2.56) from being the true eigenstates of the system. Also, which particles are detected, in this case bare modes (photons or excitons), determines greatly the dispersive contribution: the further the particles emitted are from the eigenstates, the more relevant interference becomes. In the case of very strong coupling, the Hamiltonian dynamics are the most important and the dispersive part disappears. In the opposite limit of WC, the dispersive part also disappears because the particles emitted are those that rule the dynamics (photons or excitons) although other kind of interference arises in the system, as we will see, due to the coupling, weak, but still present.

This decomposition of each peak in Lorentzian and dispersive parts is, therefore, entirely clear and expected. More quantitatively, and following the notation of our general expression for the spectra in Eq. (2.105), the peaks, are placed at the frequencies $\omega_{1,2}=\omega_a-\frac{\Delta}{2}\mp R_\mathrm{r}$ and have FWHM given by $\gamma_{1,2}=2(\Gamma_{+}\pm R_\mathrm{i})$ . As $R_\mathrm{r}>0$ , the peaks and correspond to the lower (``L'') and upper (``U'') branch emissions, respectively. The weights are given by $L_{1,2}=(1\mp \Im{\{W\}})/2$ and $K_{1,2}=\pm\Re{\{W\}}/2$ . The limit of bare modes at energies $\omega_a$ and $\omega_b=\omega_a-\Delta$ , broadened with the bare parameters $\Gamma_{a/b}$ (FWHM), is recovered at large detunings. The bare cavity mode will be taken as a reference for the energy scales in the rest of the Chapter (we set $\omega_a=0$ ). Again, we find that the real (resp. imaginary) part of the complex Rabi frequency, $R_\mathrm{r}$ ( $R_\mathrm{i}$ ), contributes to the oscillations (damping) in the correlator and therefore, to the positions (broadenings) in the spectrum.

By defining new complex frequencies

$\displaystyle \Omega_\pm=\omega_{2,1}-i\frac{\gamma_{2,1}}{2}=\omega_a-\frac{\Delta}{2}-i\Gamma_+\pm R\,,$

(3.38)

the normalized and integrated first order correlation function can also be written, directly from Eq. (3.29), as:

$\displaystyle g^{(1)}(\tau)=\frac{\int_0^\infty\langle\ud{a}(t)a(t+\tau)\rangle... ...e (t)dt}=\frac{1+iW}{2}e^{-i\Omega_-\tau} +\frac{1-iW}{2}e^{-i \Omega_+\tau}\,.$

(3.39)

The general expression for the spectrum in Eq. (3.37) therefore takes, thanks to the new parameters, the less physically meaningful but more compact form:

$\displaystyle S(\omega)=\frac{1}{2\pi}\Re\Big\{ \frac{i-W}{\omega-\Omega_-}+ \frac{i+W}{\omega-\Omega_+}\Big\}=\Re\{A(\omega)\}\,.$

(3.40)

where $A(\omega)$ is a complex function of the real frequency $\omega$ . In the same way, the normalized and integrated cross correlator reads:

$\displaystyle g_{ab}^{(1)}(\tau)=\frac{\int_0^\infty\langle\ud{a}(t)b(t+\tau)\r... ...(t)dt}=\frac{1-iW'}{2}e^{-i\Omega_-\tau} +\frac{1+iW'}{2}e^{-i \Omega_+\tau}\,.$

(3.41)

in terms of a parameter which is the counterpart of

$\displaystyle W'=\frac{\Gamma_-+i(\frac{\Delta}2+g/D)}{R}\,.$

(3.42)

Finally, the cross correlation spectral function that we defined in Eq. (2.84) reads:

$\displaystyle S_{ab}(\omega)=\frac{1}{2\pi}\Re\Big\{ \frac{i+W'}{\omega-\Omega_-}+ \frac{i-W'}{\omega-\Omega_+}\Big\}\,.$

(3.43)

So far, all the results hold for both cases of SE and SS. This shows that the qualitative depiction of SC is robust. This made it possible to pursue it in a given experimental system with the parameters of the theoretical models fit for another. This has indeed been the situation with semiconductor results explained in terms of the formalism built for atomic systems.

To be complete, the solution now only requires the boundary conditions that are given by the quantum state of the system. They will affect the parameter , Eq. (3.40), that is therefore the bridging parameter between the two cases. In the next two sections, we address the two cases and their specificities.

Subsections