First order correlation function and power spectrum

In the LM, the symmetry $a\leftrightarrow b$ allowed to focus exclusively on the cavity-emission without loss of generality, as the direct exciton emission could be obtained from the cavity emission by interchanging parameters. Here, the exciton (fermion) and photon (boson) are intrinsically different, and no simple relationship links them. They must therefore be computed independently. In order to apply the QRF (2.99), four indices are required to label the closing operators, namely $\{\eta\}=(mn\mu\nu)$ in $C_{\{\eta\}}=\ud{a}^ma^n\ud{\sigma}^\mu\sigma^\nu$ with , $n\in\mathbb{N}$ and $\mu$ , $\nu\in\{0,1 \}$ . The links established between them by the Liouvillian dynamics are given the rules:

$\begin{subequations}\begin{align}&M_{\substack{mn\mu\nu\\ mn\mu\nu}}=i\omega_a(m... ...tack{nm\nu\mu\\ n+1,m,\nu,1-\mu}}=2ig\nu(1-\mu)\,, \end{align}\end{subequations}$

and zero everywhere else. We are interested in $\Omega_1=\ud{c}$ with

and $\{\eta_a\}=(0,1,0,0)$ on the one hand, to get the equation for $\langle\ud{a}(t)a(t+\tau)\rangle$ that will provide the cavity emission spectrum, and $c=\sigma$ with $\{\eta_\sigma\}=(0,0,0,1)$ on the other hand, to get the equation for $\langle\ud{\sigma}(t)\sigma(t+\tau)\rangle$ for the QD direct emission spectrum. Contrary to the LM, this procedure leads to an infinite set of coupled equations. The equations for both $\langle \ud{a}(t)C_{(0, 1, 0, 0)}(t+\tau)\rangle$ and $\langle\ud{\sigma}(t)C_{(0,0,0,1)}(t+\tau)\rangle$ involve the same family of closing operators $C_{\{\eta\}}$ , namely with $\eta\in\bigcup_{k\ge1}\mathcal{N}_k$ where $\mathcal{N}_1=\{(0,1,0,0),(0,0,0,1)\}$ the manifold of the boson case, and for

$\displaystyle \mathcal{N}_k=\{(k-1,k,0,0),(k-1,k-1,0,1),(k-2,k,1,0),(k-2,k-1,1,1)\}\,.$

(5.15)

The links between the various correlators tracked through the indices $\{\eta\}$ , are shown in Fig. 5.11. It is very interesting to compare this schema with those of the LM (Fig. 3.2), the two coupled 2LSs (Fig. 4.2) and the AO (Fig. 5.1). We can easily distinguish the models that can be solved analytically from the fact that a manifold of the schema does not ``call'' higher ones . In the LM, this is due to a natural truncation. The first manifolds $\mathcal{N}_1$ and $\tilde{\mathcal{N}}_1$ , in green, are enough to compute the spectra and populations. In the two 2LSs, the reason of truncation is the saturation of both dots, that reduces the number of nonzero correlators to a few (all nonzero one-time correlators are in the graph). In this case, the first but also the second manifold are involved in the spectra. $\mathcal{N}_2$ and $\tilde{\mathcal{N}}_2$ are smaller than for the JC, again reduced by saturation of the second mode. The second manifold in general differs for each model. As we know, the models only converge in the first manifold that corresponds to the linear regime. The AO and the JCM both require an external truncation to close their equations. In the AO, it is the interaction who links the manifolds to higher ones while in the JCM, it is the coupling. SE imposes a truncation at the highest manifold that the initial state involves.

To solve the differential equations of motion in Eq. (2.99), the initial value of each correlator is also required, e.g., $\langle\ud{a}(t)a(t+\tau)\rangle$ demands $\langle(\ud{a}a)(t)\rangle$ , etc. The initial values of $\langle a(t)C_{\{\eta\}}(t+\tau)\rangle$ (resp., $\langle \sigma(t)C_{\{\eta\}}(t+\tau)\rangle$ ) can be conveniently computed within the same formalism, recurring to $\Omega_1=1$ and $C_{\{\tilde\eta\}}$ with $\{\tilde\eta\}=\{m+1,n\mu\nu\}$ (resp., $\{mn,\mu+1,\nu\}$ ). This allows to compute also the single-time dynamics $\langle C_{\{\tilde\eta\}}(t)\rangle$ , and their steady state, from the same tools used as for the two-time dynamics through the QRF. The indices $\{\tilde\eta\}$ required for the single-time correlators form a set--that we call $\tilde{\mathcal{N}}=\bigcup_{k\ge1}\tilde{\mathcal{N}}_k$ --that is disjoint from $\bigcup_{k\ge1}\mathcal{N}_k$ , required for the two-times dynamics. The set $\tilde{\mathcal{N}}$ has--beside the constant term $\{\eta_0\}=(0,0,0,0)$ --two more elements for the lower manifold (of the LM). This is because $\{\eta_a\}=(0,1,0,0)$ and $\{\eta_\sigma\}=(0,0,0,1)$ invoke and for the cavity spectrum on the one hand, and and for the exciton emission on the other. At higher orders , all two-times correlators $\mathcal{N}_k$ otherwise depend on the same four single-time correlators $\tilde{\mathcal{N}}_k$ . Independently of which spectrum one wishes to compute, these four elements , , and of $\tilde{\mathcal{N}}_1$ are needed in all cases as they are linked to each other, as shown in Fig. 5.11.

In the figure, only the type of coupling--coherent, through , or incoherent, through the pumpings $P_{a,\sigma}$ --has been represented. Weighting coefficients are given by Eqs. (5.14). Of particular relevance is the self-coupling of each correlator to itself, not shown on the figure for clarity. Its coefficient, Eq. (5.14a), lets enter $\gamma_{a,\sigma}$ that do not otherwise couple any one correlator to any of the others. This makes it possible to describe decay, at vanishing pump, with the manifold method by simply providing an imaginary part to the Energy in Eq. (5.11). The incoherent pumping, on the other hand, establishes a new set of connections between correlators. Note, however, that at the exception of $\{\eta_0\}$ , the pumping does not enlarge the sets $\bigcup\mathcal{N}_k$ , $\bigcup\tilde{\mathcal{N}}_k$ : the structure remains the same (also, technically, the computational complexity is identical), only with the correlators affecting each other differently. The addition of $\{\eta_0\}$ by the pumping terms bring the same additional physics in the boson and fermion cases: it imposes a self-consistent steady state over a freely chosen initial condition. In the LM, the pumping had otherwise only a direct influence in renormalizing the self-coupling of each correlator. In the JCM, it brings direct modifications to the Hamiltonian coherent dynamics. But its contribution to the self-coupling is also important, and gives rise to an interesting fermionic opposition to the bosonic effects as seen in Eq. (5.14a) in the effective linewidth:

$\displaystyle \Gamma_{a}=\gamma_{a}-P_a\,,\qquad\Gamma_\sigma=\gamma_\sigma+P_\sigma\,.$

(5.16)

For later convenience, we also define:

$\displaystyle \Gamma_\pm=\frac{\Gamma_a\pm\Gamma_b}4\,.$

(5.17)

Eq. (5.16), reminds us that, whereas the incoherent cavity pumping narrows the linewidth, as a manifestation of its boson character, the incoherent exciton pumping broadens it. This opposite tendencies, participating together in the dynamics, bear a capital importance for the lineshapes, as narrow lines favor the observation of a structure, whereas broadening hinders it. On the other hand, the cavity incoherent pumping always results in a thermal distribution of photons with large fluctuations of the particle numbers, that brings inhomogeneous broadening, whereas the exciton pumping can grow a Poisson-like distribution with little fluctuations. Both types of pumping, however, ultimately bring decoherence to the dynamics and induce the transition into WC, with the lines composing the spectrum collapsing into one. Putting all these effects together, there is an optimum configuration of pumpings where particle fluctuations compensate for the broadening of the interesting lines, enhancing their resolution in the spectrum, as we shall see when we discuss the results below.

As there is no finite closure relation, some truncation is in order. We will adopt the same scheme as for the AO, where a maximum of $n_\mathrm{max}$ excitation(s) (photon plus excitons) is considered at the $n_\mathrm{max}$ th order, thereby truncating by manifolds of correlators, which is the most relevant picture. This means that the last manifold considered in Fig. 5.11 is $\mathcal{N}_\mathrm{max}$ , the one with mean values indexes that fit $(m+n+\mu+\nu)/2=n_\mathrm{max}$ . The exact result is recovered in the limit $n_\mathrm{max}\rightarrow\infty$ . As seen in Fig. 5.11, the number $s_\mathrm{t}$ of two-time correlators from $\mathcal{N}$ up to order $n_\mathrm{max}$ is $s_\mathrm{t}=4n_\mathrm{max}-2$ and the number of mean values from $\tilde{\mathcal{N}}$ is $4n_\mathrm{max}$ . The problem is therefore computationally linear in the number of excitations, and as such is as simple as it could be for a quantum system. The general case consists in a linear system of $s_\mathrm{t}$ coupled differential equations, whose matrix of coefficients [specified by Eqs. (5.14)] is, in the basis of $C_{\{\eta\}}$ , a $s_\mathrm{t}\times s_\mathrm{t}$ square matrix that we denote $\mathbf{M}$ . With these definitions, the quantum regression theorem becomes:

$\displaystyle \partial_\tau\mathbf{v}_c(t,t+\tau)=\mathbf{M}\mathbf{v}_c(t,t+\tau)$

(5.18)

where $\mathbf{v}_c(t,t+\tau)=\langle \ud{c}(t)\mathbf{C}_{\{\eta\}}(t+\tau)\rangle$ . Explicitly, for the lower manifolds, e.g., for

$\displaystyle \mathbf{C}_{\{\eta\}}= \begin{pmatrix}C_{(0,1,0,0)}\\ C_{(0,0,0,1)}\\ C_{(1,2,0,0)}\\ C_{(1,1,0,1)}\\ C_{(0,2,1,0)}\\ \vdots \end{pmatrix}$ and $\displaystyle \quad \mathbf{v}_{a}(t,t+\tau)= \begin{pmatrix}\langle \ud{a}(t)a... ...le\\ \langle \ud{a}(t)(a^2\ud{\sigma})(t+\tau)\rangle\\ \vdots \end{pmatrix}\,.$

(5.19)

The ordering of the correlators is arbitrary. We fix it to that of Fig. 5.11, as seen in Eq. (5.19). With this convention, the indices of the two correlators of interests are:

$\displaystyle i_a=1,\quad i_\sigma=2\,.$

(5.20)

**Figure 5.11:** Chain of correlators--indexed by $\{\eta\}=(m,n,\mu,\nu)$ --linked by the dissipative Jaynes-Cummings dynamics. On the left (resp., right), the set $\bigcup_k\mathcal{N}_k$ (resp., $\bigcup_k\tilde{\mathcal{N}}_k$ ) involved in the equations of the two-time (resp., single-time) correlators. In green are shown the first manifolds $\mathcal{N}_1$ and $\tilde{\mathcal{N}}_1$ that correspond to the LM, and in increasingly lighter shades of blues, the higher manifolds $\mathcal{N}_k$ and $\tilde{\mathcal{N}}_k$ . The equation of motion $\langle \ud{a}(t)C_{\{\eta\}}(t+\tau)\rangle$ (resp. $\langle \ud{\sigma}(t)C_{\{\eta\}}(t+\tau)$ ) with $\eta\in\mathcal{N}_k$ requires for its initial value the correlator $\langle C_{\{\tilde\eta\}}\rangle$ with $\{\tilde\eta\}\in\tilde{\mathcal{N}}_k$ defined from $\{\eta\}=(m,n,\mu,\nu)$ by $\{\tilde\eta\}=(m+1,n,\mu,\nu)$ (resp. $(m,n,\mu+1,\nu)$ ), as seen on the diagram. The red arrows indicate which elements are linked by the coherent (SC) dynamics, through the coupling strenght , while the green/blue arrows show the connections due to the incoherent cavity/exciton pumpings, respectively. The self-coupling of each node to itself is not shown. This is where $\omega_{a,\sigma}$ and $\gamma_{a,\sigma}$ enter.
$\includegraphics[width=.9\linewidth]{chap5/JC/fig1-correlator.ps}$

To solve Eq. (5.18), we introduce the matrix $\mathbf{E}$ of normalized eigenvectors of $\mathbf{M}$ , and $-\mathbf{D}$ the diagonal matrix of eigenvalues:

$\displaystyle -\mathbf{D}=\mathbf{E}^{-1}\mathbf{M}\mathbf{E}\,.$

(5.21)

The formal solution is given by $\mathbf{v}_c(t,t+\tau)=\mathbf{E}e^{-\mathbf{D}\tau}\mathbf{E}^{-1}\mathbf{v}_c(t,t)$ . Integration of $\int e^{(-\mathbf{D}+i\omega)\tau}d\tau$ and application of the Wiener-Khintchine formula yield for the th and $i_\sigma$ th rows of $\mathbf{v}_c$ the emission spectra of the cavity, namely, $S_a=\frac{1}{\pi n_a}\Re\int\langle \ud{a}(t){a}(t+\tau)\rangle e^{i\omega\tau}\,d\tau$ on the one hand, and of the direct exciton emission, $S_\sigma=\frac{1}{\pi n_\sigma}\Re\int\langle\ud{\sigma}(t){\sigma}(t+\tau)\rangle e^{i\omega\tau}\,d\tau$ , on the other hand. We find, to order $n_\mathrm{max}$ :

$\displaystyle S_c(\omega)=\frac{1}{\pi}\Re\sum_{p=1}^{s_\mathrm{t}}\frac{L^c_{i_cp}+iK^c_{i_cp}}{D_p-i\omega}\,,\quad c=a,\sigma\,,$

(5.22)

where $L^c_{i_cp}$ and $K^c_{i_cp}$ are given by the real and the imaginary part, respectively, of $[\mathbf{E}]_{i_cp}[\mathbf{E}^{-1}\mathbf{v}_c(t,t)]_p/n_c$ :

$\displaystyle L^c_{i_cp}+iK^c_{i_cp}=\frac1{n_c}[\mathbf{E}]_{i_cp}\sum_{q=1}^{... ...}}[\mathbf{E}^{-1}]_{pq}[\mathbf{v}_c(t,t)]_q\,,\quad 1\le p\le s_\mathrm{t}\,,$

(5.23)

and $D_p=[\mathbf{D}]_{pp}$ (when we refer to elements of a matrix or a vector by its indices, we enclose it with square brackets to distinguish from labelling indices). Further defining $\gamma_p/2$ and $\omega_p$ as the real and imaginary parts, respectively, of

$\displaystyle \frac{\gamma_p}{2}+i\omega_p=D_p\,,$

(5.24)

we can write Eq. (5.22) in a less concise but more transparent way. To all orders, it reads:

$\displaystyle S_c(\omega)=\frac1{\pi}\lim_{n_\mathrm{max}\rightarrow\infty}\sum... ...-K^c_{i_cp}\frac{\omega-\omega_p}{(\omega-\omega_p)^2+(\gamma_p/2)^2}\right)\,.$

(5.25)

The lineshape, as in all the models we have studied in this thesis, is composed of a series of Lorentzian and Dispersive parts, whose positions and broadenings (FWHM) are specified by $\omega_p$ and $\gamma_p$ , cf. Eq. (5.24), and which are weighted by the coefficients $L^c_{i_cp}$ and $K^c_{i_cp}$ , cf. Eq. (5.23). The former pertain to the structure of the spectral shape as inherited from the Jaynes-Cummings energy levels. They are, as such, independent of the channel of detection (cavity or direct exciton emission). We devote Section 5.4.2 to them. The latter reflect the quantum state that has been realized in the system under the interplay of pumping and decay. They determine which lines actually appear in the spectra, and with which intensity. Naturally, the channel of emission is a crucial element in this case. We devote Section 5.4.3 to this aspect of the problem.