Vanishing pump case in the manifold picture

In this Section, we discuss the series of parameters $\omega_p$ and $\gamma_p$ that in the luminescence spectrum, Eq. (5.25), determine the position and the broadening (FWHM) of the lines, respectively, in both the cavity and the direct exciton emission. The case of vanishing pumping is fundamental, as it corresponds to the textbook Jaynes-Cummings results with the SE of an initial state. It serves as the skeleton for the general case with arbitrary pumping and supports the general physical picture. Finally, it admits analytical results. We therefore begin with the case where $P_{a},P_\sigma\ll\gamma_{a},\gamma_\sigma$ . The eigenvalues of the matrix of regression $\mathbf{M}$ , are grouped into manifolds. There are two for the first manifold, given by:

$\displaystyle D_{\substack{1\\ 2}}=\Gamma_++i\Big(\omega_a-\frac\Delta2\mp \sqrt{g^2-\Big(\Gamma_-+i\frac\Delta2\Big)^2} \Big)\,,$

(5.26)

and four for each manifold of higher order

, given by, for $4k-5\le p\le 4k-2$ :^5.4

$\displaystyle D_p=\Gamma_k+i \Big( \omega_a+ \mathrm{sgn}\big({p-(8k-7)/2}\big) R_k+(-1)^pR^*_{k-1}\Big)\,,$

(5.27)

in terms of the

th-manifold (half) Rabi splitting:

$\displaystyle R_k=\sqrt{(\sqrt{k}g)^2-\Big(\Gamma_-+i\frac\Delta2\Big)^2}\,,$

(5.28)

and of the

th-manifold (half) broadening:

$\displaystyle \Gamma_k=(2k-3)\Gamma_-+(2k-1)\Gamma_+=(k-1)\gamma_a+\frac{\gamma_\sigma}2\,.$

(5.29)

For each manifold, we have defined the

in order by increasing value of the line position $\omega_p$ .

**Figure 5.12:** Spectral structure of the JCM at resonance and without pumping. (a) Positions $\omega_p$ of the lines in the luminescence spectrum. Only energies higher than $\omega_a$ are shown (not their symmetric below $\omega_a$ ). We take $\omega_a=0$ as the reference energy. In green (thick), the first manifold, and in increasing shades of blue, the successive higher manifolds which form a pattern of branch-coupling curves that define different orders of SC. (b) Half-Width at Half Maximum (HWHM) $\gamma_p/2$ of the lines (with $\gamma_\sigma=0$ ). In both (a) and (b), the blue filled region results from the accumulation of the countable-infinite vanishing lines. (c) Eigenenergies of the Jaynes-Cummings Hamiltonian with decay as an imaginary part of the bare energies (the Jaynes-Cummings ladder). It can be directly compared with the bosonic case in Fig. 3.1. It provides a clear physical picture of panel (a) where the peaks positions arise from the difference of energy between lines of two successive manifolds. Lines of (a) stem from the emission from manifold in SC into manifold in WC (or vacuum). Lines and stem from the emission between the two manifolds in SC. Solid lines are those plotted in (a), dotted lines produce the symmetric lines, not shown. The horizontal line at 0 in (a) arises from decay between two manifolds in WC. The schema (c) also reproduces the broadening of the lines (b) with the sum of the imaginary parts of the eigenenergies involved in the transition.
$\includegraphics[width=.95\linewidth]{chap5/JC/fig2-ladder.eps}$

According to Eq. (5.24), these provide the position $\omega_p$ of the line and its half-broadening $\gamma_p/2$ through their imaginary and real parts. $\Gamma_k$ is always real, so it contributes in all cases to $\gamma_p$ only. is (at resonance) either pure real, or pure imaginary, and similarly to the LM or the two coupled 2LSs, this is what defines SC. This corresponds to an oscillatory or damped field dynamics of the two-time correlators within manifold , which lead us to the formal definition: WC and SC of order are defined as the regime where the complex Rabi frequency at resonance, Eq. (5.28), is pure imaginary (WC) or real (SC). The criterion for th order SC is therefore:

$\displaystyle g>\vert\Gamma_-\vert/\sqrt{n}\,.$

(5.30)

SC is achieved more easily, given the system parameters ( and $\gamma_{a,\sigma}$ ), with an increasing photon-field intensity that enhances the effective coupling strength. The lower the SC order, the stronger the coupling. This corresponds to the th manifold (and all above) being in SC (aided by the cavity photons), while the manifolds below are in WC. First order is therefore the one where all manifolds are in SC. Eq. (5.30) includes the standard SC of the LM and 2LSs, $g>\vert\Gamma_-\vert$ , as the first order SC of the fermion case, that is shown in green (thick) in Fig. 5.12 (see also and compare with Fig. 3.1 for bosons). The same position of the peaks $\omega_{1,2}$ and the same (half) broadenings $\gamma_{1,2}$ is also recovered (in the absence of pumping). Note that similarly to the boson case, the SC is defined by a comparison between the coupling strenght with the difference of the effective broadening $\Gamma_a$ and $\Gamma_b$ . The sum of these play no role in this regard.

The $\omega_p$ and $\gamma_p/2$ are plotted in Fig. 5.12(a) and 5.12(b), respectively, as function of $\Gamma_-$ . Note that $\omega_p$ only depends on and $\Gamma_-$ , whereas $\gamma_p$ also depends on $\Gamma_+$ (that is why we plot it for $\gamma_\sigma=0$ ).

The , Eq. (5.27), have a natural interpretation in terms of transitions between the manifolds of the so-called Jaynes-Cummings ladder. The eigenenergies of the Jaynes-Cummings Hamiltonian with decay granted as the imaginary part of the bare energies ( $\omega_{a,\sigma}-i\gamma_{a,\sigma}/2$ ), are given by $E_{\pm}^k$ with

$\displaystyle E_{\pm}^k= k\omega_a-\frac{\Delta}2 \pm R_k -i\frac{(2k-1)\gamma_a+\gamma_b}{4}\,,$

(5.31)

for the

th manifold. The four possible transitions between consecutive manifolds

and

give rise, when

, to the four peaks we found:

$\begin{subequations}\begin{align}&D_{4k-5}=i[E^k_--(E^{k-1}_+)^*]\, ,\quad &D_{4... ...+)^*]\, ,\quad &D_{4k-2}=i[E^k_+-(E^{k-1}_-)^*]\,. \end{align}\end{subequations}$

In the case

, only the two peaks in common with the LM arise, $D_{1,2}=iE^1_{\mp}$ , given respectively by Eqs. (5.32a) and (5.32b) with

The ladder is shown (at resonance) in Fig. 5.12(c) in the same way as we reconstructed it for bosons in Fig. 3.1. Let us discuss it in connection with our definition of SC in this system, to arbitrary . When $\Gamma_-=0$ , each step of the ladder is constituted by the two eigenstates of the fermion, dressed by the cavity photons, resulting in a splitting of $2\sqrt{n}g$ . This kind of renormalization already appeared in Chapter 4 when we discussed the coupled two 2LSs, that can be considered as a particular case of the JCM, where there can be no more than a photon in the system. An -dependent splitting produces quadruplets of delta peaks with splitting of $\pm(\sqrt{n}\pm\sqrt{n-1})g$ around $\omega_a$ , as opposed to the LM where independently of the manifold, the peaks are all placed at $\pm g$ around $\omega_a$ . In a more general situation with $\Gamma_-\neq0$ , there are three possibilities for a manifold :

Both manifold and are in SC. The two Rabi coefficients and $R_{k-1}$ are real. This is the case when

$\displaystyle \vert\Gamma_-\vert\le g\sqrt{k-1}\,.$ (5.33)

The luminescence spectra corresponds to four splitted lines $\omega_p\rightarrow\omega_a\pm(R_k\pm R_{k-1})$ , coming from the four possible transitions [Eqs. (5.32), shown as and in Fig. 5.12(c)] between manifolds and . The emission from all the higher manifolds also produces four lines. They are grouped pairwise around $\omega_a$ [Fig. 5.12(a)] and all have the same broadening, contributed by $\Gamma_k$ only [the single straight line in Fig. 5.12(b)].
Manifold is in SC while manifold is in WC. In this case, is pure imaginary (contributing to line positions) and $R_{k-1}$ is real (contributing to broadenings). This is the case when

$\displaystyle g\sqrt{k-1}<\vert\Gamma_-\vert<g\sqrt{k}\,.$ (5.34)

This corresponds to two lines $\omega_p\rightarrow\omega_a\pm R_k$ in the luminescence spectrum, coming from the two possible transitions [shown as in Fig. 5.12(c)] between the SC manifold and the WC manifold . Each of them is doubly degenerated. The two contributions at a given $\omega_p$ have two distinct broadenings $\gamma_p/2\rightarrow\Gamma_k\pm \vert R_{k-1}\vert$ around $\Gamma_k$ . [cf. Fig. 5.12(b)]. The final lineshapes of the two lines is the same. In this region, all the emission from the higher manifolds produce four lines and all from the lower produce only one (at $\omega_a$ ), being in WC.
Both manifold and are in WC. The two Rabi coefficients and $R_{k-1}$ are pure imaginary. This is the case when

$\displaystyle g\sqrt{k}\le\vert\Gamma_-\vert\,.$ (5.35)

This corresponds to only one line at $\omega_p\rightarrow\omega_a$ in the luminescence spectrum, coming from the transition from one manifold in WC to the other [shown as in Fig. 5.12(c)]. The line is four-time degenerated, with four contributions with different broadenings $\gamma_p/2\rightarrow\Gamma_{k}\pm(\vert R_k\vert\pm \vert R_{k-1}\vert)$ , as seen in Fig. 5.12(b).

Figure 5.12 is the skeleton for the luminescence spectra--whether that of the cavity or of the direct exciton emission. It specifies at what energies can be the possible lines that constitutes the final lineshape, and what are their broadening. To compose the final result, we only require to know the weight of each of these lines.

In the SE case, the weights and include the integral of the single-time mean values $\mathbf{v}_{a}(t,t)$ over $0\leq t<\infty$ . Therefore, only those manifolds with a smaller number of excitations than the initial state can appear in the spectrum. Each of them, will be weighted by the specific dynamics of the system. The ``spectral structure''--i.e., the $\omega_p$ and $\gamma_p$ --depends only the system parameters ( and $\gamma_{a,\sigma}$ ). Therefore, in the SE case, the resulting emission spectrum is an exact mapping of the spectral structure of the Hamiltonian, Fig. 5.12.

In the SS case, the weighting of the lines also depends on which quantum state is realized, this time under the balance of pumping and decay. But the excitation scheme also changes the spectral structure of Fig. 5.12. When the pumping parameters are small, the changes will mainly be perturbations of the present picture and most concepts will still hold, such as the definition of SC, Eq. (5.30) for nonzero $P_{a,\sigma}$ in (5.17). However, when the pump parameters are comparable to the decay parameters, the manifold picture in terms of Hamiltonian eigenenergies breaks, as it happen for the two coupled 2LS in Chapter 4. The underlying spectral structure must be computed numerically for each specific probing of the system with and $P_\sigma$ . It can still be possible to identify the origin of the lines with the manifold transitions by plotting their position $\omega_p$ as a function of the pumps, starting from the analytic limit. SC of each manifold can be associated to the existence of peaks positioned at $\omega_p\neq \omega_a,\omega_\sigma$ . We address this problem in next Sections.