The Jaynes-Cummings model

In this Section, we study saturation effects due to Pauli blocking. The material excitation follows Fermi statistics and the coupling to light is described by the Jaynes-Cummings model (JCM), that we write here again

$\displaystyle H=\omega_a\ud{a}a+\omega_\sigma\ud{\sigma}\sigma+g(\ud{a}\sigma+a\ud{\sigma})\,.$

(5.11)

Such an innocent looking Hamiltonian, with only the addition of dissipation, has been the object of countless investigations. Analytical solutions are tricky to find already in this case, and remain uncovered for most mechanisms of excitation. SE from a given initial state has been overly privileged as the case of study, starting by the work of Sanchez-Mondragon et al. (1983). Even when the emitter was modelled as a two-level system, the JCM was commonly reduced to the LM by considering a single excitation as the initial state. This is the approximation made in many papers, by Carmichael et al. (1989), Andreani et al. (1999), Cui & (2006), Auffèves et al. (2008), Inoue et al. (2008), Hughes & Yao (2009), etc. Other--even more numerous--works considered the excitation scheme at the same (Hamiltonian) level as the coupling, that is, they probe the system with coherent pumping. This is the case, for instance, of Mollow (1969), Savage (1989), Freedhoff & (1994), Clemens & (2000), Barchielli & (2002), Florescu (2006), Bienert et al. (2007), etc. There has been less considerations for the luminescence spectra under incoherent pumping, although there exist very interesting works on one-atom lasing, like those by Cirac et al. (1991), Tian & (1992), Löffler et al. (1997), Clemens et al. (2004), Karlovich & (2007), Karlovich & (2008), and some specifically on one QD in a microcavity, like that of Perea et al. (2004). In the atomic literature, Cirac et al. (1991) have considered a thermal photonic bath and were the first to our knowledge to report the characteristic Jaynes-Cumming multiplet in an exact optical spectrum. Tian & (1992) considered thermal baths for both modes, in the low excitation regime. Löffler et al. (1997) considered spectral shapes for the one-atom laser at resonance by numerical integration of the master equation. Karlovich & (2008) concentrated on strong coupling at resonance and low pump. Some of our results, in the good cavity case, recover these pioneering reports.

One of the most important current task of the SC physics in semiconductors is the quantitative description of the experiment with a theory that can provide statistical estimates to the data, in particular intervals of confidence for the fitting parameters. In this respect, there would be little need for fitting an experiment that would produce a clear observation of the Jaynes-Cummings energy levels, which is a strong qualitative effect. But no such structures have been observed so far and the deviations from a linear Rabi doublet, like those found by Hennessy et al. (2007), have been understood as non fundamental features of the problem. The most likely reason for this lack of crushing observations of the quantum regime in the PL lineshapes is that the best systems currently available in semiconductors are still beyond the range of parameters that allows the quantum features to neatly dominate. Instead, they are still at a stage where it is easy to overlook more feeble indications, as shall be seen in what follows for less ideal systems that are closer to the experimental situation of today. Another possible reason is that the models are not suitable and a QD cannot be described by a simple two-level system. Then more involved theories should take over, with, e.g., full account for electron and hole band structures and correlations, as those of Feldtmann et al. (2006), Gies et al. (2007) or Schneebeli et al. (2008). However, if a simpler theory is successful, notwithstanding the interest of its more elaborate and complete counterpart, it clearly facilitates the understanding and putting the system to useful applications (especially in a quantum information processing context). At present, there is more element of chance left in the research for quantum SC than is actually necessary. If a quantitative description of even a ``negative experiment'' (not reporting a triplet or quadruplet) could be provided, this would help tracking and probably even direct the progress towards the ultimate goal: a fully understood and controlled SC in the quantum regime.

In this Section, we shall not focus on the difference between the SE of an initial state in absence of any pumping, and the SS established in presence of this pumping, as we did with other models. SS is the most relevant case for the experimental configuration that we have in mind, while SE is amply studied in the literature. Rather than contrasting the SE/SS results, we shall therefore contrast the boson/fermion cases. For this reason and for concision, we shall not use the ``SS'' superscript and assume that which of the SE/SS case is assumed is clear from context or from the presence of the time variable .

In the LM, the quantum state of the system is not by itself an interesting quantity as most of its features are contained in its reduced density matrices, that are simply and in all cases thermal states with effective temperatures specified by the mean populations of the modes and . For this reason, $g^{(2)}$ , that measures the fluctuations in the photon numbers, does not contain any new information. In the Fermion case however, $g^{(2)}$ becomes nontrivial, because the saturation of the dot provides a nonlinearity in the system that can produce various types of statistics, from the coherent Poisson distributions, encountered in lasers (where the nonlinearity is provided by the feedback and laser gain), to Fock-state statistics, with antibunching, exhibited by systems with a quantum state that has no classical counterpart. The fluctuations in particle numbers influence the spectral shape. The full statistics itself is most conveniently obtained from the master equation with elements $\rho_{mi;nj}$ for , photons and , exciton ( $m, n\in\mathbb{N}$ , $i, j\in\{0,1\}$ ). The distribution function of the photon number is simply $\mathrm{p}[n]=\rho_{n,0;n,0}+\rho_{n,1;n,1}$ .

Rather than to consider the equations of motion for the matrix elements, it is clearer and more efficient to consider only elements that are nonzero in the steady state,

$\displaystyle \mathrm{p}_0[n]=\rho_{n,0;n,0}\,,\quad \mathrm{p}_1[n]=\rho_{n,1;n,1}$ and $\displaystyle \quad \mathrm{q}[n]=\rho_{n,0;n-1,1}\,.$

(5.12)

They correspond to, respectively, the probability to have

photons, with ( $\mathrm{p}_1$ ), or without ( $\mathrm{p}_0$ ), exciton, and the coherence element between the states $\ket{n,0}$ and $\ket{n-1,1}$ , linked by the SC Hamiltonian. Both

and

are real. It is convenient to separate

into its real and imaginary parts, $\mathrm{q}[n]=\mathrm{q}_\mathrm{r}[n]+i \mathrm{q}_\mathrm{i}[n]$ as they play different roles in the dynamics. The equations for these quantities, derived from Schrödinger equation for the Liouvillian (2.71), read:

$\begin{subequations}\begin{align}\frac{d\mathrm{p}_0[n]}{dt}=&-\big((\gamma_a+P_... ...athrm{r}[n-1]+\Delta \,\mathrm{q}_\mathrm{i}[n]\,. \end{align}\end{subequations}$

Note that, in the steady state, Eqs. (5.13) are detailed-balance type of equations. The conditional photon statistics with and without the exciton are similar, and coupled through the imaginary part of the $\mathrm{q}$ distribution (that is not a probability). At resonance, the real part of the coherence distribution, $\mathrm{q}_\mathrm{r}$ , gets decoupled and vanishes in the steady state. As a result, only Eqs. (5.13a)-(5.13c) need to be solved. When

vanishes, $\mathrm{q}_\mathrm{i}$ does not couple the two modes anymore, and their statistics become thermal like in the boson case. Through the off-diagonal elements $\mathrm{q}_\mathrm{i}$ , the photon density matrix can vary between Poissonian, thermal (superpoissonian) and subpoissonian distributions.

The rest of the Chapter is organized as follows. In Section 5.4.1, we provide the expressions for all--and only those--correlation functions that enter the problem, making it as computationally efficient as possible for an exact treatment. We provide a decomposition of the final spectra in terms of transitions of the dressed states, which gives a clear physical picture of the problem. In Section 5.4.2, we give the analytical expressions for the position and broadening of the resonances of the system at vanishing pumping. Weighting these resonances by the self-consistent dynamics of the system established by finite pumping and decay, gives the final spectral shape. We discuss in particular the notion of SC that varies from manifold to manifold, rather than holding for the entire system as a whole. In Section 5.4.3, we consider three particular points representative of the experimental situation, plus one point beyond what is currently available. We first discuss their behavior in terms of population and statistical fluctuations as imposed from the pumping conditions. In Section 5.4.4, we give the backbone of the final spectra at nonvanishing excitations. This is the numerical counterpart of Section 5.4.2, in the presence of arbitrary pumping. In Section 5.4.5, we present spectral shapes for the three points in a variety of configuration and compare them to each other. In Section 5.4.6, we investigate the situation at nonzero detuning, which is a case of particular importance in semiconductor physics.

Subsections