Introduction

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Introduction

Light-matter coupling in the linear model is described by the Hamiltonian that we introduced and discussed at length in Chapter 2:

$\displaystyle H=\omega_a\ud{a}a+\omega_b\ud{b}b+g(\ud{a}b+a\ud{b})$

(3.1)

where

and

are the cavity photon and material excitation field operators, respectively, with bare mode energies $\omega_a$ and $\omega_b$ , coupled linearly with strength

. In the linear model (LM), both the photon and exciton operators are Bose operators, satisfying the usual commutation rule $[a,\ud{a}]=1$ , $[b,\ud{b}]=1$ . The linear model is an important case for two reasons. The first one is that in many relevant cases, the matter-field is indeed bosonic, such as the case of quantum wells, or large quantum dots, at low density of excitations. The second reason is that this case provides the limit for vanishing excitations (linear regime) of all the other cases, and is fully solvable analytically. In Chapter 5, we investigate the case of fermionic behavior at large pumping, more relevant when dealing with small QDs that confine excitations, and more prone to involve genuine quantum mechanics as one quantum of excitation can alter the system's response.

The master equation in the absence of pump and at zero temperature,

$\displaystyle \frac{d\rho}{dt}=i[\rho,H]+\sum_{c=a,b}\frac{\gamma_c}2(2c\rho\ud{c}-\ud{c}c\rho-\rho\ud{c}c)\,,$

(3.2)

has been extensively studied, however not in its most general settings. The typical restrictions have been to consider the case of resonance, $\omega_a=\omega_b$ , with only one particular initial condition, namely, the excited state of the emitter in an empty cavity, and to detect the emission of the emitter itself. All together, they describe the spontaneous emission of an emitter placed into a cavity with which it enters into SC. This has been the topical case for decades as this was the case of experimental interest with atoms in cavities.

With the advent of SC in other systems, other configurations start to be of interest. With a QD in a microcavity, the detuning $\Delta$ between the modes, Eq. (2.55), is a crucial experimental parameter, as it can be easily tuned and to a great extent, for instance by applying a magnetic field or changing the temperature. Also in this case, the detection is in the optical mode of the cavity, rather than the direct emission of the exciton emission, because the latter is awkward for various technical reasons of a more or less fundamental character.^3.2 If both modes are bosonic, symmetry allows to focus on the cavity emission without loss of generality, as we can obtain the leaky excitonic emission by simply exchanging the cavity and excitonic parameters, that is, by exchanging the indexes ``'' and ``'' in the formulas. The spectrum could also have photon-exciton crossed terms that are computed in a similar way, as we will see.

Regarding the initial condition, more general quantum states can now be realized, at least in principle, by coherent control, pulse shaping or similar techniques. Additionally and more importantly, the type of excitation of a cavity-emitter system in a semiconductor is typically of an incoherent nature and brings many fundamental changes into the problem that go beyond the mere generalization of Eq. (3.2). Pure states do not correspond to the experimental reality. Instead, the system is maintained in a mixed state with probabilities to realize the th excited state. In all cases, a steady state is imposed by the interplay of pumping and decay. Explicitly, the complete master equation (2.71) reads:

$\displaystyle \frac{d\rho}{dt}=i[\rho,H]+\sum_{c=a,b}\frac{\gamma_c}2(2c\rho\ud... ...\rho\ud{c}c)+\sum_{c=a,b}\frac{P_c}2(2\ud{c}\rho c-c\ud{c}\rho-\rho c\ud{c})\,.$

(3.3)

The Rabi oscillations of the populations are always washed out, regardless of the photon-like, exciton-like or polariton-like (eigenstate) character of the density matrix.

There have been naturally many efforts and a large output in the literature to describe theoretically light-matter coupling in a semiconductor microcavity. A huge majority addressed the Spontaneous Emission case, partly because of the precedent set up by the atomic case. First results were obtained for polaritons in planar cavities, where SC was first realized by Weisbuch et al. (1992). Pau et al. (1995) described the spectra of microcavity polaritons in the very strong coupling regime (in a Lorentzian limit). Savona et al. (1995) outlined the importance of which measurement is being performed in assessing a Rabi splitting, deriving different expressions for the observed splittings in reflexion, transmission, absorption and photoluminescence, that are ultimately related to the channel of detection in the 0D problem. In the Quantum Dot case, Andreani et al. (1999) opened the field, relying on the atomic theory if Carmichael et al. (1989). Their major contribution was the analysis of the coupling strength and the prediction of QDs in microcavities as successful candidates for SC physics. However, the expression for the luminescence spectrum, that was taken straight from the atomic literature, concerned the configuration of direct exciton emission, which is not the canonical case of a semiconductor microcavity where photons are detected through their leakage in the cavity mode. This was addressed by Cui & (2006), who computed the spectra both in the forward and the side emission. They also focused on the role of pure dephasing, which role out of resonance was highlighted after their model by Naesby et al. (2008) or with a master equation by Yamaguchi et al. (2008). Yamaguchi et al. (2008) proposed dephasing as a possible origin for the anomalously large cavity intensity found by Reithmaier et al. (2004), that we also discuss. Laucht, Hauke, Villas-Bôas, Hofbauer, Böhm, Kaniber & Finley (2009) made an extension of the model presented here to include dephasing in the linear regime and successfully fitted their experimental data. Based on a Green function approach, Hughes & Yao (2009) also computed the spectral lines in both geometries, but accounting for their interferences, that, interestingly, can give rise to a triplet structure in the cavity emission. Let us finally mention the works by Auffèves et al. (2008) and Inoue et al. (2008), who gave an insightful description of the resonances that appear in these systems, prone to interferences in peculiar configurations. All these results correspond to the spontaneous emission of one excitation.

In this Chapter, I address both the emission spectra obtained in a configuration of spontaneous emission (SE)--where an initial state is prepared and left to decay--under its most general setting, and the case of luminescence emission under the action of a continuous and incoherent pumping that establishes a steady state (SS). We bring all results under a common and unified formalism and show how none of the cases fully encompasses the other. I focus especially on the continuous pumping case which endows the problem with self-consistency in view of its initial state. The Chapter is organized as follows. In Section 3.2, I analyze the single-time dynamics. In Section 3.3, we obtain fully analytically the main results in both of the cases explicated above, this time focusing more on the two-time dynamics, which Fourier transform gives the luminescence spectra. In Section 3.4, I discuss the mathematical results we derived in the two previous sections, accentuating the physical picture and relying on particular cases for illustration. In this Section, we consider specifically the case of resonance, where all the concepts manifest more clearly. In Section 3.5, I show how the expressions obtained for the SS spectra allow for a successful global fitting of the semiconductor experimental data of Reithmaier et al. (2004), providing an estimation for the system parameters and the pumping conditions. In Section 3.6 we briefly look into the second order correlation function. Finally, in Section 3.7, I contemplate possible future directions based on this model, and in Section 3.8, I give a summary of the main results and provide an index of all the important formulas and key figures of this Chapter.

Elena del Valle 2009-10-11