Microcavity QED (mcQED)

The physics of cQED has naturally attracted the attention of the semiconductor community, if only in view of the possible technological applications. Semiconductor heterostructures are the state of the art systems for this purpose. They allow to engineer, with an ever rising control, the solid state counterpart of the atomic system to match or isolate their excitation spectra.

At the most basic level of description, an heterostructure is a man-made, microscopic edifice of different semiconductors. A typical heterostructure is the alternating sequence of fine semiconducting slabs with different refractive index, such as gallium arsenide (GaAs), the compound of gallium and arsenic, and aluminium arsenide, AlAs, with almost the same lattice constant as GaAs but a wider bandgap. A pile of typically twenty such pairs produce a microscopic mirror for light, because of Bragg's reflection in this periodic crystal (such a structure is called a ``Bragg mirror'', or a ``distributed Bragg mirror''). Two such mirrors face to face produce a microcavity.

The different bandgaps result in energy band offsets that produce potentials for the carriers. In effect, this can be used to confine the semiconductor excitations, namely, electrons and holes, or their bound state, called an exciton. Confinement can thus be achieved in a plane when, e.g., a slab of semiconductor is sandwiched between two others with a wider bandgap, giving rise to a so-called Quantum Well (QW), for instance again with GaAs and AlAs, used above to build a mirror. Confinement can also be achieved in 1D with Quantum Wires, and, more importantly for our discussion, full confinement in all dimensions can be achieved in so-called Quantum Dots (QDs). A QD, for that reason, is sometimes referred to as an ``artificial atom'', because the full space quantization means that its excitations consist in discrete excited states, much like in the Bohr (and modern) picture of the atom. Growing a sequence of alternating layers to produce a first mirror, then growing QDs, and finally toping it off with another sequence of alternating layers to produce another mirror, completing the cavity, yields a micron-scale cQED system, opening the way forward to microcavity QED.

For historical purposes, and to give the proper background to microcavity QED, one must start with QWs as the active element,^1.11 not QDs. I will give only the most elementary overview of this huge field.^1.12

In the same way that atomic physics found it hard to achieve SC with a single atom, it was not at the reach of technology to yield strong exciton-photon coupling until the year 2004 (interestingly, at the same year than the experiment by Boca et al. (2004) in atomic cQED, that, by retaining one and the same atom, provides the closest counterpart to the QD cQED case, where at least the problem of the wandering excitation is controlled to its best possible extent). QWs are more easily grown and controlled, multiple QWs can be placed inside the cavity and at the antinodes of the field (where it is maximum) to achieve the best coupling.

The most significant finding in this field has been the observation by Weisbuch et al. (1992) of anticrossing in absorption of a planar heterostructure,^1.13 whose schema is shown on the left part of Fig. 1.4. The stack explicated above is here spelt out; the QWs also involve Indium. The anticrossing itself is shown on the right: at resonance, two symmetric dips show a splitting of the structure modes. This splitting is universally called the Rabi splitting. We pause to provide the proper context to this terminology.

**Figure 1.4:** Schema of a semiconductor heterostructure, by Houdré et al. (1994), and its observed anticrossing, by Weisbuch et al. (1992), launching the field of microcavity-polaritons physics.
$\includegraphics[width=.45\linewidth]{introduction/houdre94.ps}$ $\includegraphics[width=.35\linewidth]{introduction/weisbuch92a.ps}$

Weisbuch et al. (1992) described their seminal finding as a ``solid state QED effect'' and linked it to the vacuum Rabi splitting (VRS) with many references to the atomic QED achievements. They were well aware, however, of the alternative, ``more classical'' picture. Which one should be favored has been a topic of debates, that is still not completely settled to this day. Khitrova et al. (2006), in their review of SC with QDs in microcavities, speak of a ``genuine'' or ``real'' SC, to distinguish it from the 2D-polaritons SC what they would prefer to be called ``normal mode coupling'' (a term that we find in the original Weisbuch et al. (1992)'s paper along with VRS, but that has been completely ignored by the polariton community thereafter). Normal mode coupling essentially refers to a fully classical analog to the phenomenon of anticrossing that, because it evokes mode repulsion, is often favorably understood as quantum in origin. Instead, the mere appearance of new modes in coupled (classical) oscillators, is an exact depiction of the phenomenon without the need to recourse to quantum physics. This point has been made very clear by Zhu et al. (1990), who insist that their Rabi splitting stems from classical physics. The main issue against 2D-polaritons to perform cQED physics is the vast number of excitations involved, as opposed to QDs where the physics can be brought to the quantum limit, in the sense of involving a few quanta, one--or none if there is a way to evidence effects of the quantum vacuum--in a regime where one quantum more or less changes drastically the behaviour of the system. In the 2D case, the idea of Rabi splitting goes with that of linear response, but not with that of field quantization. The most elaborate and accurate descriptions of these systems have been in terms of continuous fields. Also, in the atomic QED problem, interaction is between two modes only (in this sense it is zero-dimensional), but with 2D-polaritons, although there is still a one-to-one matching between a photon and an exciton, this extends over a whole range of in-plane wavevectors.^1.14 This provides polaritons with a rich dynamics of scattering, but in a fundamental picture, this brings complications and somewhat blurs the picture.

This is not to say that quantum physics should be excluded as a whole from these systems. Probably the second strongest input to the polariton field was the report by Savvidis et al. (2000) of stimulated scattering of polaritons. This evidenced bosonic statistics in the system, and opened the way to the field of polaritons BEC, with its recent experimental claim by Kasprzak et al. (2006). One of the most actively sought after signature of the quantum degeneracy of 2D-polaritons is currently superfluidity, and some preliminary experimental evidence have been reported by Amo (2009a) and Amo (2009b). In any case, those are still manifestations of macroscopic coherence where large numbers of microscopic particles exhibit the behavior of a continuous field (classical or not). At the ultimate quantum limits where single polaritons matter in a quantum way, there is still some activity but not with quite the same attention. Let us mention however the pioneering proposal of Ciuti (2004)^1.15 to generate entangled photons from 2D polaritons, or the investigations of vacuum radiations, that he proposed with De Liberato (2007). In a cleverly designed ``triple cavity'', Diederichs & (2005) have proposed another promising mechanism for generating entangled pairs, later announced by Diederichs et al. (2006). Finally, Savasta et al. (2005) have also discussed the genuine quantum nature of polaritons. However, those results have elicited a limited interest (as compared to their significance) and their recognition is unclear. There is not as yet an explicit demonstration of full-field quantization such as violation of some Bell inequalities. The opposition between macroscopic quantum phenomena versus nonlinear classical optics is still important, and the necessity of the quantum picture at the one-quantum limit for 2D-polaritons is even more debatable.

Clearer manifestations of single quanta were foreseeable in the QD picture. Brunner et al. (1992) were already able to measure the PL of single dots, a feat repeated by Marzin et al. (1994) with self-organized (or self-assembled) QDs. Placing them in a cavity would seem to lead to an easy, direct and explicit manifestation of full-field quantization.

**Figure 1.5:** SEM image of pillar microcavities, from the LPN laboratory in Paris.
$\includegraphics[width=.66\linewidth]{introduction/BQM-lpn.ps}$

**Figure 1.6:** Various pillars as grown by the University of Sheffield, demonstrating the great control of shapes and sizes that can be obtained. Ellipticity of the right pillar is used to control polarization of the emitted light, as discussed by Whittaker et al. (2007).
$\includegraphics[width=\linewidth]{introduction/pillars.ps}$

To be in SC with the single-mode QD, the cavity should also sustain an isolated single mode, otherwise the mechanism of Weisskopf of non-reversible leakage of the excitation into the many cavity modes would lead to exponential decay rather than Rabi oscillations. There are various but mainly three types of designs to achieve this goal of a zero-dimensional semiconductor microcavity. Fig. 1.5 shows a stunning view of so-called pillar cavities. These are obtained from etching a conventional planar stack of Bragg mirrors. The lateral confinement is then provided by the refractive index. Each pillar is about 10 $\mu$ m of height. They would typically be selected one by one for experimental investigation, until a system evidences a sufficient interest. Chance has it that, as of today, many, and probably most, of these pillars do not present a single QD in SC. There is therefore a strong element of chance in isolating an interesting system. To the best of my knowledge, there is however no quantitative estimate for this factor. I shall discuss this question in more details later. Fig. 1.6 shows close-up views of pillars, this time grown in Sheffield. There is an impressive control in the size and shapes with which to shape these structures. Sanvitto et al. (2005) have reported high values of with such structures,^1.16 but could not observe SC. I will propose a tentative explanation for this shortcoming. In Fig. 1.7, finally, we get the closest view on the pillar system, with an image from Würzburg, where are shown both a close view of the Bragg mirror (compare with the schematic view of Fig. 1.4) as well as the particular QDs that have been used in this work. I shall describe this aspect in more details later when I come back to the theoretical description of these dots, as the pillar structure is the one to which I will devote more direct attention.

**Figure 1.7:** SEM image from Löffler et al. (2005) of the two Bragg mirrors, forming the microcavity (that can be seen near the center) and of the QDs that are embedded inside.
$\includegraphics[width=.5\linewidth]{introduction/SEM-pillar.ps}$

Another important realization of a single microcavity mode is the photonic crystal. The concept has been put to technological use only recently, after the topic was itself brought to the limelight by Yablonovitch (1987) and John (1987) (in two consecutive letters to the Physical Review). The principle is based on the same physics that leads to bandgaps in semiconductors:^1.17 the propagation of photons in the periodic structure is forbidden for continuous regions of wavelengths because of destructive interferences of the same type as those of Bragg physics. These regions form bands that can be separated by a photonic bandgap.

**Figure 1.8:** SEM image of a microdisk, from Kippenberg et al. (2006), sitting on a Si post erected on the silicon wafer. The silica microdisk cavity is 30 $\mu$ m in radius. The wedge has been etched intentionally to push the whispering gallery modes inward, to protect them from scattering-induced cavity boundaries. See also the discussion by Armani et al. (2003).
$\includegraphics[width=.66\linewidth]{introduction/mc-CALTECH.ps}$

Yablonovitch et al. (1991) reported the first 3D PC, obtained by a complex drilling of holes in a slab at three different angles, leading to a full bandgap in the microwave range. Krauss et al. (1996) reported a 2D PC near the optical spectrum. We shall be more interested in this case, that has in fact attracted more attention, if only because of the much easier fabrication (drilling holes on a plane rather than inside a volume: the crystal is directly etched into a slab of semiconductor). Noda et al. (2000) demonstrated that a cunningly introduced defect in the 2D crystal structure, is forming a cavity for photons (that they injected through a lateral waveguide close to the cavity). The of the cavity thus formed was then of about 400. They then realized that much better confinement was possible if the cavity, rather than coming merely from a defect in the crystal, would be engineered to allow the electric field distribution to vary slowly, ideally as a Gaussian. This is realized ``simply'' by shifting the position and reducing the radius of neighboring holes to the one that has been skipped to form the cavity. Now, factors of more than $2.5\times10^6$ have been reported by Takahashi et al. (2007) and Tanaka et al. (2008) speculate on designs that promises figures up to . A review of PC structures is given by Noda et al. (2007).

**Figure 1.9:** Purcell effect, as observed by Chang et al. (2006) with a QD (1) in resonance with a single mode of a PC, suffering enhancement of its decay, a QD (2) out of resonance, slowing down its SE and a QD in bulk, setting up the scale of ``normal'' decay.
$\includegraphics[width=.6\linewidth]{introduction/chang06a.ps}$

Finally, a last structure prone to rich cQED, is the so-called microdisk, where light is trapped in whispering gallery modes. An example system is shown in Fig. 1.8. A review of microdisks is given by Nosich et al. (2007).^1.18

Embedding the dots in a pillar cavity, Gérard & (1999) reported the Purcell effect of shortening, or lengthening, of the lifetime in the SE problem. The first report came with a Purcell factor of 5, and there has been since a considerable number of similar observations in various systems (e.g., Solomon et al. (2001) and Bennett et al. (2007) also in pillars, Kiraz et al. (2001) in microdisks, Chang et al. (2006) in PC, etc...). In Fig. 1.9, Purcell effect is nicely demonstrated by Chang et al. (2006) who observe the time-resolved observation of the luminescence of a dot in a bulk semiconductor (with a lifetime of $\tau\approx0.65$ ns). When a similar dot is placed inside the photonic bandgap of this semiconductor, its lifetime is extended to ns, while still another dot--this time in resonance with a cavity mode etched into the semiconductor--sees its lifetime drop to 0.21ns. Another experiment with the Purcell effect that I want to highlight is that of Bayer et al. (2001), in a pillar whose sides had been coated or not, demonstrating the impact of leaky emission on the Purcell effect. In subsequent chapters, I will discuss at length leaky emission in the physics of SC.

**Figure 1.10:** SEM image of a Noda cavity formed in a photonic crystal by ``removing one hole'' in the periodic structure, by Badolato et al. (2005). On the right, calculated electric field, with maximum in dark. Remarkably, Badolato et al. (2005) can place a single QD at a location of their choice, so they could put it at precisely one of these maxima, namely at the point marked by the red cross. The dot is indeed visible on the SEM picture!
$\includegraphics[width=.6\linewidth]{introduction/pc-badolato.ps}$

A continuous progress has been made towards a better quantum coupling with dots in 0D MC, as well as towards its external control. The inclusion in the cavity of QDs lowers the quality factor, but new progresses have been made in the engineering of the heterostructures, allowing to deterministically position a QD inside a photonic crystal to within 25nm accuracy, and thus place the dot at a maxima of the light intensity, along with an etching technique of the holes of the photonic crystal to match spectrally the QD and cavity mode emission (see Fig. 1.10). At the same time, the density of self-assembled QDs in the active medium has been successfully reduced over the years, with figures of $10^{10}$ cm $^{-2}$ , and the possibility to grow large dots (with lens shape of $\approx30$ nm) so as to provide a large oscillator strength. Nevertheless, reaching the SC has been long and tedious, and the number of reports has not been overwhelming ever since. Anyway, in late 2004, in two consecutive letters to Nature, Yoshie et al. (2004) and Reithmaier et al. (2004), reported SC of QDs with a single-mode microcavity, in a PC and a pillar, respectively. At about the same time, but published later, Peter et al. (2005) reported SC in microdisks. Their results, which consist in an anticrossing of the modes at resonance claimed as a true VRS, are shown in Fig. 1.11. We will discuss at length in the rest of the text up to which point is pertinent to identify a splitting with SC and with VRS in the presence of decoherence.

**Figure 1.11:** The seminal observations of SC of a QD with a single-mode of light, by Yoshie et al. (2004), Reithmaier et al. (2004) and Peter et al. (2005). All observe a clear anticrossing of the modes as they are brought to resonance by tuning the temperature. Because the data of Reithmaier et al. (2004) appeared the cleanest to us, we decided to focus on this particular experiment. Subsequent realizations of SC with incoherent excitation appear in this plot as well.
$\includegraphics[width=0.8\linewidth]{introduction/SC.ps}$

In contrast with my superficial overview of the situation with Purcell effect, I shall attempt to be exhaustive in the list of SC reports (not counting multiple publications of the same experiment) following the three above mentioned. We shall see that there is ample room for them even in this short introduction. They are displayed in Fig. 1.12.

**Figure 1.12:** Later reports of SC in 0D semiconductor microcavities after the seminal breakthroughs of Fig. 1.11: by Hennessy et al. (2007), Press et al. (2007), Laucht, Hofbauer, Hauke, Angele, Stobbe, Kaniber, Böhm, Lodahl, Amann & Finley (2009) and Nomura et al. (2008).
$\includegraphics[width=\linewidth]{introduction/all-SC.ps}$

Despite the steady progresses made in all areas relevant to SC physics, the reports of SC have been rare. One of the latest one, from Laucht, Hofbauer, Hauke, Angele, Stobbe, Kaniber, Böhm, Lodahl, Amann & Finley (2009), comes with an interesting setup. The authors have realized an electrically controlled device, shown on Fig. 1.13, that is operated with a mere bias voltage, using the quantum confined Stark effect to detune the dots in SC with the mode of a L3 photonic crystal.^1.19 They excited the system with an off-resonant laser pulse, and observed the SE from the fully integrated (at the exception of the pumping) chip, allowing for quick, convenient and reversible control. Their work opens the road towards on-chip control of SC. In their subsequent work, Laucht, Hauke, Villas-Bôas, Hofbauer, Böhm, Kaniber & (2009) have not only improved their estimation of the SC parameters (by fitting their experiments with an extension of the model we present in this thesis), but also studied the influence of pure dephasing when increasing power or temperature.

**Figure 1.13:** Laucht, Hofbauer, Hauke, Angele, Stobbe, Kaniber, Böhm, Lodahl, Amann & (2009)'s on-chip device, controlled by an applied bias-voltage, opening the road towards fully-integrated cavity QED devices.
$\includegraphics[width=.7\linewidth]{introduction/laucht.ps}$

The SC reported by Hennessy et al. (2007) has a strange feature, the appearance of a triplet at resonance! The authors motivate the scenario that the triplet is in fact the superposition of a normal Rabi doublet of SC on top of a single line of the system in WC, and that the system is either in WC or SC depending of some irrelevant microscopic detail.^1.20 After all, the emission is collected from millions of realizations of SC, and if a fraction of them does not succeed but remain in WC, the net impression will be that of this perplexing triplet structure. In this sense their SC is not robust to its environment, but this is still better than no SC at all. The main value of this work was, anyway, not so much in this spectral line, but in its photon-counting statistics. Measuring $g^{(2)}$ (I shall explain its exact meaning in next Chapter), they could support the double result that: the middle peak was indeed not correlated with the doublet, showing that it is just an addition, plausibly indeed an irrelevant one (but the exact nature of which is not completely certain), and that the total emission is antibunched. This second result is the most important one. It is possibly the first tangible argument to support full field-quantization of the QD-MC system, that we have discussed previously. This result has been confirmed by Press et al. (2007). This is not completely conclusive, however, as although it proves that the dynamics involves a single quantum of excitation between two isolated modes (by itself already a considerable achievement), it does not instruct on the modes themselves (consider the vacuum Rabi problem of two harmonic oscillators, that gives the same result). After all, dimming classical light until single photons remain, would exhibit antibunching, but this says nothing about the emitter itself.^1.21

A genuine, or quantum, SC, should culminate with a direct, explicit demonstration of quantization, with one quantum more or less changing the behavior of the system. In the ideal picture where the QD system can be described accurately by a 2LS (that the QD cannot accommodate more than one fermionic exciton), the JC physics should apply.^1.22 Such a quantum sensitivity would then be strongly manifest, as is well known from its Hamiltonian structure (Shore & (1993) have given an authoritative review of this textbook system.) In particular, the so-called Jaynes-Cummings ladder, built up from the light and matter states dressed by their strong interaction, would provide such a direct, unarguable proof of cQED regime at its apogee. Such nonlinearities have been more or less directly observed by Brune et al. (1996) and Meekhof et al. (1996) in atomic cQED. Recently, direct spectroscopic evidence has been reported for atoms and superconducting circuits, in elaborate experiments by Schuster et al. (2008) and Fink et al. (2008) that remind the heroic efforts of Lamb to reveal the splitting of the orbitals of hydrogen. Even more recently, very clear transitions from up to the fifth step of the ladder have been unambiguously observed in circuit QED in very strong-coupling, with the Rabi splitting more than 260 times the vacuum linewidth Bishop et al. (2009)!

To the best of my knowledge, no such nonlinear features have been reported in microcavity QED. The panorama laid down by Figs. 1.11 and 1.12 show that observation of the VRS is already a difficult task. In Fig. 1.14, we have put together the SC parameters, as estimated by the authors, of all the experimental SC reports with incoherent continuous excitation. Here, we can compare the different realizations although, almost in all cases (Münch et al. (2009); Laucht, Hauke, Villas-Bôas, Hofbauer, Böhm, Kaniber & (2009) are the exceptions), the estimation of the parameters is not rigorous.

**Figure 1.14:** Graphical representation of the SC parameters, $\gamma_a/g$ and $\gamma_b/g$ , in the SC reports as estimated by the authors. The circles correspond to photonic crystals, the triangles to micropillars, the ellipse to a microdisk and the square to the case of atoms in optical cavities. Among all these experimental groups, only Laucht, Hauke, Villas-Bôas, Hofbauer, Böhm, Kaniber & Finley (2009) and Münch et al. (2009) carried out a rigorous estimation procedure (global fitting with a model). In all other cases, parameters are obtain from simple Lorentzian fittings of the spectra.
$\includegraphics[width=\linewidth]{introduction/new/SC.ps}$

In this work, I shall endeavor to take up the situation where it has been left by Carmichael et al. (1989) with regard to the theoretical description of the lineshapes of the SC system. My principal theme will be that the semiconductor case differs in at least one fundamental respect with the paradigm set up by atomic cQED. Namely, in its excitation scheme. In the canonical semiconductor case, a steady state is established by the presence of an incoherent pumping. This pumping itself is quite different in character than the coherent excitation that is typically used in the atomic or circuit QED case, where it enters in the Hamiltonian. Extending the previous description and solving the system, we will be rewarded by a beautiful generalization of the SE problem. As far as concrete, experimental physics is concerned, I will show that my considerations bear huge importance for understanding the data. In particular, it could explain why SC reports have been so scarce, and if the description is right, it would greatly help to correct this shortcoming and to benefit from a quantitative description of the result. As far as the problem of full-field quantization is concerned, I will propose that, despite figures of merit much lower in semiconductors than other systems (atomic and circuit QED), clear qualitative features could still be observed, but still thanks to the same proper understanding of the excitation scheme. I will show that, indeed, merely increasing the pump intensity in the hope of probing nonlinearities, could more likely bring the regime to the classical realm.