Photoluminescence from a quantum dot-cavity system

Photoluminescence from a quantum dot-cavity system

10 Photoluminescence from a quantum dot-cavity system G. TAREL and V. SAVONA, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, M. WINGER,...

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10 Photoluminescence from a quantum dot-cavity system G. TAREL and V. SAVONA, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, M. WINGER, T. VOLZ and A. IMAMOGLU, Eidgenössische Technische Hochschule Zürich (ETHZ), Switzerland

Abstract: This chapter describes the photoluminescence properties of a solid-state cavity quantum electrodynamics (CQED) system, consisting of a single quantum dot (QD) coupled to a photonic crystal (PC) defect cavity. It addresses in particular the phenomenon of cavity feeding, that is, the efficient luminescence at the cavity frequency even for large detunings. The chapter presents a theoretical model of cavity feeding that takes into account the coupling to wetting-layer states. We find excellent agreement between simulations based on this model and experimental observations in the case of large detuning. In addition, the influence of acoustic phonons for the small-detuning case is discussed. Key words: cavity feeding, semiconductor, quantum dot (QD), photonic crystal (PC) nanocavity, wetting layer (WL), cavity quantum electrodynamics (CQED).

10.1

Introduction: solid-state cavity quantum electrodynamics (CQED) systems with quantum dots (QDs)

During the last two decades, increasing scientific effort has been devoted to the study of the basic system of cavity quantum electrodynamics (CQED) (Raimond et al., 2001): a two-level atom coupled to an optical cavity in the regime of single or few quanta of excitation. The basic CQED parameters (see Fig. 10.1) are the coherent light-matter coupling constant g, the relaxation rate of the atom due to spontaneous emission into electromagnetic field modes other than the cavity mode γ and the photon loss rate of the cavity mode κ. Depending on the relative values of these parameters, two regimes exist. The first one, characterized by γ, κ ≫ g, is called the weakcoupling regime. Here, the cavity typically enhances the spontaneous emission rate of the atom and leads to the so-called Purcell effect (Purcell, 1946). In the opposite limit, with g ≫ |γ − κ|/4, the system is in strong coupling, the signature of which is the so-called normal-mode or vacuum-Rabi splitting (Khitrova et al., 2006). In this regime, the eigenstates of the total light-matter 332 © Woodhead Publishing Limited, 2012

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g κ γ

10.1 The basic system of cavity-QED. A single atom is coherently coupled to a cavity mode, with g denoting the coupling rate. The system is subject to optical losses via the cavity mode (at rate κ) and via spontaneous emission of the atom into radiation-field modes other than the cavity mode (at rate γ).

Hamiltonian are linear combinations of the eigenstates of the two-level system and of the light-field (Mabuchi and Doherty, 2002). In the time domain, this corresponds to Rabi oscillations: an initially excited two-level system will experience transitions to the ground state accompanied by the creation of a cavity photon, and subsequently will be re-excited a number of times before the excitation is lost either through decay via the cavity or the atom. As a consequence, the emission spectrum of the coupled system is split into two peaks that are separated by the vacuum-Rabi splitting. CQED has a huge potential impact in the field of photonics. Photons can be used to reliably transmit information and recent years have already witnessed the trend, for example in telecommunication applications, to replace electronic signal processing through technologies based on optics (Soljacic and Joannopoulos, 2004). Recently, the use of photons as carriers of information has become even more attractive within the framework of quantum information, which uses concepts such as entanglement, to outperform classical information technology (Steane, 1998). Single photons also play a central part in quantum-key distribution schemes (Nielsen and Chuang, 2002). However, photons do not interact with each other. A many-body interaction (a non-linearity in classical terms) is needed to gain control of the quantum state of one and two photons, as required for applications in quantum photonics and quantum information. CQED holds promises in this respect, since the strong coupling between light and matter induces an effective non-linearity at the single-photon level. All these attractive properties led to the implementation of CQED systems in many domains such as those using atoms (Raimond et al., 2001) and Josephson junctions (Dicarlo et al., 2009). In the case of semiconductor material, the two-level system very often used in practice is the discrete excitonic state in a semiconductor quantum

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dot (QD) (Bimberg et al., 1999). QDs, unlike for example quantum wells, have discrete lowest energy levels, due to the three-dimensional nature of the confinement. Using cavities with embedded QDs, the strong coupling regime was realized a few years ago by several groups (Reithmaier et al., 2004; Yoshie et al., 2004; Peter et al., 2005; Hennessy et al., 2007). Producing cavities that exhibit large quality factors and that contain QDs maximally coupled to the field of the cavity is desirable but technically challenging (Hennessy et al., 2004). However, mastering this technique opens the possibility for studying these objects in a well-controlled way and access the quantum optics of this mesoscopic system. It turns out that many properties are remarkably well described using the simple two-level artificial atom model. However, there are important effects that are due to the mesoscopic confinement of the QD and for which no analogy can be found in the atomic-physics context. Hence, the focus of the scientific community studying these systems has been on the deviations of QD–CQED systems from the ideal picture. Indeed, given the advantages of solid-state CQED systems over their atomic counterparts (integrability, scalability, etc.), it is essential to understand the behaviour of QD-based CQED systems. In particular, the effect of ‘offresonant coupling’ or ‘cavity feeding’ has puzzled and kept the community busy over quite a few years: the central issue here is what happens in terms of luminescence if the QD-resonance ω0 and the cavity mode at frequency ωc are detuned by a substantial amount. If a single transition of an ideal CQED system is excited, no emission is expected at the cavity frequency for δ = |ωc – ω0| exceeding a few cavity linewidths (in practice a few tens of µeVs), that is, when the spectral overlap of cavity mode and QD line is negligible. In this case, there is no physical mechanism able to compensate the energy difference δ. However, efficient emission at the cavity frequency is still observed for all kinds of solid-state QD-cavity systems, for detunings far exceeding δ. This is for example the case in single self-assembled QDs in photonic crystal nanocavities (Hennessy et al., 2007; Kaniber et al., 2008; Ota et al., 2009; Majumdar et al., 2010) and in micropillars (Suffczynski et al., 2009). Similar behavior can be observed in the case of pyramidal QDs (Dalacu et al., 2010; Calic et al., 2011). This feeding effect is clearly a feature of solid-state CQED with QDs and cannot be observed in atomic systems. The phenomenology of cavity feeding strongly underlines the influence of the solid-state nature of these systems. For example, acoustic and optical phonons in the solid-state matrix of self-assembled QD interact rather efficiently with the QD exciton, and provide a cavity feeding channel. In addition, such QDs are grown on top of a two-dimensional layer (the wetting layer (WL)). This allows many electron-hole states to co-exist, including states above the QD barrier. It leads to a non-trivial influence on the emission spectrum.

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The cavity feeding mechanism has been recently discussed by many groups, mainly focussing on the influence of phonons (Auffèves et al., 2008; Milde et al., 2008; Naesby et al., 2008; Hohenester et al., 2009; Ota et al., 2009). The influence of WL states, on the other hand, was only qualitatively discussed in the context of recent experimental works (Chauvin et al., 2009; Laucht et al., 2010), but a theoretical model was still lacking. Recently, we have developed a kinetic model for the QD that includes the influence of WL states, and explains qualitatively the experimental observations at large detuning (i.e. power dependences, emission spectra and two time correlations) (Winger et al., 2009). In this chapter, we will give an overview of these results and discuss them along with experimental results we have obtained using our samples.

10.1.1 The QD-based solid-state CQED system QD and cavity fabrication In this chapter, we present experimental results obtained with a CQED system consisting of a single self-assembled QD embedded in a photonic crystal defect cavity. At the moment, no deterministic fabrication method, such as lithography, for obtaining homogeneous good-quality QDs exists. Typically, the QDs used in the experiment are produced by an MBE growth technique called Stranski–Krastanow growth mode: two lattice-mismatched materials, in our case GaAs and InAs, are grown epitaxially on top of each other (Bimberg et al., 1999). The lattice mismatch leads to the spontaneous formation of small InAs islands through relaxation of elastic strain that under specific conditions can give rise to QDs exhibiting full confinement of the electronic states. The resulting discrete states give rise to characteristic narrow optical transitions (Brunner et al., 1992; Marzin et al., 1994) that resemble the emission spectra of real atoms – a feature at the origin of the widespread idea that a QD can be considered as an artificial atom. In our experiments, we use photonic crystal (PC) defect cavities. While there exists a large variety of possible PC defect geometries, the design we have adopted is that of an L3 cavity. Here, linear defect of three missing holes is introduced into a triangular lattice of cylindrical holes in a dielectric slab (Noda et al., 2007). The figure of merit for achieving strong emittercavity coupling in CQED systems is given by Q / V (or g/κ), where Q is the cavity quality factor, and V the mode volume (Khitrova et al., 2006). Note that γ, the spontaneous emission rate of the QD, is typically much smaller than κ, which sets the main dissipation rate. In order to achieve strong coupling, the key technical challenge is the implementation of a cavity structure that confines photons for a sufficiently long time, that is, the Q factor of the cavity should be large. At the same time, a small mode volume

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V ensures a large intracavity electric field and therefore a large coherent coupling strength g. Spectral and spatial tuning A difficulty with self-assembled QDs is their inherently random nucleation site in the Stranski–Krastanow growth mode of elastic strain relaxation. The route followed by most research groups working in the field is random positioning of a multitude of cavities on a substrate containing QDs with a density of several 10 µm−2. Subsequently, the devices that show significant coupling between the cavity mode and some QD transition are selected and used for further experiments (Reithmaier et al., 2004; Yoshie et al., 2004; Peter et al., 2005). Typically, this approach does not lead to optimal spatial alignment between the cavity and the QD, and even for a system that shows significant coupling, the distinction of the spectral features stemming from different QDs within the cavity-mode volume can be difficult. In contrast, we follow a deterministic coupling approach that relies on active positioning of PC cavities around pre-selected QDs. This approach has been thoroughly described by Hennessy et al. (2007), and we refer to this work for all details. In addition, control over the mode wavelength of the nanocavity is needed. In a standard Fabry–Pérot resonator, one of the mirrors is typically mounted on a piezo actuator, thus allowing for external control of the cavity length and thereby of its resonance frequencies. In contrast, monolithically integrated cavities usually do not have mechanical degrees of freedom. A widely used strategy in solid-state cavity QED is therefore the control of the QD-cavity detuning by controlling the temperature of the host material, either locally (Englund et al., 2007) or by heating the entire chip (Reithmaier et al., 2004; Yoshie et al., 2004). Since both the bandgap energy and the refractive index depend on temperature, this leads to a red shift both of the QD excitons and of the cavity mode. For GaAs-based PCs the temperature dependence of the former is stronger, such that changing the temperature mainly affects the QD energy levels, leading to a net red shift of the QD spectral lines with respect to the cavity mode. In a solid-state environment, however, raising the temperature in general leads to increased phonon scattering and thereby to increased decoherence rates – an undesired effect. An alternative method is based on the electric control of the QD exciton energy via the quantum confined Stark effect (Hofbauer et al., 2007; Laucht et al., 2009). Ideally, a tuning mechanism is desirable that leaves the QD transitions unaffected. In our experiment we use a thin-film deposition method for this purpose (Strauf et al., 2006): we inject hot nitrogen gas into the vacuum chamber, which is then adsorbed on the cold surface of the photonic crystal membrane leading to a red shift of the cavity resonance frequency.

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10.2

337

Cavity feeding: influence of multiexcitonic states at large detuning

In this section, we will focus on the situation of large QD-cavity detuning, for which a corresponding PL spectrum is depicted in Fig. 10.2. Here, the neutral QD line and the cavity mode are detuned by about 10 meV, that is, hundreds of cavity linewidths. Nevertheless, significant luminescence from the cavity mode is observed – a behaviour that has been reported by several groups (Hennessy et al., 2007; Srinivasan and Painter, 2007; Kaniber et al., 2008). In this situation, the large detuning prevents the acoustic phonons (see Section 10.6) from playing any role. Hence, they can be safely neglected. Feeding from neighbouring QDs is excluded as well, since there is only a single QD present in and coupled to the cavity. One might evoke additional pure dephasing mechanisms, for example one mediated by optical phonons (Muljarov and Zimmermann, 2007), in order to explain feeding in terms of a very large and non-Lorentzian QD transition lineshapes. It turns out, however, that we can exclude this possibility as well based on photoncorrelation measurements revealing the complex feeding dynamics in the QD-cavity system. We first measure the cross-correlation function revealing directly the interplay between the QD and the cavity ( 2) gCav − X

0

(

)=

: I cav ( t )I X 0 ( t + I cav

):

[10.1]

I X 0

PL count rate (a.u.)

104 X 1+

X 1–

X0 XX0

Cavity 102

100 946

950

954

958

Emission wavelength (nm)

10.2 PL spectrum under non-resonant excitation conditions on a semilogarithmic scale. Even though the cavity mode is far-detuned from the fundamental neutral QD transition, cavity-mode emission is very pronounced. Moreover, a fairly strong QD background emission is visible togther with the discrete QD emission lines such as the neutral exiton (X0), biexciton (XX0) and charged states (X1+, X1−). It is this background emission that is Purcell enhanced by the cavity and gives rise to efficient emission at the cavity frequency, as will be explained in detail in the theoretical part of this article.

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(a)

APD 1

TAC

Grating monochromator

A corresponding measured cross-correlation histogram is plotted in Fig. 10.3: the strong antibunching at τ = 0 confirms the single-QD nature of the feeding. In addition, the curve is strongly asymmetric, while feeding from a single two-level system with large broadening would produce a perfectly symmetric curve. A distinct feature of the correlation histogram in Fig. 10.3 is the slight bunching peak at small positive time delays. This bunching feature directly reflects the feeding mechanism from higher manifolds and will be discussed and explained in more detail later on. In addition to the X0-cavity cross-correlation trace, we took additional auto- and cross-correlation traces displayed in Fig. 10.4. While the temporal correlation between photons emitted at the energies of the neutral exciton (X0) and biexciton (XX0) is the expected one (Fig. 10.4c), it comes as a surprise that the emission from the cavity exhibits slight bunching and at higher power (not displayed) shows pure Poissonian statistics. But as we will see later, all these features can naturally be explained within our model of cavity feeding. Additional insight can be obtained from pump-power dependent measurements, as presented in Fig. 10.5. Here, X0/XX0 lines have the expected linear/super-linear pump-power dependence (which originates from their

APD 2 (b)

2 τ = 23.6 ns

g(2) cav,X0(τ)

1.5 1 0.5 0 –40

–20

0 20 Time delay (ns)

40

10.3 Photon cross-correlation measurement. (a) Schematic of the setup: the spectrometer grating separates the cavity mode from the X0 emission. The imaging system directs the two spectral windows onto two separate avalanche photo-diodes. (b) Cross-correlation histogram between cavity and X0 emission. Positive time delay corresponds to X0 detection upon observation of a cavity photon.

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(a) 1

0 –40

0

40

0

20

g2(τ)

(b)

1 0 –20 (c)

1 0 –40

0 Time delay τ (ns)

40

10.4 (a) Measured cross-correlation histogram for photon pairs emitted at the exciton and at the cavity frequency, respectively (similar to Fig. 10.3). (b) Measured auto-correlation histogram of cavity photons. (c) Measured cross-correlation histogram between photons emitted at exciton and biexciton frequencies.

Cavity mode X0

PL count rate (c.p.s.)

105

XX0 4

10

103 102

101

101

102 103 Excitation power (nW)

104

10.5 Measured PL intensity as a function of excitation power for the cavity mode, the neutral exciton and the biexciton emission. Over a wide range of excitation powers, the cavity-mode emission follows that of the biexciton.

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one/two electron-hole pairs’ nature). From the data, we see that the cavitymode emission has a super-linear pump-power dependence over a wide range of pump powers, and moreover, in contrast to the two QD emission lines, does not display any signature of saturation over the power range considered here. The scenario that we propose is the following: due to the QD’s very small size, the typical excitation lasers needed to obtain luminescence forces the system to be most likely in a highly excited state with more than one electron-hole pair. In addition, usual experiments are performed using non resonant excitation. In other words, not only confined levels in the QD are part of its dynamics, but also low energy states in the WL, as already discussed in (Vasanelli et al., 2002). These states provide a continuous spectrum and therefore can contribute to cavity feeding. A first indication supporting this hypothesis comes from Fig. 10.2. On the semi-logarithmic scale, a QD emission background is present together with the discrete QD lines. This background emission is due to the hybridization with WL states. Even though the background is relatively weak compared to the discrete emission lines, it is Purcell enhanced by the cavity and leads to efficient emission at the cavity frequency. In what follows, we develop a theoretical model that provides very strong support for this picture since it successfully reproduces all observed experimental features. The PL background can be explained in terms of cascaded emission from n-exciton (n–X) states. In this case, emission from a low energy n–X (i.e. n electron-hole pairs) state to the continuum of excited (n − 1)–X states produces a background that is enhanced by the cavity mode through the Purcell effect. We will show that this explanation accounts for the observed emission spectrum, the power dependence and the photon correlations. A very suggestive picture of the physical mechanism is the following. Due to the large detuning, the process of a confined electron-hole pair relaxing into a cavity photon is in principle energetically forbidden. But this kind of approach neglects the influence of Coulomb correlations, which is of main importance (Baer et al., 2006): if the excess energy is carried away by an additional electron-hole pair through Coulomb interaction (shake-up), then the total energy is conserved and the emission occurs. We will argue that the cascaded emission from n–X states in fact coincides with the shake-up picture (Kaniber et al., 2008).

10.3

Model for a QD-cavity system

The first step in modelling the cavity feeding mechanism consists in developing an accurate model of multi-exciton (MX) states in the QD, including those lying in the WL, at energies above the QD barrier. For this, we will first compute single-particle electron and hole states within the effective

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mass approximation, for a model QD confining potential. Then, for each given number n of electron-hole pairs present in the system, we will compute Coulomb matrix elements between noninteracting n-electron-hole configurations, and diagonalize the resulting matrix. This truncated configuration interaction approach is expected to give close-to-exact Coulomb-correlated states, as the number of configurations is increased. A necessary feature of the model is that the WL states form a spectral continuum. In reality, given the numerical approach that we apply to a system of finite area, the actual spectrum will be made of a set of discrete closely spaced levels. We must take care that this energy spacing is smaller than all relevant energy scales in the system, in order for the model to be a good representation of a continuous spectrum. Since, however, the treatment of the Coulomb correlations has a complexity that grows fast with the number of electronhole pair states, a trade-off between an accurate description of our system and the numerical feasibility of the simulations has to be found. We also neglect the excitation manifolds with an odd number of particles (charged states), which, however, contribute to the emission process (Hartmann et al., 2000). We argue that charged states affect the final result only quantitatively, without qualitatively changing our conclusions about the feeding mechanism and the shape of the two-photon correlation functions.

10.3.1 QD model The QD is modeled using a truncated parabolic potential in 2-D and accounting for the WL continuum (see Fig. 10.6). The confinement potential along z is approximated by a Dirac delta function, implying a single confined state. In this way, the motion along the z-direction is assumed to be ;frozen’. This assumption is justified by the fact that both QD and WL are very thin and the energy distance to excited levels of the z-motion is significantly larger than any other relevant energy scale in the processes under study. This model is the best compromise between the most detailed description of the single-particle levels and the simplicity needed to perform the heavy numerics required in the next steps. Thus we do not include the actual shape of the QD, and the strain given by the surrounding semiconductor matrix. Using a more refined model for the QD would lead to a different – though analogous – energy spectrum, without implying qualitative changes to our conclusions. For the description of single-particle states, we use a one-band effective mass model to compute a set of Nh = 32 heavy hole and Ne = 18 electron states (including spin). In a second quantized scheme, they can be † † written as ei = c i 0 for the electron and hi = d i 0 for the hole, where |0〉 is the vacuum state (ci† di† is the creation operator for the electron/hole in the ith state). The result is depicted in Fig. 10.6 and features an ensemble of confined and extended states for electrons and holes.

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60 meV Egap

10.6 Model for the single-particle states in the QD, based on a truncated parabolic potential.

The actual n-electron-hole pair states of the QD are the MX (denoted by n–X), namely eigenstates of the Coulomb Hamiltonian. They are obtained starting from the basis of single-particle levels. We limit our investigation to the case of neutral MX states, neglecting processes involving charged states. Before including Coulomb correlations, the basis elements for the n–X manifold are of the form n, i1



in j1 … jn = c i1





c in d j



d jn 0

[10.2]

with {i1…in, j1…jn} ∈ {1…Ne} × {1…Nh}. These are configurations of n noninteracting electron-hole pairs. We will use the ensemble formed by all |n, i1… in, j1…jn〉 as a basis for the actual, that is, Coulomb-correlated MX levels. For example, a state in the biexcitonic manifold (not to be confused with XX0, which is only the ground state of this manifold) will be written as Ne

bx =

Nh

∑ ∑

i1 i2 = 1 j1 j2 = 1

α bbx i1 i2 , j1

j2

[10.3]

2, i1 i2 , j1 j2 ,

where α i1 i2 , j1 j2 are complex coefficients derived from the diagonalization of the Coulomb Hamiltonian. We must calculate all the matrix elements of of the many-body Hamiltonian, that is, for every manifold n–X, the elements 〈2, i1, i2, j1, j2|Hcc|2, i′1, i′2, j′1, j′2〉 with Hcc given by bx

Hcc

∑ ⎛⎝E

(e) c  †j c j j

j

1 + 2j

1



j2 , j3 j4

† 1 + E (j h) d j d j ⎞ + ⎠ 2 i









Vi1 ,ii2 ,i3 ,i4 c i1 c i2 c i3 c i4

1 i2 , i3 i4

† † Vj1 j2 , j3 j4 d j1 d j2 d j3 d j4







Vi1, j1 , j2 i2 c i1 d j1 d j2 c i2

i1 i2 , j1 j2

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[10.4]

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E[eV] n = 4 (3000) n = 3 (2500)

n = 2 (1500) E(XX0) 100 meV

n = 1 (700)

E(X0) E=0

10.7 Energy scheme of computed n–X states, with n = 0 to n = 4. Only neutral states are computed. In brackets, the number of configurations retained for the diagonalization of the Coulomb interaction is indicated.

We perform this configuration interaction calculation (Barenco and Dupertuis, 1995; Hawrylak, 1999; Biolatti et al., 2002; Sheng et al., 2005) up to n = 4. It is important to note that this is an approximate method, in the sense that we use a truncated basis. The convergence of the computed states toward the actual states of the system increases when increasing the size of the single-particle set, with a better convergence for lowest energy states. For the purpose of accurately describing most of the MX states, it is therefore necessary to use the largest single particle basis possible. In our case, we compute more than 2000 MX states, including the spin degrees of freedom. The accuracy of our method is estimated by evaluating the binding energies. Values of the order of 30 meV for X0 and 3 meV for XX0 agree satisfactorily with well established theoretical and experimental results (Langbein et al., 2004; Rodt et al., 2005). The energy spectrum of the MX states computed for the specific system parameters used throughout this work is depicted in Fig. 10.7, for manifolds up to n = 4 electron-hole pairs. The excited states of each manifold form a spectral quasi-continuum that plays the essential role in determining the cavity feeding mechanism, as explained in the next section.

10.3.2 Joint-optical density of states and radiative rates The rates for the radiative processes, going from a state X in in a n–X multiplet to the X jn−1 state of multiplet (n − 1)–X, are computed using Fermi’s golden rule, as −1 τ rad = ,X n → X n−1 i

j

2π X in H rad X jn − 1 

2

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[10.5]

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Here, H rad

ρC ( E )

∑I

αβ

αβ

⎛c † d † + h c.⎞ ⎝ α β ⎠

[10.6]

is the radiative Hamiltonian,

ρC ( E ) =

ω 4 π κ 2 V nefff (E Ec )2 + κ 2

[10.7]

is the density of optical states in the cavity that leads to the perturbative modification of the spontaneous emission rate called the Purcell effect (Purcell, 1946; Noda et al., 2007), with Ec the cavity resonance and κ the linewidth of the cavity mode. Iαβ are the matrix elements expressing the amplitudes of the recombination process I αβ

μ cv



α e

()

β h

( ) Ecav ( ) dr

[10.8]

µcv is the Bloch part of the interband dipole matrix element, V is the mode volume, neff is the effective refractive index. Finally Ecav(r) is the cavity-mode wave function. The Hamiltonian Equation [10.6] is to be considered as an effective Hamiltonian, appropriate to provide the correct transition rates if used within Fermi’s golden rule. In a microscopic description of the process, the coupling to the actual continuum of electromagnetic modes should instead be accounted for, and the Purcell enhancement factor would appear at a later stage when computing the transition rates. For the numerical calculations we will use κ = 100 µeV and V = 0.07 µm3, typical of a photonic crystal nanocavity system. In order to understand how a multiply excited QD can give rise to cavity feeding, we introduce the jointoptical density En. For transitions from the n-th to the (n − 1)-th manifold, this is defined as J En

∑τ i j

1 rad, X i → X jn

1

(

δ EX n − EX n − i

j

)

[10.9]

The results for the joint-optical density of states J En are presented in Fig. 10.8, for n = 1 to n = 3. We can see that because of the energy distance between the ground exciton state X0 and the first excited exciton state, no PL line other than the X0 is expected for transitions from the n = 1 to n = 0 manifolds. On the other hand, the existence of a continuum of single-particle states is the origin for a large background of transitions that, when enhanced by the cavity, will lead to the cavity peak in the PL spectrum. These transitions

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Detuning from X0 (nm) Joint optical density of states (a.u.)

–10 n=3

–5

0

5

10 2 10

n=2 101 10–2

n=2

n=1 100 10–2

n=1

n=0 100 X0

XX0

C 10–2

10.8 Joint-optical density of states JnE for a QD. (a) 1 − X to ground state (n = 1) (b) 2 − X to 1 − X (n = 2) (c) 3 − X to 2 − X (n = 3). Dashed are at X0/XX0 and cavity mode energy.

are mostly given by the recombination from a low energy state of the n-th manifold to an excited state of the (n − 1) manifold. The quasi-continuous nature of the spectrum of these final states is what determines the broad background. The joint-optical density of states alone, however, is not sufficient to explain the actual shape of measured spectra, where a few peaks dominate over the broad background, sitting at the wavelengths of the X0-toground-state transition, cavity mode, and XX0-to-X0 transition. The kinetics of excitation and decay processes inside the QD also plays an important role in determining this spectrum. In particular, at moderate excitation intensity, the X0 line originates mostly from the X0-to-ground-state transition, although the joint-optical density of states would allow for excited-manifold transitions at the same wavelength. The dominance of this particular process also explains the asymmetric shape of the cross correlation between exciton and cavity lines, shown in Fig. 10.4a. The cavity line on the other hand is pronounced simply because of the Purcell enhancement mechanism, which selects that particular spectral region out of the broad spectrum. Finally, the XX0-to-X0 spectral signature is pronounced – in spite of the broad spectrum predicted by the joint-optical density of states – because the relaxation kinetics within the n = 2 manifold strongly favors the ground state, as results from the Monte Carlo simulations of the kinetic process.

10.3.3 The capture process The following we have denoted as a capture process: the excitation laser creates a large number of electron-hole pairs lying in spatially extended high

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energy states of the WL. Then, pairs of states having a significant overlap with the QD can relax down to the levels described in our model. The corresponding rates are computed to first order in the matrix elements of a phenomenological Hamiltonian, and applying Fermi’s golden rule −1 τabs = , X n → X n+ 1 i

j

2π X in Habs X jn + 1 

2

,

[10.10]

With Habs =

abs

∑ ⎛⎝c αβ

† †  α dβ

h.c.⎞, ⎠

[10.11]

where γabs is a phenomenological constant that we model as linearly dependent on the pump intensity. As already mentioned, with this assumption we are neglecting the capture process of single electron and holes, namely the occurrence of charged MX complexes both in the WL and in the QD. Single carrier capture processes are known to play an important role in determining the spectrum of QDs and QD-cavity systems: this can be seen in the pronounced charged exciton peaks in the spectrum of Fig. 10.2. The present model, however, is aimed mainly at proving the relevance of the MX cascaded emission process in determining far-off resonance cavity feeding. To this purpose, for the sake of simplicity we decided to restrict to neutral states only. As a consequence, we expect the excitation power needed in our model to observe cavity feeding to be overestimated with respect to the experimental one, and the power dependence of the process to depart from the measured one. In Section 10.4 we lay down the main steps of a slightly different approach that includes charged states, and that we expect to be more reliable for quantitative predictions.

10.3.4 Relaxation, a linear process involving polarons In a recent paper , Grange et al. (2007) addressed the problem of the phonon bottleneck in the intradot relaxation of QD states. In first view, a phonon bottleneck is expected in the intradot relaxation due to the discrete nature of the electron spectrum, which is poorly matched to the narrow dispersion of optical phonons. However, recent experimental investigations (see e.g. Hameau et al., 1999) have proven actual single-particle states in QDs to be polarons (arising from the strong coupling of electron and LO phonons). By accounting for the presence of these polaron states and the different channels leading to the polaron decay, they were able to prove that no bottleneck exists for the relaxation within the QD. As shown by Grange et al., the strong longitudinal optical phonon coupling leads to a very fast

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intradot relaxation, even for an energy step exceeding 10 meV. Similar conclusions are drawn also by other studies in which the non-perturbative electron–phonon coupling is investigated within the independent boson model (Stauber et al., 2000) or via a quantum kinetic approach beyond the Markov approximation (Seebeck et al., 2005). We adopt for our n–X level scheme the phonon decay rates computed in Grange et al. (2007), assuming that the electron and hole spin are conserved in the process. These rates were computed by assuming polaron states that are linear combinations of zero- and one-LO-phonon states, obtained by diagonalizing the Fröhlich Hamiltonian describing the interaction between electrons and LO-phonons. In the present analysis, the rates were introduced phenomenologically, directly at the level of the Monte Carlo kinetics.

10.3.5 Spin-flip process Another important process involving n–X states is the spin-flip process. The dominant spin-flip mechanism in semiconductor nanostructures is the spin-orbit interaction (Bulaev and Loss, 2005) whose two contributions are the Dresselhaus mechanism (Dresselhaus, 1955) and the Rashba coupling (Bychkov and Rashba, 1984). As for other processes, we prefer to model the process in a phenomenological fashion, as a microscopic model would not improve in a relevant way the reliability of the model in explaining the basic features of the PL spectrum and of the correlation functions. When considering the interplay between different scattering processes in the QD-cavity system, it turns out that spin-flip processes play an important role in determining the cavity feeding mechanism. In fact, the phononassisted energy relaxation process within the QD competes against the emission of cavity photons from higher n–X manifolds. Without considering the spin, the polaron process would be largely dominant. It would quickly bring the system to the ground state of the manifold, thus suppressing the occupation of high-energy states and preventing the cavity feeding process from occurring. The spin therefore appears as an essential ingredient of our model. A sizeable spin-flip rate will effectively slow down the phonon-assisted energy relaxation process, in a way similar to the motional narrowing process known from nuclear magnetic resonance (Bloembergen et al., 1948). In simple words, if spin-flip and relaxation rates are comparable, then each time the system will randomly choose (with comparable likelihoods) whether to relax or to undergo a spin flip, thus halving the average time before the next relaxation process. Another important effect of spin flips is to compete with the radiative decay at the cavity wavelength from higher MX manifolds. This happens whenever the spin difference between initial and final states in the radiative decay is different from ±1. Our simulations have shown that this slowdown of the emission at the cavity wavelength is

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essential to reproduce experimental data, as otherwise the cavity feeding signature would be considerably overestimated. We account for a spin-flip process by including in the model a quasi-resonant phenomenological spin-flip mechanism with a rate of 100 ps (Grange, 2008).

10.4

Radiative processes revisited

In the approach developed previously, there is only one way to compute radiative rates: these are direct transitions from a given n–X state (involving WL states or not) to another. In this scheme, WL and QD states are taken into account on the same ground. The drawback is the very demanding numerical procedure required to compute the configuration interaction. The essential features of the MX picture can be described within a simplified approach that, by distinguishing the roles of QD and WL states, allows applying perturbation theory and restricting the configuration interaction scheme to the QD states only. The key assumption of this second approach is that the capture of an electron-hole pair in a QD state can be described within second-order perturbation theory, without the need to model the WL MX states explicitly. We will rather describe extended states as an external bath. We assume that the excitation laser creates a distribution of electrons and holes in extended states. The distribution can be assumed as non-degenerate if the density is not too large. The general idea behind this description was suggested by Kaniber et al. (2008) and Winger et al.( 2009). We start with an initial/final n–X state |ci〉/|cf〉, and consider an initial/final electron state in the WL |ki〉/kf〉. They will V , where L is the light-matter interacinteract through the Hamiltonian L the Coulomb interaction between extended and confined states. tion, and V Indeed, the Coulomb interaction is already included within confined state (n–X states are Coulomb correlated), and neglected within extended states (WL is treated as an uncorrelated electron-hole plasma). A schematic view of this process is given in Fig. 10.9: the excess energy needed to emit at ωc despite the detuning is carried away through Coulomb interaction by a WL electron. In second-order perturbation theory the transition rate ω ki i kff , mediated by the WL states |ki〉 and |kf〉, can be written as a sum over intermediate states |dp〉 where d is the confined part and p the plane wave component (i.e. WL state). k kf f

ωi i

=

2π 

V dp dp L V c k ci ki L f f



×δ

ΔE

dp

(

i

f

)

ω =

2π mi 

2

[10.12] 2 f

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kf ki

WL

QD



10.9 Radiative process assisted by an electron: the excess energy to emit at the cavity wavelength is carried by a WL electron through Coulomb interaction.

where hω is the energy of the emitted photon and ΔE = Eci+Eki−Edp = E0−Edp. We then separate mi→f in two parts and keep only mixed terms in the interaction as the two other terms do not allow the transition we wish to describe. mi

f

=

∑ dp

⎡ ck V d c k d c k dp dp L ci ki L dp dp V f f f f ⎢ i i + ⎢ E0 Edp E0 Eddp − Eph ⎢⎣

⎤ ⎥ ⎥, ⎥⎦

[10.13] The first term in this expression does not involve photons in the intermediate state, while there is one in the second term. In the next step, we include does not modify the momentum the fact that the light-matter interaction L of the WL state. In addition, the matrix element of the Coulomb interac should not be taken between two confined states, which are already tion V Coulomb correlated. This leaves only one term in the sum, namely ⎡ ck V ck c c c k ci L ci L c f ki V i f f f f f ⎢ i i mi f = ⎢ + Eki Ek f Eci Ec f − Eph ⎢⎣ which in turn leads to k kf f

ωi i

=

2 π −1 τ rad,ci →c f 

ck ci ki V i f Ei

Ef

+

c k c f ki V f f Ei

E f − ω

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⎤ ⎥ ⎥ , [10.14] ⎥⎦

2

,

[10.15]

350

Quantum optics with semiconductor nanostructures −1

where τ rad,ci →c f is the Purcell transition rate, and Ei/Ef is the energy of the initial/final n–X state. From this we can see that the rate of the assisted process is the product of the direct rate and a term depending on the strength of the Coulomb interaction between WL and confined states. This is then summed over all possible initial and final states for the WL electron. This sum can be expressed as an integral over the distribution of WL states. The resulting total radiative rate Ωi→f is then the sum of the contribution of elece− e− trons Ωi f and holes Ωi f . For example,

Ωei

f

ne



ki f (ki ) Γ cp →c i

ck ci ki V i f f

Ei

Ef

+

c k c f ki V f f Ei

E f − ω

2

, [10.16]

where f(ki) is taken as the Maxwell–Boltzmann distribution (we assume a nondegenerate electron bath) of electrons in WL states. This rate can take significant values in typical situations. As an example, by assuming ne− = 1010 cm−2, the radiative rate of a biexciton state into the cavity mode, computed from Equation [10.16], is of the order of 0.5 ns−1, that is, comparable to the direct radiative rate of an exciton confined in the QD. Similar considerations might be applied to the relaxation process within one MX manifold, where a Coulomb process assisted by the scattering of a carrier in the WL might occur. We, however, expect the polaron relaxation to be dominant at low and moderate density of carriers in the WL, while Coulomb-assisted relaxation should become important at higher excitation power. Finally, it should be noted that this perturbative approach does not account for a screening of the Coulomb interaction induced by the finite carrier density in the WL. Introducing such a screening would result in time-dependent rates for the kinetics, which are computationally very challenging within a Monte Carlo approach. We thus limit it to the bare Coulomb interaction, keeping in mind that this constitutes a crude approximation.

10.5

Cavity feeding: Monte Carlo model

In the previous sections, we have derived or presented the rates of all essential processes taking place in a QD–Monte Carlo system under nonresonant excitation. They are all closely related to the complex nature of the semiconductor system. For example, phonons play an important role in the intradot relaxation. In addition WL states contribute to capture and to the existence of a broad background in the QD PL spectrum. This has,

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as a consequence, the occurrence of very fast rates for processes in which a detuned cavity photon is emitted from a MX transition. After having characterized the rates of all relevant processes, we model the kinetics of the QD excitation, relaxation and emission processes within a classical Monte Carlo simulation, that we present in Winger et al. (2009). Monte Carlo simulations can in general be successfully applied to systems of QDs. In particular, a Monte Carlo algorithm can be used to simulate the kinetics of the transitions between different ‘microstates’ (Grundmann and Bimberg, 1997), generated by the various interaction processes. Here, we will assume a steady state pump process that drives the capture process in the QD. All results are computed using the full model of MX states, including Coulomb-correlated WL states, as discussed in Section 10.3. A similar procedure can, however, be applied to the simplified model presented in Section 10.4, by including the second-order Coulomb-assisted rates among the possible Monte Carlo steps. It is not the purpose of the present work to apply the Monte Carlo simulation to this second scheme, but we have carefully checked that all results presented below are reproduced by the alternative approach. The Monte Carlo algorithm that we apply strictly follows the prescriptions described in Jacoboni and Reggiani (1983) and we refer to that work for all details about the method. A typical Monte Carlo walk consists of a time series of events – capture, radiative decay, phonon relaxation or spin flip – occurring between pairs of n–X levels.

10.5.1 First results In the following, all the results we show are computed for a large 5 nm X0-cavity detuning, for which the influence of phonons can be safely neglected. Let us recall Equation [10.10], which describes the capture process. The rates of the electron-hole capture processes are computed by calculating the matrix elements of a phenomenological Hamiltonian where γabs is the phenomenological pump intensity. Relating the actual pumping intensity in experiments with this parameter needs to be done carefully. In what follows, we define P0 as a reference pump intensity. Its value is chosen such that the system is still below the saturation of the emitted intensity at the exciton linewidth. To start, we plot in Fig. 10.10 an example of a Monte Carlo walk computed for a pump intensity equal to P0. To illustrate the relaxation involving optical phonons (i.e. the polaron effect), in Fig. 10.11 we magnify a region of the biexcitonic manifold (i.e. two electron-hole pairs). Here, an excited electron-hole pair is created and subsequently relaxes down to the XX0 state (ground state of the manifold).

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Quantum optics with semiconductor nanostructures

XX0 X0 Cavity

Energy (eV)

352

E(X0)

0 0

500 Time (ns)

10.10 An example of a Monte Carlo walk. The lines are the n–X levels, as in Fig. 10.7. Line shapes label emissions at the cavity/X0/XX0 frequency. The pump is equal to the reference value P0.

Energy

Spin-flip

E(XX0) 0

100 Time (ps)

10.11 A region of a Monte Carlo walk, magnified to illustrate the decay process involving phonons. The lines are the levels as in Fig. 10.7. In this plot, the pump intensity was set to our reference P0.

10.5.2 Computation of emission spectra and correlation curves The PL spectrum can be modeled as a sum of Lorentzian lineshapes, associated to each radiative decay process in the Monte Carlo walk. For the purpose of the present analysis, the Lorentzian linewidths have been chosen all equal to a fixed value of 10 µev, which is a reasonable assumption to qualitatively extract PL spectra. An example of computed PL spectrum is

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PL intensity

Photoluminescence from a quantum dot-cavity system

X0

–2

XX0

0

353

Cavity

2

4

6

λ–λx0 (nm)

10.12 Computed PL spectrum. QD emission lines are labeled (neutral exiton (X0), biexciton (XX0)) together with emission at the cavity wavelength (C). The pump is set to P0.

plotted in Fig. 10.12. It mainly features three main peaks at the linewidths of the cavity, X0 and XX0 respectively. Smaller structures originate from other radiative decays among the computed n–X states, which still have a sizeable radiative rate to contribute to the spectrum. Similar features are present in all measured PL spectra. From this spectrum it is clear that the system profoundly differs from an ideal CQED system, for which no cavity peak would appear for detuning beyond a few linewidths, as there would be no process available to compensate the excess energy. We should point out that these spectra are still obtained by neglecting charged states. However, even without charged states, our model catches the essential features of the feeding mechanism. More information can be extracted from the model, as the Monte Carlo walk provides us with the time record of all emitted photons. We can then select transitions corresponding to the three main peaks and, in particular, label cavity, X0 and XX0 photons. Photons are attributed to a given peak with a tolerance of 0.5 nm in wavelength, which is typical of the experimental setup. Then we have access to the two time correlations g2(τ), as in a Hanbury-Brown and Twiss experiment. The results are presented in Fig. 10.13. The shape of these correlation curves can be explained with the help of Fig. 10.14. The cavity-X0 cross correlation is not symmetric with respect to zero delay, contrarily to what would be expected in the case of phonon-assisted feeding. Indeed there is a possible cavity-X0 cascade which gives the sharp increase for τ > 0, while the soft decrease for τ < 0 is due to the time needed to re-excite the QD after the emission at X0, before the system can again feed a photon at the cavity wavelength. The cavity auto-correlation has no antibunching, as a direct consequence of the

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Quantum optics with semiconductor nanostructures (a) 1 0 –40

0

40

0

20

g2(τ)

(b)

1 0 –20 (c)

1 0 –40

0 Time delay τ (ns)

40

10.13 Computed two-photon correlation curves for (a) exciton-cavity, (b) cavity–cavity and (c) exciton–biexciton photon pairs. The pump is set to P0.

XX0 X0 Cavity (a)

(b)

(c)

Energy (ev)

354

0

500 Time (ns)

10.14 Sketches illustrating the processes that can give rise to the three correlation curves of Fig. 10.13. (a) After emitting at the exciton wavelength the system has to be excited at least twice before being able to emit a photon at the cavity wavelength. (b) Cavity photons can be emitted at arbitrarily short time intervals, through a cascaded emission from a high n–X manifold. (c) The X0-XX0 correlation curve has a shape similar to that of the X0-C curve, because it arises from an analogous sequence of emission and excitation processes.

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multiexcitonic cascaded emission: when starting from a high n–X manifold, it is possible to undergo a cascaded emission where all emitted photons match the cavity wavelength. Finally, the XX0/X0 cross correlation is asymmetric, as is usually expected from experiments on single QDs.

10.5.3 Power dependent results One quantity that can be easily varied in experiments is the non-resonant pump. In Figs 10.15 and 10.16, we present computed data obtained at four different pumps. The first observation one can make is that for P = 20 * P0 (P0 is the reference pump) we are in a regime where our model is no longer valid. In this limit, a suitable model should include manifolds with more than four electron-hole pairs. For the three other cases, one sees that when increasing the pump, the system is more likely to find itself in a higher n–X manifold. In the corresponding PL spectra, the build-up of the cavity peak is clearly visible, while X0 and XX0 tend to saturate, as expected. In addition, we present in Fig. 10.16 the computed X0-cavity crosscorrelation curves for the four cases depicted in Fig. 10.15. They show a decrease of the width of the central dip with increasing power. Again, they feature non-physical results for too large pump.

10.5.4 Comparison with experimental data The results of this model can now be compared to our experimental data. A first argument that speaks in favor of our model is the super-linear pumppower dependence of the cavity peak that is pointed out in the experimental results. This is a direct consequence of the MX origin of this peak, and it can be verified using our Monte-Carlo approach, as can be seen in Fig. 10.15. We can then compare the computed cross correlations with experimental data (see Figs. 10.13 and 10.4). The agreement is qualitatively very good. In particular, the bunching observed for the cavity auto-correlation curve is reproduced – although somewhat overestimated – mainly due to the absence of charged states in the model. Indeed, cavity feeding can in reality also originate from charged QD states, namely for a radiative process involving the lowest (one electron or one hole) manifold.

10.5.5 Further experimental evidence As we have just discussed, the theory presented here is confirmed by comparison with experimental data. With the further support of two recent experimental works (Chauvin et al., 2009; Laucht et al., 2010), the influence

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Quantum optics with semiconductor nanostructures (a)

XX0 X0 Cavity

E(X0) 0 (b)

Energy

E(X0) 0 (c)

E(X0) 0 (d)

E(X0) 0 0 T = 10 ms

2 4 λ–λX0 (nm)

6

10.15 Monte Carlo walks, for varying pump intensity (left) and corresponding computed emission spectra (right). The pump is set to: (a) P0, (b) 2P0, (c) 4P0 and (d) 20P0.

of WL states on the emission properties of QD-cavity systems at large detuning is now an established fact. In the first of these works, Chauvin et al. (2009) investigate the system of a non-resonantly pumped QD-cavity system, with an additional electric field parallel to the WL. The effect of this field can be understood as follows: WL states are strongly depopulated, thus bringing the density of states back to the one of a macroatom. The consequence is a vanishing intensity at the cavity wavelength for increasing electric field. In the second experiment by Laucht et al. (2010), the same system is studied but under pulsed excitation, and the time dynamics of the observed photons is recorded. As a result, a large initial population of electron-hole pairs is created. Immediately after, the system decays mostly through the emission of cavity photons, while the X0 intensity is negligible and only increases

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Photoluminescence from a quantum dot-cavity system (a)

(c)

(b)

(d)

357

g2(τ)

1 0

1 0 –20

0

20

–20

0

20

Time delay τ (ns)

10.16 Computed cavity-X0 correlations, for varying pump intensity (a) P0, (b) 2P0, (c) 4P0 and (d) 20P0.

with increasing delay after the excitation pulse. Indeed, in the time-resolved data the cavity peak appears at early times and fades at the same time as the X0 peak gets stronger. This is perfectly compatible with our model in which cavity feeding originates from the radiative decay from higher n–X manifolds.

10.6

Cavity feeding: influence of acoustic phonons at small detuning

In the previous section, we discussed the influence of electronic degrees of freedom in the WL at large X0-cavity detuning (∼ 10 meV). We will now consider the case of small exciton-cavity detuning, where, in addition to WL states, the confined electrons and holes interact efficiently with the vibrational degrees of freedom of the semiconductor environment (Besombes et al., 2001; Borri et al., 2001; Vasanelli et al., 2002; Krummheuer et al., 2005; Muljarov and Zimmermann, 2007). The coupling of the QD to an external reservoir of phonons turns out to be another cause of strong deviations from the ideal CQED behaviour. As a general trend, electronic and vibrational degrees of freedom are described as separate to a first approximation (the Born–Oppenheimer approximation), and their coupling is then accounted for perturbatively. When an electron state is spatially extended, its effect in modifying the local vibrational properties of the atomic lattice is very weak. For localized excitons, however, an electron can significantly affect the local vibrational modes depending on its state, thus giving rise to strong electron–phonon

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coupling. This gives rise to hybrid states of electron and phonon degrees of freedom that can only be described beyond the Born–Oppenheimer picture. For a single electron the problem admits an analytical solution, called an independent boson model (Mahan, 1981). Similar effects are known to affect the behaviour of molecules and of impurity states in crystals. It is thus natural to expect these effects for QDs as well. Indeed, non-perturbative electron–phonon coupling has been characterized both for optical and acoustic phonons. For acoustic phonons, the independent boson model leads to broad sidebands originating from the coupling to longitudinal acoustic phonons (Besombes et al., 2001; Borri et al., 2001; Zimmermann and Runge, 2002; Favero et al., 2003; Milde et al., 2008). A similar effect arises from the coupling to optical phonons (Verzelen et al., 2002; Stauber and Zimmermann, 2006; Muljarov and Zimmermann, 2007).

10.6.1 A simple model Simple two-level system In Tarel and Savona (2010), we develop a semi-classical formalism for modeling the PL spectrum of a QD embedded in a nanocavity that features a resonant mode at the frequency ωc. The QD is modelled as a two-level emitter, which couples to the electric field via its linear susceptibility tensor. In a first step, we derive the linear spectrum of the QD, which exhibits a Lorentzian line at frequency ω0 that is characterized by a constant broadening γ0. In this case, one recovers the result that is well known from CQED (Carmichael et al., 1989) S (ω ) = =



4g2

(

ω − i (γ

Ω+ − ω0 + i (

ω − Ω+

)2 4 ( )) ( ω c )−

c

i (κ

))

(

)

)) − g 2

ω i (κ



− ω0 +

[10.17]

2

ω − Ω−

,

expressed in the resonant case (ω0 = ωc), with Ω± = ω0 −

i ( + 4



⎛ γ −κ⎞ g2 − ⎜ ⎝ 4 ⎟⎠

2

[10.18]

The emission spectrum features either one or two peaks, depending on the relative values of the CQED parameters g, κ and γ.

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Influence of acoustic phonons A microscopic derivation of the bare QD self-energy in presence of electron–phonon coupling is then used, in order to study how the QD-cavity spectrum is affected as a function of detuning by non-perturbative coupling to acoustic phonons. We restrict ourselves again to the linear spectral response of the cavity-QD system valid at low excitation power. But instead of a Lorentzian lineshape we assume a QD spectrum arising from the coupling to longitudinal acoustic phonons (Fig. 10.17), with the corresponding self-energy modelled within a second-order Born approximation (Krummheuer et al., 2005). The coupling of one exciton to the LA-phonon band is described exactly through the solution of the independent Boson model (Mahan, 1981; Zimmermann and Runge, 2002). It has, however, been shown (Krummheuer et al., 2005) that a very good account of the exciton spectrum can already be obtained at the level of the second-order Born approximation, with the advantage of having a simple expression for the exciton-phonon self-energy. Within second-order Born approximation and taking into account only one phonon band, the exciton-phonon self-energy reads



(ω ) =

∑ q

⎡ gqx ⎢ ⎢ ω + i (γ ⎢⎣

2

(

+

( ))

) − ω0

ω(

)

+

gqx

ω + i (γ

2

( ( ))

) − ω0 + ω (

⎤ ⎥ ) ⎥⎥ ⎦ [10.19]

X, 0c LA phonon g, 1c X0 photon Cavity photon

g, 0c

10.17 The cavity feeding phenomenon mediated by LA phonons. Instead of emission of an X0 photon, a cavity photon is emitted with the excess energy being transferred to an acoustic phonon.

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360

Quantum optics with semiconductor nanostructures 102 Cavity mode T = 10 K 0 Phonons

Im [χ(ω)]

101

100

10–1

10–2 1060

1062

ω (meV)

10.18 Imaginary part of the quantum dot susceptibility in the presence of LA-phonon coupling (full) and without phonons (dashed), computed at T = 10K. As an illustration, we plot the optical density of the cavity mode at a positive detuning of 1 meV (dotted).

Here, n(q) is the Bose–Einstein equilibrium phonon occupation at temperature kBT. We consider the case of deformation potential coupling with acoustic phonons of dispersion ωq = cs q, where cs is the sound velocity, as in Zimmermann and Runge (2002). The imaginary part of the QD susceptibility displays pronounced sidebands compared to the spectrum of an ideal QD exciton (see Fig. 10.18). The sidebands are more pronounced on the high-energy side, where they are determined by acoustic phonon emission. Intuitively, the emission intensity at the cavity-mode frequency depends on the optical density of the underlying exciton spectrum. Hence, the presence of acoustic-phonon sidebands is expected to enhance the PL intensity when the cavity is detuned from the exciton. We use the QD susceptibility to compute the emission spectrum, which reads

S (ω ) =

(

4gg 2

ω

ω − i (γ

( )

)2 / 4 ( c ( ω )) ( ω c

)) i ( κ )) − g 2

i (κ

ω

2

[10.20]

As expected, the exciton-phonon coupling results in a modified emission spectrum (see Fig. 10.19). In particular, the exciton-phonon self-energy is responsible for a modified intensity at the cavity-mode frequency and a small polaron shift of the exciton frequency. As a consequence, the peak at frequency ω = ωc is enhanced with respect to the simple CQED model,

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2 2 ⏐ε (r, ω) ⏐/⏐ε 0⏐

Photoluminescence from a quantum dot-cavity system

4

361

0 Phonons T=4K T = 10 K T = 40 K

2

0

0

1 ω–ω0 (meV)

10.19 Computed emission spectrum for different values of the temperature, ωc − ω0 = 1 meV.

provided the detuning is not larger than the energy extent of the phonon bands (less than a few meVs at typical temperatures in the experiment). In Fig. 10.19 we see the influence of the temperature on the emission spectrum S(ω). The phonon sidebands grow with temperature and result in an increased emission through the cavity mode. As discussed below, the acoustic-phonon mediated cavity feeding has been well characterized in different systems and exhibits particular features in the PL spectrum, the power dependence, and the intensity correlations that substantially differ from those observed in the WL-mediated cavity feeding process.

10.6.2 Other theoretical approaches A highly relevant theoretical work discussing the influence of phonons on a semiconductor CQED system is the one by Wilson-Rae and Imamoglu (2002), who study the influence of pure dephasing in the strong coupling regime. In their model, the QD and the cavity are both treated as two-level systems. They prove that the emission spectra are broadened and that the single-photon performance is reduced in the presence of pure dephasing. This is addressed as well in the works by Cui and Raymer (2006) and Milde et al. (2008). This approach is further extended by Naesby et al. (2008) for the non-resonant case. In that work, the existence of large pure dephasing rates leads to cavity feeding, similar to the discussion in the present work. Additional confirmation is given by the work of Auffèves et al. (2008, 2009), where the QD linewidth γ0 is replaced by γ0+γ*, where γ* is a phenomenological pure

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dephasing term. Similarly, in Tawara et al. (2009) an effective exciton decay rate is used, including pure dephasing. In this latter work, computed two time correlation curves show clear antibunching for both QD and cavity emission, as expected for a simple two-level system. In our approach, we describe the coupling to phonons microscopically. However, we do not have access to the time dependence of the emitted light. A combination of the microscopic description and the density matrix approach was recently proposed in Hohenester (2010). The results presented there agree well with those obtained using the independent boson model (Hohenester et al., 2009), and also with the ones obtained with our approach. Signatures of the two cavity feeding channels In the solid state, the number of available platforms for experimental CQED is very large. On the emitter side, two main classes of QDs exist, namely the self-assembled QDs (Bimberg et al., 1999) and the pyramidal ones (Hartmann et al., 2000). In addition, many different cavity geometries are used in experiments. More importantly, however, QD-cavity systems can be excited in different ways: either by quasi-resonant excitation of the QD exciton or the cavity mode or by non-resonant excitation of the QD using an above-bandgap laser or electrical injection of carriers. As a consequence, the identification of the dominating effect for emission at the cavity wavelength – either coupling to acoustic phonons or to WL states – might be subtle. When only coupling to phonons determines the PL spectrum at the cavity wavelength, two experimentally available quantities show a distinct behavior. Firstly, there is the intensity cross-correlation function g(2)(τ) between the exciton (X0) and the cavity peak (see Equation [10.1]). Even in presence of phonon sidebands, the emission still originates from a single electronic transition between the QD exciton and the semiconductor ground state. In this case, the statistics of the emitted light is sub-Poissonian, and g(2)(τ) exhibits an antibunching dip at τ = 0 (Ates et al., 2009; Englund et al., 2010). For the same reason, the power dependence of cavity and QD exciton coincide at low powers. There, both lines are expected to show a linear pump-power dependence (see Fig. 10.20). While the QD line saturates at higher powers, the cavity-mode emission also has some feeding from higherlying states and therefore does not saturate. Ideally, if there was only a single exciton line coupled to the cavity, the cavity would be expected to saturate at roughly the same power as the exciton. The experimental observations in the phonon-feeding regime are therefore substantially different from the ones obtained in the regime when feeding via WL states dominates. With our capability to in situ tune the cavity

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PL count rate (c.p.s.)

106

104

Cavity mode X0

102

XX0 0

10 101

102

103

104

Excitation power (nW)

10.20 Power dependence of QD exciton, biexciton and cavity-mode emission for a detuning of 0.62 nm, a regime in which phonons give a major contribution to cavity feeding as confirmed by the linear pumppower dependence of the cavity mode over a large power range (compare to Fig. 10.5 for the case of cavity feeding via WL states).

resonance frequency, we were able to access both regimes in our experiments and study the two regimes in detail. This led to a new and profound understanding of the limits of the ‘artificial’ atom picture. The observations reported here for both regimes were recently confirmed by the Finley group in Munich (Hohenester et al., 2009).

10.7

Conclusions

In this chapter, we studied the limits of a QD-cavity system in terms of ideal CQED behaviour. In particular, we were interested in the effect of off-resonant cavity feeding, which is unknown in the context of atomic-physics experiments. Depending on the detuning between QD exciton and cavity mode, we could identify two different regimes with two completely different mechanisms leading to off-resonant cavity feeding. While the closeto-resonance case has been extensively studied before, both in theory and experiment, the investigation and explanation of the far-off resonant case constitutes the main result of the current chapter. In this regime, the Poissonian statistics of cavity photons implies the existence of many independent emitters. The phenomenology is best explained by the addition of extended (WL) states. The complexity implied by taking into account a multitude of levels, however, prevents an analytical treatment, and therefore our approach relies first on heavy configuration interaction numerics, and then on a Monte Carlo random walk involving thousands of states. We prove qualitative agreement with all the main experimental observations. In particular, our results solve the long-standing mystery of far-off

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resonant cavity feeding and add the missing piece enabling a complete and deep understanding of the intricate dynamics in a QD-cavity system.

10.8

Acknowledgements

The authors would like to thank Michiel Wouters for fruitful discussions concerning the QD model, Wolfgang Langbein for having inspired many of the ideas that we developed in this work, and Kevin Hennessy, Antonio Badolato and Evelyn Hu for their contributions to the early stages of this work.

10.9

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