Photon correlations in semiconductor nanostructures

5 Photon correlations in semiconductor nanostructures M. AßMANN and M. BAYER, Technische Universität Dortmund, Germany

Abstract: This chapter discusses the characteristics of semiconductor nanostructures as light emitters in terms of their coherence properties. These are evidenced by investigating the emitted photon statistics. Special emphasis is placed on second- and higher-order correlation functions. Several experimental approaches to measure them on different timescales for stationary as well as for non-stationary fields are introduced and discussed. Three model systems – one in the weak-coupling regime, one in the strong-coupling regime and one that shows a transition from one to the other – are discussed in detail and their equal-time correlation functions are analyzed. It is shown that transitions from predominant spontaneous emission towards coherent light emission can be identified by changes in the correlation functions. Key words: photon statistics, coherence, semiconductor light sources.

5.1

Introduction

Over the last few decades there has been an ever increasing interest in the field of quantum optics. From the first pioneering experiments on intensity interferometry to modern experiments on antibunching and entangled photons, the quantum nature of light has been a heavily investigated topic. In particular cavity quantum electrodynamics (QED) has given rise to several landmark experiments like realization of the single-atom maser (Meschede et al., 1985) and deterministic single-photon sources (Kuhn et al., 2002) because atom cavity systems allow the study of thefundamental processes of a quantized mode of the light field to a quantized two-level system without disturbing influences. Aimed at real everyday applications instead of proof-of-principle experiments, it is necessary to switch from atom cavity QED to solid state cavity QED. However, performing cavity-QED experiments in a solid state environment requires precise control over the electronic and optical properties of the solid state system. Since the early seventies (Esaki and Chang, 1974) the properties of semiconductor heterostructures have been explored. Advances in growth techniques allowed the confinement and control of electrons in one or more dimensions in those structures. The fabrication of semiconductor 154 © Woodhead Publishing Limited, 2012

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microcavities allowed the same for photons. In microcavities it is possible to reproduce many of the experiments from atom cavity QED. By placing quantum wells (QWs) or quantum dots (QDs) inside them, it becomes possible to enhance or suppress the interaction between the exciton and the cavity mode. Even the strong-coupling regime where the coupling between the light and the excitons becomes stronger than the relaxation mechanisms can be realized (Weisbuch et al., 1992; Reithmaier et al., 2004; Yoshie et al., 2004), underlining the close analogy between atom and solid state cavity QED. On the other hand, there are also some severe differences, calling the close analogy between solid state and atom cavity-QED systems into question. Although QDs are often termed artificial atoms due to their discrete density of states, they cannot be modeled as a simple two-level system in the same way as atoms. The differences become even larger when modeling QWs. In solid state systems, many-body effects become significant. Starting from the basic difference that the carriers of interest in semiconductors are excitons – bound electron-hole pairs – instead of electrons, many-body effects play a prominent role in solid state cavity QED. Coulomb correlations between carriers cause nonlinearities, Pauli-blocking effects may arise and optical excitation will generally produce many-particle states. Adding the spin degrees of freedom, semiconductor microcavities have become a prototype system for controlled studies of many-body systems, including collective behavior. However, many-body systems are also subject to decay, decoherence and dephasing on fast timescales. Accordingly ultrafast timeresolved spectroscopic techniques need to be developed and applied to study the many-body physics in cavity QED. This chapter discusses ultrafast techniques to study the emission of semiconductor heterostructure and focuses especially on coherence properties of appealing candidates for photonic devices on ultrafast timescales, namely low-threshold lasers and polariton condensates.

5.2

Theoretical description of light-matter coupling

Before discussing the techniques used to study the emission from novel semiconductor light sources, it is worthwhile reviewing the basics of lightmatter coupling in semiconductors and the fundamental photonic device concepts that utilize it. This section discusses the signatures of the weakand strong-coupling regimes and introduces some of the most frequently realized photonic device layouts based on these regimes.

5.2.1 Light-matter coupling in microcavities The properties of light traveling through a material differ from light traveling through a vacuum. In a vacuum the description is straightforward

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as light is a transverse electromagnetic field in this case and the quanta of this field is bare photons. The interaction of light with matter can be described at two levels. The first is a perturbative treatment known as weak coupling. In this case, the excitation of the matter, in this case excitons, and the electromagnetic field are independent, but interacting entities. Accordingly, these excitations exchange energy at a rate smaller than the inherent decay rates, thus rendering the energy exchange irreversible. If on the other hand the energy exchange rate is increased such that the energy exchange rate between the photon and the exciton becomes larger than the inherent decay rates, the energy exchange becomes reversible and the eigenstates of the strongly coupled system are best described as polaritons, collective excitations formed from a coherent interaction of photons and excitons. A quantum mechanical description in terms of a two-coupledoscillators model yields the following results for the eigenmodes of the coupled system:

ω 1, 2 =

ω X + ωC i − (γ 2 2 ±

(ω

ω 4

+γ

)

)2 + g 2 − ( γ

γ 4

)2 + i

2

[5.1]

(ω

−ω

) (γ X

γ C ),

where ωC − iγC and ωX − iγX are the complex eigenfrequencies of the bare cavity photon mode and bare exciton modes, respectively, and g denotes the exciton–photon coupling constant, which is proportional to the square root of the oscillator strength. Weak and strong coupling correspond to two limiting cases, namely g < |γX − γC|/2 and g > |γX − γC|/2, depending on whether the square root gives an imaginary or real term. In the weak-coupling case, at resonance (ωX = ωC) the eigenstates can be expressed approximately as follows:

ω 1,w

ω0 − iγ C ,

ω 2 ,w

ω0 − iγ X

[5.2] i

4g2 . γC

[5.3]

Both modes are degenerate in energy, but a splitting is present in the imaginary part that is directly linked to the decay rate. While the first mode can be identified as the cavity mode, the second mode can be interpreted as the exciton mode having a modified spontaneous emission rate due to the presence of the cavity, the so-called Purcell effect (Purcell et al., 1946). Here, γX represents the exciton decay rate without light-matter coupling whereas the

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second term denotes the decay of the exciton due to coupling to the cavity mode. While spontaneous emission into free-space accounts for a continuum of modes, this model considers only a single cavity mode. Nevertheless, the Purcell effect also applies to this situation. The decay rate into this mode clearly depends on the cavity parameters via the effective light-matter coupling g and the decay rate γC in Equation [5.3]. The same term will also appear in a fully quantum mechanical treatment as the spontaneous decay rate. While the spontaneous emission rate is enhanced on resonance, in general the spontaneous emission rate can be suppressed or increased depending on the detuning between the two modes. In the strong-coupling case, 2 at resonance a splitting Ω R = 2 g 2 − X − γ C ) / 4) between the energies of the two modes occurs, where both have an identical average linewidth. This energy splitting is called the vacuum Rabi splitting and is an indicator for the formation of dressed states. These eigenstates can be interpreted as polaritons, quasiparticles that are partially excitonic and partially photonic in nature and are called lower polariton (LP) and upper polariton (UP), respectively. The corresponding eigenfrequencies of these quasiparticle modes are given by ωL and ωU in the following. The energies of the bare photons and excitons are not fixed, but show a dependence on the in-plane wavevector k||. Both of these dispersions are basically parabolic, but the cavity photons show a much steeper dispersion that can be treated in terms of a small effective mass. Therefore, all quantities in Equation [5.1] are dependent on k||. As a result, the polariton dispersion is also parabolic at small k|| and it is possible to assign an effective mass to the polaritons. As shown in Fig. 5.1 the actual steepness of the dispersions of the two polariton branches depends strongly on the detuning Δ = ωC − ωX between the bare excitons and cavity photons forming the polaritons. This opens up the possibility of tailoring the properties of the polaritons and their relative excitonic and photonic fractions in terms of the squares of the so-called Hopfield coefficients 2

C =

X

2

=

ωU ω X − ω L ωC ωC +

X

Δ2 + 4g2

ωU ω X − ω L ωC ωC +

X

Δ2 + 4g2

,

[5.4]

,

[5.5]

giving the relative photonic and excitonic fraction of the LP. The excitonic fraction of the LP is also the photonic fraction of the UP and vice versa. Generally speaking, LPs have high photonic fractions at negative detunings and high excitonic fractions at positive detunings. The polariton lifetime can

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(b)

(c)

Energy (eV)

1.64

1.62

1.60 –8

–4

0

4

8 –8

–4

0 4 kll (μm)–1

8 –8

–4

0

4

8

5.1 Exciton-polariton energies at detunings of ħΔ is (a) −5 meV, (b) 0 meV and (c) +5 meV. Solid lines correspond to the lower and upper polariton branches. Dashed lines give the eigenmodes of the bare exciton and cavity photon.

also be expressed as a combination of the exciton and cavity photon lifetime and is given by 1

τ LP

=

C

2

τC

+

X

2

τX

[5.6]

.

All of these quantities depend on k||. Exemplary lifetimes and Hopfield coefficients for different detunings are shown in panels (i) and (ii) of Fig. 5.2, respectively.

5.2.2 Photonic devices As has been pointed out before, both the strong- and the weak-coupling regimes offer the possibility of creating photonic devices. The central quantity identifying the efficiency of a laser is the β-factor, which is defined as the ratio of the spontaneous emission rate guided into the lasing mode 1/τ1 divided by the total spontaneous emission rate 1/τsp into all modes, including non-lasing and leaky ones:

β=

τ l−1 τ −sp1

=

τ l−1

τ l−1 + τ n−1l

.

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[5.7]

Photon correlations in semiconductor nanostructures 1.0 0.8 (a) 0.6 0.4 0.2 0.0 –10 1.0 0.8 (b) 0.6 0.4 0.2 0.0 –10 1.0 (c) 0.8 0.6 0.4 0.2 0.0 –10

(ii) |X(kll)|2

100 |C(kll)|2

–5

0

5

10 –10

10

–5

0

5

10

–5

0

5

10

–5

0

5

10

100 τ LP

Hopfield coefficients

(i)

159

–5

0

5

10

10 –10

100

–5

0

5

10

10 –10 –1

kll (mm)

5.2 (i) Hopfield coefficients as a function of k|| for 2hΩR = 14.4 meV. The coefficients are displayed for detunings of (a) +5 meV, (b) 0 meV and (c) −5 meV. (ii): corresponding lower polariton lifetimes for the same detunings assuming a cavity lifetime of 5 ps and an exciton lifetime of 500 ps.

The most efficient laser theoretically possible is the so-called thresholdless laser for which β equals unity. The weak-coupling regime allows tailoring of the spontaneous emission into a mode of interest by making full use of the Purcell effect and is thus ideal for realizing efficient lasers. Ideally the emitters used should also exhibit a discrete spectrum which renders QDs microcavity vertical-cavity surface-emitting lasers (VCSELs) ideal candidates for low-threshold lasing. Common gas or diode lasers offer β-factors on the order of 10−7–10−5 only (Yamamoto et al., 1991), while values up to 0.85 have been realized using QDs in photonic crystal cavities (Strauf et al., 2006). On the other hand characterizing such efficient devices becomes increasingly more difficult as the lasing transition becomes harder to identify. While a significant shift of the threshold pump rate towards lower thresholds occurs with increasing β-factor, the step-like jump of the emitted intensity at the threshold also decreases as shown in Fig. 5.3. For atom lasers the magnitude of the jump can be estimated as β−1, which complicates identification of the onset of lasing via the emitted intensity alone (Rice and Carmichael, 1994). For VCSELs, semiconductor-specific effects like Pauli-blocking, different

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Quantum optics with semiconductor nanostructures 106

β = 10–5 –4

β = 10 β = 10–3

5

Mean photon number

10

104 103 102 1

10

β = 10–2 β = 10

–1

β=1

100 10–1 10–2 10–3 10–4 10–5 10–1

100

101 102 103 104 –1 Pump rate (ps)

105

106

5.3 Input–output curves for lasers with varying β–factors decreasing from top to bottom. The curves were calculated from rate-equation models and do not take semiconductor-specific processes and reabsorption into account.

relaxation times for electrons and holes and modified spontaneous emission terms cause a deviation from the β−1 behavior in the intensity curve (Gies et al., 2007), but the kink is still small at large β. An alternative way of identifying the lasing transition lies in monitoring the coherence properties of the emission as will be discussed in detail in Section 5.3.1. The strong-coupling regime also offers the possibility of creating efficient photonic devices. As the LP state with k|| = 0 shows the lowest energy, one might imagine this state to act as a polariton trap with the major part of the emission originating from this state under non-resonant excitation. Experimental results show a different situation with the major part of the emission stemming from the so-called bottleneck region (Tartakovskii et al., 2000), especially at negative detunings. These findings can be explained by taking the relaxation mechanisms towards the ground state into account. The relaxation process is schematically shown in Fig. 5.4. Non-resonant optical excitation creates a thermal reservoir of electrons and holes. Polaritons with large k|| are formed from the thermal reservoir via acoustic- or opticalphonon emission. Inside the polariton branches, acoustic-phonon scattering and radiative recombination are the most important effects at low densities. The maximum energy transferred by acoustic-phonon scattering is on the order of approximately 1 meV (Tassone and Yamamoto, 1999). In the bottleneck region the polariton dispersion changes very fast with k||, resulting in a small density of states and also in a reduced acoustic-phonon scattering rate. Additionally the polariton lifetime decreases strongly in this region due to

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Photon correlations in semiconductor nanostructures EH

1.63

Energy (eV)

LO

1.62

161

UP AC

1.61 LP 1.60 SC 1

BR 10 k ll (mm)–1

TR 100

5.4 Schematic representation of the polariton dispersion and the possible polariton formation and relaxation processes by means of optical (LO) and acoustic (AC) phonon emission as discussed in the main text. Please note the logarithmic scale of the k||-axis. Dashed lines give the bare cavity and exciton dispersions. LP and UP label the lower and upper polariton branch, respectively. The dotted line labeled EH marks the thermalized free-carrier reservoir. SC, BR and TR denote the strongcoupling, bottleneck and thermal regions.

increasing photonic contents as shown in Fig. 5.2, causing the multiple phonon scattering processes needed to realize relaxation to the ground state to be very unlikely. This relaxation bottleneck becomes more pronounced for larger negative detunings because the density of states in the low-k|| region decreases strongly with increased photonic content of the LP. Stimulated polariton–polariton scattering processes provide a way for more efficient relaxation towards the ground state as the maximum amount of energy transferred in a single scattering process is larger. Also, polaritons are composite bosons and therefore subject to bosonic final state stimulation. Here, the low density of states around k|| = 0 is even beneficial as the low density of states causes a rather large occupancy per state in this region and final state stimulation depends critically on the occupancy of the final state. Starting from a ground state occupancy of unity, a coherent degenerate population of the ground state should build up and the polariton system should turn into a so-called polariton condensate that is in many aspects a dissipative and driven analogue of a Bose–Einstein condensate (BEC) acting as a laser without inversion. However, such processes become efficient only at moderately large polariton densities. For high polariton densities the Coulomb interaction between the excitonic fractions of the polaritons increases and finally leads to a bleaching of the exciton oscillator strength (Houdré et al., 1995) that in turn causes a reduction of the Rabi splitting and finally breaks the strong-coupling regime. Therefore, one can basically distinguish two types

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of emitters in the strong-coupling regime: polaritonic diodes, which rely on spontaneous scattering processes and for which the strong-coupling regime is broken before the ground state occupancy reaches unity; and polariton condensates for which bosonic final state stimulation can occur at densities smaller than the transition density into the weak-coupling regime. From the device design point of view, there are several strategies to overcome the problem of bleaching and achieve large polariton densities and efficient polariton–polariton scattering. In GaAs, which is among the materials most commonly used for growth of QWs and microcavities as it allows for precise sample growth with small defect densities, typical Rabi splitting achievable for a single QW in a microcavity are on the order of three to four meV, only. Correspondingly, there have been significant efforts to realize larger Rabi splittings in the strong-coupling regime using materials with larger exciton oscillator strength like CdTe (typical ΩR up to 25 meV (Dang et al., 1998)), GaN (ΩR exceeding 50 meV (Christopoulos et al., 2007)) and ZnO (ΩR up to 50 meV (Schmidt-Grund et al., 2007)). Another strategy used for GaAs based systems lies in increasing the number of QWs embedded in the cavity (Bloch et al., 1998). As ΩR increases with the square root of the exciton oscillator strength, adding more QWs corresponds to an effective coherent addition of the single QW oscillator strengths while simultaneously the polariton density per QW at fixed total polariton number scales with the inverse of the number of QWs. A common technique is to position three stacks of four QWs each at the central antinode of the cavity and the first antinode in each cavity mirror. By doing so, Rabi splitting as large as 15 meV have been realized for GaAs systems, allowing achievement of stimulated scattering into the LP ground state. Finally, there are several different excitation schemes able to reach a degenerate macroscopic population of the ground state. However, not all of them are suitable for every kind of experiment. The easiest method is direct resonant excitation of the k|| = 0 ground state. While being rather efficient, this excitation scheme is not suitable for proof-of-principle experiments aimed at demonstrating features such as spontaneous build-up of coherence as the coherence properties of the ground state could have been inherited directly from the resonant optical pumping process. The same problem arises for resonant pumping at the so-called magic angle. The magic-angle state k||,M marks the point of the LP dispersion from which a resonant polariton–polariton scattering process to the states k|| = 0 and 2k||,M conserves both total energy and in-plane wave vector. However, this is a resonant scattering process and it is still possible that the coherence of the pump beam is carried over to the condensed state. Basically, two excitation schemes remain to demonstrate spontaneous build-up of coherence non-resonant pumping and resonant pumping of the LP under large angles where direct resonant scattering to the ground state is forbidden. In both cases several polariton– polariton or polariton–phonon scattering processes are necessary before

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carriers reach the ground state. During these processes the initial coherence is lost. Both excitation schemes have different pros and cons. Non-resonant excitation is very efficient. Here, the initial excitation pulse creates free electrons and holes. For those, the density of states is significantly higher than for polaritons in a certain state. Accordingly threshold densities are at least a factor of ten smaller compared to resonant excitation of polaritons at large k||. On the other hand, non-resonant excitation also creates a large number of background carriers that interact with the polariton gas. Therefore, the exact properties of the polariton BEC will depend strongly on the density and spatial distribution of the residual carriers. Resonant excitation of polaritons at large k|| is less efficient and results in large threshold excitation densities, but also offers well defined experimental conditions. Polaritons can be created directly with a desired polarization and there is no excitation of a huge number of background carriers interacting with them. In the following, the spontaneous build-up of coherence will be discussed in terms of a hierarchy of correlation functions that will be introduced in the next section.

5.3

Photon statistics

Aside from obvious parameters like frequency, polarization or intensity, there are also further characteristics of light fields that manifest in their coherence properties, which can be described by a hierarchy of correlation functions starting with the field–field correlation function:

g

−

( )

  (r1 , t1 , r2 , t2 ) =

E





(r1 , t1 ) E + (r2 , t2 )

2  E ( r1 , t1 )

2  E ( r2 , t2 )

.

[5.8]

+

E and E denote the negative and positive frequency parts of a mode of the light field, respectively. g(1) is a measure of phase correlations of a light field and reflects in the contrast of interference patterns of the electromagnetic field. The two common quantities deduced from g(1) are the coherence time τcoh and the correlation length lcoh, which give the time and distance over which phase correlations are maintained, respectively. Still, a complete characterization of electromagnetic fields that is also able to identify nonclassical states requires consideration of correlation functions of at least second order. Neglecting any spatial dependencies the normal-ordered second-order photon number correlation function is given by: †

g(

)

(t1t2 ) =

†

a (t1 )a (t (t2 )a (t1 )a (t (t 2 ) †

a (t1 )a ((tt1 )

†

a (t2 )a ( 2 )

,

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[5.9]

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Quantum optics with semiconductor nanostructures

where ↠and â are the photon creation and destruction operators for the photon mode of interest, respectively. The normal-ordering assures that the change of the state of the em field introduced by the detection of a photon is taken into account. For stationary light sources g(2) will only depend on the relative delay τ between two photon detections: †

g

( )

(τ) =

†

a (t )a (t

)a (t )a (t (t

n (t ) n (t

) [5.10]

, )

where the averages are time averages and n denotes the photon number of the mode of interest. g(2) can be considered as the conditional detection rate of photon pairs with a delay τ between the detections of the two photons, normalized to the photon pair count rate at the same delay for photons that are emitted at the same mean photon count rate, but statistically independent of each other. For very large delays, τ → ∞, the photon emission events are uncorrelated in any realistic state of the light field, so g(2)(τ → ∞) = 1. It is possible to distinguish three basic kinds of states of the light field, namely thermal, coherent and non-classical light, by comparing g(2)(τ = 0) = 1 to g(2)(τ → ∞) = 1 depending on whether the probability for simultaneous detection of two photons is increased, unaltered or decreased. The enhanced or decreased photon pair detection probability relaxes back towards 1 on a timescale depending on the coherence time of the light. Theoretical curves displaying this effect for stationary light fields are shown in Fig. 5.5. While in the simplest case for thermal light g(2)(τ) is simply related to the first order correlation function via the Siegert relation as g(2)(τ) = 1 + |g(1)(τ)|2, The general calculation of this quantity as a function of the delay is quite sophisticated and can be found in Loudon (2000). Accordingly, the value of the equal-time second-order correlation function n(t )(n(t ) − 1) 1 2 g ( ) ( 0) = = 1− + 2 n ( t) n(t )

(

nt

)2

n(t )2

[5.11]

is a good characterization of the state of the light field. It is composed of three terms: the first term is a unity valued constant. The negative second term describes the change of the state of the light field induced by the detection of the first photon. The positive third term takes the intrinsic noise of the photon emission process into account in terms of the relative photon number variance. Correlation functions can be generalized up to arbitrary

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2.0 Thermal state Coherent state Nonclassical Fock state

g(2)(τ)

1.5

1.0

0.5

0.0

0

1

2 τ /τcoh

3

4

5.5 Second-order correlation function g(2)(τ) for continuous wave thermal (solid line), coherent (dashed line) and non-classical (dotted line) states of the light field. τ is measured in multiples of the coherence time τc of the light field.

order to describe the probability of n-photon detections. The most general definition of a nth order correlation function is given by n

∏ n (t ) :

: n g ( ) (t

tn ) =

i

i =1 n

∏

.

[5.12]

n (ti )

i =1

The double stops denote normal ordering of the underlying photon creation and annihilation operators. The statistical properties of coherent, thermal and non-classical light will be discussed in more detail in the following sections.

5.3.1 Statistical properties of coherent, thermal and Fock states This section discusses the statistical properties of different light fields. The expected outcomes in measurements of g(2) are discussed. Thermal light is emitted if the system under investigation can be considered as an ensemble of emitters in thermal equilibrium with a radiation field and is showing the corresponding emission and absorption rates. Examples of these kind of emitters are incandescent light bulbs or even stars. The corresponding photon number distribution follows Bose–Einstein statistics. In this case, the

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probability of detecting n photons for a light field of mean photon count rate 〈n〉 can be described by a geometric distribution:

(

)

Pth n, n =

n

(

+ n

n

)1+ n

[5.13]

.

One of the characteristic features of this distribution lies in the fact that the most probable photon number is always n = 0 regardless of the mean photon number as shown in Fig. 5.6. As the distribution also decreases monotonically it is obvious that the photon number variance 〈(Δn(t))2〉 must be rather large for large mean photon numbers. In fact, the variance is given by 〈(Δn(t))2〉 = 〈n〉2 + 〈n〉. As can be seen from Equation [5.11], this results in g(2)(0) = 2. This result also applies for pseudo-thermal light sources like Martienssen lamps (Martienssen and Spiller, 1964). Coherent states are minimum uncertainty states with uncertainty equally distributed between photon number and phase and also eigenstates of the photon annihilation operator. It can be shown that the latter requirement results in the photon number distribution being Poissonian:

(

Pcoh n, n

e

− n

n

n

n!

[5.14]

.

= 1

(a) Pcoh(n,)

)

= 5

= 10

0.5 0.4 0.3 0.2 0.1 5

10

15

20 0

5

10

15

20 0

5

10

15

20

5

10

15

20 0

5

10

15

20 0

5

10

15

20

5

10

15

20 0

5

10 n

15

20 0

5

10

15

20

PFock (n,)

Pth (n,)

0 (b) 0.5 0.4 0.3 0.2 0.1 0.0 (c) 1.0 0 0.8 0.6 0.4 0.2 0.0

0

5.6 Photon number distributions for (a) coherent, (b) thermal and (c) Fock states for different mean photon numbers.

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As can be seen from Fig. 5.6, this distribution is peaked at 〈n〉. The variance of this distribution is given by 〈(Δn(t))2〉 = 〈n〉, resulting in g(2)(0) = 1. This is a remarkable result as it reflects several properties of coherent states. First, it indicates that the joint detection probability factorizes into the product of the mean photon detection rates and therefore shows that the emitted photons are statistically independent. Second, it shows that the detection of one photon does not alter the state of the light field in this case, which demonstrates that coherent states are indeed eigenstates of the photon annihilation operator and also the closest analogue to classical light fields there is. Finally, Fock states are non-classical eigenstates of the photon number operator and have a simple photon number distribution:

(

)

PFock n, n = δ n, n .

[5.15]

This fact reflects in a vanishing variance 〈(Δn(t))2〉 = 0 and a photon number dependent 1 2 g( ) ( ) = 1 − . n

[5.16]

Regardless of the value of g(2)(0), g(2)(τ) will return towards unity with increasing delays with a decay constant on the order of the coherence time τcoh for all stationary light fields as shown in Fig. 5.5.

5.4

Experimental approaches to photon correlation measurements

There is a wide variety of experimental techniques applied to measure g(2)(τ). The most direct method consists of keeping track of the times of detection events of a single-photon detector followed by a straightforward calculation of g(2)(τ) according to Equation [5.10]. However, directly determining photon statistics in this manner is a difficult task because such measurements are limited to timescales longer than the detector dead time. For detectors with single-photon sensitivity like single-photon avalanche photodiodes (SPADs) the dead time is typically on the order of several tens of ns. However, in the regime of predominant spontaneous emission, the coherence times of the emission are on the order of a few tens of ps only. Several experimental approaches to overcome this limitation have been proposed and realized. This section gives a brief overview of the most common experimental techniques.

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5.4.1 Hanbury Brown–Twiss setups Hanbury Brown–Twiss type setups (Hanbury Brown and Twiss, 1956) overcome the dead time limitation using at least two independent detectors. The signal is split using a beam splitter and distributed evenly among the detectors. In such a scenario the second detector can still be active during the dead time of the first and a histogram of the delay times between consecutive photon detection events can be recorded by using the signals from the two diodes as start and stop events for a time-to-amplitude converter. If the dark count rate is negligible and the mean photon detection rate is much smaller than the inverse detector dead time, the resulting histogram is a measure of G(2)(τ), the correlation function without normalization, for long integration times. The upsides of this technique are the easy implementation, high detector efficiencies and the flexibility to perform auto- or crosscorrelation measurements with only minor changes to the setup. Drawbacks are the SPAD temporal resolution, which is usually on the order of a few hundred ps for efficient SPADs and therefore limits the timescales that can be studied, the need to measure the normalization separately and the growing complexity of the setup when trying to measure higher-order correlations as each additional order involves the usage of another detector.

5.4.2 Two-photon absorption setups Two-photon absorption (TPA) in semiconductors is a well-known effect that can be visualized as an electronic transition from the valence band to the conduction band via an intermediate virtual state in the energy gap Eg. The TPA transition rates are proportional to 〈â†(t)â†(t)â(t)â〉 and therefore a measure of G(2)(0). In order for such a transition to occur, two photons must be absorbed within a time interval given by the Heisenberg lifetime ħ/Eg, which is on the order of a few fs for Eg corresponding to the visible light spectral range, allowing for excellent temporal resolution. TPA occurs for any photon energy between 0.5 Eg and 1 Eg, usually resulting in reasonable detection bandwidth (Boitier et al., 2009). To obtain the time-dependent G(2)(τ), interferometric methods can be applied, for example by placing a Michelson interferometer in the beam path and scanning the relative delay between the two paths. In this case G(2)(τ) can be determined by the following relationship: I (τ) I1

I2

= 1 + G( 2) ( τ )

(F (τ) e ) 2i t

2

(F (τ) e ), 1

iωt

[5.17]

where I(τ) is the TPA signal at the detector (usually a photocathode operated in photon counting mode) and I1 and I2 are the signals obtained using

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the two arms of the interferometer separately. The phase interference terms corresponding to F1 and F2 oscillate rapidly and can be filtered out at low frequency. The remaining term allows G(2)(τ) to be determined. Measurements of higher-order correlations are possible by interferometric means (Hayat et al., 2010) or by using materials suitable for multi-photon absorption instead of TPA (Nevet et al., 2011). Advantages of this technique are the unmatched temporal resolution and the efficiency in the infrared spectral region. Drawbacks lie in the complicated nature of extending this measurement scheme to non-stationary sources and the indistinguishability between two photons and several photons being present as both cause a single detection event on a photocathode operated in photon counting mode.

5.4.3 Streak camera setups Streak cameras convert the incoming light field to photoelectrons using a photocathode, accelerate them and then deflect them using electric fields synchronized with the excitation laser repetition rate applied orthogonally to the photoelectron direction of motion. They then hit a phosphor screen on a position depending on the relative time of arrival of the initial photon. The resulting afterglow is the signal that can be recorded using a charge coupled device (CCD) camera and gives the time dependence of the light field with a temporal resolution of down to two ps. Using a streak camera for correlation measurements requires further customization. Usually the streak image is recorded by integrating over many repeated streak cycles. However, information about correlations is only present in single pictures and gets washed out by this integration. It is therefore necessary to record images of single signal pulses. This prerequisite significantly reduces the possible data acquisition rate due to the limited readout rates achievable for CCDs. State-of-the-art CCDs with sufficient quantum efficiency for recording single streak pictures can usually be operated at 100 Hz at best. However, it is possible to add a second slow deflection voltage orthogonally to the other one in order to record several pulses on one screen. Doing so allows the recording of up to 40 pulses on one screen and allows for an effective repetition rate of 4 kHz. From those pictures it is then possible to calculate G(2)(τ) by counting all the photon pairs in these pulses and creating a histogram. The normalization constant can easily be determined by averaging over all the single pulses, which yields the average count rates at the respective times (Aßmann et al., 2010). Advantages of this technique lie in the possibility of determining correlations between all photons in one pulse – not just consecutive ones – with ps temporal resolution and the ability to measure time-resolved correlations even for non-stationary fields. Also, it is possible to extract second- and higher-order correlations

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from a single measurement run (Aßmann et al., 2009). Drawbacks are the low quantum efficiency of this approach and the need to reconstruct singlephoton detections from the pattern recorded by the CCD in real time.

5.5

Correlation measurements on semiconductor nanostructures

Studying the correlation functions defined in Section 5.3 allows for a more detailed characterization of the properties of the emission from semiconductor nanostructures than an analysis of the emitted intensity, the spectrum or the line widths alone can. However, from an experimental point of view, such studies are rather complicated as correct identification of the correlation functions can require detectors with temporal resolution in the picoseconds range. In the following we use the streak camera approach presented in Section 5.4.3 to examine the correlation functions of three model systems: one showing a degeneracy transition in the weak-coupling regime, one where the degeneracy threshold and the transition from strong- to weak-coupling coincide and one that is capable of building up a degenerate ground state occupancy in the strong-coupling regime.

5.5.1 The polariton diode–vertical-cavity surface-emitting laser (VCSEL) transition In this section the streak camera technique is applied to a strongly coupled QW-microcavity system. This system makes full use of the advantages of the streak camera measurement technique. The polaritons resulting from the strong coupling between cavity photons and QW excitons are expected to be in a thermal state with a coherence time of a few tens of ps at low pump powers. In this regime the system acts as a polariton diode. At sufficiently high pump powers, the Coulomb interaction between the excitonic fractions of the polaritons increases and finally leads to a so-called bleaching of the exciton oscillator strength, which manifests as a reduction of the exciton oscillator strength caused by many-body screening effects and phase-space filling at high carrier densities. This bleaching in turn leads to a reduction of the Rabi splitting and finally breaks the strong-coupling regime. The polaritonic diode then transits into the weak-coupling regime and turns into a VCSEL. This transition also reflects in the emission photon statistics. One expects to see photon bunching in the polaritonic diode regime and coherent emission in the VCSEL regime. The QW VCSEL sample used consists of a GaAs/AlGaAs microcavity grown by molecular beam epitaxy. It contains one 10-nm-wide QW placed in the electric field antinode of a slightly wedged λ-cavity especially designed

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to avoid charge accumulation in the QW. The sample displays a vacuum Rabi splitting of 3.9 meV. The polariton dispersion for different excitation densities shows an apparent bleaching of the strong-coupling regime with increasing excitation power. Additionally, the LP ground state was found to be only weakly populated far below the lasing threshold. Therefore, polariton–polariton scattering is also weak in this regime. The far-field emission of the LP branch was investigated at a negative detuning of −2 meV. The Fourier plane of the emission was either imaged onto the entrance slit of a monochromator for measuring the dispersion or onto the entrance slit of a streak camera for photon counting measurements. Photons that are emitted at an angle of θ directly correspond to polaritons with energy E and an inplane wave vector k|| = (E/ħc)sinθ. Thus, in the first case the entrance slit of the monochromator selects a narrow strip with kx, || = 0. In the second case only the k|| = 0 state of the LP branch is selected with an angular resolution of ≈ 1° by using a pinhole. Additionally an interference filter with a spectral transmission width of 1 nm is used to ensure that only one mode contributes to the signal. With increasing excitation density, the filter is tuned so that the central transmission wavelength follows the blue shift of the polariton dispersion as shown in Fig. 5.7. As can be seen, there is a smooth and continuous blue shift starting at the onset of the nonlinear region in the input–output

106

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1.5476 104

1.5474 1.5472

Lower polariton energy (eV)

Emitted intensity (arbitrary units)

1.5482

103 0.1

1 Excitation power (mW)

10

1.5470

5.7 Characteristics of the ground state emission along the diode-toVCSEL transition. Filled black dots give the emitted intensity at normal incidence as a function of the non-resonant excitation power at a detuning of −2 meV. Open circles denote the emission energy around normal incidence as a function of the excitation power. Dashed lines indicate the linear dependence of the emitted intensity on the excitation power below and above threshold. (Source: Adapted from Aßmann et al. (2009).)

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3!

g(n)(τ = 0)

5 4 3 2

2!

1 0.5

1

5

10

Excitation power (mW)

5.8 Second- (triangles), third- (circles) and fourth-order (open circles) intensity correlation function versus excitation power. The solid line marks the value of unity expected for all orders of the correlation function when a coherent light field is present. Dashed and dotted lines mark values of two and six as expected for the second- and third-order correlation functions, respectively, when a thermal light field is present. (Source: Adapted from Aßmann et al. (2009).)

curve. This is a clear sign of increasing interactions between particles and bleaching of exciton oscillator strength. The measured time-averaged normalized intensity correlation functions g(n)(τ = 0) up to the fourth order (Fig. 5.8) show that for high excitation densities all orders approach the value of unity expected for lasing operation. With decreasing excitation density, a smooth transition towards the thermal regime occurs, which is accompanied by the onset of photon bunching. At an excitation power of ≈ 1.5 mW, the bunching effect saturates at values of approximately two and six, which are the expected values of n! for the second and third orders of g(n)(τ = 0). The fourth order also shows an increase in the joint detection rates, but the number of detected four-photon combinations is too small at low excitation densities to give statistically significant results in the thermal light regime. In summary, the lasing transition of a QW VCSEL and the accompanying bleaching of the strong-coupling regime could be demonstrated.

5.5.2 Polariton condensates and highly photonic states In the previous section the coherence properties of a planar VCSEL were discussed. In such systems the strong-coupling regime is broken before the population of the LP ground state reaches unity. Next, it is worthwhile to compare these results to a sample where it is possible to create a quantum

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degenerate ground state with a population exceeding unity while preserving the strong-coupling regime. The sample we used consists of 12 GaAs/ AlAs QWs embedded in a microcavity with 16 (21) AlGaAs/AlAs mirror pairs in the top (bottom) distributed Bragg reflector. Reflectivity measurements give a value of 13.8 meV for the Rabi splitting, allowing us to create larger polariton densities before the strong-coupling regime is broken. It is quite difficult to characterize such a degenerate polariton state. On the one hand it bears many similarities to a BEC as has been demonstrated by the appearance of signs of superfluidity (Amo et al., 2009; Sanvitto et al., 2010), quantized vortices (Lagoudakis et al., 2008, 2009) and modified excitation spectra (Utsunomiya et al., 2008; Aßmann et al., 2011). On the other hand, BECs are equilibrium states from the point of view of thermodynamics and, due to the photonic polariton content constantly leaking out of the cavity, polaritons are subject to dissipation and decay and the polariton system is never in true equilibrium. One might therefore expect some similarities to common lasers to appear. However, the polariton system consists basically of three different regions: polaritons at large k|| have large excitonic fractions and are usually thermalized with the lattice. Polaritons at intermediate k|| in the bottleneck region are usually always far from equilibrium. Polaritons near the ground state are also usually never in thermal equilibrium with the lattice, but depending on their photonic fractions may be in a state of local self-equilibrium, which means that the low-energy part of the polariton system can be considered as a bath that can be described by a temperature, but is out of equilibrium with any other bath in the system. Usually the polariton lifetime becomes shorter than the time needed to reach even local selfequilibrium at negative detunings and large photonic fractions of the LP (Deng et al., 2006). It is now worthwhile to study whether the properties of the degenerate polariton state change when going from positive to negative detunings and therefore also from local self-equilibrium to a non-equilibrium system. To avoid ambiguity we will call the local self-equilibrium case polariton condensate and the non-equilibrium state highly photonic quantum degenerate (hi-p) in the following. In the experiments, resonant excitation of the LP at k|| = 5.8 µm−1 using pulses with a duration of approximately 1.5 ps was used. The spot diameter on the sample was estimated to be 30 µm. In this geometry the coherence properties of the ground state emission also depend strongly on the polarization of the excitation. As can be seen in Fig. 5.9, g(2)(0) does not reach the expected thermal value of two for any detuning as in this case there is no single-polarization mode singled out, as in the other experiments described before, but both realizations of the spin-degenerate ground state are detected simultaneously. Polaritons with different spin states will not interfere with each other and can be considered statistically independent. The expected value of g(2)(0) in the thermal regime is therefore two if polaritons

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2

1.00

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(0)

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g Circularly polarized excitation

(b)

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g(3)(0)

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1.75 –10 meV

–7 meV

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1.50

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1.00

1 100

250 400100

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250 400 100 Pexc (mW)

250400100

250 400 100

250 400

5.9 Measured g(2)(0) (dots) and g(3)(0) (hollow triangles) determined by simultaneous two-photon and three-photon detections of the whole fundamental mode emission for a wide range of excitation powers and detunings of −10 meV (|C|2 ≈ 77%), −7 meV (|C|2 ≈ 70%), −4 meV (|C|2 ≈ 62%), −2 meV (|C|2 ≈ 56%), 0 meV (|C|2 ≈ 50%) and +2 meV (|C|2 ≈ 44%) under (a) linearly polarized (upper panel)and (b) circularly polarized (lower panel) excitation. Solid horizontal lines denote the expected limit for coherent and two thermal modes, respectively. Vertical dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion. (Source: Adapted from Aßmann et al. (2011).)

sharing the same polarization are detected and one if polaritons of different polarization are detected. In experiments that do not single out one of these polarizations the measured values of g(2)(0) in the thermal regime will correspond to 2 ppol +1 (1 − ppol) where ppol gives the probability that the two detected photons share the same polarization. The results of higher-order correlation measurements are modified in a similar manner. Accordingly the values of g(2)(0) and g(3)(0) expected in the thermal regime of two equally intense superposed modes are 1.5 and 3, respectively as ppol equals 0.5 in this case. In this figure the results for linearly polarized and circularly polarized excitation are shown in the upper and lower panel, respectively. Under linearly polarized excitation all detunings between Δ = +2 meV and Δ = −10 meV show a degeneracy threshold that is evidenced by a decrease in g(2)(0) and g(3)(0) and agrees well with the position of the threshold (shown as vertical dashed lines in Fig. 5.9), evidenced in measurements of the input– output curve. Here, the threshold is defined as the point where the LP emission and the emission from the k|| = 0 condensed state are equally strong. In

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the case of strong negative detuning of Δ = −10 meV no differences from a common photon lasing transition are observed under linearly polarized pumping. Additional measurements of the polariton dispersion also show that above threshold the emission indeed stems from the bare cavity mode in this case. Under circularly polarized pumping the threshold is not even reached for a detuning of Δ = −10 meV. This result is in accordance with earlier experiments showing that polariton relaxation towards the ground state is less efficient if only polaritons of one spin are injected, which in turn leads to a higher threshold excitation density (Roumpos et al., 2009). For all other detunings, significant deviations from a simple photon laser behavior emerge. Even at high excitation densities the ground state emission has lower energy compared to the bare cavity mode with an energy difference of at least 4 meV, indicating that the strong-coupling regime is still intact. The second- and third-order correlation functions give further evidence that the system is not a simple photon laser in the range of detunings between +2 and −7 meV, corresponding to the polaritons having photonic contents in the range between 44% and 70%. Although at first a decay towards unity is seen for linearly polarized excitation, especially for a detuning of −7 meV, an increase is evidenced for larger excitation densities. Depending on the detuning, g(2)(0) can reach values even higher than the expected value of 1.5 in the thermal regime. For an even further increased excitation density, a smooth decrease back towards one is observed. For circularly polarized excitation the general behavior of the correlation functions is similar to the linearly polarized case as a decrease and reoccurrence of the degenerate mode quantum fluctuations can be identified for detunings between +2 and −7 meV. However, in this case the correlation functions can also show increased fluctuations slightly above threshold as can be nicely seen for a detuning of −7 meV. This increase is caused by the build-up of polarization. The thermal regime value of g(2)(0) = 1.5 is just as valid for unpolarized twomode emission. As the excitation density reaches the threshold, the emission will also start to polarize and the two modes will not contribute equally to the correlation functions anymore. In the case of a superposition of two noninterfering modes A and B, the resulting measured g (AB) ( ) is given by: g (AB) ( )

( ) g (A ) ( )RA

( ) gB ( )RB( )

2 RA RB ,

[5.18]

where RA and RB are the relative intensity ratios of mode A and B to the total intensity. Therefore, it is possible to calculate (A ) ( ) from g (AB) ( ) if the ( ) relative intensity ratios and g B ( ) are known: g (A ) ( ) =

( ) g (AB) ( ) g B ( )RB2 2 RA

2 RA RB

.

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Comparing results of single-mode g(2)-measurements to the values obtained by measurements including both modes shows that for circularly polarized excitation the two modes are indeed independent and the mode polarized cross circularly to the excitation stays thermal. When crossing the threshold this effect will lead to an increase in g(2)(0) while the build-up of coherence will lead to a decrease. These results are in reasonable agreement with mean-field and reservoir-based model calculations of the second-order correlation function of a polariton BEC (Sarchi et al., 2008; Schwendimann and Quattropani, 2008) where the increased fluctuations arise as a consequence of depletion of the condensate ground state and parametric scattering between ground state polaritons and non-condensed polaritons with opposite wave vectors. Nevertheless, they are not sufficient evidence for deviations from a photon laser as the non-monotonous behavior could also be a result of the interplay of the build-up of coherence and polarization. To ensure this is not the case, g(2)(0) was also studied under circularly polarized excitation for the co-circularly polarized emission only. As shown in Fig. 5.10, here the expected value of g(2)(0) = 2 is approximately reached in the limit of low excitation densities for all detunings except Δ = +2 meV. At this detuning the intensity emitted below threshold is too small to perform sensible measurements using our setup. Above threshold the shape of g(2)(0) still

–7 meV

–4 meV

–2 meV

0 meV

+2 meV

2.00

g(2)(0)

1.75

1.50

1.25

1.00 100 250 400 100 250 400 100 250 400 100 250 400 100 250 400 Pexc (mW)

5.10 Measured g(2)(0) of the co-circularly polarized fundamental mode for a wide range of detunings and excitation powers under circularly polarized excitation. Solid lines denote the coherent and thermal limit, respectively. Vertical dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion. (Source: Adapted from Aßmann et al. (2011).)

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shows reasonable agreement with the theoretical results mentioned before (Sarchi et al., 2008). For 0 meV detuning it is apparent that full coherence is not reached within the available excitation density range. Instead g(2)(0) decreases monotonically towards a value between 1.3 and 1.4. There is a trend towards further decrease at high excitation densities; however, the dependence on pump power is rather small. Going to more negative detunings, the dip already seen without polarization sensitive detection still occurs. Apparently the excitation density corresponding to the occurrence of the dip takes on smaller values compared to the threshold excitation density for larger negative detuning. However, also the rise in g(2)(0) seen for further increase of the excitation density grows in magnitude. This can be clearly seen for the most negative detuning of −7 meV where almost complete coherence is reached at an excitation density of approximately 1.1 Pthr and a steep rise to g(2)(0) > 1.6 is evidenced at 1.5 Pthr. Calculations of the second-order correlation function are generally performed using one of two common approaches (Sarchi et al., 2008): meanfield calculations predict a decrease of g(2)(0) towards approximately 1.2 at the threshold, followed by a short rise for increasing excitation density until a constant value of roughly 1.3 is reached. This prediction is in reasonable agreement with our results for no or small negative detuning. A two-reservoir model predicts a sharp decrease of g(2)(0) at the threshold, followed by a strong recurrence of photon bunching up to values of g(2)(0) = 1.6 and a slow drop for even higher excitation densities. This model better reproduces our results for large negative detunings. We conclude that due to the increasing relaxation bottleneck and decreased scattering rates expected for large negative detunings, the two-reservoir model appears to be a valid description in the hi-p regime and the dip seen for several detunings can be interpreted as a sign of inefficient scattering between degenerate ground state polaritons and those in excited or non-condensed states. As the emission photon statistics depend strongly on the detuning, one might also expect to find variations depending on the Rabi splitting and the photonic and excitonic decay constants. We have shown that – apart from thermodynamic properties – hi-p states are not too different from polariton condensates. Although this behavior might seem surprising at first glance, it is in fact not. Theoretical analysis in terms of Keldysh Green’s functions has predicted that the effect of pumping can be interpreted as introducing an effective chemical potential (Keeling et al., 2005; Szymanska et al., 2007), which is given by the boundary of a spectral region in which gain occurs. In an equilibrium system, condensation occurs as soon as the chemical potential reaches the bottom of the band. In analogy to the equilibrium case, condensation will occur in the nonequilibrium system if the effective chemical potential reaches the bottom of the band. In contrast to the lasing case, this scenario does not necessarily

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involve inversion of the pumping bath, thus making it possible to distinguish between lasing and condensation. This distinction can also explain why it is possible to achieve condensation, although the polariton distribution is clearly not even a local equilibrium one and different from a Bose–Einstein distribution: any distribution function that diverges at the effective potential just like the Bose–Einstein distribution diverges at the chemical potential is sufficient to realize condensation, even if the distribution is far from equilibrium.

5.5.3 Quantum dot (QD) micropillar lasers We now turn to a different situation, namely a study of a lasing system that operates in the weak-coupling regime below and above threshold. Efficient low-threshold lasers require high quality mirrors, low mode volume and good temperature stability, which makes QD micropillar lasers ideal candidates for low-threshold lasing and observation of cavity-QED effects. A detailed discussion of the basics of QD lasers is out of the scope of this book chapter, so we refer the reader to a detailed review article (Reitzenstein and Forchel, 2010). Three different QD micropillar lasers with different characteristics were studied. All of them have nominally cylindrical shape. One sample based on a II–VI material system and two III–V material system based samples were used. The II–VI based sample was grown by molecular beam epitaxy. The distributed Bragg reflectors consist of 15 upper and 18 bottom layers in which ZnS0.06Se0.94 (48 nm) layers were used as high-index material and a 25.5 period MgS (1.7 nm) ZnCdSe (0.6 nm) superlattice was used as low-index material. The central λ cavity contains a single sheet of CdSe/ZnSe QDs with an approximate density of 5 × 1010 cm−2. A pillar with 1.5 µm diameter was used for the measurements. The cavity quality factor Q was estimated to be approximately 1850. The III–V micropillar samples were grown by molecular beam epitaxy on a GaAs substrate. The distributed Bragg reflectors consist of 20 upper and 23 lower alternating layers of AlAs (79 nm)/GaAs (67 nm) λ/4 pairs for the low-Q micropillar and 26 upper and 33 lower alternating layers of AlAs (74 nm)/GaAs (68 nm) λ/4 pairs for the high-Q micropillar. The central λ cavity contains one layer of self-assemble InGaAs QDs with a density of approximately 3 × 1010 cm−2 in the low-Q case and one layer of self-assembled AlGaInAs QDs with a density of approximately 6 × 10−9 cm−2 in the high-Q case, from which cavities with diameters of several micrometers were fabricated by means of high-resolution electron beam lithography and plasma-induced reactive ion etching. Micropillars with diameters of 5 µm (low-Q) and 8 µm (high-Q) were used for the experiments. The Q-factors

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were deduced from the line widths of the fundamental modes seen in the modal spectra. Taking the finite resolution of the spectrometer into account, they are estimated to be 9000 and 19 000, respectively. A typical mode spectrum taken at high excitation power under non-resonant, pulsed optical excitation of 1 mW for the low-Q III–V micropillar is shown in Fig. 5.11. The fundamental mode is located at 893.53 nm. The first excited mode can be seen at an emission wavelength of 892.85 nm. This difference is large enough to single out the fundamental mode using an interference filter with 1 nm spectral width. The main aim of the correlation measurements was to investigate the basic emission properties of the three samples under varying excitation power. At low excitation densities, a broad emission peak from the QD ensemble is seen, superimposed by a series of narrow high-intensity peaks marking the microcavity modes. With increasing excitation density, the integrated fundamental mode intensities of all pillars show a characteristic slope change in double logarithmic plots as shown in the lower panel of Fig. 5.12. This nonlinearity marks the onset of stimulated emission in the microlaser structures. The nonlinear region is apparently broadened over a range of excitation densities. For all three samples, the width in excitation powers of this broadened region roughly equals the excitation power at its onset, which complicates the definition of a clear-cut lasing threshold. Determining the β factor is another nontrivial task as the samples operate in a regime where the kink in the input–output curve does not scale with β−1. Theoretical analysis reveals β factors on the order of approximately 0.1 for the III–V cavities

50000

Intensity (a.u.)

40000 30000 20000 10000 0 890

892

894

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Emission wavelength (nm)

5.11 Longitudinal emission mode spectrum of the low-Q micropillar laser. The fundamental mode is twofold degenerate and shows emission at 893.53 nm. The polarization splitting cannot be resolved. The first four excited modes can also be seen.

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2.0

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and a slightly higher β factor on the order of approximately 0.13 for the II–VI cavity (Wiersig et al., 2009). Complementary measurements of the equal-time second-order photon correlation function using the experimental setup presented in Section 5.4.3 allow for a more detailed analysis of the properties of the emitted light. Results are shown in the upper panel of Fig. 5.12. At high excitation densities far above the lasing threshold all samples show lasing emission identified by values of g(2)(0) of approximately one, which are clear evidence for the Poissonian nature of the underlying photon statistics. The low-Q III–V and the II–VI sample are still subject to some small amount of excess fluctuations that manifest in values of g(2)(0) = 1.1 and 1.2, respectively. The origin of this small amount of excess noise is not completely clear. Possible reasons include relevant contributions from spontaneous emission from early and late times in the pulse and efficient cavity feeding effects (Ates et al., 2009; Winger et al., 2009; Laucht et al., 2010). Below threshold the behavior is rather different for the three samples. None of the samples shows a saturation of g(2)(0) at a value of two as would be expected for a classical low-β laser. For the II–VI cavity g(2)(0) saturates for low excitation powers at a value of ≈ 1.9–1.95. The small difference from the expected value for a low-β laser is a manifestation of a limited number of emitters being present.

108 107 106 105 104 20

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5.12 g(2)(0) (upper panel) and corresponding input–output curves (lower panel) for three different QD lasers. The left column shows results for a 1.5 μm diameter II–VI cavity. The other columns show results for III–V cavities of diameters 5 μm (middle column) and 8 μm (right column). Dashed lines in the upper panel denote the coherent limit. (Source: Adapted from Wiersig et al. (2009).)

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A detailed treatment shows that thermal radiation emitted by a system consisting of a fixed number, N, of emitters already shows an emitter-number dependent second-order correlation function (Hennrich, 2003) without even considering further cavity-QED effects: ( 2) gth ( 0 )

1⎞ ⎛ 2 1− ⎟ . ⎝ N⎠

[5.20]

Values between 1.9 and 1.95 are therefore expected if 20–40 QDs contribute to the emission in the II–VI cavity. Considering the sample QD density, this number corresponds to roughly 8–17% of the QDs inside the micropillar coupling to the cavity mode. Taking the spectral overlap between the distribution of the QD emission energies and the cavity mode and possible cavity feeding effects into account, this value is reasonable. Interpretation of the results below threshold is more complicated for the III–V samples. For these structures, there is no resolvable saturation of g(2)(0) below threshold. The quantum efficiency of the photocathode inside the streak camera is about one to two orders of magnitude lower for the wavelength range around 900 nm where the III–V samples operate compared to the 500 nm range where the II–VI sample operates. Correspondingly background noise contributes more strongly and the necessary integration times become much longer in the threshold region for the III–V samples investigated here. Under these circumstances, it becomes impossible to monitor the photon statistics below threshold. The highest measured values for g(2)(0) are about 1.7 for the low-Q sample and approximately 1.4 for the high-Q sample. A detailed theoretical analysis of the expected photon statistics for parameters as given by the samples used in our experiments, and which also takes cavity-QED effects into account, shows that the measured values are not far from the modeled saturation values (Wiersig et al., 2009) of approximately 1.9 for the low-Q sample and 1.45 for the high-Q sample. Inside the threshold region the II–VI and the low-Q III–V sample show the behavior expected classically: g(2)(0) undergoes a smooth transition towards unity. The high-Q sample shows different properties. Here, g(2)(0) drops to values below one around the lasing threshold, giving clear evidence for the emission of non-classical light. This behavior is accompanied by an even smoother input–output curve. This dip is reproduced in calculations (Wiersig et al., 2009) and can be traced back to the small number of emitters coupling to the cavity mode. This non-classical behavior vanishes with rising excitation density as the intracavity photon number increases and coherence starts to build up. Similar transitions from bunching to antibunching have been seen for atoms in a cavity (Hennrich et al., 2005) for a variable number of emitters.

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5.6

Future trends and conclusions

In this short review we have discussed the emission from several semiconductor light sources in terms of their statistical properties. We have shown that the coherence properties of the emission depend strongly on whether the occupation of the emitting state is degenerate or not, but we have also seen that the magnitude of the light-matter interaction strength plays an important role by comparing three model systems: one in the weak-coupling regime below and above the degeneracy threshold, one where the transition from strong- to weak-coupling coincides with the degeneracy threshold and one that stays in the strong-coupling regime across the degeneracy transition. These results also help to understand the distinction between polariton condensation and lasing. Driving a polariton condensate from positive detunings, where the low-momentum polaritons are able to thermalize during their lifetime, to negative detunings, where the polariton lifetime becomes shorter than the thermalization time does not show a crossover from condensation to conventional lasing behavior. Instead condensate signatures such as increased fluctuations and a linearized excitation spectrum persist, indicating that those features of condensation are not intimately linked with thermodynamic equilibrium. All three model systems presented in this review are prospective candidates for efficient photonic devices. Significant progress on the physics of semiconductor light sources was very often linked to corresponding advances in nanofabrication technologies. It is therefore probable that future developments will also be driven by the rapidly evolving epitaxial techniques. It seems very likely that improved micro- and nanolasers allow researchers to approach the ideal case of thresholdless lasing. Among the most promising candidates for future applications in terms of polariton condensation are wide-bandgap materials (Christmann et al., 2008) and structures of different dimensionality (Wertz et al., 2010; Trichet et al., 2011), which might allow for realizing spontaneous coherence at room temperature. Theoretical proposals suggesting interesting applications such as polariton-mediated roomtemperature superconductivity (Laussy et al., 2010) are already present.

5.7

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