Hamiltonian dynamics

The lowest-order observables of interest are the carrier populations, as well as the mean number of photons. Higher order operator averages appear in the derivation

Figure 3.3: QD model with the considered ratesγfor electrons in the conduction and valence band, describing scattering into and out of the s-shell. Carrier generation is modeled by a transition process between the p-levels at rate P. Light-matter coupling leads to recombination processes between the s- and the p-states due to spontaneous emission (dashed arrows).

of EoM. The arising hierarchy is finite in the carrier degrees of freedom, while the photonic hierarchy is truncated at the Mtrunc= 2 level, cf. Sec. 3.2.2.

The contribution of the light-matter interaction HLM to the Ehrenfest EoM for the conduction band carrier population f_i^c=hc^†_icii, are given by

d dtf_i^c

H_LM =−2 Re X

ξ

g^∗_ξΠξ,i. (3.16)

The real part of the photon-assisted polarizationΠ_ξ,i=δ(b^†_ξv_i^†c_i)describes transition amplitudes between QD levels, and is proportional to the light-matter coupling strength gξ. In order to solve Eq. (3.16) additional dynamical equations for the photon-assisted polarization, which are one step up in the hierarchy with respect to photon operators, are required and evolve as

d dt Πξ,i

HLM

=gξf_i^c(1−f_i^v) +gξ

X

α

C_αiiα^x +gξNξ(f_i^c−f_i^v) +gξNξ,i^c −gξNξ,i^v .

(3.17)

The recombination of a QD excitation described by Π_ξ,i does not only require the presence of a conduction-band carrier, but also the non-occupancy of a valence-band state, which ends up in an emission rate proportional to hc^†_iviv_j^†cji (cf. Ref. [Kira

et al., 1999]). In Eq. (3.17) enters the decomposition of this EV in a contribution of two uncorrelated carriers ∝f_i^c(1−f_i^v) in the upper and lower state, plus interband carrier correlations∝C_ijji^x :=δ(c^†_iv_j^†c_jv_i)according to Eq. (3.2)³. Thus, the first two terms in Eq. (3.17) can be identified as the source term of spontaneous emission, naturally appearing within this formalism due to quantization of the light field.

The remaining terms of Eq. (3.17) arises from mixed EVs hb^†_ξb_ξc^†_ic_ii and hb^†_ξb_ξv^†_iv_ii, of which the uncorrelated contribution is proportional to the photon number N^ξ :=

hb^†_ξbξiand can be attributed to stimulated emission and absorption, whereasNξ,i^c :=

δ(b^†_ξbξc^†_ici) and Nξ,i^v := δ(b^†_ξbξv_i^†vi) represent carrier-photon correlations. Note that hb^†_ξiand hv_i^†cjivanish in the incoherent regime. Throughout this chapter we do not account for correlations between different optical modes. This is well justified in the presence of a microresonator, where a single cavity mode strongly dominates over all other leaky and far detuned cavity modes. In chapter 5 correlations between two slightly detuned high-quality cavity modes are included, and are crucial for the mode-coupling effects. Also when free space emission is considered, mode coupling effects may play a role and an evaluation of such terms can be considered. One must be aware, however, that the inclusion of continuum mode-coupling effects severely increases the numerical effort, and is in fact not feasible in a straightforward manner for higher-order CFs.

Significant contributions of higher-order correlations with respect to photons can be expected, e.g., if one of the considered QD transitions is resonant with a cavity mode, thereby providing feedback of the emitted photons. However, for QD emission into a continuum of free-space modes, where photons disappear once emitted, cor-rections to the dynamical evolution of the photon-assisted polarization, introduced by higher order photon correlations are negligible. Nevertheless,carrier correlations can still play an important role [Baer et al., 2006]. Especially in the regime of few emitters these correlations strongly dictate the carrier-photon dynamics and are indispensable for the description of single-QD luminescence.

N

0 1 2

0 / hc^†ci,hv^†vi δ(c^†v^†cv), δ(c^†c^†cc), δ(v^†v^†vv) M 1 / δ(b^†v^†c) δ(b^†c^†v^†cc), δ(b^†v^†v^†cv)

2 hb^†bi δ(b^†bc^†c), δ(b^†bv^†v) δ(b^†bc^†v^†cv),δ(b^†bv^†v^†vv), δ(b^†bc^†c^†cc), δ(b^†b^†v^†v^†cc)

... ... ... ...

Table 3.1: Overview of all relevant CFs for the semiconductor luminescence model discussed in this chapter.

3The abbreviate notations for certain CFs likeC_ijji^x is introduced to facilitate the comparison to the EoM presented in [Baer et al., 2004,Baer et al., 2006], this notation will be dropped in later chapters.

The light-matter part of the Hamiltonian (A.5) yields the time evolution d

dtC_ijkl^x

H_D = (3.18)

−X

ξ

δilδjk

g^∗_ξ(f_i^v−f_i^c)Πξ,j+gξ(f_j^v −f_j^c)Π^∗_ξ,i

+X

ξ

gξΠ^c,_ξ,lkji^∗ +g^∗_ξΠ^c_ξ,ijkl−gξΠ^v,_ξ,lkji^∗ −g_ξ^∗Π^v_ξ,ijkl .

The first bracket contains the factorized contributions of the EVs hb^†_ξc^†_iv_j^†ckcli and hb^†_ξv^†_iv_j^†ckvli. The remaining correlation contributions Π^c_ξ,ijkl := δ(b^†_ξc^†_iv_j^†ckcl) and Π^v_ξ,ijkl := δ(b^†_ξv^†_iv_j^†ckvl) appear in the second bracket. Specifically, Π^c/v_ξ,ijij = −Π^c/v_ξ,ijji describe the correlated process of a photon-assisted polarization in presence of an additional carrier in the conduction- or valence-band, respectively. In a similar manner equations for the intraband carrier correlations C_ijkl^c := δ(c^†_ic^†_jckcl) and C_ijkl^v :=δ(v_i^†v_j^†v_kv_l) can be obtained and are provided in Appendix A.2.

The before-mentioned ’natural’ truncation of the hierarchy of carrier operators becomes apparent in the time evolution of the mixed CFs Π^c_ξ,ijkl and Π^v_ξ,ijkl. To facilitate a better understanding, we provide a schematic explanation using the notation introduced in Section 3.2. The quantitiesΠ^c_ξ,ijkl andΠ^v_ξ,ijklare CFsδ(M, N) of the order M = 1and N = 2. The time evolution with respect to the dipole part of the Hamiltonian is given by

d

dt δ(1,2)

HD

=h0,3i+h2,2i (3.19)

− d

dt (δ(1,1))h0,1i −δ(1,1) d

dt h0,1i .

Each term in this schematic representation may correspond to several contributions.

The time derivative of the factorization is subtracted in the last line in order to obtain a CF (cf. Eq. (3.2)). Due to the limitation to two carriers in the QD states the first term drops out, because it describes processes where three carriers are created or annihilated. Enforcing this property of the system requires the strict fulfillment of the boundary condition

δ(0,3) =−δ(0,2)h0,1i − h0,1i h0,1i h0,1i , (3.20) which means that, in fact,all CFs up toNmax= 2must be taken into account. This is the explicit manifestation of what we referred to earlier as the enhancement of correlations due to the limited size of the system.

The remaining hierarchy in the photon operators is truncated at the desired level Mmax. All terms appearing in the CE up toMmax = 2are summarized in Table 3.1.

Applying the CE to the remaining second term in Eq. (3.19) yields d

dt δ(1,2)

HD

= (3.21)

+δ(2,2) +δ(2,0)δ(0,2) +δ(1,1)δ(1,1) +δ(2,1)h0,1i

− d

dt (δ(1,1))h0,1i −δ(1,1)d

dt h0,1i .

Explicitly performing the calculation behind this schematic representation leads to the following EoM

d

dt Π^c_ξ,ijkl

H_D =h

g_ξf_i^cf_j^cf_j^v −g_ξ^∗Π_ξ,iΠ_ξ,j (3.22) +g_ξNξ,i^c (f_j^c−f_j^v)−g_ξf_i^cX

α

C_αjjα^x i

(δ_ilδ_jk−δ_ikδ_jl) +gξ(1 +N^ξ)C_ijkl^c +gξN^ξ(C_ijlk^x −C_ijkl^x )

+gξδ(b^†_ξbξc^†_ic^†_jckcl) +gξδ(b^†_ξbξc^†_iv_j^†clvk)

−gξδ(b^†_ξbξc^†_iv^†_jckvl)−g_ξ^∗δ(b^†_ξb^†_ξv_i^†v^†_jckcl),

and similar equations can be given for Π^v_ξ,ijkl by exploiting the symmetries of the Hamiltonian.

It is worthwhile pointing out that the restriction to a certain system size funda-mentally changes the structure of the underlying EoM. In Eq. (3.22), the uncommon product of three populations appears in the first line, originating from the subtrac-tion of the factorizasubtrac-tion in the last line of Eq. (3.21) (from Eq. (3.17) one finds that there is a contribution _dt^d δ(1,1)∝ h0,1i h0,1i). In a system where the restriction to two carriers was lifted, these terms would be compensated by the factorization of the three-particle EV h0,3i, which would, in this case, have a non-zero contribution. In fact, this compensation is also known as the linked-cluster theorem [Fricke, 1996b].

Finally, the CF δ(b^†_ξb^†_ξv_i^†v_j^†ckcl) in the last line of Eq. (3.22) can be attributed to spontaneous two-photon emission, recently demonstrated for a single-QD in a high-Q photonic crystal nanocavity [Ota et al., 2011].

Eqs. (3.16)–(3.22), together with the additional equations given in Appendix A.2, form a closed set of coupled nonlinear equations for the dynamics determined by the Hamiltonian. Before we turn to numerical results, we discuss the scattering and dephasing contributions to these equations.

Im Dokument Theory of many-particle correlations and optical properties of semiconductor quantum dots - photons and quantum dots (Seite 38-42)