3.4 Equations of Motion for single QD PL
3.4.2 System-bath interaction
Explicitly performing the calculation behind this schematic representation leads to the following EoM
d
dt Πcξ,ijkl
HD =h
gξficfjcfjv −gξ∗Πξ,iΠξ,j (3.22) +gξNξ,ic (fjc−fjv)−gξficX
α
Cαjjαx i
(δilδjk−δikδjl) +gξ(1 +Nξ)Cijklc +gξNξ(Cijlkx −Cijklx )
+gξδ(b†ξbξc†ic†jckcl) +gξδ(b†ξbξc†ivj†clvk)
−gξδ(b†ξbξc†iv†jckvl)−gξ∗δ(b†ξb†ξvi†v†jckcl),
and similar equations can be given for Πvξ,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 dtd 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†ξvi†vj†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.
method on the Lindblad formalism. As a benefit, delocalized states enter the system dynamics only in the determination of the Lindblad rates, which can be obtained phenomenologically or from independent quantum-kinetic calculations [Bockelmann and Egeler, 1992,Vurgaftman et al., 1994,Inoshita and Sakaki, 1997,Braskén et al., 1998,Braskén et al., 1998, Jiang and Singh, 1998,Stauber et al., 2000, Lorke et al., 2006,Steinhoff et al., 2012,Schuh et al., 2013]. In principle the structure of the Lind-blad formalism allows for a high degree of sophistication. For example, occupation-induced energy renormalizations due to the Coulomb interaction can be explicitly taken into account in the calculation of scattering, which then leads to different scattering rates for different configurations [Steinhoff et al., 2012]. The influence of this effect is strongly entwined with the dynamics of the system. Especially at low WL carrier densities when screening is weak, the rates may differ significantly, and a study of the impact on the emission dynamics of a single QD in a microcavity could prove interesting for future studies.
We now turn to the EoM. By evaluating the second line of Eq. (3.14) we can calculate the contributions for scattering, pumping and cavity losses.
Intraband scattering
For the scattering between the bound QD states µ and ν in the conduction band, we obtain
d dt hAi
scatt= X
µ6=ν µ,ν∈{s,p}
γµνcc 2
hh[c†νcµ, A]c†µcνi+hc†νcµ[A, c†µcν]ii
, (3.23)
where we have identified ση by c†µcν, and the Lindblad rates γη by the intraband scattering rates γµνcc. The resulting change in the single-particle population fνc = hc†νcνi due to intraband scattering
d dtfνc
scatt = (Sνin(1−fνc)−Sνoutfνc) +X
µ6=ν
(γνµcc −γµνcc)Cνµνµc ,
takes on the form of a Boltzmann-like collision term (first term), consisting of in-and out-scattering contributions with the corresponding rates Sνin = P
µ6=νγνµccfµc andSνout =P
µ6=νγµνcc(1−fµc), as well as correlation contributions beyond the single-particle description (second term). It it evident that the total carrier number is preserved, i.e. dtd P
νfνc= 0, reflecting the trace-conserving property of the Lindblad formulation that we have already discussed in the context of Eq. (3.11).
If carrier correlationsCνµνµc are neglected in Eq. (3.24), only populations of single-particle statesfνc are taken into account. These populations are obtainedby averag-ing over all configurations containing a carrier in the stateν. Due to this averaging, the single-particle description is not able to distinguish between different configu-rations with an occupation of the state ν and can, thereby, account for the Pauli exclusion principle only in an averaged sense. Consider for example the carrier re-laxation in the conduction band: The configurations |Xpi, |0piand |XXi are valid initial configurations for a p-to-s electron scattering process, although the Pauli exclusion principle forbids a carrier transition for the latter, because the s-shell
already contains one carrier. Thus, in this case the single-particle description al-lows for relaxation and attributes for dephasing, whereas an exact configuration-based treatment does not. Especially for few-emitter systems, this deficiency of the single-particle description (sometimes called ‘collision approximation’ of carrier correlations) should be avoided by considering the carrier correlations in Eq. (3.24).
The inclusion of scattering processes introduces a source of dephasing for all CFs.
The scattering contribution to the EoM of the photon-assisted polarization, d
dtΠξ,ν
scatt =−ΓνΠξ,ν− 1 2
X
µ6=ν
(γνµcc −γµνcc)Πcξ,µννµ, (3.24) includes a population-dependent dephasing rate Γν = 12(Sνin +Sνout), instead of a constant rate Γ that is frequently used in the literature. To study the influence of correlations, it is crucial to account for their proper dephasing, because it determines the timescale on which correlations are damped out. It is shown in Ref. [Baer et al., 2006] that the influence of the carrier correlations on the luminescence dynamics can be strong if no dephasing is used at all, while a small constant dephasing of the interband CF in the µeV regime already leads to a complete damping towards an uncorrelated system on a timescale of several 100 ps. To make quantitative predic-tions, a consistent treatment of dephasing with correct rates for the different CFs is important. The dynamics of the interband carrier correlations due to intraband scattering introduced by the Lindblad term (3.23) is given by
d dtCijklx
scatt =− X
µ6=ν µ,ν∈{s,p}
γµνcc 2
n
Cijklx (δiµ+δkµ) (3.25)
−2
fµcfjv(δiµδkµ−δiνδkν)δjl+Cµjµlx δiνδkν
−2
(fµc(1−fic)−Cµiiµc )δiν
−(fic(1−fνc)−Ciννic )δiµ]fjvδikδjl
o,
from which we obtain, using Cspspc = Cpspsc = −Csppsc = −Cpsspc , the following sum rule
d
dt(Cssssx + 2Cpspsx )
scatt= 0. (3.26)
Thus, both CFs Cssssx and Cpspsx are not independent quantities, but are linked by the scattering process they represent. Obviously this property cannot be fulfilled by a simpler approach, where equal and constant rates Γ are used to describe the dephasing of both CFs.
Pumping
In a typical situation for incoherent pumping carriers are excited in the barrier states and subsequently captured into the QD states. We describe this by a simultaneous generation/annihilation of carriers in the conduction/valence band p-state, which is assumed to persist during the pump pulse and to rapidly disappear afterwards.
Specifically, we consider a time-dependent capture rate P(t) following a Gaussian-shaped pump pulse. The corresponding Lindblad contribution to the EoM reads
d dt hAi
pump= P(t) 2
h[vp†cp, A]c†pvpi+hvp†cp[A, c†pvp]i
. (3.27)
This treatment of the pump process leads to an automatic built-up of CFs, e.g. for correlations between conduction- and valence-band carriers in the p-shell
d
dt Cppppx
pump =P(t)(fpv−fpc)(Cppppx + (1−fpc)fpv). (3.28) This way, initial conditions for correlations do not have to be calculated separately by considering a quasi-equilibrium initial-state population that is defined by a total carrier density and a temperature [Baer et al., 2006,Feldtmann et al., 2006], which is a great practical and conceptual advantage of our approach.
It is worth noting that it is particularly the pair-wise generation of electrons and holes that leads to the generation of certain electron-hole correlations in the system. Furthermore, since the recombination also destroys electrons and holes pair-wise, only configurations with an equal number of electrons and holes appear in the system dynamics. Thereby, charged excitonic configurations are excluded.
Alternative pump schemes can be considered [Gies et al., 2011,Gies et al., 2012], in which electrons and holes are captured independently. The implications are worth a separate discussion, which can be found in [Florian et al., 2013b].
Cavity losses
In Sec. 3.6 we show results for a QD in a microcavity. The latter provides a three-dimensional confinement of the electromagnetic field, leading to a spectrum of well-separated cavity modes. This allows for the situation, in which only a single mode ξ¯is resonant with the s-exciton transition of the QD. Nevertheless this resonant cavity mode couples to a continuum of modes outside the cavity which introduces dissipation on a nanosecond timescale. To account for a finite lifetime of the resonant mode, we introduce the Lindblad contribution
d dt hAi
cav = κξ¯
2
h[b†ξ¯, A]bξ¯i+hb†ξ¯[A, bξ¯]i
, (3.29)
where the photon loss rate κξ¯is directly connected to the quality factor Q=ωξ¯/κξ¯
of the cavity mode ξ¯at the energy ωξ¯.
Note that this contribution has a similar structure as the one for the carrier scattering (3.23), but now contains system operators acting only on the photonic degrees of freedom and leading to transitions between states involving n and n − 1 photons in the mode ξ¯. The contribution to the EoM leads to a damping of correlations at a rate M κξ¯/2,
d
dtδ( b†ξ¯p
bξ¯
q
C)
cav =−Mκξ¯
2δ( b†ξ¯p
bξ¯
q
C) , (3.30)
where M =p+q is the order of the corresponding CF with respect to the photon operators, independent of further carrier operators C contained in the CF. Thus,
photon correlations get more strongly damped in the presence of a lossy cavity, and even more so with increasing order. This plays an important role in the truncation of the hierarchy within the CE approach and is demonstrated in Sec. 3.6.