The total emission rate into the laser mode, Γem, can be obtained from the EoM for the mean photon number
d
dthb†bi=−2κhb†bi+ 2X
α
gαRe(hb†QαGXsi+hb†QαXpXXi)
=−2κhb†bi+ Γem . (C.15)
To simplify the discussion in the main text, an approximate expression of Γem has been used there. This facilitates a comparison to laser theories that consider only one bright configuration, i.e. [Rice and Carmichael, 1994]. In the following, we derive the exact expression that is used to compute the curves in Fig. 6.3. To this end
we solve the EoM for the photon-assisted polarization adiabatically and insert the solution into Eq. (C.15). The adiabatic solution for the photon-assisted polarization of the s-exciton and biexciton can be written as
hb†QαGXsi=gτDephX X
β6=α
hQαGXs(QβXsG+QβXX,Xp)i +gτDephX hb†b(QαXsXs −QαGG)i+hQαXsXsi
, (C.16)
hb†QαXp,XXi=gτDephXX X
β6=α
hQαXp,XX(QβXsG+QβXX,Xp)i +gτDephXX
hb†b(QαXX,XX −QαXpXp)i+hQαXX,XXi
(C.17) respectively, with the exciton and biexciton dephasing times
τDephX = 1
κ+γspont+ 2P, and
τDephXX = 1
κ+ 2γr+ 3γspont
.
For the sake of simplicity of our discussion, we choose the light-matter interaction gα = g, and relaxation and spontaneous loss rates γspontα,{s,p} =γspont, γr,α{s,p} =γr to be equal for all QDs. Comparison to Eq. (C.15) leads to an expression of the total emission rate in terms of the right-hand side of Eqs. (C.16) and (C.17)
Γem = 2g2τDephX X
β6=α
hQαGXs(QβXsG+QβXX,Xp)i + 2g2τDephX X
α
hb†b(QαXsXs −QαGG)i+ 2g2τDephX X
α
hQαXsXsi
+ 2g2τDephXX X
β6=α
hQαXp,XX(QβXsG+QβXX,Xp)i + 2g2τDephXX X
α
hb†b(QαXX,XX −QαXpXp)i+ 2g2τDephXX X
α
hQαXX,XXi . (C.18) To connect our results to theories that consider only one bright configuration, the assumption of equal dephasing of the bright configurations allows the definition of a common dephasing rate
τDeph−1 ≈2g2τDephX ≈2g2τDephXX .
Rewriting Eq. (C.18) in terms of the dipole operatorDα =Es†α+Esα we obtain the simple form of Eq. (6.7) for the total emission rate used in the main text:
Γem = P
αhDαDαi τDeph
+ P
β6=αhDαDβi τDeph
+ hb†bInvi τDeph
= Γspont+ Γsr+ Γstim with the total inversion operator defined as
Inv =X
α
(QαXX,XX −QαXpXp +QαXsXs −QαGG) .
Note that in all numerical calculations, the correct dephasing rates are used.
The computer algebra system FORM
In this appendix we will show how we use the language FORM [Vermaseren, 2000]
to derive the equations of motion (EoM). We will assume that the reader is familiar with the basic concepts of FORM. A very helpful introduction toFORM can be found at http://www.nikhef.nl/~form/. The advantage of FORM is that, the user has full control of what the system does, in contrast to other commonly used computer algebra systems like Wolfram Mathematica or Maple. The calculations performed byFORM are actually a combination of self defined sophisticated search and replace operations. We will describe the core procedures that are required to perform the cluster expansion (CE). These procedures are basically the implementation of the Ehrenfest EoM, and the application of the factorization and truncation operators (F, F−1,∆hNi,δ(N)) introduced in chapter 4.
D.1 Establishing standard order
In the generalized Ehrenfest EoM, the commutator in the Hamiltonian part and in the Lindblad terms, produces strings of operators deviating from a yet to define standard order. To obtain a simple form of the EoM we need to establish a prede-fined standard order of the operators. The most convenient order for our purposes is the order of operators that is closest to the normal order for operators in second quantization. We define the standard order that we want to establish in all strings of operators in three steps: (i) Separate Bose- and Fermi-operators (Bose left, Fermi right), and within each group (ii) separate creation and annihilation operators (cre-ation operators left annihil(cre-ation operators right, which is the normal order in second quantization), and within these groups (iii) alphanumeric order according to the let-ters of the operators and their indices. This (unique) standard form ensures that no terms appear in the EoM that would actually cancel each other out. The normal ordering of creation and annihilation operators also facilitates the implementation of the truncation operator ∆hNi, that is applied to systems with a limited number of particles.
(i) Separate Bose and Fermi operators: To establish the previously defined standard order FORM searches for products of two operators that deviate from this order and replaces them according to their commutation relation. The id
command is the central search and replace command in FORM. Its structure is
id search=replace, where search stands for a mathematical expression, that is to be replaced by the expression replace. The lines of code within the repeat/endrepeat -loop are executed until the output of the operation does not differ from the input.
The following lines of code separate the Fermi from the Bose operators
repeat;
id [Q](?j)*[b+]?bose(?i)=[b+](?i)*[Q](?j);
endrepeat;
To define the mathematical expressions in a general way FORM uses different types of wild-cards. The first type of wild-cards encounters in the term [Q](?j) where ?j is the argument-field wild-card for any argument (of arbitrary number) entering the function [Q].
To address (c)functionsFORM has also the function wild-card ?. This wild-card is used in the following manner: typeoffunction?set, where typeoffunction specifies the type of the function for which FORM has to search for and set can be used to further specify the function that are to be replaced by previously defined sets.
In this particular case FORM searches for all operators from the set bose=[b],[b+]. Altogether the code above means: replace all products of functions [Q] times func-tions [b+] from the set bose, regardless of their argument, by the very same Bose function [b+] times the [Q] function with their arguments respectively. In other words, the commutation relation [Q, b(0)] = 0, between photon and configuration operators is applied until (i) is fulfilled.
(ii) Normal order: In the configuration operators there are no creation and an-nihilation operators, therefore we only have to do this step for the Bose operators.
The following lines of code will establish normal order of the Bose operators:
repeat;
id [b]*[b+]=[b+]*[b]+1;
id [b]*[N](a?)=a*[N](a-1)*[b]+[N](a)*[b];
id [N](a?)*[b+]=a*[b+]*[N](a-1)+[b+]*[N](a);
id [b+]*[N](a?)*[b]=[N](a+1);
endrepeat;
In the above [N](a)=b†aba, all the performed substitutions are performed according to the standard bosonic commutation relations.
(iii) Alphanumeric order: For this example, the Fermi part of the Hilbert space is described with the configuration operators introduced in chapter 6 and appendix C. To obtain alphanumeric order in the configuration operators we use a combination of id commands:
repeat;
id disorder [Q](k?,i?,j?)*[Q](t?,l?,m?)=([Q](k,i,m)*d(j,l)-[Q](
k,l,j)*d(i,m))*d(k,t)+[Q](t,l,m)*[Q](k,i,j);
id [Q](k?,i?,j?)*[Q](k?,l?,m?)=[Q](k,i,m)*d(j,l);
endrepeat;
The first line in the above is the implementation of the commutation relation between the configuration operators Eq. (C.4) with the aim to establish numerical order with
respect to the indices. The disorder option of the id command tells FORM to substitute a matching expression only when this results in an increased numerical order of otherwise equivalent expressions. For example id would replace F21 ∗ F12 by F12∗F21 and then by F21∗F12 and so on while id disorder would replace F21∗F12 by F12∗F21 and then stop. In the second line the contracting property of the configuration operator is reflected. This line ensures that no products of configuration operators addressing the same QD are present in the EoM [Leymann et al., 2015].
The d(i,j) function is the Kronecker delta, which can be implemented by the two lines
id d(i?,i?)=1;
id d(i?,j?)=0;
that can be placed at the end of every routine using the Kronecker delta.