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for an arbitrary operator A we have 

## A. Third-order non-linear response theory 

Non-linear optical signals are related to the RDM via polarization 

In particular, for the third order non-linear signal E (3) s ( t ) one can write 

where the upper index (3) denotes that the quantity is of the third order of the perturbation theory with respect to the external electric field E ( t ) . By defining the evolution superoperator U ( t ) which fulfills Eq. (32) with E ( t ) = 0 we can write the third order perturbation term as 

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In experiment, the laser field is often prepared in a form of three incident pulses 

with different k-vectors k$_{1}$ , k$_{2}$ and k$_{3}$ . In the rest of the paper we assume Ω$_{1}$ = Ω$_{2}$ = Ω$_{3}$ ≡ Ω , A$_{1}$ ( t ) = A$_{2}$ ( t ) = A$_{3}$ ( t ) ≡ A ( t ) . The expression obtained by inserting Eq. (36) into Eq. (35) can be significantly simplified in cases where the system consists of a ground-state and a band of excited states, with the transition frequency close to reso- nance with the laser pulse frequency Ω , and by assuming the laser pulses are ultra short, i.e. A ( t ) ≈ E$_{0}$δ ( t ) . For an experiment which detects non-linear signal emitted in the direction − k$_{1}$ + k$_{2}$ + k$_{3}$ , the third order signal has a frequency ≈ Ω and it is obtained from just a handful of response functions that represent certain contributions to the triple commutator in Eq. (36). The details of the derivation can be obtained e.g. in Ref. [11]. 

If the delays between the pulses are selected such that τ denotes the delay between the first ( k$_{1}$ ) and the second ( k$_{2}$ ) pulses, and T denotes the delay between the second and third ( k$_{3}$ ) pulse (e.g. t$_{3}$ = 0 , t$_{2}$ = − T and t$_{1}$ = − T − 

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τ ) we can write for the time and the delay dependent signal field 

The absolute value in Eq. (39) originates from the fact that response functions R are defined for positive time arguments only, and negative τ is achieved by switching the order of the k$_{1}$ and k$_{2}$ pulses. The individual re- sponse functions R are listed in the Appendix A. Most importantly, they consist of series of propagation of the density matrix blocks by evolution operators obtained from the solution of equations of motion. We have e.g. 

where the evolution superoperators V ( R ) ab act on an arbi- trary operator A as a dipole operator µ$_{ab}$ from the right, i.e. V ( R ) ab A = Aµ$_{ab}$ . The superoperator V ( L ) ab is defined analogically with the action of µ$_{ab}$ from the left. The in- dices e and g denote electronic bands as denoted in Fig. 1. Thus, the above operators and the action of superop- erators on an arbitrary operator A are expressed in the basis of Hamiltonian eigenstates as 

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Eqs. (41) to (44) together with the Appendix A enable us to calculate expected non-linear signal from the knowl- edge of the matrix elements of the evolution superop- erator. This type of knowledge can be obtained from solutions of the four different equations of motion that we presented in Section III. 

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