Destructive Photon Echo Formation in Six‐Wave Mixing Signals of a MoSe2 Monolayer

Abstract Monolayers of transition metal dichalcogenides display a strong excitonic optical response. Additionally encapsulating the monolayer with hexagonal boron nitride allows to reach the limit of a purely homogeneously broadened exciton system. On such a MoSe2‐based system, ultrafast six‐wave mixing spectroscopy is performed and a novel destructive photon echo effect is found. This process manifests as a characteristic depression of the nonlinear signal dynamics when scanning the delay between the applied laser pulses. By theoretically describing the process within a local field model, an excellent agreement with the experiment is reached. An effective Bloch vector representation is developed and thereby it is demonstrated that the destructive photon echo stems from a destructive interference of successive repetitions of the heterodyning experiment.


Destructive photon echo formation in six-wave mixing signals of a MoSe 2 monolayer (Supplementary Information)
Note, that we have again omitted all terms that are irrelevant for the final SWM signal, especially constant terms for the occupation that do not carry any phase information. Next, we consider the time-evolution after the third pulse, where we can already sort for the SWM phase 2φ 3 −2φ 2 +φ 1 . In Fig. 5 in the main text we schematically depict the origin of the contributions with two local-field mixing processes for O(V 2 ). We disentangle the different contributions with O(V 1 ) and the SWM polarization from the pure TLS without local-field mixing with O(V 0 ) by the help of the flow charts in Fig. S1.
Starting with the polarization in (S5a) that already carries the SWM phase, their signal contribution reads The first part represents the SWM signal from the pure TLS as it is independent of V . Its origin is depicted in Fig. S1 on the left side (i) and we directly see that the coherence is detected when scanning the delay between the second and third pulse as mentioned in the main text. The second term (ii) is proportional to V τ , therefore it stems from the local-field mixing between φ2 p + 2 and |∆21| n + 2 during the delay propagation, depicted is the most right path in Fig. S1.
From the FWM-polarizations created by the third pulse, we get the two SWM contributions (iii) and (iv) by a single local field mixing process in the real time propagation. According to Eq. (5) we have to calculate p(t) = −iV tp 0 n 0 for single and p(t) = −(V t) 2 p 0 n 2 0 /2 for double mixing. (iii) is created by mixing the three-pulse FWM term ∆32+φ1 p + 3 = φ3−φ2+φ1 p + and |∆32| n + 3 resulting in The second one (iv) is created by the two-pulse FWM term 2φ3−φ2 p + 3 mixed with |∆21| n + 3 leading to Finally, contribution (v) is created by local field mixing φ3 p + 3 with |∆23−∆12| n + 3 resulting in For completeness we also give the three SWM contributions with O(V 2 ) here. As illustrated in Fig. 5 in the main text they are retrieved by double local field mixing between φ1 p + 3 and ∆32 n + where the factor 2 stems from the sum of the two occupations, and finally the single local field mixing between φ3 p + 3 and ∆32−∆21 n + Collecting all the expressions finally yields for the SWM signal in the χ (5) -regime.
From Eq. (7)/(S8) we see that in the lowest order of the light-matter coupling, the signal amplitude scales with the fifth power of the pulse area. Therefore, operating in the low excitation regime the SWM dynamics do not depend on the choice of θ. To confirm this in Fig. S2 we plot the same SWM dynamics as in Fig. 4(b) in the main text but with halved (a) and doubled (b) pulse area. There are no obvious differences between the two simulations. This confirms that the chosen pulse area  does not affect the signal dynamics and that the determined local field coupling is a reasonable quantity to characterize the dynamics.
S2 Impact of the local field strength From the analytical solution, we find the destructive echo only in the highest possible order, i.e., V 2 . To understand, how the local field strength influences the destructive echo, in Fig. S3 we show the SWM dynamics for different values of V increasing from (a) to (d). While for V = 2 ps −1 (a) and V = 5 ps −1 (b) no destructive echo is visible, for V = 10 ps −1 (c), the depression is already visible. For comparison Fig. S3(d) shows the case with V = 100 ps −1 from the main text. This again confirms, that the V 2 -order has to dominate the signal to observe the destructive echo. For V = 10 ps −1 in Fig. S3(c) we see that the destructive echo follows the dynamics of the minimum approximated by Eq. (9) shown as dashed curved line. This confirms our finding that this behavior is not affected by the strength of the local field coupling V . In addition to the local field coupling which shifts the transition energy depending on the exciton occupation, we can also take excitation induced dephasing (EID) into account [1], where the dephasing grows with the occupation β total = β + W (n + n ). For an EID on the same order of magnitude as the local field the destructive echo effect is strongly suppressed as illustrated in Fig. S4(a) for W = 50 ps −1 . As reference in Fig. S4(b) we show the case without EID (W = 0) from the main text [ Fig. 4(b)]. Obviously, the EID prohibits the development of a destructive photon echo.   Fig. 7(a).
In Fig. S5 we plot the same SWM amplitude dynamics as in Fig. 7(a) in the main text but in the simulation we consider a EID of W = 10 ps −1 , which is much smaller than the applied local field coupling of V = 100 ps −1 . We see that the destructive photon echo minimum is significantly less pronounced when taking the EID into account. Therefore, to maintain the good agreement between theory and experiment, we have excluded EID from the discussion. We can conclude that the EID should not have a significant impact in our system.

S4 Impact of intervalley scattering
In the context of the approximation in Eq. (4b) we estimated that the intervalley scattering should only have an impact for small delays. To confirm this we calculate the SWM amplitude dynamics for λ = 0 as depicted in Fig. S6(a). Compared to Fig. 4(b) in the main text we cannot find significant deviations. To highlight the changes when setting λ = 0 in Fig. S6(b) we plot the difference with respect to the case with λ = 4 ps −1 . We indeed find, that the deviations are on the order of only ≈ 10% and mainly restricted to delays |τ | < 0.5 ps. Interestingly, we find that the amplitude difference has a change of sign that exactly follows the approximated evolution of the destructive echo minimum marked by the dashed line [see Eq. (9) in the main text]. Note, that the depicted color maps were calculated numerically employing the full model and taking a nonvanishing pulse duration into account.  Fig. 4(b) in the main text.

S5 Impact of a non-vanishing delay τ 12
In the main text we have considered a vanishing delay between the two pulses labeled with φ 1 and φ 2 . As a reference the corresponding SWM dynamics are shown in Fig. S7(a) as functions of τ 23 . To identify the impact of a non-vanishing delay between these two pulses, in Fig. S7(b) we plot the SWM dynamics for τ 12 = 0.35 ps. As depicted in the inset, for this delay the two pulses are almost entirely separated. The corresponding SWM amplitude dynamics are identical to the ones with τ 12 = 0 in Fig. S7(a) but have an overall smaller amplitude. This demonstrates that the destructive echo is not caused by pulse overlap effects. The reason for the decreased amplitude is, that the coherence φ1 p + 1 after the first arriving pulse declines due to the considered dephasing. Consequently, the following processes, i.e., optical excitations and local-field mixing, are the same but start from a smaller coherence. Therefore, the final SWM signal is weaker.

S6 Impact of inhomogeneous broadening
The experimental investigations of the sample do not show a significant inhomogeneous broadening, which should have manifested in the development of a photon echo in FWM and SWM.
To demonstrate the impact of inhomogeneous broadening on the visibility of the destructive echo, we plot the destructive photon echo dynamics in Fig. S8(a), where the curved dashed black line indicates the minimum of the signal. In Fig. S8(b) we simulate the same SWM signal but set the local field coupling to V = 0 and consider an inhomogeneous broadening of σ = 4 ps −1 which leads to the development of a traditional constructive photon echo [2]. As marked by the two black dotted lines, the signal stretches along the diagonal τ = t.