Influence of local fields on the dynamics of four-wave mixing signals from 2D semiconductor systems
Thilo Hahn, Jacek Kasprzak, Pawe? Machnikowski, Tilmann Kuhn, Daniel Wigger
IInfluence of local fields on the dynamics of four-wavemixing signals from 2D semiconductor systems
Thilo Hahn , , Jacek Kasprzak , Paweł Machnikowski ,Tilmann Kuhn , Daniel Wigger Institut für Festkörpertheorie, Universität Münster, 48149 Münster, Germany Department of Theoretical Physics, Wrocław University of Science and Technology,50-370 Wrocław, Poland Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble,FranceE-mail: [email protected]
Accepted manuscript of New J. Phys. (2021) 023036 (CC BY 4.0)https://doi.org/10.1088/1367-2630/abdd6cpublished: 19 February 2021 Abstract.
In recent years the physics of two-dimensional semiconductors was revivedby the discovery of the class of transition metal dichalcogenides. In these systemsexcitons dominate the optical response in the visible range and open many perspectivesfor nonlinear spectroscopy. To describe the coherence and polarization dynamics ofexcitons after ultrafast excitation in these systems, we employ the Bloch equationmodel of a two-level system extended by a local field describing the exciton-excitoninteraction. We calculate four-wave mixing signals and analyze the dependence ofthe temporal and spectral signals as a function of the delay between the excitingpulses. Exact analytical results obtained for the case of ultrafast ( δ -shaped) pulsesare compared to numerical solutions obtained for finite pulse durations. If two pulsesare used to generate the nonlinear signal, characteristic spectral line splittings arerestricted to short delays. When considering a three-pulse excitation the line splittings,induced by the local field effect, persist for long delays. All of the found features areinstructively explained within the Bloch vector picture and we show how the excitonoccupation dynamics govern the different four-wave mixing signals. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b
1. Introduction
With the observation of an extraordinarily strong light emission from monolayers oftransition metal dichalcogenides (TMDCs) in 2010 [1, 2], this material class came intothe focus of semiconductor optics. Since then, the research on the fundamental physicsof these materials and the potential applications of them has flourished and is stillgrowing. The optical properties of TMDCs are strongly dominated by excitons which,due to the strong Coulomb interaction in these materials, exhibit binding energies on theorder of 500 meV [3], which is orders of magnitude larger than in typical III-V or II-VIsemiconductors. While initially the observed spectral lines were rather broad, it turnedout that by encapsulating the monolayer in hexagonal boron nitride the inhomogeneityof the structure could be strongly reduced resulting in linewidths of the excitonictransitions approaching the homogeneous limit [4, 5, 6]. Among other techniques, alsofour-wave mixing (FWM) spectroscopy has been applied to these materials [7, 5, 8],which gave access to the coherence and density dynamics of the excitons after ultrafastexcitation. These studies revealed FWM signals even for negative delay times [8], whichare clear hints for contributions to the signals resulting from exciton-exciton interactions.These features revived the interest in local field models for the description of FWMsignals.Exciton dynamics in semiconductor nanostructures have been thoroughly exploredover many years using tools of nonlinear spectroscopy [9, 10, 11], like differentialtransmission [12, 13], spectral hole burning [14, 15] or FWM [16, 17]. In thecorresponding theoretical description it is often sufficient to apply two- or few-levelmodels [18, 19]. For systems like quantum dots (QDs), in particular in samples witha low QD density, it is generally accepted that each excitonic few-level system canbe treated individually [20, 21] and a collection of QDs is then treated by ensembleaveraging [22]. In spatially extended systems like GaAs quantum wells, however,many body effects become relevant [23]. It turned out that an effective mean fieldtreatment of the exciton-exciton interaction successfully reproduces the experiments insuch samples [24, 25]. This model describes the influence of all other excitons on thetwo-level system (TLS) of a single exciton in the form of a so-called local field , whichallows one to analyze the resulting FWM signals in the lowest contributing order ofthe exciting fields, i.e., the χ (3) -regime [23]. This local field contribution to a TLS hassubsequently been derived from more fundamental theoretical approaches. Starting froma microscopic density matrix description, it can be interpreted as an interaction withthe exciton-exciton scattering continuum [26, 27]. In the case of a resonant excitation ofthe 1s exciton, such local field contributions can also be obtained from a simplificationof the semiconductor Bloch equations [28]. Recently, also in TMDC systems simplifiedmodels derived from a microscopic theory have been found to show terms which can beinterpreted as local field effects [29, 30].Motivated by these theoretical works and the experiments in Ref. [8], but notlimited to TMDC systems, we build on the original model and thoroughly study theinfluence of the local field effect on the spectral dynamics of different FWM signals.In practice we focus on different excitation scenarios, namely a two- and a three-pulseexcitation scheme, which allows us to probe different aspects of the system dynamics [20].We show that an analytic treatment in the limit of ultrafast laser pulses is possibleeven without restrictions to a perturbative treatment, which allows us in particularto study the dependence of the FWM signals on the intensities of the exciting pulses.An instructive explanation of the involved signal dynamics based on the Bloch vectordescription clarifies their physical interpretation. In turn, the numerical simulation ofFWM signals generated by laser pulses with realistic durations in the sub-picosecondrange allows us to make predictions for actual experiments and to study the effect oftemporally overlapping pulses.The paper is organized as follows: After the introduction to the model in Sec. 2an analytic solution of the equations of motion for ultrafast optical excitations is givenin Sec. 3. Two-pulse FWM signals are discussed in Sec. 4 and three-pulse signals inSec. 5, where first the δ -pulse limit and then non-vanishing pulse durations are analyzed.Finally, in Sec. 6 we draw some conclusions.
2. Model
The optical driving of a collection of two-level systems, each consisting of the groundstate | g (cid:105) and the excited state | x (cid:105) separated by an energy (cid:126) ω , is in general describedby the optical Bloch equations [18], ∂p∂t = i (1 − n )Ω( t ) − βp , (1 a ) ∂n∂t = 2Im[ p Ω ∗ ( t )] − Γ n . (1 b )The central quantities are the polarization p = (cid:104)| g (cid:105)(cid:104) x |(cid:105) and the occupation n = (cid:104)| x (cid:105)(cid:104) x |(cid:105) .Transitions between the two states are induced by the classical light field E ( t ) , expressedhere in terms of the instantaneous Rabi frequency Ω( t ) = M · E ( t ) / (cid:126) . M is the dipolematrix element and E ( t ) is taken to be in resonance with the transition energy (cid:126) ω .Therefore, both E ( t ) and Ω( t ) are described only by their envelopes. Note that wehave applied the standard rotating wave approximation (RWA) and the equations aregiven in the frame rotating with the transition energy (cid:126) ω .The RWA is well justified for aresonant or close-to-resonant excitation with pulse durations in the hundred femtosecondrange and moderate pulse powers [31]. In the context of TMDC monolayers, therestriction to a two-level model is expected to be applicable, when the dynamics arerestricted to excitons in a single valley, which is the case in a resonant co-circularexcitation scheme. For large exciton densities the scattering into other states mightbecome relevant [30]. Nevertheless, we will study the behavior of the TLS also inthe regime of larger pulse areas beyond the third order regime, keeping in mind thatin actual experiments effects from other excitons might contribute to the signals. Inthis way, when comparing to measured signals, our results may act as a reference toestimate at which excitation powers the TLS model loses its validity. We have includeda phenomenological dephasing rate β and a decay rate of the excited state Γ .In the local field model the optical field is supplemented by a contribution due tothe field generated by the polarization of the TLSs themselves [23]. The full opticalfield then reads Ω( t ) = Ω ext ( t ) + V p , with the external optical laser field Ω ext ( t ) and thecoupling parameter V that results from the Coulomb interaction among the excitonsand that determines the strength of the self-interaction [28, 26]. This substitution leadsto the Bloch equations with a local field contribution, ∂p∂t = i (1 − n )[Ω ext ( t ) + V p ] − βp , (2 a ) ∂n∂t = 2Im[ p Ω ∗ ext ( t )] − Γ n . (2 b )Note that the local field is characterized by a real value of V [23], therefore itdoes not appear in Eq. (2 b ). Considering an imaginary part for V in Eq. (2 a )also describes an excitation induced dephasing (EID) effect [32], as can be derivedfrom a microscopic theory including exciton-exciton scattering [33]. Note, however,that Eq. (2 b ) remains unchanged because exciton-exciton scattering does not lead togeneration or recombination of excitons and therefore does not change the excitonoccupation. To include both features, in the following we will treat V as a complexconstant. For simplicity we will continue to refer to V p as a local field effect. Whendiscussing the analytical results we will explicitly address the influence of real andimaginary part of V .The local field obviously leads to an additional nonlinearity ∼ np in the opticalequations. By reformulating Eq. (2 a ), this term can be interpreted as an effectiveoccupation-dependent shift of the transition energy from (cid:126) ω to (cid:126) ω eff = (cid:126) ω + (cid:126) ω loc ( n ) and an effective occupation-dependent dephasing β eff = β + β loc ( n ) according to ∂p∂t = i (1 − n )Ω ext ( t ) − [ β + β loc ( n )] p − iω loc ( n ) p , (3)(4)with ω loc ( n ) = − (1 − n )Re( V ) ,β loc ( n ) = (1 − n )Im( V ) . (5)When the system is in its ground state with n = p = 0 , the effective transition frequencyis reduced to ω eff = ω − Re( V ) . It increases linearly with growing occupation, reaching ω eff = ω + Re( V ) in the excited state n = 1 , p = 0 . Consequently, for n = 1 / thefrequency remains unchanged ω eff = ω . To obtain EID, i.e., a dephasing which increaseswith increasing density Im( V ) < is required. This, in turn, requires β > − Im( V ) tostill have a dephasing in the linear regime.
3. Solution for ultrashort pulses
Despite the nonlinearity, Eqs. (2) can be solved analytically in the limit of ultrashortpulses, i.e., for δ -pulses. Note that while mathematically we use δ -pulses, physicallythe ultrashort pulse limit is reached if the pulse duration is much shorter than thecharacteristic timescale of the system’s dynamics. This is still well within the validityof the RWA. As will be seen later, the obtained results will be helpful for understandingthe specific behavior of the considered FWM signals. To reach the δ -pulse limit, weassume a rectangularly shaped, resonant pulse of duration ∆ t centered around the time t with total pulse area θ and phase φ as Ω ext ( t ) = θe iφ t , t − ∆ t/ ≤ t ≤ t + ∆ t/ , , otherwise , (6)and at the end perform the limit ∆ t → . To solve the differential equations, we use aparametrization of the instantaneous pulse area [34], A = θ ∆ t ( t − t + ∆ t/ , (7)instead of the time t . Clearly, A is restricted to the interval [0 , θ ] corresponding tothe beginning of the pulse and its end, where the total pulse area is reached. Utilizingthe pulse area instead of the time avoids divergences in Eqs. (2) connected with anultrashort optical pulse. The transformed equations during the pulse read ∂p∂A = i ∆ tθ (1 − n ) (cid:18) θe iφ t + V p (cid:19) − ∆ tθ βp, (8 a ) ∂n∂A = Im( pe − iφ ) − ∆ tθ Γ n . (8 b )In the limiting case ∆ t → all contributions from the local field, dephasing and decayof the TLS vanish and Eqs. (8) have the same solution as the ordinary optical Blochequations given by [34], n + = n − + sin (cid:18) θ (cid:19) (1 − n − ) + sin( θ )Im( p − e − iφ ) , (9 a ) p + = cos (cid:18) θ (cid:19) p − + i − n − ) sin( θ ) e iφ + sin (cid:18) θ (cid:19) p −∗ e iφ , (9 b )where p + , n + ( p − , n − ) denote the polarization and occupation directly after (before) thepulse, respectively.In the absence of an external field, Ω ext = 0 , the dynamics of the occupation are onlysubject to the decay. For an initial occupation n at t = 0 it simply drops exponentially n ( t ) = n e − Γ t . (10 a ) FWM time t Figure 1.
Pulse alignment for two-pulse FWM with positive delay τ > . TheFWM signal is launched by the second pulse which also defines the time t = 0 . This time dependence contributes to the polarization as a time-dependent phase shiftintroduced by the local field coupling which therefore reads p ( t ) = p exp (cid:20) i V Γ n ( e − Γ t − (cid:21) exp( iV t − βt ) . (10 b )In most semiconductor systems, the polarization dephases on a much shorter timescalethan the occupation decays [20]. In this case we can take Γ → and the polarizationdynamics simplify to p ( t ) = p exp [ iV (1 − n ) t − βt ] , (11)where the polarization simply rotates with the effective transition frequency ω eff ( n ) = ω − (1 − n )Re( V ) and it decays with the effective dephasing rate β eff = β + (1 − n )Im( V ) , as given in Eq. (3). However, note that for completeness in the analyticalresults presented in the following this limit has not been performed and the parameter Γ is still included.
4. Two-pulse FWM
The most basic FWM experiment utilizes the two-pulse sequence depicted in Fig. 1. Tworesonant laser pulses are applied with a variable delay τ with respect to each other.The pulse areas θ and θ of the first and second pulse are kept fixed while the pulsephases φ and φ are varied during the repetition of the experiment. In the simulationthe FWM signal is extracted by a phase-selection scheme from the microscopic excitonpolarization as explained later. The second arriving pulse generates the FWM signaland sets the time t = 0 . δ -pulse limit Based on the analytic results for a single ultrashort pulse derived in Eqs. (9) and (10),we start our discussion by calculating the two-pulse FWM signal analytically in thatlimit. The polarization after the second pulse depends on the time t after that pulseand the delay τ within the pulse pair and reads p ( t, τ ) = (cid:104) ae i ( φ +Φ) + be iφ + ce i (2 φ − φ − Φ ∗ ) (cid:105) e − βt × exp (cid:16) i (cid:8) α + ηe − Im(Φ) cos[Re(Φ) + φ − φ ] (cid:9) (cid:17) , (12 a )with a ( τ ) = i θ ) cos (cid:18) θ (cid:19) exp ( − βτ ) , (12 b ) b ( τ ) = i θ ) (cid:20) − (cid:18) θ (cid:19) e − Γ τ (cid:21) , (12 c ) c ( τ ) = − i θ ) sin (cid:18) θ (cid:19) exp ( − βτ ) , (12 d ) α ( t, τ ) = V t − V Γ (cid:20) sin (cid:18) θ (cid:19) + cos ( θ ) sin (cid:18) θ (cid:19) e − Γ τ (cid:21) × (cid:0) − e − Γ t (cid:1) , (12 e ) Φ( τ ) = − V Γ sin (cid:18) θ (cid:19) (cid:0) − e − Γ τ (cid:1) + V τ , (12 f ) η ( t, τ ) = − V Γ sin ( θ ) sin ( θ ) e − βτ (cid:0) − e − Γ t (cid:1) . (12 g )Note, that the polarization is not oscillating with the transition frequency because weare working in the rotating frame of the laser field. Directly after the second pulseat t = 0 , the polarization consists of three parts depending on the phases φ , φ and φ − φ . In the field-free propagation after the second pulse, the polarizationreceives an additional contribution from the occupation via the local field coupling. Thecorresponding term is the last exponential function in Eq. (12 a ), which also contributesto the phase dependence via the difference φ − φ and its complex conjugate phase.This will be discussed in detail below. From Eqs. (12) we find that in the limit Γ → an imaginary part of the local field parameter only gives rise to an additional damping,depending on the pulse areas and the delay time. Thus, we expect that EID manly givesrise to a faster decay of the signals and a broadening of the corresponding spectra. Sincehere we are mainly interested in characteristic features in the signals and the spectra inthe following we will neglect an imaginary part and assume that V is a real parameter.In the Appendix B we will show results of calculations including EID, which will confirmthat this expectation is indeed fulfilled.The FWM dynamics are obtained by filtering the polarization in Eq. (12 a ) withrespect to the inverse FWM phase − φ (2)FWM = − (2 φ − φ ) and we retrieve the two-pulseFWM polarization p (2)FWM ( t, τ ) = π (cid:90) p ( t, τ ) e − i (2 φ − φ ) d φ d φ (2 π ) . (13)Hence, the FWM dynamics consist only of that part of the polarization which carries theFWM phase as all other phase-dependencies vanish. This procedure models the typicalheterodyne detection in FWM experiments [11] Without local field interaction there isno additional phase in the free propagation. Then the FWM polarization consists onlyof the term ∼ c in Eq. (12 a ) which reads p (2)FWM ( t, τ ) = ce − βt ∼ e − β ( t + τ ) . (14) freepropagationlocal fieldinducedlaserinduced mixingorder Figure 2.
Flow chart of the pulse sequence and phase selection leading to the FWMsignal. Black solid line: Free propagation without pulse interaction. Green wave: laserinduced propagation. Violet dotted line: Local field induced mixing of the occupationinto the polarizations. The numbers mark the possible mixing orders.
With local field interaction, the resulting FWM dynamics are p (2)FWM ( t, τ ) = e i ( α − Φ) − βt [ cJ ( η ) + ibJ ( η ) − aJ ( η )] . (15)We find that each term of the polarization contributes to the FWM dynamics with aBessel function of a different order. The full expression, after inserting Eqs. (12 b )-(12 g )into Eq. (15), becomes quite involved, but we will now have a more detailed look onthe creation of the FWM signal to understand the origins of its different contributions.A flow chart of the pulse sequence and phase selection resulting in the FWM signal isdepicted in Fig. 2. Initially the system is in the ground state with n = p = 0 . Thefirst pulse creates an occupation n +1 and a polarization with the phase of the first pulse φ p +1 , marked as green wave for a laser pulse induced step. For clarity, we give the phasedependence of each quantity on the left side of the symbol and the number of the pulseon the right side, where + is directly after and − directly before the respective pulse.During the field-free propagation before the second pulse two things happen. On theone hand both quantities simply propagate in time indicated by black arrows. On theother hand the occupation contributes to the phase through the local field coupling [ Φ in Eq. (12 f )], indicated by a violet dotted arrow. The second pulse creates an occupationconsisting of a part independent of the pulse phases n +2 and a part depending on thepulses’ phase difference φ − φ φ − φ n +2 . Because the occupation is a real quantity, this quantityhas to depend on the phase difference and its conjugate, therefore two terms appear onthe left side of the symbol. As mentioned above, the polarization created by the lastpulse consists of parts depending on the phases of the exciting pulses φ p +2 , φ p +2 and theFWM phase φ − φ p +2 . After the second pulse, φ − φ φ − φ n +2 contributes to the polarizationvia the local field resulting in additional wave-mixing possibilities (violet dotted arrows). time t FWM
Figure 3.
Pulse alignment for negative delays τ < . The pulse with φ launchesthe FWM signal via the local field contribution and sets t = 0 . We illustrate this by one example:The simplest wave-mixing process between the occupation φ − φ φ − φ n +2 and thepolarization φ p +2 that results in the FWM phase is φ FWM = φ (cid:124)(cid:123)(cid:122)(cid:125) p + ( φ − φ ) (cid:124) (cid:123)(cid:122) (cid:125) n , (16)where the occupation enters once. It is possible to generalize this to larger numbers ofadmixtures of φ − φ φ − φ n +2 giving φ FWM = φ + k ( φ − φ ) + ( k − φ − φ ) , (17)with k ∈ N , where the occupation is used in total k − times. In the flow chart thisorder of the phase difference mixing is annotated on the violet dotted lines. Consideringthe limit of weak local field coupling | V | (cid:28) β , i.e., | η | (cid:28) the mixing orders enter as η k − . This directly determines the order of the corresponding Bessel function, whichis one in this case. Applying this procedure also to the other polarizations we find theBessel functions of 0th, 1st, and 2nd order.In the case of a negative delay, the pulse alignment is inverted as depicted in Fig. 3.The chronologically first (second) pulse has the pulse area θ ( θ ) and the phase φ ( φ ).To obtain the FWM signal for negative delays, Eq. (12) can be re-used where the pulseareas, pulse phases and the sign of the delay have to be switched θ ↔ θ , φ ↔ φ , τ → τ = − τ . The FWM polarization then reads p (2)FWM ( t, τ <
0) = e i ( α +2Φ) − βt [ iaJ ( η ) − bJ ( η ) − icJ ( η )] . (18)In contrast to the FWM dynamics with positive delay, the orders of the Bessel functionsare different as the wave-mixing orders of φ − φ φ − φ n +2 change. In particular, as both pulseschange their ordering, in the pure TLS only the polarization φ − φ p +2 is created, whichresults in a vanishing FWM signal. Therefore, to obtain a signal with the FWM phase φ − φ at least a third order mixing via the local field is necessary. Thus, the lowestorder of wave-mixing is one, which is the contribution of φ p +2 mixing with the phasedifference φ − φ of the occupation once. To study FWM signals with non-vanishing pulse durations ∆ t > , the Bloch equationswith local field coupling in Eqs. (2) and the phase filtering are evaluated numerically [35].0The optical field is modeled as resonant Gaussian pulses centered at t = − τ and t = 0 , Ω ext ( t ) = 1 √ π ∆ t (cid:26) θ exp (cid:20) − ( t + | τ | ) t + iφ (cid:21) + θ exp (cid:18) − t t + iφ (cid:19) (cid:27) , (19) for τ ≷ . To account for the inverted pulse ordering, for τ > the upper indices in θ and φ are used, while the lower ones refer to τ < . For the following discussion we keep asmany system parameters as possible fixed to keep the analysis as instructive as possible.For the exciton dephasing we choose β = 0 . ps − which is a typical value for hBN-encapsulated TMDC monolayer [36]. The exciton decay time is usually much longerthan the dephasing [37] and we choose Γ = 0 . For a non-vanishing value the impact ofthe local field effect would additionally decay on time in a non-trivial way. To avoid thiscomplication we disregard the decay. For the pulse duration we assume ∆ t = 0 . ps asa typical value for pulse trains in FWM experiments [8]. In addition we set θ = 2 θ which is usually applied in FWM experiments [38]. As local field coupling we consider V = 6 ps − which is an order of magnitude larger than the dephasing rate β . In theliterature similar proportions have been used to describe semiconductor quantum wellsystems [25, 24]. In monolayer TMDCs, the dephasing and decay rates differ betweensamples. Additionally, the strength of the local field is generally not known and maydepend on the ambient conditions. For simplicity, it is fixed to a value such that localfield effects are clearly visible. Future comparisons to experiments will help to find thecorrect values for the local field strength.We begin the discussion by analyzing the influence of the local field coupling onFWM signals in the regime of small pulse areas. Exemplarily, in Fig. 4, the FWMdynamics for θ = 0 . π of the pure TLS with V = 0 and with a local field coupling V = 6 ps − are shown in (a) and (b), respectively.In the pure TLS without local field interaction, the FWM dynamics in Fig. 4(a)directly represent the polarization dynamics which are given by exponential decays inreal time t and delay time τ with the dephasing rate β [Eq. (14)]. In the direction ofnegative delays the dynamics drop sharply when going to τ < as there is no FWMpolarization created when the pulse with phase φ arrives before the pulse with phase φ [20]. The corresponding FWM spectrum, defined as S FWM ( ω, τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) −∞ p (2)FWM ( t, τ ) e i ( ω − ω ) t d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20)is a Lorentzian line for positive delays, i.e., a single peak at the bare transition energy (cid:126) ω = (cid:126) ω (not shown).Upon including the local field interaction, the FWM dynamics in Fig. 4(b) changequalitatively. On the one hand, the signal extends to τ < and on the other hand, themaximum of the signal is shifted to larger times t > . Both aspects can be understood1 time t (ps) − d e l a y τ ( p s ) a V = 0 0 2 4 6 time t (ps) b V = 6 ps − . . . . . . n o r m . F W M a m p l. Figure 4.
FWM dynamics as a function of real rime t and delay τ for small pulseareas θ = 0 . π . (a) Without a local field coupling V = 0 and (b) with a local fieldcoupling V = 6 ps − . with the analytic solution for δ -pulses. For low excitation powers, the lowest ordercontributing to the FWM signal is called the χ (3) -regime which for t ≥ reads [23] p (2)FWM ( t, τ ) ≈ κ e iV ( t − τ ) − β ( t + | τ | ) (21) × (cid:8) Θ( τ ) + 2 iV t (cid:2) Θ( τ ) + e ( − β + iV ) | τ | Θ( − τ ) (cid:3)(cid:9) , with κ = − iθ θ / and the Heaviside function Θ( t ) . For positive delays, the dynamicsconsist of a part decaying as e − β ( t + τ ) which reflects simply the FWM dynamics withoutlocal field interaction and a part that rises for short times ∼ V te − β ( t + | τ | ) . Thiscontribution results from the mixing of the polarization φ p +2 with the phase-dependentoccupation φ − φ φ − φ n +2 , i.e., mixing order 1 in Fig. 2. In contrast, for negative delays, onlythe last term contributes, as the FWM polarization is not created in the pure TLS butthe polarization φ p +2 can still mix with the occupation. Here, the pulse with the phase φ arrives first, therefore both φ p +2 and φ − φ φ − φ n +2 are reduced due to dephasing before thesecond pulse with φ . This results in the additional decay of the signal with e − β | τ | inEq. (21). So, in total, the signal decays twice as fast for negative delays than for positiveones. The FWM spectrum also consists of a single Lorentzian peak that is shifted by − (cid:126) V due to the energy renormalization from the local field coupling (not shown).For larger pulse areas beyond the χ (3) -regime, the FWM dynamics changeremarkably. In Fig. 5(a) the FWM dynamics for θ = 0 . π are shown. As a first strikingdifference, they exhibit oscillations, which stem from the Bessel functions known fromthe δ -pulse solution in Eq. (15). For increasing absolute delays, the minima of thismodulation shift to larger t . Their functional evolution τ (min)12 ( t ) in the plot can beestimated from the equations. It is determined by the curve for which the argument η of the Bessel functions [Eq. (12 g )] is constant, i.e., its differential d η vanishes: d η = e − βτ d t − βte − βτ d τ
12 ! = 0 ⇒ dτ d t = 1 βt ⇒ τ (min)12 = 1 β ln( t ) + τ . (22)2 − d e l a y τ ( p s ) a θ = 0 . π b θ = 0 . π time t (ps) − d e l a y τ ( p s ) c θ = 0 . π − − energy ¯ h ( ω − ω ) (meV) d θ = 0 . π . . . . . . n o r m . F W M a m p l. . . . . . . n o r m . F W M a m p l. Figure 5.
FWM signals for larger pulse areas. (a, b) θ = 0 . π and (c, d) θ = 0 . π .FWM dynamics in (a) and (c) and the respective FWM spectra as a function of energy (cid:126) ( ω − ω ) and delay in (b) and (d). The dashed line in (a) indicates the time dependenceof the minima as predicted by Eq. (22). This curve is included as dashed line in Fig. 5(a). Therefore, the exponential loss ofcoherence between the pulses leads to the logarithm-shaped evolution of the minima.In Fig. 5(b), the corresponding FWM spectra are shown as a function of τ . Whilefor small absolute delays | τ | (cid:28) /β , the spectrum is broad it becomes narrow forlarge delays, which can be explained by considering the FWM dynamics. Because theoscillations vanish for large τ , the single maximum at t > leads to a single line inthe spectrum. This behavior of the FWM spectra is similar for positive and negativedelays and the spectral positions of large | τ | coincide. However for negative delays thesignal is weaker and decays faster, as already seen in the χ (3) -regime in Eq. (21). Thephysical meaning of the spread and position of the spectrum will be discussed below.Also the FWM dynamics for higher pulse areas of θ = 0 . π , shown in Fig. 5(c),exhibit similarities with the previous example like the oscillatory behavior. However,one difference is that the signal decays much faster. In addition, when looking at smalldelays | τ | ≈ ∆ t , the minima deviate from the logarithmic shape. This indicates,that there is a difference between the action of two separate and two simultaneouspulses. The reason is that the pulse area and duration have a non-trivial impact onthe resulting state when the system deviates from the pure TLS [38, 39]. Therefore,the pulse area theorem [18] does not hold for extended pulses anymore [39] and thepulse area θ does not agree with the rotation angle of the Bloch vector. This makes thedynamics of the signal more involved. Details about this pulse area renormalization aregiven in Appendix A. In addition, during the pulse overlap complex dynamics stemming3 . . . . . pu l s e a r e a θ / π τ (cid:28) /β a N (2) − N (2)+ − − energy ¯ h ( ω − ω ) (meV) . . . . . pu l s e a r e a θ / π τ (cid:29) /β c N (2) ∞ . . . . . . norm. FWM ampl. pos.delay neg.delay | g |x smalldelay bd large delay | gN + (2) (2) N – | x dephasinglaser pulsesocc. after2nd pulse1st pulse2nd pulseBloch vector (2) N ∞ Figure 6.
Limiting cases for the FWM spectra. (a) FWM spectra for small delays τ = 2∆ t as a function of pulse area θ . (b) Bloch vector illustration of smallest andlargest energies in the spectral distribution in (a). (c) FWM spectra for long delays τ (cid:29) /β . (d) Bloch sphere illustration for (c). from Rabi oscillations contribute to the FWM signal for large enough pulse areas [40].Qualitatively also the FWM spectra, shown in Fig. 5(d), have a similar form as the onesat lower pulse area. The most striking difference is however, that the final peak energyfor long delays moves from ω < ω to ω > ω . A discussion of the differences betweenthe simulations and the δ -pulse limit is presented in Appendix B, where the respectivesignals in the ultrashort pulse limit are shown. As expected, some deviations betweenthe simulations with non-vanishing pulse duration and δ -pulses appear for delays thatare shorter or of the order of the pulse duration, i.e., in the regime τ (cid:46)
200 fs , wherethe overlap has a significant influence on the generated signal. For longer delays, the δ -pulse limit turns out to be a very accurate approximation.In the following we analyze the spread and position of the FWM spectra at smalland large delays in detail. A systematic study for small delays is shown in Fig. 6(a),where the spectra are depicted for a continuous variation of the pulse area θ . Thedelay is fixed to twice the pulse duration τ = 2∆ t , such that the two pulses are mainlyseparated. At low pulse areas the spectrum is very narrow as we have discussed for the χ (3) -regime. We have already seen in the exemplary FWM spectra that the spectralwidth increases with growing pulse area. This spread of the spectra can be tracedback to the fact, that the transition energy is modified by the occupation [see Eq. (3)].So, in order to develop an intuitive picture which occupations can be reached in the4FWM experiment, we consider the Bloch vector representation of the systems state inFig. 6(b). The Bloch vector is given by its coordinates [2Re( p ) , p ) , n − and itpoints to the surface of the Bloch sphere for pure states and is shorter than unity whendephasing happens. Laser pulse excitations lead to rotations of the Bloch vector aroundan axis that depends on the phase of the pulse and, in general, also on the laser pulsedetuning. However, here we assume a resonant excitation throughout this study.Keeping in mind that all possible combinations of pulse phases φ and φ thatmatch the FWM phase are realized during the repetition of the experiment, we searchfor the smallest and largest possible occupation after the two pulses. Following Eq. (3)the corresponding renormalized energies can be calculated to (cid:126) ω eff which connects thespread of the FWM spectrum with the extent to which the system can be addressed by itscoherences. We can interpret this reasoning as a kind of coherent control experiment [41].In particular, such a manipulation of the occupation by varying the phases of opticalpulses is performed in Ramsey fringe experiments [42].As schematically shown in the picture the largest occupation N (2)+ is reached whenthe two pulses rotate the Bloch vector into the same direction effectively adding uptheir pulse areas. The smallest occupation N (2) − appears if the two pulses act in oppositedirections. So for δ -pulses the maximal (+) and the minimal ( − ) occupation are givenby N (2) ± = sin (cid:18) | θ ± θ | (cid:19) . (23)The corresponding energies (cid:126) ω eff ( N (2) ± ) are marked in Fig. 6(a) as dashed black lines.They describe the boundaries of the spectrum very well. But we find a slight mismatchbetween the dashed line and the edge of the spectrum. This deviation can be tracedback to a renormalization of the pulse area due to the local field interaction, as we havementioned before. While the pulse area is defined via its rotation angle in the pure TLS,the local field interaction changes this rotation angle in a non-trivial way (see AppendixA). The FWM spectra for large delays are depicted in Fig. 6(c). As we have alreadyseen in Figs. 5(b, d), the spectrum becomes a single narrow line. Its energy dependson the pulse area and is the same for positive and negative delays. These findings canagain be traced back to the behavior of the Bloch vector depicted in Fig. 6(d). Afterthe first Rabi rotation the polarization dephases as indicated by a dotted line, i.e., theBloch vector moves onto the Bloch sphere’s z -axis. Afterwards, the second pulse leadsto another Rabi rotation, which is reduced because the length of the Bloch vector issmaller than one. The spectral position of the FWM signal is then determined by theoccupation after the second pulse N (2) ∞ . For ultrafast pulses the final occupation reads N (2) ∞ = sin (cid:18) θ (cid:19) + cos( θ ) sin (cid:18) θ (cid:19) = 14 [2 − cos( θ − θ ) − cos( θ + θ )] (24)and is the same for both positive and negative delays as also shown in the schematic.5This is the reason why the spectra lie at the same position for positive and negativedelays. Because N (2) ∞ does not depend on the pulse phases anymore the spectrum consistsof a single sharp line. In the context of coherent control this means that for delays muchlonger than the dephasing time the system cannot be addressed coherently anymore.Naturally, the loss of coherence also leads to a decay of the signal strength.
5. Three-pulse FWM
While in the bare TLS the two-pulse FWM signal carries information on the coherenceof the system, it is possible to measure the occupation dynamics in a three-pulseconfiguration with φ (3)FWM = φ + φ − φ additionally [20]. As for the influence of thelocal field the occupation plays a crucial role we expect interesting features to appearfor this pulse sequence. δ -pulse limit With three pulses there are two delays, τ between the first and second pulse and τ between the second and third pulse. Like in two-pulse FWM, the pulse areas θ , , are kept fixed while the pulse phases φ , , are scanned to generate the FWM signal.Usually, the delay τ is kept small, such that the pulses slightly overlap, and it staysfixed while the delay τ is varied.As explained before in the limit of ultrashort pulses all properties of the systemcan be calculated analytically. Following the results in Eq. (12) and applying Eqs. (9)we end up with the following phase dependencies immediately after the third pulse:The occupation n +3 carries the phases φ − φ , φ − φ , φ − φ + φ and their complexconjugates, the polarization p +3 carries the phases φ , φ , φ , φ − φ , φ − φ , φ − φ ,and φ − φ + φ . In the following free propagation additional wave mixing betweenthe occupation and the polarization can contribute to the FWM signal. As this largenumber of possible phase combinations complicates the situation massively, we will onlyanalyze the limiting case of large delays τ (cid:29) /β analytically. So we assume thatbefore the interaction with the third pulse the polarization has dephased entirely andwe can set p − = 0 . Choosing further τ = 0 , the polarization after the three pulsesreads p ( t, τ ) = exp { i [ ϑ + ν cos( φ − φ ) + φ ] − βt }× [ u + 2 w cos( φ − φ )] , (25 a )with ϑ ( t, τ ) = V t − V Γ (cid:0) − e − Γ t (cid:1) (cid:110) sin (cid:18) θ (cid:19) + e − Γ τ cos( θ ) × (cid:20) sin (cid:18) θ (cid:19) + cos( θ ) sin (cid:18) θ (cid:19)(cid:21) (cid:111) , (25 b ) ν ( t, τ ) = − V Γ sin( θ ) sin( θ ) cos( θ ) e − Γ τ (cid:0) − e − Γ t (cid:1) , (25 c )6 u ( τ ) = i θ ) (cid:104) − (cid:18) θ (cid:19) − θ ) sin (cid:18) θ (cid:19) (cid:105) e − Γ τ , (25 d ) w ( τ ) = − i θ ) sin( θ ) sin( θ ) e − Γ τ . (25 e )Note that these equations are valid for a complex local field parameter V , i.e., EID isincluded. However, in the following we will again concentrate on a real parameter V .Using the FWM phase φ (3)FWM = φ + φ − φ , the three-pulse FWM dynamics areobtained by the phase integration p (3)FWM ( t, τ ) = π (cid:90) p ( t, τ ) e − i ( φ + φ − φ ) d φ d φ d φ (2 π ) . (26)A flow chart of the single steps leading to the FWM signal is shown in Fig. 7. Analogousto Fig. 2 the first and second pulse create an occupation which consists of a partindependent of the pulse phases n +2 and a part dependent on the pulses’ phase difference φ − φ φ − φ n +2 . These two terms remain even for long delays because we assume that theexcited state decay is much slower than the dephasing. The polarization which consistsof φ p +2 , φ p +2 and φ − φ p +2 vanishes due to the long delay τ which is indicated by blackdotted arrows. As a consequence the last pulse creates polarizations exclusively fromthe occupations before this pulse. Therefore, the final polarization consists of a partwith the phase of the third pulse φ p +3 , indicated by the green wave from n +2 , and a part φ + φ − φ φ − φ + φ p +3 which adapts the phase difference from the occupation φ − φ φ − φ n +2 . There is alsothe possibility that the occupation does not receive an additional phase dependence andsimply becomes φ − φ φ − φ n +3 .Without the local field coupling, only φ − φ + φ p +3 contributes to the FWM signaland we see that it carries the information of the previous occupation. Through the localfield coupling, the polarization mixes with φ − φ φ − φ n +3 in the following field-free propagationwhich is marked by violet dotted arrows. This again results in Bessel functions describingthe dynamics of the FWM signal. In direct analogy to the discussion of two-pulse FWM,the minimal number of mixing processes involved determines the order of the Besselfunction. The possible mixing orders are again annotated on the violet arrows.Finally, in case of long delays τ (cid:29) /β , the FWM dynamics read p (3)FWM = e iϑ − βt [ wJ ( ν ) + iuJ ( ν ) − wJ ( ν )] . (27)The dynamics have a similar form as the analytical result from the two-pulse FWM inEq. (15). This is because the occupation n +2 has the same phase dependence as in thetwo-pulse case and lowest mixing orders are also 0, 1, and 2 for φ + φ − φ p +3 , φ p +3 , and φ − φ + φ p +3 , respectively.7 To discuss non-vanishing pulse durations ∆ t > , we simulate the FWM signalsnumerically with Gaussian laser pulses. We set all pulse areas to the same value θ = θ = θ and fix the delay between the first two pulses to τ = 0 . ps. The otherparameters agree with the two-pulse case, i.e., β = 0 . ps − , Γ = 0 , and ∆ t = 0 . ps.Beginning with small pulse areas, the FWM dynamics for θ = 0 . π without localfield interaction V = 0 and with a local field coupling of V = 6 ps − are shown inFigs. 8(a) and (b), respectively. For V = 0 in (a) the FWM signal decays in real time t as the FWM polarization dephases. The signal does not change when varying positivedelays τ because we did assume a vanishing decay rate and the signal is probing theoccupation dynamics. For negative delays however, the pulse with φ arrives first. Afterthis pulse only the polarization is phase-dependent and the next two pulses with φ and φ create the FWM signal. Therefore the FWM signal reflects the polarization dynamics.This quantity decays ∼ e − β | τ | due to the dephasing in the field-free propagation afterthe single pulse.With the local field interaction in Fig. 8(b), the signal changes qualitatively. Similarto two-pulse FWM, the maximum of the FWM dynamics is shifted to t > due to thewave-mixing in the free propagation after all three pulses. In addition the signal isstrongest for short delays τ (cid:28) /β . These aspects can be explained by calculating the fulldephasing , , , ... , , , ... , , , ... freepropagationlocal fieldinducedlaserinduced mixingorder Figure 7.
Flow chart of the generation of the three-pulse FWM signal. The delay τ is supposed to be long enough that all polarizations can fully dephase. The differentarrows have the same meaning as in Fig. 2. time t (ps) − d e l a y τ ( p s ) a V = 0 0 2 4 6 time t (ps) b V = 6 ps − . . . . . . n o r m . F W M a m p l. Figure 8.
Three-pulse FWM dynamics for small pulse areas θ = 0 . π . (a) Withouta local field coupling V = 0 and (b) with a local field coupling V = 6 ps − . FWM dynamics in the χ (3) -regime to p χ (3) FWM = κ e iV t − βt (cid:110) Θ( τ ) (cid:2) iV t (cid:0) e − βτ (cid:1)(cid:3) (28) + Θ( − τ ) e − β | τ | + iV | τ | (1 + 2 iV t ) (cid:111) , with κ = − iθ θ θ / . Independent of the sign of the delay, the energy is renormalized.Hence, the FWM spectrum consists of a single line at ω = ω − V (not shown).For positive delays the first term in the curly brackets in Eq. (28) contributes.There, the first contribution simply stems from the pure TLS without local fieldcoupling. The second contribution is created by the interaction with the local field andconsists of two parts. The first one does not depend on the delay and stems from themixing between the polarization after the third pulse φ p +3 and the occupation φ − φ n +3 .The second part decays with e − βτ and stems from the polarization φ p +3 which mixeswith φ − φ n +3 . As both of these quntities stem from the polarization after pulses 1and 2, they suffer from the dephasing in the field-free propagation after the pulse pair.Therefore this contribution decays with twice the dephasing rate.For negative delays the second term in the curly brackets in Eq. (28) contributes.The pulse with phase φ arrives first and after the delay τ , the two pulses with φ , φ excite almost simultaneously. Consequently the only contributing quantity after the firstexcitation is φ p +3 and the entire signal decays with e − β | τ | . The FWM dynamics consistof a contribution from the pure TLS and a part from the local field interaction. Thelatter consists of the mixing between φ p +3 and φ − φ n +3 and between φ p +3 and φ − φ n +3 resulting in the term ∼ V t .Going to larger pulse areas, the FWM spectra first change slightly. For anintermediate pulse area of θ = 0 . π in Fig. 9(a) the FWM spectrum is initiallysignificantly broadened and becomes narrower for larger delays but it does not evolveinto a single symmetric line and remains broader than the Lorentzian in the χ (3) -regime.A detailed discussion of the linewidth at large delays will be carried out later.At still higher pulse areas, e.g., θ = 0 . π depicted in Fig. 9(b), the FWM spectrachange significantly. Initially, the FWM spectrum is broad and evolves into two separatelines for large τ which is a striking difference to the previous picture. Both lines are9 − − energy ¯ h ( ω − ω ) (meV) − d e l a y τ ( p s ) a θ = 0 . π − − energy ¯ h ( ω − ω ) (meV) b θ = 0 . π . . . . . . n o r m . F W M a m p l. Figure 9.
Three-pulse FWM spectra for larger pulse areas. (a) θ = 0 . π and (b) θ = 0 . π . The local field coupling is V = 6 ps − . separated by a minimum of the signal at ω = ω . This characteristic behavior will beexplained in the context of coherent control later.For negative delays τ < the details of the FWM spectra change significantlywhen comparing Figs. 8(a) and (b) and become quite involved. However, in all casesthe intensity decays rapidly due to the dephasing as discussed for the χ (3) -regime.While the spectra are all broadened and quite involved for small delays τ ≈ psthey become very clean for long delays τ ≈ ps. The reason for this behavior is themultitude of different possible phase combinations for short delays and the vanishinginfluence of the polarizations for long delays as explained before. We find pulse areasthat result in a single or in two separate lines. This effect is further highlighted inFig. 10(a), where the FWM spectra for large delays τ (cid:29) /β are shown as a functionof the pulse area θ . For small pulse areas in the χ (3) -regime, the spectrum appears asa single line at ω = ω − V . Then the spectrum broadens for an increasing pulse area.At θ ≈ π/ , the spectrum splits into two parts with a pronounced minimum at ω = ω in agreement with Fig. 9(b). For θ ≈ π/ , the two lines merge again into a single peakat ω = ω . Increasing the pulse area further, the single line splits into two lines again.For the explanation of this behavior we consider the Bloch vector picture again.Figure 10(b) shows the schematic construction of the smallest and largest possibleoccupation after the three pulses and a full dephasing after pulse two. The first Rabirotations (blue and orange) can act destructively, i.e., the respective rotations of theBloch vector compensate each other. The state of the TLS after the second pulse thenpoints to the south pole. The third pulse rotates from there and reaches the finaloccupation N (3) − = sin (cid:18) θ (cid:19) , (29 a )which is the smallest possible. Note, that strictly speaking this phase combination leadsto a vanishing FWM signal because both, polarization and occupation are zero after thesecond pulse. But at this point we are just interested in the hypothetically reached shiftof the transition energy. In the other extreme case the two first pulses act constructivelyand the pulse areas essentially add up. Before the third pulse the state dephases entirely,0 − − energy ¯ h ( ω − ω ) (meV) . . . . pu l s e a r e a θ / π τ (cid:29) /β a N (3) − N (3)+ . . . . . . n o r m . F W M a m p l. largedelay b |g N + N – | x dephasinglaser pulsesocc. after3rd pulse1st pulse2nd pulse3rd pulseBloch vector (3)(3) Figure 10. (a) Three-pulse FWM spectra in the limit of longe delays τ (cid:29) /β .(b) Sketch of the Bloch vector to illustrate the generation of the smallest and largestpossible energy in the spectrum. which means that it is projected onto the z -axis. From there the third pulse rotates theBloch vector on a smaller circle and reaches the final occupation N (3)+ = sin (cid:18) θ (cid:19) + cos( θ ) sin (cid:18) θ + θ (cid:19) . (29 b )The effective transition energies corresponding to these two limiting cases are depictedin Fig. 10(a) as dashed black lines. We find that they again very accurately follow theboundaries of the calculated spectra. The slight deviation again stems from the pulsearea renormalization due to the local field coupling (see Appendix A).Also the reduction of the signal at ω = ω (seen most clearly for pulse areas around θ = 0 . π ) can be explained in this picture. We consider all possible combinations of φ and φ that make the Bloch vector point to the equator after the second pulse. Thefollowing dephasing ends in the center of the sphere, which is the balanced mixture ofground and excited state. In this point the system is called transparent because thethird pulse cannot create a polarization from there. Consequently also no FWM signalcan be generated.For the special case θ = 0 . π , where the spectral peaks in Fig. 10(a) merge at ω = ω , we remember that after the full dephasing for long delays τ all possibleBloch vectors lie on the z -axis of the Bloch sphere. From there the Bloch vector isrotated by the third pulse into the equatorial plane no matter where on the z -axis itwas. Therefore the only possible occupation is n +3 = 1 / which results in a renormalizedenergy of (cid:126) ω loc = 0 [Eq. (3)], i.e., a single line in the spectrum at ω = ω . Note, thatin this scenario the equator is reached after the third and not the second pulse, so theargument is not in conflict with the previous discussion of the spectral minimum at ω = ω .
6. Conclusions
Motivated by recent nonlinear FWM measurements on TMDC monolayers we revivedthe Bloch equation model extended by a local field effect. This effect takes exciton-1exciton interactions into account on a mean field level. After introducing analyticsolutions for the optical driving and the field-free propagation between two laser pulsesin the limit of ultrashort excitations we calculated the FWM signal after a two-pulseexcitation in this limit. This result was later used to explain the numerically simulatedspectral dynamics for excitations with laser pulses of non-vanishing duration. We foundthat the local field effect leads to oscillations in the FWM signal dynamics whichtranslate into a spectral broadening and even a line splitting for short pulse delays. Wehave explained these aspects with the pulses’ action on the TLS’s occupation, visualizedby means of the Bloch vector, which determines the FWM signal via the local field. Inthe case of three-pulse FWM signals, the occupation dynamics plays the central roleand we found that line splittings even persist for long delays between pulses two andthree. Utilizing the illustrative Bloch vector again, we showed that the variations of theFWM spectral dynamics strongly depend on the applied pulse areas.Overall, this work develops a basic understanding of nonlinear optical signals fromsystems where exciton-exciton interaction can be modeled by a local field approximation.Motivated by the first experiments on TMDCs indicating local field effects, this workproposes how the fundamental physics can be explored by investigating the spectraldynamics in different FWM scenarios. Especially promising is the utilization of the localfield model for situations where funneling effects lead to increased exciton densities andtherefore to pronounced exciton-exciton interaction.
Acknowledgments
T.H., P.M., T.K., and D.W. acknowledge support from the Polish National Agency forAcademic Exchange (NAWA) under an APM grant. T.H. thanks the German AcademicExchange Service (DAAD) for financial support (No. 57504619). D.W. thanks NAWAfor financial support within the ULAM program (No. PPN/ULM/2019/1/00064).
Appendix A. Rabi rotations for non-vanishing pulse durations
In the discussion of non-vanishing pulse durations, we have noted qualitatively that therotation angle of the Bloch vector is altered in the presence of a local field. In Fig. A1this behavior is depicted quantitatively for the parameters used in the simulations, i.e., β = 0 . ps − , Γ = 0 , and ∆ t = 0 . ps. In comparison to V = 0 (blue line), the rotationangle for θ < π is smaller when a local field is taken into account (red line). Thisdifference is largest for θ ≈ π/ . In addition, the assumed dephasing of the polarizationprohibits a full occupation of the excited state in both cases. Therefore, the reductionof the effective pulse area becomes eminent for the discussion of Figs. 6 and 9, where thespectral features for extended pulses lie slightly above the δ -pulse results. Both wouldagree when taking the effective pulse area into account.2 . . . . . . . . θ/π . . . . . . h | x ih x | i equatorial plane V = 0 V = 6ps − Figure A1.
Occupation after a single pulse without V = 0 (blue) and with local fieldinteraction V = 6 ps − (red). − d e l a y τ ( p s ) a θ = 0 . π b θ = 0 . π time t (ps) − d e l a y τ ( p s ) c θ = 0 . π − − energy ¯ h ( ω − ω ) (meV) d θ = 0 . π . . . . . . n o r m . F W M a m p l. . . . . . . n o r m . F W M a m p l. Figure B1.
FWM signals for δ -pulses. (a, b) θ = 0 . π and (c, d) θ = 0 . π . FWMdynamics in (a) and (c) and the respective FWM spectra as a function of energy (cid:126) ( ω − ω ) and delay τ in (b) and (d). The dashed line in (a) indicates the timedependence of the minima as predicted by Eq. (22). Appendix B. FWM signals for ultrashort pulses
In the main text we discuss FWM signals from non-vanishing pulse durations andinterpreted the signals with the help of calculations in the δ -pulse limit. To get aclearer view on the impact of the pulse duration, the FWM-dynamics and spectra fromFig. 5 are calculated for δ -pulses via Eqs. (15) and (18). Regarding the case of θ = 0 . π ,whose FWM dynamics and spectra are depicted in Figs. B1(a) and (b), respectively, theanalytical solution matches the numerically simulated signal almost perfectly for bothpositive and negative delays.For the larger pulse area θ = 0 . π the dynamics shown in Figs. B1(c) and (d)3 − d e l a y τ ( p s ) a θ = 0 . π b θ = 0 . π time t (ps) − d e l a y τ ( p s ) c θ = 0 . π − − energy ¯ h ( ω − ω ) (meV) d θ = 0 . π . . . . . . n o r m . F W M a m p l. . . . . . . n o r m . F W M a m p l. Figure B2.
FWM signals in the ultrashort pulse limit ias in Fig. B1 but includingexcitation-induced dephasing (EID). (a, b) θ = 0 . π and (c, d) θ = 0 . π . FWMdynamics in (a) and (c) and the respective FWM spectra as a function of energy (cid:126) ( ω − ω ) and delay τ in (b) and (d). The dashed line in (a) indicates the timedependence of the minima as predicted by Eq. (22). EID is described by an imaginarypart Im( V ) = − . − and β is adjusted to obtain the same linear dephasing as inFig. B1. exhibit more prominent differences with respect to Figs. 5(c) and (d). On the onehand, the oscillating signal follows the derived logarithmic behavior also for small delaysbecause the influence of the pulse overlap vanishes for δ -pulses. On the other hand, thetransition from positive to negative delays is very abrupt. Nevertheless, the qualitativeaspects of the spectra, i.e., width, line splitting, and narrowing are well-reproduced.This demonstrates that overall an interpretation of the spectra via the Bloch vectorretrieved in the δ -pulse limit is well justified.For all the results presented up to now we have assumed a real-valued local field,i.e., a real value of V . As discussed in the main text, an imaginary part can be addedto model the phenomenon of excitation-induced dephasing (EID). This gives rise toan additional exponential decay of the polarization depending on the pulse areas ofthe exciting pulses, as discussed in connection with Eq. (12). In Fig. B2 we showthe same two-pulse FWM signals as in Fig. B1, but now with EID described by animaginary part of Im( V ) = − . − . To obtain the same linear dephasing, we set β + Im( V ) = 0 . − . In the case of weak excitation θ = 0 . π in Figs. B2 (a) and (b),we observe a slightly enhanced decay both as a function of the real time t and the delaytime τ and correspondingly a slightly broader spectrum. However, the differences arerather small. This changes strongly when increasing the pulse area to θ = 0 . π in4Figs. B2 (c) and (d). Now the decay is indeed much faster and there is a considerableadditional broadening of the spectrum. The enhancement of the decay is particularlystrong in the case of negative delay times, which can be understood from the fact thenow the stronger pulse with θ = 0 . π arrives first and leads to a higher occupationbetween the pulses than in the case of positive delays.The results depicted in Fig. B2 also show that despite the faster decay of the signalsand the corresponding broadening of the spectra, the overall shape of the signals andspectra remains remarkably unaffected. In particular, the minima in the signals asa function of real time and delay time (panels (a) and (c) in Figs. B1 and B2) areessentially unchanged, except for a reduction in the contrast due to the faster decay.This confirms our motivation to neglect EID in the main text, where we concentratedon the characteristic temporal and spectral features introduced by a real local field.Let us briefly comment on the order of magnitude of the parameters chosen inour studies. Our value of V = 6 ps − leads to shifts in the spectra of the order of afew millielectronvolts. In Ref. [29], based on a microscopic theory, local fields of thisorder have been obtained for exciton densities of the order of cm − . According toRef. [43] the threshold for lasing in TMDC materials, which is related to the presenceof an inversion, is expected for carrier densities of the order of a few times cm − ,indicating that densities of several cm − are indeed already close to inversion. TheEID parameter has been calculated to be in the range of about 1 meV for densities ofthe order of cm − [33]. Therefore we conclude that our parameters have a realisticorder of magnitude for TMDC materials. References [1] Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C Y, Galli G and Wang F 2010
Nano Lett. Phys. Rev. Lett.
Phys. Rev Lett.
Phys.Rev. X ACS Nano ACS Appl.Electron. Mater. [7] Hao K, Specht J F, Nagler P, Xu L, Tran K, Singh A, Dass C K, Schüller C, Korn T, Richter M,Knorr A, Li X and Moody G 2017
Nat. Commun. [8] Boule C, Vaclavkova D, Bartos M, Nogajewski K, Zdražil L, Taniguchi T, Watanabe K, PotemskiM and Kasprzak J 2020 Phys. Rev. Mater. (3) 034001[9] Shah J 2013 Ultrafast spectroscopy of semiconductors and semiconductor nanostructures vol 115(Springer Science & Business Media)[10] Mukamel S 1995
Principles of Nonlinear Optical Spectroscopy (Oxford University Press)[11] Langbein W 2010
Riv. Nuovo Cimento Soc. Ital. Fis. [12] Stievater T H, Li X, Steel D G, Gammon D, Katzer D S and Park D 2002 Phys. Rev. B Science
Phys. Rev. Lett. (19) 2074–2077[15] Borri P, Langbein W, Hvam J M, Heinrichsdorff F, Mao M H and Bimberg D 2000 IEEE J. Sel.Top. Quantum Electron. Nature
Opt. Lett. Optical resonance and two-level atoms vol 28 (Courier Corporation)[19] Voss T, Rückmann I, Gutowski J, Axt V M and Kuhn T 2006
Phys. Rev. B Optica Materials for Quantum Technology Phys. Rev. B Phys. Rev. A (9) 5675–5683[24] Kim D S, Shah J, Damen T C, Schäfer W, Jahnke F, Schmitt-Rink S and Köhler K 1992 Phys.Rev. Lett. (18) 2725–2728[25] Mayer E J, Smith G O, Heuckeroth V, Kuhl J, Bott K, Schulze A, Meier T, Bennhardt D, KochS W, Thomas P, Hey R and Ploog K 1994 Phys. Rev. B (19) 14730–14733[26] Victor K, Axt V M, Bartels G, Stahl A, Bott K and Thomas P 1995 Z. Phys. B Phys. Rev. B Semiconductor Optics and Transport Phenomena (Springer BerlinHeidelberg)[29] Katsch F, Selig M and Knorr A 2019
2D Mater. Phys. Status Solidi B
Phys. Rev. A (4) 043402[32] Wang H, Ferrio K B, Steel D G, Berman P R, Hu Y Z, Binder R and Koch S W 1994 Phys. Rev.A R1551[33] Katsch F, Selig M and Knorr A 2020
Phys. Rev. Lett.
Phys. Rev. B (16) 165312[35] Haug H and Jauho A P 2008 Quantum kinetics in transport and optics of semiconductors vol 2(Springer)[36] Martin E W, Horng J, Ruth H G, Paik E, Wentzel M H, Deng H and Cundiff S T 2020
Phys. Rev.Appl. (2) 021002[37] Robert C, Lagarde D, Cadiz F, Wang G, Lassagne B, Amand T, Balocchi A, Renucci P, TongayS, Urbaszek B and Marie X 2016 Phys. Rev. B (20) 205423[38] Wigger D, Mermillod Q, Jakubczyk T, Fras F, Le-Denmat S, Reiter D E, Höfling S, Kamp M,Nogues G, Schneider C, Kuhn T and Kasprzak J 2017 Phys. Rev. B (16) 165311[39] Slepyan G Y, Magyarov A, Maksimenko S A, Hoffmann A and Bimberg D 2004 Phys. Rev. B (4) 045320[40] Wigger D, Schneider C, Gerhardt S, Kamp M, Höfling S, Kuhn T and Kasprzak J 2018 Optica Phys. Rev. Lett. (13) 2598–2601[42] Noordam L D, Duncan D I and Gallagher T F 1992 Phys. Rev. A (7) 4734–4737[43] Lohof F, Steinhoff A, Florian M, Lorke M, Erben D, Jahnke F and Gies C 2018 Nano Lett.19