Emergence of electron coherence and two-color all-optical switching in MoS2 based on spatial self-phase modulation

Significance Generating electron coherence is nontrivial in that most sophisticated electronic experimental methods are noncoherent or cannot be used to induce and detect collective states. By using coherent spatial self-phase modulation (SSPM) (a nonlinear optical property) we observed the emergence of electron coherence in a gapped quantum material, MoS2. By observing gap-dependent SSPM we discovered that it is a ubiquitous property of two-dimensional layered quantum materials. Furthermore, we demonstrate that this ac electron coherence can be harnessed to realize two-color all-optical switching with superb performance. Generating electron coherence in quantum materials is essential in optimal control of many-body interactions and correlations. In a multidomain system this signifies nonlocal coherence and emergence of collective phenomena, particularly in layered 2D quantum materials possessing novel electronic structures and high carrier mobilities. Here we report nonlocal ac electron coherence induced in dispersed MoS2 flake domains, using coherent spatial self-phase modulation (SSPM). The gap-dependent nonlinear dielectric susceptibility χ(3) measured is surprisingly large, where direct interband transition and two-photon SSPM are responsible for excitations above and below the bandgap, respectively. A wind-chime model is proposed to account for the emergence of the ac electron coherence. Furthermore, all-optical switching is achieved based on SSPM, especially with two-color intraband coherence, demonstrating that electron coherence generation is a ubiquitous property of layered quantum materials.

Generating electron coherence in quantum materials is essential in optimal control of many-body interactions and correlations. In a multidomain system this signifies nonlocal coherence and emergence of collective phenomena, particularly in layered 2D quantum materials possessing novel electronic structures and high carrier mobilities. Here we report nonlocal ac electron coherence induced in dispersed MoS 2 flake domains, using coherent spatial self-phase modulation (SSPM). The gap-dependent nonlinear dielectric susceptibility χ (3) measured is surprisingly large, where direct interband transition and two-photon SSPM are responsible for excitations above and below the bandgap, respectively. A windchime model is proposed to account for the emergence of the ac electron coherence. Furthermore, all-optical switching is achieved based on SSPM, especially with two-color intraband coherence, demonstrating that electron coherence generation is a ubiquitous property of layered quantum materials. electron coherence | transition metal dichalcogenide | self-phase modulation | optical switching | emergent phenomena R ecently 2D layered quantum materials have attracted tremendous interest since the discovery of graphene a decade ago (1). Various layered materials, ranging from boron nitride sheets to transition metal dichalcogenides and from topological insulators to high-temperature superconductors, have been intensively investigated (2)(3)(4)(5)(6)(7)(8)(9)(10)(11). Strict 2D atomic crystals can now be produced at a macroscopic scale, using a variety of methods (11,12). Among them molybdenum disulfide (MoS 2 ) and related layered quantum materials are particularly interesting due to their novel optical properties and potential valleytronics applications (4,5,8,9) at a thickness of monolayer and few layers (2,10,13). Layered materials share common physical properties rooted in their ubiquitous 2D quantum nature, for which achieving pure coherence among electrons (lattices) is of particular interest (14)(15)(16)(17)(18)(19). The presence of multiple domains is ubiquitous in many known 2D quantum materials, ranging from stripe-order cuprate superconductors to polycrystalline strongly correlated systems. For example, phase locking between different layers of stripe orders is crucial for enhancing the superconducting phase in layered hightemperature superconductors (20,21).
In this work we demonstrate unambiguously that nonlocal and intraband ac electron coherence, of which the electronic wave function oscillates at an optical frequency of 10 14 Hz, can be generated in separate MoS 2 flakes, using spatial self-phase modulation (SSPM). The SSPM is a coherent third-order nonlinear optical process systematically investigated decades ago (22), where the nonlinear optical susceptibility χ (3) is uniquely determined by the laser-intensity-dependent refractive index n = n 0 + n 2 I, where n 0 and n 2 are linear and nonlinear refractive indexes, respectively. If this effect is strong enough in a material, the phenomenon of selffocusing can be directly observed. The SSPM is also frequently referred to as the optical Kerr effect or the ac Kerr effect (note the difference from the regular Kerr effect). It is parallel to the other third-order nonlinear optical processes, such as third harmonic generation (THG) and four-wave mixing (FWM).
Our investigations show that gap-dependent SSPM is a general method for inducing electron coherence in 2D layered materials. Monolayer and few-layer MoS 2 (and other similar layered materials) have finite bandgaps, which lift the obstacle to versatile electronic and optical applications (2,23,24). Because SSPM using below-gap photons has not been reported, MoS 2 provides an ideal test platform for observing SSPM in finite-bandgap materials. A model has been developed to account for the coherence emergence process and theoretical calculations have been carried out to unravel the underlying mechanism. Moreover, we demonstrate all-optical switching based on SSPM, which employs intraband electron coherence. This optical switch has multiple advantages, including weak-control-strong performance, cascade-possible, and high-contrast two-color switching. 1 D-F we show the SSPM patterns obtained using 400-nm ultrafast laser pulses, 532-nm continuous wave (cw) laser beams, and 800-nm ultrafast laser pulses, respectively. The SSPM is a nonlinear optical process in that the phase (rather than the intensity) of the output beam depends on the input intensity (Methods).

Significance
Generating electron coherence is nontrivial in that most sophisticated electronic experimental methods are noncoherent or cannot be used to induce and detect collective states. By using coherent spatial self-phase modulation (SSPM) (a nonlinear optical property) we observed the emergence of electron coherence in a gapped quantum material, MoS 2 . By observing gap-dependent SSPM we discovered that it is a ubiquitous property of two-dimensional layered quantum materials. Furthermore, we demonstrate that this ac electron coherence can be harnessed to realize two-color all-optical switching with superb performance. To unambiguously verify that we have observed SSPM, we performed intensity-dependence measurements. Fig. 2 shows that the number of rings N increases nearly linearly with the laser intensity, until the saturation is reached. By this linear dependence the optical nonlinear coefficient n 2 , hence χ (3) , can be obtained directly ( Fig. 2 and Methods). Because nearly all of the light is diffracted off the incidence direction and the Rayleigh scattering is low, ideally it is a purely coherent third-order nonlinear optical process. This pure coherence must originate from the coherence within the sample-the electron coherence, where the electronic wave function preserves a definite phase (see discussion below). In the case that photoexcited carriers become out of phase due to collisions, impurity scatterings, and boundary reflections, they reassume their coherence transiently (at a timescale of femtoseconds) under the external light pulse.
Next we consider nonlocal electron coherence. In our sample, each flake is a domain. Charge carriers in any form (including photocarriers, excitons, free electron-hole pairs, and intrinsic electrons and holes) in different and the same domains are initially completely out of phase. Besides, each domain has an arbitrary orientation. Upon light interaction, photoinduced quasiparticles interact with the electric field ( Fig. 3A) and assume its local phases.
We propose a wind-chime model to describe the emergence of electron coherence from the nonlocal domains driven by SSPM (Fig. 3A). Initially there exists an arbitrary orientation angle θ between a MoS 2 flake (thus the electrical polarization it contains) and the electric field. Due to energy relaxation, the electric field reorients the flakes so that each domain contains an axis parallel to the polarization of the external field, as if each domain is "hung" by a vertical "thread." This image mimics that of a wind chime (Fig. 3A). Once the wind chime is formed, the electron coherence is completely set up within and among each of the different domains. This scenario depicts the emergence of nonlocal electron coherence as a complex collective behavior.
We first show that the model is reasonable. Assuming a circular disk shape for the MoS 2 flake domains, we derived (SI Text, section S1) that the rotation torque generated by the laser pulse M = R jP × Ej dV has a magnitude of where R is the disk radius, h is the disk thickness, and τ is the pulse width. Taking the Newton fluid with constant viscosity coefficient η and linear velocity v at the interface, the rotation torque V due to the viscous force is where Ω is the rotation velocity and ξ is the portion of the fluid globe that is rotating together with the disk. Hence the rotation angle accumulated due to each single light pulse is (SI Text, section S1) where T is the repetition period. The time needed for the pattern formation is (SI Text, section S1) where we have used the values « r = 3.33 for MoS 2 , η = 3.2 × 10 −4 Pa•s for acetone, R ∼ 1 μm, h ∼ 10 nm, I = 250 W/cm 2 , and ξ ∼ 0.03 in our experiment. As a result, T does not depend on T.
Thus, for both pulsed and cw excitations the results are identical, which has been verified in our experiment. To compare with the experiment, we recorded the formation processes of the ring patterns (see snapshots in  time for the ring formation is a strong experimental support to the wind-chime model. We then discuss the emergence of the ac electron coherence. Writing the electronic wave function at r A as being the local electron density, the phase ϕðr A Þ = k · r A − ω t + ϕ 0 ðr A Þ of the electronic wave function is completely determined by the external light field (Fig. 3A). The wave function oscillates at a frequency of ∼10 14 Hz (i.e., optical frequency). Classically, this is a forced oscillation, where electrons have to assume the enforced local phase by the external field, even surviving scatterings and collisions. Here ϕ 0 ðr A Þ represents the phase lag in response to the external field. It is not the initial random phase, which is wiped off by dissipation. The same applies to the electronic wave functions at r B in a different (or the same) flake. Because the two wave functions are both coherent with the light wave, they are coherent to each other too. This is a dynamic or ac electron coherence, which is different from the commonly known steady-state or dc electron coherence in transport measurements. The electrons do not move far away from their equilibrium positions. After the wind-chime formation the electron coherence becomes stronger because more electrons get correlated and fewer collisions get involved, owing to the aligned geometry. At the far field, the intensity is Iðr C Þ = E p ðr C Þ · Eðr C Þ, where E(r C ) is the electric field of the light wave at r C . It becomes Eðr C Þ = ςðχ ð3Þ ÞΨðr C Þ, with rB being the optical phase except for a normalization factor. Here k ⊥ = dΔϕ=dr is the wavevector generated due to the nonlinear response term of n 2 (Methods). The exact coefficient ςðχ ð3Þ Þ is determined by the optical nonlinearity χ (3) , which is comprehensively determined by the carrier properties in the material.
We found that SSPM can be used to measure easily the χ ð3Þ onelayer value for many of the layered quantum materials (we have also observed SSPM in dispersion of MoSe 2 flakes and obtained the χ (3) value). In other coherent methods (e.g., THG or FWM), the signal is much weaker than the input beam (25). Instead, in SSPM the diffracted beam is strong, owing to the intrinsic electron coherence. Significantly, SSPM can be used to measure χ (3) from near IR to UV, compared with THG working only for the region where the 3 ω signal can be easily detected. This is essential for materials with a relatively large gap. In this sense we have found a powerful and ubiquitous (in some cases unique) way to measure χ (3) for many of the 2D gapped quantum materials.
To quantitatively investigate the gap-dependent property, we obtained χ (3) values for excitations at multiple wavelengths (Fig. 4A). For example χ ð3Þ onelayer at 532 nm is 1.6 × 10 −9 electrostatic units (e.s.u.) (i.e., 2.23 × 10 −17 m 2 /V 2 ; Methods), which is orders of magnitude larger than that of conventional semiconductors, owing to the high carrier density and mobility in 2D layered materials. In parallel we calculated the absorption as a function of wavelength for few-layer MoS 2 , where the dipole transition is proportional to the occupation probabilities of valence and conduction bands. As shown in Fig. 4A, the experimental results and theoretical calculations compare well. A sharp change in χ (3) and threshold values can be seen at 1.72 eV (722 nm), which is exactly the gap value of MoS 2 . This gap-dependent property is a manifestation of the electronic band structure of MoS 2 . Thus, SSPM can be used to investigate band structures of layered quantum materials and can be conveniently compared with other experimental methods such as absorption and photoluminescence. The SSPM method applies especially to those materials where large single crystals are not available and absorption is hard to measure.
In Fig. 4A the experimental results compare well with the calculated absorption along x and y, but not that along the z direction (SI Text, section S2). This is strong evidence that, in our experiment, light propagation is perpendicular to the x or y but not the z axis, which confirms the validity of the wind-chime model. Note that this is an even stronger restriction than the wind-chime model. Considering the plasmonic interaction due to free carriers (26), it is indeed possible that the collective behavior leads to a "frozen" wind chime, where all domains become parallel to each other and perpendicular to the direction of laser propagation. The gapdependent nature displayed in Fig. 4 shows again that SSPM is mainly due to the various charge carriers. The below-gap SSPM has never been reported before. Three mechanisms might be responsible for it (Fig. 4 B and C): Ⓐ singlephoton SSPM with Fermi distribution smearing, Ⓑ two-photon SSPM, and Ⓒ phonon-assisted indirect-gap SSPM. For Ⓐ, it can be seen that Fermi smearing relevant for significant photoabsorption occurs only at zones near the H and K points in momentum space. For Ⓑ, the carriers generated through two-photon absorption build up an electrical polarization, which diffracts not only those photons above the gap, but also those below the gap (Fig. 4B). As such, even though it is a two-photon nonlinear process, one expects a linear power dependence relation because the change in refractive index is proportional to the light intensity I. Heuristically, the excited carriers diffract both excitation photons (i.e., two output photons). In comparison there is only one output photon in second harmonic generation and two-photon photoluminescence. Two-photon SSPM is the most likely mechanism responsible for below-gap SSPM. For Ⓒ, the indirect gap photon excitation requires assistant phonon injection (Fig. 4C), which usually has a very small probability. In-gap defect states are relatively sparse (resonant in-gap states are more sparse) and localized (whereby charge carriers cannot move freely and scattering occurs frequently, which ruins the acquired uniform phase), so they are unlikely a major reason for SSPM.
We further demonstrate that, in forming SSPM, free carriers generated by photons at one wavelength can diffract photons at another wavelength, as if they were generated by the latter photons. The experimental setup is illustrated in Fig. 5C, which demonstrates a two-color (473 nm and 532 nm) SSPM. When we fix the power of one beam (even below threshold) and increase that of the other, the ring number and diameter for each beam both increase simultaneously. The brightness decreases, with the total intensity remaining unchanged, and energy conservation is preserved (Fig. 5D). In Fig. 5B the ring number of the 473-nm pattern exhibits linear dependence on the sum of the two beam powers, although the intensity of the 473-nm beam is fixed. The same is true for the 532nm ring (SI Text, section S3). Once the coherence is correlated, each beam experiences the SSPM-as if the free carriers are created by the beam itself. Fig. 5A is an overall illustration of this property. This indistinguishability helps explain why in two-photon SSPM one can still have linear dependence and still obtain a relatively large χ (3) .
Significantly, our experiment is to our knowledge the first demonstration of all-optical switching based on SSPM. Because the switching beam is controlling the phase of the other one, a small change in intensity will result in the shift of the strong signal pattern in real space, where a detector is placed. The detector can be made to detect the whole rings. Thus, a weaker beam can switch on and off a stronger beam (27,28). Quantitatively taking the intensity of the signal beam (strong beam) at 473 nm to be I strong = 120 W/cm 2 , the intensity of the control beam (weak beam) at 532 nm needs only to be I weak = 2.0 W/cm 2 to change the phase of the strong beam by π (i.e., ΔN = 0.5, Fig. 2). Hence to make an all-optical switch with full contrast ratio, I weak :I strong = 1:60. This is characteristic of weak-control-strong all-optical switching, which is apparently cascade possible (27,28).
It is interesting that in the experiment intraband electron coherence has been correlated between photocarriers of 473-nm and 532-nm excitations. We note that similar coherence is also induced among the holes, the excitons, and the free electron-hole pairs. The two-color switching can further enable high-contrast-ratio operations, because in two-color operation optical filters can be used to block the signal beam. Two-color switching also makes it possible for multichannel mixed switching and thus enables very broad-band optoelectronic devices. We leave additional details of the SSPM-based all-optical switching to future investigations. In summary, our experiment demonstrates an all-optical switching based on SSPM, with the advantages including a weak-controlstrong, cascade-possible, high-contrast-ratio, room-temperature device; broadband functioning; condensed state; and particularly broad-band-accessible quantum materials.

Conclusions
In conclusion, nonlocal ac electron coherence among different flake domains of MoS 2 has been optically correlated based on SSPM. A wind-chime model is proposed to account for this emergent collective behavior, which is verified by experimental observation of the pattern formation time. Significantly, gapdependent SSPM has been observed, where the χ (3) value and the threshold I th both show abrupt changes across the MoS 2 bandgap. Mechanisms including two-photon SSPM have been discussed. Furthermore, we have experimentally demonstrated alloptical switching based on SSPM, which employs the intraband electron coherence. Our investigation signifies a ubiquitous property of 2D layered quantum materials and extends their potentials in optoelectronic applications by realizing all-optical switching.

Methods
Ultrafast and cw Experiment of SSPM in MoS 2 . Multiple ultrafast and cw laser beams were used to investigate the SSPM from suspended MoS 2 flakes. In the ultrafast experiment, 800-nm light pulses of 200-fs pulse width and 80-MHz repetition rate were focused onto the sample. The solvent is acetone and the container is a 1-cm-thick cuvette. The distance between the focusing lens (f = 200 mm) and the center of the cuvette is kept fixed at 150 mm. The beam spot size after focusing is 0.2 mm for the 1/e 2 intensity radius. In the cw experiment, single transverse-mode 532-nm and 473-nm laser beams were used instead, having a similar beam spot size on the sample. The relation between the fringe thickness and the flake concentration is investigated and discussed in SI Text, section S4. Micrometer-sized powder of MoS 2 crystals (LSKYD, www.kydmaterials. com/en/index.php) was added into the solvent and then ultrasonicated for 30 min before the measurement. The optimum concentration is 0.14 g/L (i.e., 8.5 × 10 −4 mol/L). From our Raman and absorption characterization there are very few monolayer MoS 2 flakes in our sample (SI Text, section S5). From Fig. 1B, it can be seen that our sample is mainly composed of MoS 2 flakes with a thickness of 30 ∼ 55 layers. Being different from the linear optical phenomena of Newton's rings, the SSPM pattern has a finite number of fringes and the thickest fringe has the largest diameter. The critical proof of observing a SSPM pattern is that the number of rings, N, increases linearly with the input laser intensity.
Fundamentals of SSPM and the Calculation of the χ (3) onelayer in MoS 2 . The nonlinear optical response SSPM is characterized by n = n 0 + n 2 I, where n 0 and n 2 are the linear and nonlinear refractive indexes, respectively. Assuming a Gaussian beam, the optical phase accumulated after propagation length l is ΔψðrÞ = ð2πn 0 =λÞ R l=2 −l=2 n 2 Iðr, zÞdz, where z is the propagation direction of the incident laser beam. Thus, we have Δψ(0) − Δψ(∞) = (2πn 0 /λ)n 2 ·2I·l, with I(0, z) = 2I. The number of rings N is proportional to the phase change in the output beam (Δψ(0) − Δψ(∞) = 2πN ), leading to n 2 = (λ/2n 0 l)(N/I), where n 0 = 1.36 and N is proportional to I for a nearly constant n 2 . By the slope in Fig. 2 we can straightforwardly obtain n 2 from the experimental data. With n 2 = ð12π 2 =ðn 2 0 cÞÞ10 3 χ ð3Þ (in the SI unit) we estimate that χ (3) = 1.44 × 10 −4 e.s.u. for the 532-nm cw laser beam excitation. The number of effective single layers, M, that the laser beam passes through is estimated to be about 300, using the concentration of the sample (SI Text, section S6). As I total ∼ M 2 I onelayer , we estimate that χ Conveyance of the ac Electron Coherence to the Optical Coherence. The diffraction of the beam off the z axis is due to a finite perpendicular wavevector, which is, for a Gaussian beam ΔψðrÞ = Δψð0Þexpð−2r 2 =a 2 Þ, explicitly written as k ⊥ = dΔψðrÞ=dr = ð−16πn 0 n 2 Il=a 2 λÞr · expð−2r 2 =a 2 Þ. After the wind chime is formed, the optical coherence forming the fringes is conveyed from the electron coherence: i .
Here we have taken that ϕ 0 is identical for r A and r B , and only identical k ┴ can produce interference rings.