S emiconducting van der Waals (vdW) materials, e.g., transition metal dichalcogenides (TMDCs), have emerged as a new material platform to explore various quantum degrees of freedom including spins and pseudospins (valley and layer index) for information processing. [1][2][3] Defect-bound singlephoton emitters have been identified in TMDC monolayers and heterostructures. [4][5][6][7][8][9] Moreover, they are highly sensitive to mechanical strain, 10,11 suggesting that TMDCs are an interesting platform for studies of coupling between phonons with spins and photons in the coherent regime. To achieve this goal, one also needs to understand decoherence associated with different phonon modes. Although spontaneous Raman spectra have been routinely used to characterize layer thickness, chemical composition variations, and strain in TMDCs, [12][13][14] little effort has been devoted to understanding phonon coherence.

Optical phonons couple to light efficiently and subsequently decay to generate acoustic phonons. It is well-known that coherent phonons can be generated by a pulsed laser with a sufficient bandwidth, i.e. exceeding the phonon oscillation frequency [15][16][17][18] in a stimulated Raman process. However, it is not widely appreciated in the 2D material community that spontaneous Raman scattering also represents a coherent lightmatter interaction process within the phonon coherence time. 19,20 This spontaneous Raman process has been used to generate the motional entanglement between vibrational states of two millimeter-sized diamonds at room temperature. 21 This scheme is just one specific example of the widely applied longdistance quantum communication protocol with linear optics. 22 To extend this approach of entangling vibrational states to new materials such as TMDCs, one needs to search for phonon modes with a long dephasing time.

In this work, we investigate intrinsic phonon dephasing dynamics in bulk and monolayer MoS 2 using a unique spectroscopy technique, transient coherent ultrafast phonon spectroscopy (TCUPS). 23 There are two commonly observed optical phonon modes, A 1g and E 2g mode in TMDCs. 12,[24][25][26][27][28][29][30] Although stimulated Raman scattering experiments have been used to probe the coherent dynamics of the A 1g mode previously, the E 2g mode cannot be excited in those nonlinear spectroscopy experiments [31][32][33] because of the displacive phonon excitation mechanism and selection rules constrained by symmetry. 34 In contrast, TCUPS relies on the spontaneous Raman scattering signal generated by a pair of phase-locked laser pulses to detect the E 2g mode. By recording how the interference contrast decays as a function of the delay time between the two excitation pulses, our experiments reveal intrinsic phonon dephasing dynamics of the E 2g mode in the MoS 2 monolayer. We find that the phonon dephasing time of the E 2g mode is significantly longer than that of the A 1g mode. Our first-principles calculations suggest that the phonon dephasing involves two types of three-phonon processes in which a pair of acoustic phonons with opposite momentum are generated. The knowledge of phonon decoherence provides valuable guidance for designing phonon-photon and phononspin interfaces in layered materials.

The key concepts of TCUPS are illustrated in Figure 1. In contrast to the conventional Raman spectroscopy using a narrow-band continuous-wave (CW) laser, the excitation source used in TCUPS is a pair of phase-locked ultrafast laser pulses prepared using a Michelson interferometer as illustrated in Figure 1a (see the Supporting Information section 1 for details). The probability of each excitation laser pulse generating a Stokes-shifted photon and optical phonon is significantly less than one. If the time delay of the laser pulses is within the coherence time of the excited phonon, the time-integrated detection cannot provide information about which pulse generates the detected Stokes photon. In analogy to the double-slit experiment, the detected Raman signal is a superposition of Stokes photons generated by the laser pulse pair. This lack of "which-pulse" information leads to the interference pattern recorded in the subsingle-photon limit. 35,36 Following the excitation and sending the generated Stokes signal through a spectrometer, a spectrum is measured using a charge-coupled device camera, as illustrated in Figure 1b. In the frequency domain, the Stokes photons from one pulse and the incident laser pulses have very similar spectra, except that the Stokes photons are shifted toward lower energy by the phonon energy. When a Stokes photon from a pair of pulses is measured, the time-integrated interference spectrum can be written as

where E total is the total electric field, ω is the measured frequency, E Si (ω) is the Stokes electric field (i = 1, 2), and τ is the time delay between the pulse pair; ⟨. . .⟩ represents the shot-to-shot average, and Γ is the reverse of coherence time. 23 The interference is related to the phonon coherence due to the definite phase relation between the excitation laser, the phonon oscillations, and the Stokes photons, a feature that has been explored in a limited number of previous experiments. 19,23,35,37 As the delay τ increases, both the fringe contrast and spacing (2π/τ) decreases, as described by e -Γ|τ| and cos(ωτ) in the interference term in eq 1, illustrated in Figure 1b. The decay of the fringe contrast as a We first take a conventional Raman spectrum with excitation wavelength at 532 nm from a MoS 2 monolayer prepared by exfoliation method at room temperature as shown in Figure 2a. The spectrum features two prominent optical phonon modes, A 1g and E 2g modes at 404 and 384 cm -1 , corresponding to the out-of-plane and in-plane vibrational modes, respectively. Although the line width of the A 1g mode is measured as 1.85 cm -1 , corresponding to a phonon dephasing time 5.73 ps, the line widths of the E 2g mode approach the instrument response function of the spectrometer. Thus, we are unable to extract reliable information about phonon dynamics from the conventional Raman method.

TCUPS spectra from a MoS 2 monolayer are shown in Figure 2b. The blue curve is the sum of spectra generated by the pump pulses individually, i.e., |E S1 (ω)| 2 + |E S2 (ω)| 2 , whereas the red curve is the interferogram observed upon excitation by a pair of phase-locked pulses. The fringe spacing Δλ is determined by the delay between the two pulses and can be derived from the last term in eq 1 as Δλ = λ 2 /cτ, where λ = 2πc/ω is the free space wavelength and c is the speed of light. We tune the center wavelength of the excitation laser pulse to be 793 nm, which is below the bandgap and exciton resonance in MoS 2 to avoid exciting free carriers or electron-hole pairs. By doing so, we largely eliminate electron-phonon and exciton-phonon interaction as dephasing channels. [38][39][40] Thus, the phonon dephasing time measured here is mostly determined by intrinsic three-phonon processes due to the anharmonic coupling as we further discuss later. By choosing proper polarization, one can isolate either E 2g or A 1g modes or excite both modes simultaneously. 41,42 We focus on the E 2g mode in the current study because its longer dephasing time makes it more interesting for quantum applications. In addition, this mode is inaccessible by the ultrafast pump-probe technique.

To explore the dynamics of the E 2g mode and to exclude the contribution from the A 1g mode in the spectra, we use a linear polarizer in the detection path orthogonal to the incident laser polarization. [43][44][45] The A 1g phonon Stokes signal is thus removed because its polarization is parallel to that of the incident laser. However, the E 2g phonon Stokes signal is partially retained because the E 2g mode is doubly degenerate, with both parallel and perpendicular polarization components with respect to the incident laser polarization. We note that additional phonon modes may contribute to Raman signal when the excitation wavelength is changed to 793 nm in TCUPS measurement. 46,47 These modes either have weak intensity or have very fast dephasing rate. Thus, they have minimal contribution to the interference fringes and the extracted decay time.

The fringe contrast, or the visibility of the spectral interference, quantifies the degree of phase coherence of the excited phonon mode. A series of TCUPS spectra at various delays are taken. The interference term displayed in Figure 2c is isolated by subtracting the sum of the Stokes spectra excited by each individual pump pulse from the interferogram taken with the phase-locked pulse pair. As the time delay between the two pulses increases, the visibility of the spectral interferogram shown in Figure 2c decreases (from bottom to top), offering a direct measurement of the phonon coherence time. See SI section 3 for detailed data analysis.

We plot the amplitude of the interference term as a function of delay τ and fit the data with a single exponential function in Figure 3a (see SI section 3-4 for detailed data analysis). We assume that the dephasing arises entirely from phonon population decay and that pure dephasing processes are absent. The measured coherence time is inversely proportional to the phonon line width measured in the frequency domain. Our measurements yield an E 2g phonon dephasing time of 7.1 ± 0.7 ps or a line width of 1.5 ± 0.1 cm -1 in a MoS 2 monolayer at room temperature, consistent with previous CW Raman measurements. 48,49 We repeat these measurements on both a suspended monolayer and a bulk crystal (Figure 3b). The dephasing time of the E 2g does not change appreciably in a suspended monolayer and increases slightly in the bulk to 8.6 ± 0.8 ps (1.3 ± 0.1 cm -1 ) at room temperature. The similar dephasing time in these samples may originate from the nature of in-plane relative motions of the Mo and S atoms associated with the E 2g mode.

Having established that a monolayer and a bulk MoS 2 have similar coherence times, we focus on the temperature-dependent measurements on the bulk sample due to its better signal-tonoise ratio. In addition to a systematic temperature-dependent frequency shift of the E 2g mode observed by using conventional Raman spectroscopy, we also observe a systematic temperaturedependent change in phonon coherence time in TCUPS experiments, with a longer phonon dephasing time (narrower line width) 20.8 ps (0.51 cm -1 ) at 15 K. The retrieved temperature-dependent line width from dephasing time is shown in Figure 3c. Previous Raman experiments were not able to capture the temperature dependence of the E 2g phonon line width, likely due to the limited spectral resolution (∼1 cm -1 ) of commonly accessible spectrometers. The lack of a clear temperature-dependent phonon line width was used as an evidence for impurity-scattering dominated phonon dephasing. 50 In contrast, our TCUPS experiments reveal that the dephasing time of the E 2g mode is significantly longer at low temperature than previously thought possible. 51 The temperature dependent frequency shifts and line widths can be well captured by Klemens model (curved lines in Figure 3c), suggesting that a three-phonon process is the leading mechanism for phonon dephasing (see SI section 5 for details).

To reveal the microscopic processes of the phonon dephasing, we further perform first-principles density functional theory (DFT) calculations of a bulk MoS 2 . The calculated phonon dispersion is shown in Figure 4a. The phonon dephasing rates are determined by two factors, i.e., the availability of the scattering phase space governed by conservation rules, and the coupling strength between phonon modes characterized by anharmonicity (see details in SI section 6). Anharmonicity does not change with temperature, but the scattering phase space, where the phonon population is a factor, is temperaturedependent. As can be seen in Figure 4b, the scattering phase space increases as a function temperature, which agrees with the increasing trend of the measured line width in Figure 3c. We further divide the scattering processes into two types based on the final states: 2L and 1H1L processes. Here, the low frequency phonon modes (L) are the modes below the phonon bandgap (between 233 and 275 cm -1 ) and the high frequency phonon modes (H) are those above this bandgap. In the 2L process (red dash-dot line in Figure 4a), one optical phonon decays into two low-frequency acoustic phonons. The 2L phonon decay process is reminiscent of the spontaneous parametric down-conversion process for the generation of correlated photon pairs via a second-order nonlinear optical process. We note that the 2L process does not necessarily require the two low-frequency phonons to have the same energy as constrained in the Klemens model. In fact, E 2g phonon decaying into two low-frequency phonons with the same energy is found negligible in the 2L process. In the 1H1L process (blue dashed arrow lines in Figure 4a), the E 2g mode decays into one high-frequency and one lowfrequency phonon.

The calculated phase space in Figure 4b provides an insight into the different dephasing process between A 1g and E 2g mode. First, the smaller scattering phase space of E 2g mode can explain its smaller scattering rates than that of A 1g mode in a bulk MoS 2 at all temperatures (see temperature-dependent dephasing time of A 1g mode determined by conventional Raman spectroscopy in SI section 8). We note that a small phase space difference can lead to a large lifetime change. 52 In addition, the scattering phasing space indicates that the dominant decay channels for A 1g and E 2g modes are different. From Figure 4b, we see that at all temperatures, the 2L process is the dominant decay channel for the E 2g mode, whereas the 1H1L process is negligible. In contrast, for the A 1g mode, the 1H1L process contributes significantly to the overall scattering. The different decay channel between E 2g and A 1g modes is mainly due to the energy conservation rule. For the relatively low frequency E 2g mode in MoS 2 , the number of possible H modes is very limited, and therefore, the E 2g mode mainly decays into two different L modes. A 1g mode has a higher frequency, which makes it more likely for this mode to decay through the 1H1L process.

In conclusion, by using the TCUPS method and firstprinciples calculation, we revealed a long dephasing time associated with the E 2g phonon in MoS 2. The dephasing time is measured to be ∼7 ps in both a monolayer and a bulk crystal at room temperature and extends to ∼21 ps at 15 K in a bulk crystal. This dephasing time is comparable to single-photon emitters identified in TMD monolayers, [4][5][6][7][8][9] suggesting phonon-exciton coupling as one of the main dephasing channels for those single-photon emitters. In addition, we show it is possible to generate collective excitations in 2D materials with lifetimes similar to that found in diamond. 23 Thus, our experiments suggest that the E 2g phonon can be used for generating the motional entanglement between vibrational states of two macroscopic 2D materials, 21 an interesting application in quantum information science. In the case that circularly polarized light is used, the Raman signal associated with the E 2g mode completely reverses the helicity of the incident photon. 41 In TMD materials, such chiral phonons at the center and corners of the honeycomb lattices may find unique applications in valleytronics. 42,53 The TCUPS technique we employ here has unique advantages in studying the coherence of all Raman-active phonon modes. Future studies may investigate the strong coupling regime between phonon-excitons or phonon-spins in vdW materials. One can envision a hybrid quantum system in vdW materials by taking advantage of coherent coupling between these degrees of freedom.