Quantum Beats between Spin-Singlet and Spin-Triplet Interlayer Exciton Transitions in WSe2–MoSe2 Heterobilayers

The long-lived interlayer excitons (IXs) of semiconducting transition metal dichalcogenide heterobilayers are prime candidates for developing various optoelectronic and valleytronic devices. Their photophysical properties, including fine structure, have been the focus of recent studies, and the presence of two spin states, namely, spin-singlet and spin-triplet, has been experimentally confirmed. However, the existence of the interaction between these states and their nature remains unknown to date. Here, we demonstrate the presence of coherent coupling between the spin-singlet and spin-triplet IXs of a WSe2–MoSe2 heterobilayer utilizing quantum beat spectroscopy via a home-built Michelson interferometer. As a clear signature of coherent coupling, the quantum beat signal has been observed for the first time between closely spaced transitions of IXs. The observed strong damping of the quantum beat signals with fast dephasing times of 270–400 fs indicates that fluctuations giving rise to inhomogeneous broadening in the photoluminescence emission of these states are uncorrelated.

inside the cryostat was used to conduct magneto-PL studies in the range of 0 T to 9 T in Faraday geometry.The dephasing time measurements are done by sending the emission through the homebuilt Michelson interferometer using a single-mode fiber.According to the interlayer exciton PL characteristics, the PL emission was filtered either by an 850 nm long-pass filter (Thorlabs FELH0850) for the quantum beating interferogram, or 880 nm (Thorlabs FBH880) and 905 nm hard coated (FBH905-10) band-pass filters for individual spin singlet and triplet PL interferograms, respectively.

Quantum Beat Spectroscopy via a Michelson Interferometer
To measure the dephasing time T2 of individual IX states and observe the quantum beating between these states, a home-built Michelson interferometer with two moveable retro mirrors was constructed.Both the retro mirror on the DC motorized stage (Thorlabs DDS050) with a step size of 5 μm and the retro mirror on the piezo motor stage (Thorlabs PIA13) with a step size of 20 nm could be controlled using a LabVIEW program via appropriate control boxes (Thorlabs KBD101 and TIM101).First-order correlation function g (1) () measurements were done by using the piezo stage that can resolve the fringes and the data collection was done by using PicoQuant HydraHarp 400 along with a silicon single photon avalanche diode (SPAD) with a photon timing resolution of 50 ps (Micro Photon Devices (MPD)-PDM series).Using the maxima and minima of various fringes, the visibility as a function of delay time was obtained.Due to the inhomogeneous decay from IXT and IXS, a Gaussian fit g (1) () ~ exp (− 2 / (  2 √/2 ) 2 was used to fit the data and their T2 values were obtained.For the quantum beating interferogram, the decay fit and T2 values were obtained by using g (1) () = I0(1 + Aexp (-||/ T2)).

Supplementary Note 1: Temperature dependence of the PL spectrum of IX spin states
Figure S1a shows the temperature dependence of the IXT and IXS PL emission lines.As the temperature increases, the high-energy IXS state is thermally populated and gets more pronounced at high temperatures.Also, both emission peaks show a redshift with the increased temperature.Similar trends were also observed in the previous studies [3][4][5][6] .In Figure S1b, we plot the extracted PL intensity ratio of IXS to IXT at various temperatures.As can be seen from the figure, this ratio increases with increasing temperature and takes the value of ~1 at 90 K. shows the best fit of the decay that corresponds to the dephasing time for the spin-singlet and spintriplet IX coherence.The fit is obtained by using the expression g (1) () = I0(1 + A exp (-||/T2)).
The dashed lines show the error interval when fitting the beat interferogram via the above expression, which is given by ± 0.05 ps, ± 0.04 ps, ± 0.03 ps, ± 0.03 ps, and ± 0.03 ps, from 90 K to 3.5 K, respectively.shows the best fit of the decay that corresponds to the dephasing time for the spin-singlet and spintriplet IX coherence.The fit is obtained by using the expression g (1) () = I0(1 + A exp (-||/T2)).
The dashed lines show the error interval when fitting the beat interferogram via the above expression, which is given by ± 0.03 ps, ± 0.03 ps, ± 0.04 ps, ± 0.03 ps, and ± 0.03 ps, from 90 K to 3.5 K, respectively.dephasing time and T1 is the radiative lifetime of the IXs.As can be seen from Figure S7, the radiative lifetimes of IXT and IXS at 90 K (~600 -700 ps) are almost three orders of magnitude larger than their dephasing times (~300 -400 fs), meaning that contributions from the radiative lifetimes to the spectral line width are negligible.Therefore, from the above relation, approximation of T 2 * ~ T 2 can be made and used in the spectral line width calculation.The calculated linewidth of coupled state is given by:  Red and blue-lined Gaussian fits show the individual peak fits of IXT and IXS, respectively.Data are recorded under 532 nm CW laser excitation with an incident power of 400 µW.

Figure S1 :
Figure S1: Temperature dependence of spin-singlet and spin-triplet IX PL emission.(a) Temperature-dependent PL spectra of the spin states of IXs.(b) Temperature dependence of the PL intensity ratio of spin-singlet to spin-triplet states of the IXs.Data are recorded under a 532 nm CW laser with 400 µW pump power.

Figure S3 :
Figure S3: Temperature dependence of the quantum beat interferometry of spin-singlet and spin-triplet IXs of WSe2-MoSe2 heterobilayers under 730 nm laser excitation.(a) Temperature-dependent PL spectra of the IXT and IXS recorded under 100 µW pump power.(b) Temperature-dependent quantum beating interferograms of the IXT and IXS, produced by sending both emissions simultaneously into the Michelson interferometer.The red-lined beat envelope .582 × 10 −16 .)√82 0.29 × 10 −12  = 9.47  The line widths of the IXS and IXT can be calculated from their experimentally measured individual dephasing times.As indicated in the main text, their dephasing times are given by 0.535 ps and 0.575 ps, respectively.Using them in the above line width equation, their homogenous line widths can be calculated as follows.582 × 10 −16 .)√82 0.575 × 10 −12  = 4.77 meV Γ IX S−T ~ Γ IX S + Γ IX T → 9.47 meV ~ 9.89 meV The calculated homogeneous linewidth of coupled state 9.47 meV is almost equal to the sum of the homogeneous linewidths of IXT (4.77 meV) and IXS (5.13 meV) and indicates that spectral fluctuations giving rise to inhomogeneous broadening in each line widths are uncorrelated 10 .The strong damping of the quantum beats in Figure 4b of the main text can be attributed to these uncorrelated fluctuations similar to the one observed for the inhomogenously broadened exciton and trion spectra of MoSe2 in four-wave mixing experiments 11 .Figure S8 displays the PL spectral line widths of the IXS (9.97 meV) and IXT (11.64 meV) obtained from the time-integrated PL spectrum.It clearly shows that inhomogeneous broadening contributes to the PL line widths and can be attributed to the charge fluctuations in the vicinity of the IXs.

Figure S8 :
Figure S8: PL spectrum and spectral line widths of spin-singlet and spin-triplet IXs at 90 K.

Supplementary Note 7 :
Calculation of the energy splitting between the IXT and IXS via measured beat periodThe energy splitting between IXT and IXS states can be calculated using the expression ∆E IX S−T = 2ℏ/T IX S−T .Here, T IX S−T is the average quantum beat period and can be directly determined from the difference between the fringe maxima or minima in the quantum beat interferogram.We determined the average beat period as T IX S−T = 195 ± 17 fs, which can be used to calculate the energy splitting as follows.The calculated energy difference and the energy difference obtained directly from the time integrated PL spectrum can be compared to further confirm the nature of the coupling between the IXS and IXT.FigureS9shows the PL spectrum of the both spin states of the IX at 90 K.The separation between the centers of the IXT and IXS PL peaks is given by ∆E IX S−T ~ 23.13 meV.It is evident that the calculated and measured energy separation values are in close agreement, confirming the coherent nature of the coupling between these IX states.

Figure S9 :
Figure S9: PL spectrum of the spin states of IX and the experimental energy separation between them at 90 K. Data are recorded under 532 nm CW laser excitation with an incident power of 400 µW.