A tunable time-resolved spontaneous Raman spectroscopy setup for probing ultrafast collective excitation and quasiparticle dynamics in quantum materials

The last decade has seen a surge of experiments where light is exploited as a strong external stimulus to manipulate quantum materials.^1–3 The light-matter interaction can be a fully coherent process as in the case of the Floquet state in photon-dressed materials,⁴ or drive quantum materials into a strongly non-thermodynamic state by disturbance of the balance between electronic, orbital, spin, and lattice degrees of freedom. Seminal cases of the latter phenomenon include the photo-induced insulator-metal transition in VO₂,⁵ optical control of colossal magnetoresistivity in manganites,⁶ and transient signatures of photo-induced superconductivity in a stripe-ordered cuprate.⁷ 

Integral in the description of the emanating dynamics in the optically driven non-equilibrium state is the creation and annihilation of electronic quasiparticles, collective excitations associated with the various degrees of freedom and their interaction. Their transient annihilation and creation may, for instance, induce symmetry changes across photo-induced phase transitions⁸ and dictates nonequilibrium temperatures,⁹ while their interaction underlies the coupling, and therefore the equilibration between the various degrees of freedom.¹⁰ A scala of partially complementary pump-probe techniques have been developed to study quasiparticle and collective excitation dynamics, and their interaction. For instance, the coupling between low-energy collective excitations and high-energy electronic excitations may be mapped into the time-domain and quantified, through all-optical coherent fluctuation spectroscopy.^11,12 A direct view on momentum-dependent electron and hole quasiparticle population dynamics is provided by means of the matured technique of time-resolved angle-resolved photoemission spectroscopy.^8,13 The situation for direct probing of collective excitation populations is perhaps more ominous.¹⁴ Rather novel photon-probes of dynamics of collective excitations such as phonons, magnons, and orbital excitations are time-resolved diffuse x-ray scattering¹⁵ and time-resolved resonant inelastic x-ray scattering.¹⁶ Plasmon dynamics in quantum materials has, for instance, successfully been probed with femtosecond electron energy loss spectroscopy.^17,18 However, a quantification of occupation numbers and corresponding effective temperatures with these techniques still seems a stretch.

Time-resolved Raman scattering is a reasonably well-established probe for vibrational dynamics in organic materials,^19–23 and phonon population dynamics in semiconductor materials^24–26 and carbon allotropes.^27–30 With this technique, the transient evolution of the spontaneous Raman spectrum after photo-excitation is studied.³¹ The pump excitation mechanism only needs to induce incoherent dynamics. There are thus no special prerequisites to the pulse duration, as is the case with coherent pump-probe techniques.^11,12 In the time-domain spectra, transient changes in excitation energies, line-widths, and scattering intensities of Raman-active excitations can be obtained. This has the advantage over coherent excitation techniques that one can follow the true time evolution of the system through its incoherent response, rather than the time evolution of coherent excitations. Through detailed balance of the anti-Stokes to Stokes scattering intensity ratio, the temporal evolution of selected effective mode temperatures can be directly determined,^31,32 as in contrast to techniques relying on a comparison of the dynamical response with the thermodynamic temperature evolution. This makes time-resolved spontaneous Raman spectroscopy a truly unique technique in the study of ultrafast processes and, for instance, allows testing of the validity and limitations of widely applied multiple effective temperature models.^9,33 An interesting situation is expected on the shortest time-scales, where truly non-thermal dynamics is present and the fluctuation dissipation theorem may not hold anymore. Phase transitions can be optically induced when a quantum material is driven far out-of-equilibrium. The associated symmetry changes across the phase transition can be deduced from the changes in selection rules of the transient Raman spectrum.³⁵ Since the transient excitation dynamics is directly measured in the frequency domain, the renormalization of excitation energies and linewidths across the phase transition (or partial melting of a phase) may be measured with large accuracy. This should be compared and contrasted with time-domain techniques which rely on the detection of a frequency chirp in the coherent excitation dynamics.³⁴ 

Major factors to the limited use of time-resolved spontaneous Raman spectroscopy in the study of ultrafast dynamics in quantum materials are energy-time resolution limitations, the relevant excitation energy scales,³⁷ and low inelastic light scattering cross sections. In time-resolved Raman spectroscopy, the time resolution and spectral resolving power are Fourier transform related. This asks for a proper choice of design parameters as we recapitulate in this article. The Fourier transform limit can, in principle, be overcome with the femtosecond stimulated variant of the technique.³⁸ A recent success with time-resolved femtosecond stimulated Raman spectroscopy (tr-FSRS) is the observation of a modification of the Heisenberg exchange in an antiferromagnet on the femtosecond timescale.³⁹ In FSRS, the stimulated gain and loss intensities both have contributions from anti-Stokes and Stokes processes. The gain and loss ratio therefore cannot be easily interpreted in terms of an occupation number or equivalent temperature.⁴⁰ Time-resolved spontaneous Raman therefore stays an appealing technique to study quasiparticle and collective excitation population dynamics, and optimization of signal detection remains a central issue. Recent advances in high repetition rate amplified laser systems, tunable light sources, and detection techniques can nowadays bring the sensitivity of time-resolved spontaneous Raman spectroscopy close to that encountered in a continuous wave laser based steady-state spontaneous Raman scattering experiment, even without relying on resonances. A resurgence of the technique as a tool to probe quasiparticle population dynamics, photo-induced symmetry changes, and energy and momentum-transfer in quantum materials is thus foreseen.

In this paper, we present a flexible and efficient ultrafast time-resolved spontaneous Raman spectroscopy setup to study collective excitation and quasiparticle dynamics in quantum materials. The setup has a broad energy tuning range extending from the visible to near infrared spectral regions for both the pump and probe pulse energies. It thereby allows to selectively excite materials and to probe under off- and on-resonant Raman conditions. The balance between energy and time-resolution can be controlled, allowing few picosecond time-resolution studies with an energy resolution down to Δν ∼ 7 cm⁻¹ or femtosecond studies with Δν ∼ 50 cm⁻¹ energy resolution. A high light collecting efficiency is realized by high numerical aperture collection optics and a high-throughput flexible spectrometer. We demonstrate the functionality of the setup with three different case studies. We first focus on the zone-center longitudinal optical phonon (LO-phonon) and hole continuum dynamics in photo-excited silicon and discuss the role of the Raman tensor by simultaneously tracking the Stokes and anti-Stokes spectra. Changes in the Raman tensor resulting from photo-induced symmetry changes have been discussed previously.³⁵ Here, we show that changes in the Raman tensor can additionally appear due to resonance effects. This can be used as an alternative probe to track electronic population dynamics. Higher order Raman processes can serve as an all-optical momentum dependent probe. We show evidence for dissimilar electron-phonon scattering rates at different high symmetry points in the Brillouin zone (BZ) of silicon by detecting pump-induced changes in the two-phonon overtone spectrum. This measurement underscores the unique resolving power of measuring transient changes in excitation energies in the frequency domain. Demagnetization dynamics of helimagnetic order in the chiral insulator Cu₂OSeO₃ is studied by observing softening and broadening of a magnon excitation after photo-excitation. We thereby illustrate the technique's feasibility to probe photo-induced phenomena in quantum materials, such as melting of phases and photo-inducing phase transitions. With the high stability and sensitivity of the described setup, new avenues in ultrafast dynamical studies of quantum materials are foreseen. These include the tracking of order parameter evolution and detection of symmetry changes across photo-induced phase transitions, the qualitative determination of energy and angular momentum transfer rates in the relaxation of non-equilibrium states, and time- and momentum-resolved scattering.

Typical design parameters for a time-resolved spontaneous Raman setup include energy resolution, time-resolution, the laser linewidth, and laser repetition rate.³¹ Other design factors include tunability of the pump excitation and Raman probe light, and suitable schemes to filter the Raman scattered light from the elastically scattered light. The energy- and time-resolution parameters are of special importance in time-resolved spontaneous Raman spectroscopy. For a transform-limited Gaussian pulse, the bandwidth-time relation is ΔνΔτ ≈ 14.7 cm⁻¹ ps, with Δν being the frequency (energy) bandwidth of a pulse, and Δτ is the pulse duration. Here, Δν and Δτ refer to the full width at half maximum (FWHM). A trade-off in time and energy resolution is thus inherent to the technique, and ideally a setup should allow control of energy and temporal resolution. The experimentally accessible region, limited by the Fourier transform relation, is indicated in Fig. 1. In materials with only a few Raman-active modes, such as IV and III-V semiconductors,^24,41 or well-separated high-energy modes in, for instance, graphite allotrope materials,^27–29 a pulse bandwidth of Δν ≈ 100 cm⁻¹ (FWHM) suffices to resolve individual modes, and thus sub-ps time-resolution can be realized (see Fig. 1). This energy resolution, however, generally is not sufficient to study dynamics of collective excitations in quantum materials such as magnons⁴² and phonons, or the more complex types such as electromagnons,⁴³ Cooper-pair breaking,³⁶ and charge-density wave modes⁴⁴ since these modes generally are low energy excitations with ΔE < 120 meV. This energy scale corresponds to Raman shifts of ∼1000 cm⁻¹ and lower and thus requires a higher spectral resolving power. This holds especially in the lowest energy region (Ω < 100 cm⁻¹) where, for instance, soft modes in ferroelectrics are observed.⁴⁵ In this region, stray light from spectrally broad laser pulses can hinder the observation of the modes of interest. Another case where high resolving power is necessary is materials of low crystallographic symmetry where more interesting modes of electronic and magnetic origin might overlap with a multitude of phonon modes.

In spontaneous Raman spectroscopy, the inelastically scattered light needs to be separated from the Raman excitation light. For higher energy modes such as vibrational excitations in carbon-based materials, this can, for instance, be realized with notch filters. When tunability in probe wavelength is of importance, a triple subtractive Raman spectrometer is still favored. In time-resolved spontaneous Raman spectroscopy, an additional complication appears since the pump-excitation and the pump-induced Raman spectrum need to be rejected. Different filtering schemes have been applied for this.³¹ In a degenerate pump-probe experiment, polarization optics are used to reject the pump-induced elastic and inelastic scattered light. The downside is that only the parallel Raman polarization geometry can be studied.³⁵ In addition, unwanted polarization leakage also ends up in the Raman spectrum, and the lowest energy excitations are difficult to access. In a two-color setup, however, the pump-induced elastic (and inelastic) scattering can be conveniently rejected by spectral filtering.^21,31

Amplified laser systems allow for tunable two-color experiments, as opposed to MHz-oscillator experiments which are limited to the fundamental and double wavelength for the pump and probe beams. In addition, with an amplifier, the problem of average heating is avoided. For low repetition rate systems, the average laser power can, however, bring challenges. The Raman intensity scales with average laser power. For detectable average laser powers with low repetition rate systems, the pulse peak intensities can thus easily get too large, which may lead to nonlinear effects and sample damage. The ideal situation between these laser system limits thus exists in the form of high repetition rate amplifiers.^21,31

The time-resolved Raman system consists of three main parts: (1) the amplified laser system and optical parametric amplifiers to generate pulses for selective pumping, and narrow-bandwidth pulses for Raman probing, (2) the table optics for pulse cleaning, polarization control, and the delay line, and (3) the confocal Raman microscopy interface, the high efficiency spectrometer, and the charge-coupled device detector. A layout of the setup is shown in Fig. 2.

A chirped pulse amplification based laser system (Light Conversion PHAROS) was selected for the setup. The PHAROS laser system consists of a Kerr-lensing mode locked oscillator module, a regenerative amplifier (RA), and the stretcher-compressor units which are all embedded in a compact module. The laser uses diode-pumped Yb:KGW (Ytterbium-doped potassium gadolinium tungstate) as the active medium. The emitted pulses have λ_c = 1024 nm central wavelength. An internal pulse picker allows to set the repetition rate up to f_max = 100 kHz. After the regenerative amplification stage, the 1024 nm stretched pulse beam is split into two beams. One beam is directly emitted as an 11 W output of non-compressed highly chirped 150 ps long pulses. The other beam is compressed, resulting in a 7 W average power beam of 0.3 ps duration pulses.

The uncompressed 11 W beam is routed to a second harmonic bandwidth compressor (Light Conversion SHBC).⁴⁶ Inside the SHBC, a 1:1 beamsplitter divides the beam. A high-intensity grating gives the split beams an opposite chirp, after which the beams are overlapped on an β-BBO (barium borate) crystal. The SHBC converts the 1024 nm pulses of full-width at half maximum (FWHM) Δν ≈ 50 cm⁻¹ into 512 nm transform limited pulses with Δν ≈ 10 cm⁻¹ FWHM and Δτ ≈ 1.5 ps temporal width. The SHBC output can directly be used as Raman probe light or used to pump a three stage white-light seeded picosecond optical parametric amplifier (Light Conversion PS-OPA). The white light seed is generated with 1 W of 0.3 ps 1024 nm compressed pulses. The PS-OPA can continuously tune the Raman probe wavelength from 630 nm to 950 nm. The signal and idler of the PS-OPA output can be externally doubled to tune the probe wavelength in the range of 320–600 nm.

A total power of P = 6 W of the 0.3 ps 1024 nm compressed beam is routed to a double-pass white-light seeded femtosecond optical parametric amplifier (Light Conversion ORPHEUS FS-OPA) to generate pump pulses. The signal wavelength is continuously tunable from 620 nm to 1000 nm, where the idler runs from 1060 nm to 2500 nm. An external β-BBO crystal can be used to extend the pump wavelength range from 320 to 600 nm.

The time-resolved system operates in the green oval region in the (Δν, Δτ)-plane of Fig. 1 when probing is realized with the SHBC or PS-OPA output. High time-resolution, with lower energy resolving powers, can be realized by only working with the FS-OPA. The FS-OPA's signal output is in this case still used for pumping. The residual output of the OPA's 0.3 ps, λ_c = 512 nm pump beam is used for probing. For the FS-OPA based operation mode, the system works in the blue sphere in the (Δν, Δτ)-plane.

Narrow-band laser line filters are placed in the pump beam to remove unwanted spectral components which may otherwise lead to spurious signals in the Raman spectra. The polarization state of the pump beam can be controlled by a Berek compensator. For the probe beam, the unwanted spectral components are either removed by laser line filters or by a folded grating-based pulse shaper,⁴⁷ where the latter results in the cleanest pulse, as required for probing small Raman shifts (Ω < 100 cm⁻¹). In addition, the pulse shaper allows to narrow down the spectral pulse bandwidth at least to about Δν ≈ 7 cm⁻¹ FWHM at the expense of time-resolution and average probing power. A Berek compensator is used to control the polarization state of the probe beam.

The temporal delay between the pump and probe pulse is controlled via a mechanical delay line of 30 cm length (Physikalische Instrumente M-531.EC) placed in the pump beam path. The step size resolution is ∼0.5 μm, corresponding to a time-resolution of approximately ∼3 fs. The time-resolution is thus effectively determined by the pump-probe pulse cross-correlation. A silver-coated retro-reflector is mounted on the delay line. The pump and probe beams are made collinear before the Raman microscopy interface with beam combiner BS1.

The collinearly propagating pump and probe beams enter the Raman microscopy interface via a periscope. The collinear pump and probe beams are reflected from beam splitter BS2 (R/T = 0.2/0.8) into a high numerical aperture microscope objective, MO (Olympus LMPLFLN-series long working distance (WD) objectives are used, where the NA = 0.4, 20×, working distance, WD = 12 mm is the standard choice). The backscattered light is collected by the microscope objective, transmitted through beam splitter BS2, and focused on the spectrometer entrance slit.

The maximum pump pulse energy lies around ∼150 nJ at the sample position for 650 nm pump excitation. The maximum probe pulse energy lies around ∼20 nJ at the sample position for 512 nm probe excitation when the grating-based pulse shaper is used. Since the Raman scattering efficiency scales with incident laser intensity, a large average incident probe power is preferred. For diffraction-limited pump-probe spot sizes, this, however, results in a few challenges. Pulse fluences get too large, and samples damage easily. This can be solved by defocussing the microscope objective. The divergence of the scattered light in this case is corrected with a telescope system placed after BS2. This allows working with spot sizes of about ∼1 μm to ∼100 μm diameter. The samples can be placed in a Janis ST500 coldfinger cryostat with small working distance (about 2 mm).

A successful implementation of time-resolved Raman spectroscopy significantly hinges on the efficiency of the spectrometer. Raman-scattered light is an order $O∼10−8$ weaker compared to the incident excitation light. This necessitates a high stray light rejection. We use a cascade of three Czerny-Turner⁴⁸ imaging spectrographs (TriVista 555, S&I GmbH), which are operated in subtractive mode (high stray light rejection by the first two spectrometers). Different sets of holographic and ruled gratings allow to maximize the scattered light detection efficiency. Silver-coated optical elements ensure a high throughput. The last stage (spectrometer stage) has a second entrance port. This allows to bypass the subtractive stage of the spectrometer. In this case, a notch filter is used to block the elastically scattered light. The use of a notch filter can provide up to a factor of 2 to increase in throughput efficiency with respect to the subtractive filtering. A color filter may additionally be used to suppress the scattered pump light.

The used imaging spectrometer⁴⁹ contains toroidal mirrors which corrects for the astigmatism present in a spectrometer containing spherical mirrors. In the toroidal case, a smaller image size in the lateral direction is thus produced on the charge-coupled device (CCD) chip. This has the advantage that less CCD-rows need to be binned to integrate the scattered light spectrum. This significantly reduces the appearance of occasional spikes from cosmic events in the Raman spectrum and reduces the CCD-readout noise with respect to an ordinary spectrometer. The low spherical aberration of the spectrometer makes that one can resort to the use of pseudo-confocal microscopy.⁵⁰ In this case, the entrance slit rejects out-of-focus light in the horizontal direction, whereas the CCD-binning is used to digitally reject out-of-focus light in the vertical direction from the Raman spectrum. By resorting to pseudo-confocality, the use of an opto-mechanical confocal system is avoided. In the latter case, a secondary focus point through a mechanical pinhole before the spectrometer is used to obtain confocality. This would, however, lead to reflection losses for the scattered light from additional lenses. The CCD-detector is a low-etaloning PyLoN:1000BR_eXcelon, 1340 × 100 pixels CCD, with a pixel size of 20 × 20 μm². The quantum-efficiency lies above QE > 90% for the wavelength range of 450–900 nm.

Time-resolved Raman spectroscopy allows to address population dynamics of individual low-energy excitation modes after photo-excitation. Through detailed balance of the Stokes and anti-Stokes signals, the population number of different modes can be determined and calculated into effective mode temperatures.³² This direct way of measuring transient temperature evolution in the time-domain allows to test the validity and limitations of multiple effective temperature models,^9,33 which on the shortest time-scales may not hold anymore due to the presence of truly non-thermal occupation statistics. Such effective temperature models are not only widely applied to describe ultrafast dynamics in simple materials^9,33 but also in quantum materials where different spin, orbital, electronic, and lattice degrees of freedom are present with their respective excitations.^51,52

Raman tensor dynamics is a less treated aspect of time-resolved Raman scattering. It is, however, of crucial importance in the correct determination of effective mode temperatures.³² The anti-Stokes intensity scales as I_AS ∝ χ²n, where n is the mode occupation number, and χ² is the squared Raman tensor. The Stokes-intensity scales as I_S ∝ χ²[n + 1] and is thus less sensitive to transient population dynamics. This can be taken as a motivation to “neglect” dynamics of the Stokes-spectrum. However, from these relations, it directly becomes clear that assigning transient anti-Stokes dynamics solely to transient population changes may not necessarily be true since the Raman tensor χ² may also show photo-induced changes. This is expected to be of special importance under resonant probing conditions. We have recently demonstrated in silicon that for a correct transient mode temperature determination indeed a dynamical modification of the Raman tensor χ² needs to be taken into account.²⁶ The quench of χ² in silicon was discussed to originate from the photo-induced hole density. Below, we reiterate our results on the LO-phonon population and hole continuum dynamics in silicon for similar measurement conditions. A dynamical modification of the Raman tensor is expected to play a role in many other materials such as 3d/4d/5d-oxides, carbon allotropes⁵³ like graphene or Bucky-tubes, and transition metal dichalcogenides.⁵⁴ 

An intrinsic (100) oriented silicon wafer (resistivity >10 000 Ω m), at room temperature, is excited above the indirect band-gap with 650 nm pulses of 0.3 ps duration. The Raman probe excitation is narrow-band (Δν ≈ 10 cm⁻¹ FWHM) pulsed light of 512 nm central wavelength, and 1.5 ps temporal duration. The pump and probe powers were set to P_pump = 12 mW and P_probe = 1 mW for the pump and probe beam, respectively. With a pump spot size of 100 μm diameter, the photo-excitation density is about n₀ ∼ 8 × 10¹⁸ cm⁻³. Both pump and probe beams are aligned with the [110] crystallographic directions. The backscattered light is collected with the single-stage spectrometer, where a notch-filter (Kaiser Optical Systems, Inc., SuperNotch-Plus 514.5 nm) is used to block the elastically scattered light.

Figures 3(a) and 3(b) show the anti-Stokes I_AS and Stokes I_S spectra of silicon (spectra at t = −5 ps). The spectra consist of the zone-center longitudinal optical (LO) phonon at ±520 cm⁻¹ and an electronic continuum originating from hole scattering.^55,56 The silicon peak has a (Gaussian) half width at half maximum of ∼8 cm⁻¹. This is larger than the intrinsic half-width of ∼3 cm⁻¹ at room temperature,⁵⁵ since Gaussian ps-pulses were used to collect the Raman spectrum. In Figs. 3(c) and 3(d), the differential anti-Stokes scattering intensity ΔI_AS(t) = I_AS(t) – I_AS (−5 ps) and Stokes scattering intensity ΔI_S(t) = I_S(t) – I_S(−5 ps) are shown for various representative delay-times. At 0 ps to 0.5 ps, a scattering increase is observed on the anti-Stokes side. This corresponds to the creation of a transient optical phonon population. However, on the Stokes side, a negative differential feature is observed, which evidently cannot originate from a transient phonon population. In addition, a small positive shoulder in the ΔI_S(t) spectra is observed. The integrated phonon scattering intensities are shown in Fig. 4(a). The phonon response is integrated over the spectral regions ±[460–580] cm⁻¹ (indicated with black bars) for the Stokes and anti-Stokes sides, respectively. A linear background was subtracted to take into account the increased hole continuum scattering over the integration region. The cross-correlation of the pump and probe pulse is plotted in faded blue for comparison.

The Stokes intensity I_S and anti-Stokes intensity I_AS are given as³² 

and

Here, χ²(ω_L ± Ω, ω_L) gives the Raman tensor for the anti-Stokes and Stokes resonance, ω_L is the probe excitation frequency, and n(Ω) is the occupation number for the mode with energy $ℏΩ$⁠. The factors C(ω_L ± Ω, ω_L) contain the incident intensity and optical constants (absorption, transmission, and refractive index) at the incident and scattered frequencies.³² We, however, neglect the frequency dependence of C and χ², which will lead to a small error of <5% in the absolute phonon temperature determination.^32,57 The anti-Stokes to Stokes scattering ratio can be used to calculate the phonon temperature, or equivalently the particle occupation number n(Ω), through the ratio

In Fig. 4(b), the transient temperature, determined according to Eq. (3), is plotted. An average heating to 310 K is measured before time-zero. After photo-excitation, the temperature rises by ΔT ≈ 45 K to 355 K within the time-resolution, followed by a decay with a time-scale of τ ≈ 2.0 ± 0.6 ps. The rapid rise and consecutive decay are consistent with the transient creation of an optical phonon population through electron-phonon coupling, which thereafter decays into acoustic phonons through anharmonic coupling.^26,58 After 10 ps, the temperature reaches quasi-equilibrium at an increased temperature of ΔT ≈ 5 K, which does not recover within the measured time window.

Figure 4(c) shows the transient increase in hole continuum scattering in the region +[600–650] cm⁻¹ (indicated with the red bar). The transient scattering signal is well-fitted with a step function convoluted with a cross-correlation function of the pump and probe pulse. The photo-induced electron-hole-density has other effects on the Raman spectrum. Since the phonon Raman transition probability is proportional to the density of electrons in the valence band and holes in the conduction band, the photo-induced electron-hole density is expected to alter χ².^59,60 From the Stokes intensity I_S and the calculated population n, the time-evolution of χ² is calculated according to Eq. (1) and shown in Fig. 4(d). A step-wise tensor quench Δχ² ≈ −5% is observed, underlining the electronic origin of the transient decrease in the Stokes intensity I_S. The transient hole density, in addition, leads to an increase in the Fano-asymmetry, as evidenced by the ingrowing positive shoulder in the ΔI_S(t) spectra. The origin of the tensor quench and the transient Fano-asymmetry are discussed in more detail in Ref. 26.

Electron-phonon and phonon-phonon interaction strengths, or more generally the coupling between any type of quasiparticles, depend on the momentum of the interacting quasiparticles.⁶¹ The characteristic timescales associated with optical phonon creation, and relaxation through anharmonic coupling, are thus momentum-dependent.^9,62 Pump-probe techniques spanning from x-ray to the optical range allow for time- and momentum-resolved probing. Whereas the momentum-dependence is implicit for pump-probe techniques in the x-ray range, such as diffuse scattering,¹⁵ diffraction,⁶³ or resonant inelastic x-ray scattering,¹⁶ this may not directly be obvious for optical pump-probe techniques as these are k ≈ 0 momentum transfer probes. Optical inelastic scattering techniques, notably femtosecond stimulated Raman spectroscopy,³⁹ and spontaneous Raman spectroscopy, however, can be utilized to disentangle momentum-dependent dynamics through higher-order scattering processes. In a two-particle process, two excitations of opposite momenta +k and −k are created (or annihilated).⁶⁴ Two-particle scattering processes appear in the inelastic light scattering spectrum as bands of the frequency dependence of the combined density of states of the excitation pair, weighted by a scattering efficiency distribution.⁶⁵ The multi-particle scattering response thus forms a mapping of momentum-space to the energy domain and thereby allows for the study of momentum-resolved particle dynamics. After careful analysis of the line-shape, even momentum-relaxation dynamics should be, in principle, feasible to detect.⁶⁶ With time-resolved spontaneous Raman spectroscopy, we succeeded to detect photo-induced phonon softening of different high-symmetry Brillouin Zone-points (BZ-points) in the two-phonon spectrum of silicon. We find evidence that the softening at the different points occurs with dissimilar time-scales, which is indicative of dissimilar electron-phonon scattering rates.

Probing of the two-phonon spectrum on a (100) oriented silicon wafer was performed at room temperature. A 512 nm Raman probe excitation was used, with the probe polarization along the [110] axis. The two-phonon spectrum is observed between 920 cm⁻¹ and 1040 cm⁻¹ as seen in Fig. 5(a) (dark blue line). The two-phonon spectrum consists of a summation of TO-phonon overtones from four high-symmetry Brillouin Zone (BZ) points.⁶⁷ Previous reports have assigned the characteristic spectral features based on the analysis of critical points in the phonon density-of-states.^65,67,68 The sharp increase at 920 cm⁻¹ is related to scattering from the X-point, the shoulder at 940 cm⁻¹ to the W-point, and the shoulder at 970 cm⁻¹ to scattering from the L-point. The tail extending from 980 cm⁻¹ to 1040 cm⁻¹ is the overtone from the zone center, i.e., the Γ-point. The different regions are marked with bars.

The sample is excited above the indirect band-gap with 650 nm pulses of 0.3 ps duration and pump polarization along the [110] axis. The photo-induced electron-hole density is n₀ ∼ 9 × 10¹⁹ cm⁻³. Figure 5(a) shows the transient Stokes spectra I_S(t) of the two-phonon peak for various pump-probe delays. A photo-induced redshift of the two-phonon spectrum is observed, in addition to an increased hole continuum scattering background, as discussed above. Figure 5(b) shows the differential spectra ΔI_S(t) = I_S(t) – I_S(−2ps) for the respective time-delays, where the hole continuum scattering increase is subtracted (the tail of a Voight profile is fitted to the increase in electronic scattering in the ΔI_S(t) spectra). The ΔI_S(t)/I_–2ps spectra are integrated over four different wavelength regions and plotted normalized to the I_S/I_–2ps value at +2 ps. This gives the spectral weight (S.W.) as a function of time delay as shown in Fig. 5(c).

The two-phonon Stokes signal has a I_S(ω) ∝ χ²(ω)[n²(Ω) + 1] dependence [cf. Eq. (1)]. Both the Raman tensor χ²(ω) and the population factor [n²(Ω) + 1] transiently change upon photo-excitation as discussed above, where the population term can only increase. The tensor dynamics χ²(ω) thus appears to dominate the transient two-phonon response. This is also evidenced by the large redshift ΔΩ ≈ −2 cm⁻¹. The redshift is understood as a photo-induced phonon softening at the different high symmetry points in the Brillouin Zone. Extracting time-constants for the different BZ-points is not possible due the limited amount of time-delay points. However, from the ΔI_S(t) spectra in Fig. 5(b) and the spectral weight (S.W.) in Fig. 5(c), it appears that the phonon softening at the X- and L-point has a faster ingrowth than at the W- and Γ-point. This hints to dissimilar electron-phonon coupling constants for the different points. The quantitative determination of time-constants for the four BZ-points asks for a higher time-resolution. In spontaneous Raman spectroscopy, this would, however, result in a loss of frequency resolving power, and thereby momentum-resolving power. This would again lead to a complication in disentangling time-and momentum resolved scattering, however, now because of frequency resolution limitations. A qualitative disentanglement of momentum-dependent scattering in silicon thus forms an interesting study case for the time-resolved femtosecond stimulated Raman scattering variant.

The understanding of out-of-equilibrium phenomena in magnetic materials is a highly active part of condensed matter research.⁶⁹ Recent scientific advances with a high potential towards applications include all-optical magnetization switching⁷⁰ and picosecond optical writing and read-out of magnetic bits.⁷¹ With increasingly complex magnetic phases and dynamical phenomena being studied, there is a high demand for novel all-optical probes. Magnetization dynamics in finite magnetization phases, such as ferromagnetic and ferrimagnetic order, or a more exotic phase such as the skyrmion lattice,⁷² can be probed by optical techniques which are sensitive to the order parameter M which describes the (macroscopic) magnetization. Linear magneto-optical effects, for instance, the Faraday or magneto-optical Kerr effect, measure a polarization rotation θ ∝ M proportional to the magnetization.⁷³ The dynamical variant allows to detect photo-induced incoherent and coherent dynamics of the order parameter M.^52,72 All-optical detection of antiferromagnetic order dynamics already becomes more challenging. The antiferromagnetic order parameter L is proportional to the difference in magnetic sublattice magnetizations: L ∝ M_↑ − M_↓. Optical probing of L is realized by second-order magneto-optical effects, such as magnetic linear birefringence and second harmonic generation. Ultrafast variants have been used to study photo-induced dynamics in antiferromagnets.^74,75 For more complex net-zero magnetization order, such as cycloidal, helical, or spin wave order, it is more challenging to probe the magnetic order parameter. One suggestion is tracking the thermalization of a magnon mode with THz-time-domain spectroscopy after photo-excitation, as was realized in the spin-cycloid material TbMnO₃.⁷⁶ The Raman-activity of a magnon mode would, however, permit all-optical detection, as opposed to detection through its dipole activity. Here, we show that time-resolved Raman spectroscopy allows to all-optically probe melting of helimagnetic order after photo-excitation in the chiral magnet Cu₂OSeO₃.⁷⁷ This is realized by tracking the softening and broadening of a Raman-active magnon mode in the time-domain.

Cu₂OSeO₃ has a long-range magnetic ordering temperature of T_N ≈ 58 K.⁷⁷ The magnetic ground state consists of helimagnetically aligned effective S = 1 spin clusters, which by itself consist of three-up-one-down S = 1/2 spins on the Cu²⁺(3d⁹) sites.^78,79 The spin cluster ordering results in high energy ∼meV magnons, from which the magnons at the Γ-point are observable by spontaneous Raman spectroscopy.⁸⁰ A [111] oriented Cu₂OSeO₃ sample at bias temperature of 5 K in the helimagnetic phase was studied. The SHBC output at 512 nm, with Δν ≈ 10 cm⁻¹ FWHM, was used as Raman probe (0.5 mW probe power at sample position). In Fig. 6(a), part of the Stokes spectrum is shown. The M-mode at Ω ≈ 264 cm⁻¹ corresponds to a ΔS^z = +1 magnon mode, which can be understood as a spin-flip excitation within the three-up-one-down spin cluster.⁸¹ The modes labeled “P” correspond to phonons, or regions of not fully resolved overlapping phonons.

Local crystal field excitations⁸² were excited with 0.3 ps, λ = 570 nm pump pulses (with 2 mW pump power, and fluence F ≈ 2mJ/cm²), after which the thermalization dynamics is measured in the transient Raman spectra. In Fig. 6(b), the scaled differential spectra ΔI_S (t)/I_–5ps are shown for representative time-delays, where ΔI_S(t) = I_S(t) – I_S(−5 ps). A derivative-like lineshape is observed in the ΔI_S(t)/I_–5ps spectra, resulting from a spectral weight shift to lower Raman shift Ω for the magnon excitation. For the low temperatures and Stokes range of interest (Ω = 220–320 cm⁻¹), the occupation number n(Ω,T) ≪ 1. The change in occupation thus also is Δn(Ω,T) ≪ 1. We thereby assign the spectral weight transfer to a change in the Raman tensor χ²(ω). This physically corresponds to a dynamic softening and broadening of the magnon excitation. From fitting of the ΔS^z = +1 peak position in the I_S(t) spectra, it is found that the dynamic shift ΔΩ(750 ps) at late delay times lies below 1 cm⁻¹. By comparison to the temperature dependent position, we conclude that the transient magnetic temperature does not rise above 25 K. Under our pump excitation conditions, the long-range magnetic order thus gets strongly perturbed, but not fully destroyed. The phonon peak shifts are too small to be resolvable in the ΔI_S(t)/I_–5ps spectra. The ingrowing spectral weight transfer (S.W.) of the ΔI_S(t)/I_–5ps signal, corresponding to the temporal evolution of the magnetic order parameter, is shown in Figs. 6(c) and 6(d) in linear and logarithmic timescales. A typical time-scale of τ ≈ 200 ps for spin-lattice thermalization in insulators is observed.⁶⁹ This agrees well with the findings of Langner et al. where a spin-lattice thermalization time of τ ≈ 300 ps was observed by means of time-resolved resonant x-ray diffraction.⁸³ The slight discrepancy might originate from the different pump excitation conditions, and the different bias temperatures.

In this paper, we have presented a flexible and efficient ultrafast time-resolved spontaneous Raman spectroscopy setup and illustrated its strength and capabilities with different conceptual time-resolved Raman studies of collective excitation and quasi-particle dynamics. One of the strongest feats of time-resolved spontaneous Raman is the determination of transient population numbers and effective modes temperatures through detailed balance of the anti-Stokes to Stokes intensity ratio. A less studied aspect of the dynamics is the resonance enhancement in the Raman process. This, however, plays a role, as we demonstrated in the prototype test material silicon. We have shown that the photo-induced hole density leads to a quench of the Raman tensor, and in addition to an increase in electronic inelastic scattering. Dynamic changes in the electronic population and structure are of importance to photo-induced phenomena in many classes of quantum materials, such as 3d/4d/5d-materials, carbon allotropes, and transition metal dichalcogenides. Time-resolved resonant Raman can thereby provide a convenient tool to study both electronic population dynamics, as well as photo-induced changes in the electronic band structure such as band gap renormalization.

Although Raman spectroscopy is a zero-momentum transfer probe, momentum-dependent scattering can nevertheless be resolved in the higher-order Raman response. The overtone spectrum can thus serve as a momentum- and time-resolved scattering probe. As a proof of concept, we have studied photo-induced changes in the two-phonon response of silicon and found evidence for dissimilar phonon softening rates at different high-symmetry points of the Brillouin Zone. Applying this concept to slower phenomena, such as spin-lattice relaxation in 3d-antiferromagnetic materials, seems fortuitous to pursue and possibly allows us to detect momentum-dependent transient population dynamics and momentum relaxation.

Raman-active collective excitations such as magnons, charge-density wave modes, and the Cooper-pair breaking peak can serve as a time-resolved probe for the order parameter of quantum phases. We illustrated this in the helimagnet Cu₂OSeO₃, where the photo-induced melting of helimagnetic order can be tracked by the observation of a dynamic softening and broadening of a magnon peak. This principle can also be applied to probe dynamics of other net-zero magnetization order such as cycloidal, spin-density wave, and antiferromagnetic order. On a broader scope, this can be applied to photo-induced phase transitions, where the associated change of symmetry can be derived from the transient Raman spectrum. The Raman spectrum of collective low energy excitations can additionally serve as a probe to study energy and angular momentum transfer between the lattice, electronic, orbital, and magnetic degrees of freedom in quantum materials.

This project was partially financed by the Deutsche Forschungsgemeinschaft (DFG) through SFB Grossgeräteantrag INST217/782-1 and SFB-1238 (Projects A02 and B05). R.B.V. acknowledges funding through the Bonn-Cologne Graduate School of Physics and Astronomy (BCGS).